Applied Complex Flow: Applications of Complex Flows and CFD 9811977453, 9789811977459

Table of contents :
Preface
Contents
Modeling Hemodynamics of Rotary Blood Pumps and Predicting the Potential Risks
1 Introduction
2 Hemodynamics Modeling Using CFD
2.1 Types of Rotary Blood Pumps
2.2 CFD Framework
2.3 Rheological Model of Blood
2.4 Impeller Rotation
2.5 Turbulence Models
2.6 Model Validation
3 Modeling Blood Damage
3.1 Hemolysis
3.2 Thrombosis
4 Summary
References
Microfluidic-Integrated Biosensors
1 Biosensors
1.1 Basic Principals
1.2 Main Categories of Biosensors
1.3 Applications and Challenges
1.4 Summary
2 Microfluidic-Integrated Biosensors
2.1 Microfluidics
2.2 Types of Microfluidic-Integrated Biosensors
2.3 Summary
3 Numerical Approaches for Microfluidic-Integrated Biosensors Design
3.1 Computational Fluid Dynamics
3.2 Molecular Dynamics Simulation
3.3 Virtual Fluid Particles
3.4 Machine Learning
3.5 Summary
References
Droplet Microfluidics: A Multiphase System
1 Introduction
2 Fundamentals of Droplet Microfluidics
2.1 Dimensionless Numbers
2.2 Microfluidics Design Classification
2.3 Geometry Classification
2.4 Breakup Regimes
3 Effect of Operating Parameters
3.1 Surface Wettability
3.2 Viscosity
4 Complex Droplet Formation
4.1 Double Emulsions
4.2 Encapsulating Droplets
5 Summary
References
Subject Specific Modelling of Aortic Flows
1 Introduction
2 Clinical Imaging
2.1 Imaging Modalities
2.2 Multi-modal Imaging Routine
2.3 Segmentation, Geometry Reconstruction, and Flow Data Acquisition
3 Multiscale Computational Workflow
3.1 Boundary Conditions
3.2 Governing Equations
3.3 Numerical Method
4 Blood Haemodynamics in the Aortic Artery
4.1 Haemodynamic Metrics
4.2 General Characteristics of Flow in Aorta Arteries
4.3 Limitations
References
3D Printing of Polymer Composites
1 Introduction
2 Fibre Fillers
2.1 Fibre Orientation and Mechanical Properties
2.2 Thermal Properties
3 Non-fibre Fillers
4 Core–Shell Polymer Composites
5 Summary and Future Outlook
References
Magnetorheological Fluids
1 Introduction
2 Operation Modes
2.1 Valve Mode
2.2 Shear Mode
2.3 Squeeze Mode
3 MRF Applications
3.1 MRF Brakes
3.2 MRF Clutches
3.3 MRF Dampers
3.4 MRF Valves and Seals
3.5 MRF Polishing Devices
3.6 Medical Applications
4 Constitutive Models
4.1 Bingham-Plastic Model
4.2 Herschel-Bulkley Model
4.3 Casson Model
4.4 Khajehsaeid et al. Model
References
Ceramic Manufacturing for Green Energy Applications
1 Introduction
2 Shaping Stage
2.1 Rheological Behaviour
2.2 Tracking Interface
2.3 Tracking Particles
3 Drying Stage
3.1 Decoupled Drying
3.2 Coupled Approach
4 Summary
References
Rheology and Cure Kinetics of Modified and Non-modified Resin Systems
1 Introduction
2 Material and Methods
2.1 Thermosetting Polymers (Resins)
2.2 Nanotechnology-Based Thermosetting Polymers
2.3 Preparation
3 Processing
3.1 Resin Liquid Moulding Processes of Composites
3.2 Resin Impregnation of Dual-Scale Fabrics
4 Cure Kinetics and Rheology
4.1 Curing Kinetics
4.2 Rheological Behaviour
4.3 Heat Transfer and Cure Modelling
5 Conclusion and Future Trends
References

Citation preview

Emerging Trends in Mechatronics

Aydin Azizi Editor

Applied Complex Flow Applications of Complex Flows and CFD

Emerging Trends in Mechatronics Series Editor Aydin Azizi, Oxford, UK

Aydin Azizi Editor

Applied Complex Flow Applications of Complex Flows and CFD

Editor Aydin Azizi School of Engineering, Computing and Mathematics Oxford Brookes University Oxford, UK

ISSN 2731-4855 ISSN 2731-4863 (electronic) Emerging Trends in Mechatronics ISBN 978-981-19-7745-9 ISBN 978-981-19-7746-6 (eBook) https://doi.org/10.1007/978-981-19-7746-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Most fluids exhibit a complex behaviour that is directly linked to their stressdeformation relationship. Whether the aforementioned relation is simply nonlinear or linear but combined with an external force (thermal, chemical, magnetic or etc.), the liquids that are being dealt with are collectively classed as “Complex Fluids”. This book is aimed to present improved numerical techniques and applied computer-aided simulations as a part of emerging trends in mechatronics in all areas related to complex fluids, with a particular focus on using a combination of modelling, theory, and simulation to study systems that are complex due to the rheology of fluids (i.e., ceramic pastes, polymer solutions and melts, colloidal suspensions, emulsions, foams, mico-/nanofluids, etc.) and multiphysics phenomena in which the interactions of various effects (thermal, chemical, electric, magnetic, or mechanical) lead to complex dynamics. The area of applications spans around materissals processing, manufacturing, and biology. In the following, a brief overview of each chapter is presented. Oxford, UK

Aydin Azizi

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Contents

Modeling Hemodynamics of Rotary Blood Pumps and Predicting the Potential Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonardo N. Rossato, Jonathan Kusner, and Farhad R. Nezami

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Microfluidic-Integrated Biosensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fatemeh Shahbazi, Masoud Jabbari, Mohammad Nasr Esfahani, and Amir Keshmiri

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Droplet Microfluidics: A Multiphase System . . . . . . . . . . . . . . . . . . . . . . . . . Maryam Fatehifar, Alistair Revell, and Masoud Jabbari

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Subject Specific Modelling of Aortic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . Amin Deyranlou, Alistair Revell, and Amir Keshmiri

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3D Printing of Polymer Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Hamid Narei and Masoud Jabbari Magnetorheological Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Hesam Khajehsaeid, Ehsan Akbari, and Masoud Jabbari Ceramic Manufacturing for Green Energy Applications . . . . . . . . . . . . . . . 149 Masoud Jabbari, Hesam Khajehsaeid, Mohammad Souri, and Mohammad Nasr Esfahani Rheology and Cure Kinetics of Modified and Non-modified Resin Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Hatim Alotaibi, Constantinos Soutis, and Masoud Jabbari

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Modeling Hemodynamics of Rotary Blood Pumps and Predicting the Potential Risks Leonardo N. Rossato, Jonathan Kusner, and Farhad R. Nezami

Abstract The global burden of cardiovascular disease is immense. Within this broad category of diseases, heart failure is recognized as a major contributor to both adverse patient events and healthcare spending. As western populations age, epidemiologic data projects significant increases in the prevalence of heart failure. The gap between those with end-stage heart failure and the availability of heart transplants continues to widen, compelling the transition of mechanical support solutions, initially developed as a bridge to transplant therapies, to destination therapies for thousands of patients each year. Unfortunately, patients receiving these devices continue to be plagued by a range of adverse events related to incompletely optimized blood–device interactions impacting hemolysis and thrombosis. Blood pumps, the principal component of such assist devices, are the focus of ongoing study as attempts are made to optimize their performance and minimize any associated risks. Fortunately, with recent advancements in computational power and simulation methods, such as computational fluid dynamics (CFD), reliable approaches have emerged to assess the hemodynamics of blood pumps and explore their hydraulic, hemolytic, and thrombolytic challenges. These methods observe several complex properties and interactions that occur within blood pumps, including the rheological properties of blood as well as the multicomponent motion of the pump impeller. This chapter initially focuses on preparing CFD frameworks for modeling rotary blood pumps, from geometry creation and meshing through more advanced methods of blood modeling, including turbulence model implementation. This is followed by a presentation of several approaches used to validate numerical models including in vitro experiments and the use of established parametric benchmarks. Finally, we discuss how to compose available methods to quantitate the risk of hemolysis and thrombosis within rotary blood pumps. L. N. Rossato · F. R. Nezami (B) Thoracic and Cardiac Surgery Division, Department of Surgery, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA e-mail: [email protected] L. N. Rossato e-mail: [email protected] J. Kusner Department of Medicine, Duke University School of Medicine, Durham, NC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_1

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1 Introduction Cardiovascular diseases remain one of modern medicine’s most significant challenges, routinely topping the list of global causes of death in adults [1]. As populations age, the prevalence of heart disease increases. Specifically, the prevalence of heart failure (HF), once thought to be on the decline [2], has been re-evaluated with new data not only showing it to be increasing, but also associated with a devastating disease burden that is straining healthcare systems globally [3]. In the United States (US) alone, more than 6.2 million patients have a diagnosis of HF [4], driving an associated annual mortality rate of 380,000 deaths [4] and $30 billion in health care spending each year [5]. This disease burden is expected to grow steeply such that by 2030 the cost of HF-related healthcare in the US is projected to be $70 billion annually [5]. Despite this spending, survival rates of HF patients remain alarmingly low, with estimates, for those managed in the community, of 80% at one year from diagnosis, which falls to 50% at five years [6, 7]. Even with advances in medications and healthcare delivery, heart transplantation (HT) remains an important therapeutic option for patients with end-stage HF. The prevalence of end-stage HF has consistently exceeded the capacity of HT. Unfortunately, as the prevalence of HF, including end-stage HF, has increased, the annual volume of HT remains stagnant at around 3000 transplants annually in the US [8, 9]. Due to this epidemiologic mismatch, ventricular assist devices (VAD), although initially developed as bridge-to-transplant (BTT) therapies for patients awaiting HT, are necessarily becoming destination therapies (DT). For patients receiving VAD today, nearly 50% are indicated as DT, 26% constitute BTT, while the remaining 24% are used in bridge-to-decision, bridge-to-candidacy, and bridge-to-recovery scenarios [10]. Many solutions to advanced and end-stage HF are being pursued, including gene therapies, improving logistics and technics for HT, xenotransplantation, and mechanical circulatory support. For the foreseeable future, VAD will remain an important therapeutic option for patients with end-stage HF. The medical community’s experience with VADs continues to expand, with greater than 2500 devices newly implanted each year [10]. VADs are associated with systemic improvements as total body circulation is re-established [11]. This is reflected in annualized survival, with VAD patients in the US experiencing one-year survival of 81 and 70% at two years [10]. Despite these encouraging survival figures, these patients experience high rates of VAD-related adverse events (AEs), including complications related to infection, bleeding, and clotting [12]. Improved understanding of tissue-device relationships and blood circulation through VAD devices are critical in reducing AEs and improving survival in patients with VAD. The most critical and complex component of VAD is the blood pump, which is principally characterized by the actuation method into two main categories: displacement pumps and rotary blood pumps. This chapter will address rotary blood pumps alone as they are associated with better clinical outcomes and are more commonly implanted. In addition, rotary pumps have higher power efficiency, allowing for

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smaller device profiles and simpler surgical implantation. Ease of implantation and superior device performance have driven the real-world utilization of rotary pumps over displacement pumps. Rotary pumps transfer the kinetic energy into the bloodstream as they achieve physiologic cardiac outputs. Excellent hydraulic performance is needed to balance this introduction of energy while minimizing damage to the blood and supplied organs. The study of blood–device compatibility in VAD interrogates this balance with a focus on reducing hemolysis (the destruction of red blood cells) and thrombosis (blood clotting). Both hemolysis and thrombosis in VAD are associated with significant complications that drive morbidity and mortality in patients who rely on these devices. Although relevant to all patients, blood–device compatibility is critical to the increasing segment of patients receiving DT VAD. In addition to biological and biochemical aspects of such compatibility, the mechanical design and hemodynamic performance of pumps significantly impact device durability and the rates of AE. Numerical models, such as computational fluid dynamics (CFD), have become fundamental tools for designing and optimizing mechanical circulatory support pumps. Computational methods represent a cost-effective and ethical set of tools that provide unique insights into the complex fluid dynamics of VADs. These tools enable time-resolved quantification of important fluid dynamic parameters related to blood damage and hydraulic performance. CFD analysis is instrumental in characterizing complex turbulence features, such as recirculation and stagnation zones that are flow patterns implicated in hemolysis and thrombosis [13, 14]. Significant increases in computational power paired with improved models of human physiology forward the use of CFD for the study, development, and optimization of mechanical circulatory support pumps and understanding their relationship with human physiology post-implantation. This chapter will overview how CFD models are leveraged to characterize the characterization of the hemolytic, thrombotic, and hydraulic potential of rotary blood pumps. The first section will discuss the process and components of developing a CFD simulation, including mesh generation and governing equations related to pump hemodynamics. Following that, we will explain the process and significance of modeling the rheological properties of blood, how to numerically induce pump impeller motion, and how to properly account for the effects of turbulence in blood pumps. Finally, we will introduce computational approaches to quantify blood damage in the forms of hemolysis and thrombosis.

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2 Hemodynamics Modeling Using CFD 2.1 Types of Rotary Blood Pumps Rotary pumps are classified according to the positioning of the outlet(s) and flow orientation. In centrifugal pumps, the outlet is positioned tangentially at the end of the pump housing, whereby the impeller rotation linearly accelerates the blood flow. As a result, such pumps are generally characterized by a high ratio between pressure head and rotation speed, reducing the hemolytic potential of the pump due to lower blood damage levels than other pumps. Furthermore, due to the geometry of the rotor, centrifugal pumps can make use of magnetic or hydrodynamic levitating bearing suspension systems to minimize the potential risk for pump thrombosis. Finally, the reduction in stagnation zones, commonly found near impeller bearings, increases the overall durability of this variety of pumps (Fig. 1). In axial pumps, both the impeller and the outlet are collinear, resulting in linear and rotational propulsion of the flow. These pumps tend to operate at high speeds to supply the reduced pressure head; consequently, they are more likely to induce blood damage. Furthermore, axial pumps require stationary guide vanes, which have direct contact between bearings and impeller, aggravating their hemolytic and thrombotic potential and accelerating pump wear. Thus, axial pumps tend to have a shorter service life as compared to centrifugal pumps, with an average durability of less than five years. An advantage of axial pumps is their smaller size which confers greater energy efficiency, rendering them more suitable for pediatric applications.

Fig. 1 Essential components of (a) a centrifugal blood pump and (b) an axial flow blood pump. Pictures are adapted from [15] with permission from Trends in Biomaterials and Artificial Organs

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2.2 CFD Framework For the past six decades, three-dimensional CFD has experienced a dramatic breakthrough in its theoretical bases and breadth of application. The process of CFD model generation has been well documented. It is usually divided into five steps: geometrical setup and fluid domain definition, computational grid generation or meshing, physics model setup, iterative solving of discretized governing equations of the chosen physics model, and post-processing to extract relevant spatial and temporal maps for variables of interest. CFD starts with defining the geometry of interest, where the working fluid will flow. In the case of ventricular assist devices, it is necessary to use computer-aided design (CAD) software to recreate the three-dimensional pump in a virtual environment to help define this blood space. To obtain the dimensions of the pump, one may either consult the technical drawings provided by the pump supplier or reverseengineer the dimensions using images obtained from high-resolution tomography, such as micro-CT scanning. After extracting the bounding components, the blood space, which we refer to as the fluid domain, is exported in an appropriate format for the next step, meshing. Meshing, or computational grid generation, includes discretizing the fluid domain into small elements or cells, over which the governing equations will be solved, given that they otherwise do not have an analytical solution. Additional steps, such as mesh refinement and smoothing, the addition of inflation layers close to the walls, and mesh convergence investigation are commonly practiced to optimize meshes for CFD solving. Adding inflation layers into one’s mesh is critical to accurately model blood pump application, as it allows solutions to account for the viscous effects at the boundary layer between the blood and internal pump components; within these regions, the flow field is subjected to large gradients across multiple parameters. Including inflation layers near the boundary allows for such parametric spans to be accounted for accurately. The dimensionless distance from the wall, the y+ value, is one of the most important parameters to define the near-wall mesh resolution. y+ < 1 is generally recommended for CFD of blood pumps [16, 17]. This standard results in a degree of mesh refinement with high enough resolution to accurately describe flow separation within pump applications. With the fluid domain discretized into a refined mesh, the next step is to model the physics of fluid dynamics using mathematical relations. Physics model development includes identifying the governing equations for the fluid under study and assigning the material properties of the fluid, its container in some applications, and the assignment of boundary conditions. For blood pumps, defining the boundary conditions includes quantifying the velocity or pressure at the inlet and outlet, the rotational speed of the impeller, and treating the surfaces as walls or interfaces. Simultaneously, the definition of transport properties in circulatory pump applications requires assigning the rheological properties of blood to the fluid domain. Furthermore, in this step, the flow regime is identified based on preliminary estimations, and the turbulence model may be assigned if necessary. Once the entire

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physics model is defined, the next step is to iteratively solve the system of equations defined by the physics model over the fluid domain. For transient problems, it is necessary to define several timesteps and, in some cases, iteratively optimize the solution for each timestep. In the case of blood pumps, it is suggested to set a timestep equivalent to less than 1 degree of impeller rotation to avoid large gradients between timesteps and ensure solver convergence. Finally, after the equations are solved within a predefined accuracy of the variables under study, simulation outputs are post-processed and readied for interpretation. Some principal parameters obtained from CFD simulations of blood pumps include the wall shear stress, residence time, viscous shear stress, and hemolysis index. There are several open-source and commercial software programs specialized in CFD model development. The major difference between such packages is the discretization method, which employs finite volume or finite element methods. Some available programs are Ansys Fluent, CFX, STAR-CD, Abaqus, COMSOL, and ADINA. In exceptional cases, developing CFD solvers for specific problems may be necessary. However, standard packages can adequately model the hemodynamics of blood pumps.

2.3 Rheological Model of Blood The rheological properties of blood have been empirically characterized, and there is consensus about human blood’s non-Newtonian behavior in certain flow regimes, identified by viscoelasticity [18–22], shear-thinning [23], thixotropy [24, 25], and yield stress [18, 19, 26–28]. These parameters are influenced by several factors, such as the alignment of red blood cells [29], the level of aggregation and deformability of red blood cells [30], the plasma viscosity [30], fibrinogen concentration [31], flow size and geometry [32], hematocrit, shear rate, gender, temperature, and diet among several others. However, when accounted for in CFD models, blood rheological properties are typically simplified to be applied by empirical Newtonian and nonNewtonian models. In most cases, the model selection is based on the shear rate range expected by the application. There is a threshold above which the blood can be simplified as purely Newtonian. This threshold, however, varies between studies and is often determined as a function of the hematocrit [33]. Some studies consider the threshold to be 50 s−1 [34], 100 s−1 [35], or in the range of 100–300 s−1 [33]. As a rule of thumb, blood is considered Newtonian for large arterial blood vessels since their blood flow is characterized by high shear rate levels [36]. In these cases, viscosity is usually treated as constant and equivalent to 3.5 mPa. For regions of low-shear rate, the blood is assumed pseudoplastic, exhibiting shear- thinning behavior, and should be treated as a non-Newtonian fluid. Modeling the shear-thinning properties of blood is usually done by an empirical equation, such as Casson, Carreau, and power-law shear-thinning models [37] that relate viscosity

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and shear rate. In addition, several models describe the viscoelastic property of blood, including Maxwell type or Olroyd models [38, 39]. CFD models of blood pumps are characterized by high shear stress levels, making it acceptable to simplify the blood model as Newtonian. However, the working condition of the device design may invalidate the Newtonian blood assumption for blood pumps that operate with low rotational speeds, and those that include small gap clearances or are designed with contact bearings. For such cases, it is recommended to use a non-Newtonian model to describe the rheological properties of blood with non-linear relation of viscosity to shear rate, similar to what is prescribed by the Carreau model presented in Eq. (1). (n−1)/2

μ = μ∞ + (μ0 + μ∞ )[1 + (λγ˙ )2 ]

(1)

where μ∞ and μ0 are, respectively, the minimum and maximum viscosity equal to 0.0035 and 0.25 Pa s, λ is the time constant equal to 25 s, n is the power-law index equal to 0.25, and γ˙ is the scalar shear rate. It has been shown that in cases of very small gap clearances, a phenomenon of separation of the RBCs occurs in which the cells do not penetrate the gap [40], and therefore, are not subject to blood damage in those regions which are usually marked by high shear stress. To model such cases, one must consider the rheological properties of blood at the micro-scale, a field with much ongoing investigation.

2.4 Impeller Rotation One of the major challenges for simulating blood pumps is to accurately model the impeller motion. The difficulty is that the pump impeller assumes six degrees of freedom and is composed of multiple moving parts. Because these parts are in continuous contact, they may generate complexities in the grid generation and solver processes. In addition, the impeller structure varies between blood pump models, being open or closed impeller, and can be in direct contact with the bearing (first and second generation of blood pumps) or not (third generation of blood pumps). Finally, the most challenging scenario to model arises in closed impellers, wherein the impeller housing also moves, remaining in direct contact with an in-motion bearing. Such intricate designs warrant special settings for physics models of the fluid domain to accurately account for the contact between components to model their motion. During the last decades of developing numerical models for turbomachinery applications, several approaches have been proposed to properly account for the motion of impellers. Such models are performed by resorting to moving reference frames. Commercial software offers varying approaches, such as single reference frame (SRF), multiple reference frame (MRF), mixing plane (MP), and sliding mesh (SM) models. SRF is an approach that refers to the entirety of the domain as a single moving domain and thus is limited to simplified cases. For more complex scenarios,

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MRF is employed; here, the domain is differentiated into multiple zones, wherein the interfaces are required to be properly defined. In cases with transient interaction between the stationary and moving zones, SM is the most applicable approach. In MP approaches, each fluid zone is solved as a steady-state problem, while the flow data at the mixing plane interface are averaged in the circumferential direction at prespecified iteration intervals. However, MP is less frequently used due to its high computational cost and complexity, low robustness, and dependency on turbulence model choice [41]. Resultantly, MRF and SM approaches are two widespread methods due to their accuracy, ease of use, and reduced computational burden. MRF, also called a frozen rotor approach, includes splitting the fluid domain into two parts: a rotating domain and a static domain. The governing equations of this motion are modified to incorporate Coriolis and centrifugal acceleration terms. The motion of the rotating domain is added directly into the governing equations, as below. v = vr + vt + ω × r

(2)

Here, v is considered the absolute velocity (i.e., velocity observed from the stationary frame), vr is the translational frame velocity, ω is the angular velocity, and r is the vector oriented toward the center of rotation. The absolute velocity vector is obtained after the velocity and the velocity gradient from the rotating coordinate system are transferred to the stable coordinate system. The rotating domain is solved in the rotating coordinate system, while the static domain is solved in the stable coordinate system using the absolute velocity values. The transfer of the results at the interface between rotating and static domains is done by interpolation. Given that this strategy can induce numerical errors in a simulation, the region modeled with this approach should be minimized. Another drawback of MRF for blood pump applications occurs because the model is solved assuming steady-state conditions, not allowing it to consider transient effects commonly related to blood pumps. This further impacts the accuracy of simulation results. The SM approach is a robust alternative when simulating blood pumps. It also divides the fluid domain into rotating and static compartments and transfers the information between both domains by means of interpolation. However, the cell zone of the rotary domain changes over time, based on the rotational speed set by the user, equivalent to the rotational speed of the pump. It is worth noting that the impeller walls inserted in the rotary domain must have the same rotational speed as the rotary domain, so that the final deformation of the meshing grid at each timestep is zero. The movement of the boundaries is governed by Eq. (3). d dt

 ρϕd V = V

(ρϕV )n+1 − (ρϕV )n t

(3)

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The fluid density is defined by ρ, the control volume is V , n, and n+1 correspond to the current and upcoming timesteps, and ϕ is the general scalar of the defined control volume. The sliding mesh approach considers the transient effects of the model, allowing it to apply more realistic boundary conditions and more complex turbulence features. Both of these are key factors in the analysis of blood pumps, making SM-based approaches the recommended method for such applications. The drawback of SM is that it is inherently associated with higher computational costs, although parallel computing has begun to obviate these limitations.

2.5 Turbulence Models Turbulence refers to chaotic changes in the flow pressure and velocity field. It directly affects the properties of the blood pump, including its hydraulic efficiency and hemolytic and thrombotic risk. In addition, certain turbulent phenomena, such as recirculation and stagnation zones, are directly related to hemolysis and thrombotic risk. Therefore, the inclusion of turbulence in the definition of the flow regime is critical for accurate modeling of blood pumps to reliably understand their safety. Turbulent regime assignment and proper selection of its model are, however, challenging as the operating regime of pumps covers ranges of laminar, low Reynolds (Re) number regions, transitional regions, and fully turbulent regions. Most of the existing turbulent models were developed for fully turbulent, high Re number regimes. Because of this, most researchers simplify the flow as laminar or apply traditional turbulence models with proven reliability to simulate transitional regimes. The turbulent model should thus be selected for higher operational RPMs and only when there is a boundary layer close to the walls wherein the computational mesh is adequately refined to fully characterize viscous flow development. Given the importance of modeling transition to turbulence in blood pump applications, models with proven performance have emerged. The shear stress transport (SST) k-ω model for CFD models of rotary blood pumps is the most utilized model by industry and academia. This model makes use of modified formulas for the turbulent viscosity and considers the transport of the principal shear stress under conditions of adverse pressure gradient near the boundary layers. Thus, it has proven potential to accurately simulate the near-wall flow field and is recommended for blood pump simulation [42–44]. The following equations explain the model:    μt ∂k ∂ ∂ ∂   μ+ − ρu i u j − ρβ ∗ kω (ρk) + (ρku i ) = ∂t ∂ xi ∂x j σk ∂ x j    μt ∂w ∂ ∂ ∂   ω μ+ − αρu i u j − ρβω2 (ρwu i ) = (ρw) + ∂t ∂ xi ∂x j σω ∂ x j k 1 ∂k ∂ω + 2(1 − F1 )ρ σω2 ∂ x j ∂ x j

(4)

(5)

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wherein the fluid density is given by ρ, k is the turbulent kinetic energy, ω is the specific dissipation rate, μ is the dynamic viscosity, μt is the turbulent eddy viscosity, F1 is the blending function to blend the k-omega model (where F1 = 0) and k-epsilon model (where F1 = 1), and the coefficients of the model are given by: β = 0.075, β ∗ = 0.09, σk = 2, σω = 2, and σω2 = 1/0.856..

2.6 Model Validation CFD has great potential to accelerate the development of blood pumps by providing unique insights critical to design optimization. In addition to impacting development, there remains continued interest in leveraging CFD studies for regulatory evaluation of emerging blood pumps. This interest is well earned, but CFD models are just that— models that require proper validation. Due to limitations of computational processing power and accuracy of available physical models, contemporary CFD methods make assumptions to allow for the convergence of numerical models. Therefore, model validation is a fundamental step in the framework of creating useful CFD models. Validation takes many forms; the ultimate goal of this step is to evaluate the accuracy of simulation results as compared to reality. This is done by comparing the simulation results with experiments, which can be in vitro or in vivo, or with benchmarks of similar CFD models. Only after being validated are the simulation results meaningful. One of the most common ways to validate numerical models of blood pumps is through in vitro experiments. These methods often incorporate non-invasive techniques to trace flow using laser Doppler anemometry [45, 46], particle image velocimetry [44, 47– 52], and oil drop streaking techniques paired with ultrasonic sensors to measure wall shear stress [53, 54]. Such experiments are usually performed in mock circulation loops; these benchtop apparatuses are structured to replicate the human cardiovascular system, incorporating its inherent characteristics in terms of resistance, compliance, and pressure. A typical way to validate predictions from the CFD models of blood pumps is to compare the obtained pressure head in the simulation with that measured in vitro. More advanced mock circulation loops include quantifying blood damage and measuring the amount of plasma-free hemoglobin, for instance, which is compared with hemolysis parameters implemented in the numerical model. CFD models are also validated by comparing them to benchmark models, such as those provided by the Food and Drug Administration (FDA) for blood pumps [55, 56]. In the case of blood pumps, the FDA has provided the geometry for a generic blood pump and its boundary conditions. Available validating studies of this kind provide simulation results from several laboratories using diverse in-house and commercial platforms paired with results from in vitro experiments.

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3 Modeling Blood Damage 3.1 Hemolysis Hemolysis refers to the process of red blood cells (RBCs) breaking down and releasing their contents, such as hemoglobin, into the blood plasma. In the case of blood pumps, this phenomenon is usually induced due to aberrant mechanical stress caused by high shear stress levels and high residence times. Other factors, such as osmotic, chemical, and thermal processes influence hemolysis but are generally ignored in available CFD models for blood pumps [50]. Though recent generations of blood pumps have optimized their design to minimize blood trauma, sublethal damage to RBCs persists and has become relevant for medium- and long-term applications. Hemolysis is clinically relevant; lysed red blood cells release hemoglobin into the blood; hemoglobin has immense oxidizing potential and can induce kidney dysfunction [57] or, in more extreme cases, multiple-organ failure. This has motivated the development of numerical models to characterize the hemolytic potential of blood pumps. Although many methods continue to be refined, the method developed by Giersiepen et al. [58] is the most frequently used; it consists of a power-law function for the blood damage fraction. The empirical equation is parameterized with in vitro hemolysis results obtained by Wurzinger [59], wherein a Couette system was employed to subject the blood to different magnitudes of shear stress (τ ) and exposure times (t) HI(%) =

f Hb × 100 = Ct α τ β Hb

(6)

The percent blood damage or hemolysis index (HI) is quantified by the increase in plasma-free hemoglobin concentration ( f H b) and the total amount of hemoglobin (H b). Shear stress is calculated from the velocity field, obtained from the CFD simulation, commonly approximated as a scalar viscous shear stress from the below equation ⎡

∂u ∂x

1 2



∂u ∂y

+

⎢  = ⎢ ∂v τ = 2η⎢ 21 ∂u + ∂∂vx ∂ y ∂y ⎣

  1 ∂v 1 ∂u + ∂w + 2 ∂z ∂x 2 ∂z

∂v ∂x

∂w ∂y





1 ∂u 2 ∂z 1 ∂v 2 ∂z

+ + ∂w ∂z

⎤

∂w ∂x ⎥ ∂w ⎥ ∂y ⎥ ⎦

(7)

where η is the blood viscosity, and u, v, and w are the flow velocity components along the x, y, and z axis, respectively. This model is also called the stress-based method and uses the von Mises criterion [60] to calculate the viscous shear stress, as shown in the equation below

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τ=

  

2

2 1 1

2 2 2 σx x − σ yy + σ yy − σzz + (σzz − σx x )2 + σx2y + σ yz + σzx 6 (8)

This model suggests that the blood damage is caused solely by the shear-induced strain, while the strain due to volumetric changes is neglected for an incompressible flow. After determining the viscous shear stress and measuring the residence time, the hemolysis index can be calculated from the Eq. (9) ⎛

⎛

1 H I = ⎝1 − exp⎝−  Q



⎞⎞β (Cτ α ) d V ⎠⎠ 1 β

(9)

V

In which the empirical constants C, β, and α are equal to 1.745 × 10−6 , 0.0762, and 1.963, respectively [61, 62]. A recent study implemented an Eulerian scalar transport equation to evaluate the viscous shear stress and hemolysis index of three blood pumps. Therein, the hemolytic performance of those pumps was also compared (Fig. 2).

Fig. 2 Comparison of blood damage risk of three models of rotary blood pumps: Breath (left), CentriMag (middle), and Rotaflow (right), in terms of their a viscous shear stress (SSS) distribution and b hemolysis index (HI). The pictures are adapted from [63] with permission from The International Journal of Artificial Organs

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3.2 Thrombosis Thrombosis refers to the physiological process of blood clotting. This process comprises a cascade of events induced by several chemical and mechanical factors [64–66]. The main factors that trigger thrombosis are summarized by Virchow’s triad [67]: Blood stasis and flow disturbance, endothelial cell injury or dysfunction, and systemic or inborn patient characteristics inducing or inhibiting thrombosis referred to broadly as coagulopathy. Taken together, blood pumps, characterized by regions of high shear stress, zones of recirculation and stagnation, and blood contact with surfaces of low hemocompatibility, offer an ideal substrate for the development of thrombosis. Clot formation affects the pump’s functionality and if dislodged from the pump and circulated freely in the bloodstream, a phenomenon called embolism, it is capable of obstructing blood vessels supplying vital organs leading to devastating consequences. For instance, embolic obstruction of blood vessels supplying the brain is a major cause of stroke. Embolism is a significant cause of death in patients with VADs, and extensive research is being conducted to minimize this risk. Several models of thrombosis have been developed. In the one presented, an accelerated thrombosis model was developed [16] composed of three diffusion and convection equations to define the transport of nonactivated platelets (NP), activated platelets (AP), and adenosine diphosphate (ADP), as described in the following equations:  

  ∂φa + (u · ∇)φa = Da ∇ 2 φa + [Ac (ADP)]φn + A M φ f , τ (φa + φn ) (10) ∂t  

  ∂φn + (u · ∇)φn = Dn ∇ 2 φn + [Ac (ADP)]φn + A M φ f , τ (φa + φn ) (11) ∂t ∂(ADP) + (u · ∇)ADP = DADP ∇ 2 ADP ∂t  

  + RADP [Ac (ADP)]φn + A M φ f , τ (φa + φn ) (12) D with varying subscripts (a: activated and n: nonactivated) corresponds to different diffusion coefficients, A M is the mechanical activations, Ac is the chemical activation, and RADP is the amount of ADP per platelet. The activation of platelets can be induced mechanically due to shear stress as defined in Eq. 13 or chemically based on the threshold ADP concentration as presented in Eq. 14.

 1 

A M φ f , τ = 1 − φ f C β βφ f  Ac (ADP) =

ADP , (ADPt )(tADP )

0,

β−1 β

α

τβ

ADP ≥ ADPt ADP < ADPt

(13)

(14)

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Fig. 3 Scaled concentration of activated platelets in Heartmate II pumps delivering 2 L/min of blood. a Computational cells with a concentration above 5.68 are colored in red. b contour of activated platelet concentration on the impeller surface. The pictures are adapted from [16] with permission from Springer Link

φ f is given by the ratio of AP and total platelet count. The power-law model parameters, α, β, and C, are obtained from [68]. The threshold for chemical activation is defined by ADPt wherein tADP is the characteristic time of platelet activation. The scalar shear stress is calculated according to Eq. (8). Finally, the total rate of platelet activation is given by the summation of the mechanical and chemical cues, both included in the governing equation for the transport of species. The final output of the thrombosis model is the amount of AP, which represents the thrombolytic potential. The rest of the parameters of the equation can be found in [16]. An example of leveraging this advanced model of thrombosis quantification is shown in Fig. 3, based on the research of [16]. As previously described, the thrombotic potential is given by the amount of AP. In this case, the AP concentration has been rescaled to improve the generalizability of the results. In Fig. 3a, the regions of AP concentrations above 5.7 are highlighted in red, and in Fig. 3b, the contour plot of the AP concentration along the impeller surface is presented. In Fig. 3, the high AP concentration near the impeller bearing and blades is underlining the importance of these regions in blood clotting within blood pumps.

4 Summary The number of patients with end-stage HF requiring HT continues to greatly outpace the number of hearts available for transplant globally. This epidemiologic mismatch compels the study and advancement of alternative therapies for the growing segment of patients with end-stage HF. Developing new solutions is urgent; the prevalence of

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HF is on the rise globally as are its associated morbidity, mortality, and healthcare costs all of which are projected to substantially strain the resources and capacity of global healthcare systems. Initially developed as BTT devices, VADs are increasingly utilized as DT with encouraging patient outcomes. Although early survival rates are satisfactory, these complex devices continue to experience myriad complications related to blood–device compatibility. Advancing our understanding of blood–device compatibility within these devices is critical to reducing the incidence and severity of these devastating complications. CFD methods are an exciting set of tools that allow for the detailed study of mechanical circulatory support and native circulation throughout cardiovascular medicine. Specifically, CFD methods have proven to be a robust set of tools to study blood–device compatibility and have propelled the research and development of VAD and other mechanical circulatory support [69–72]. Numerical methods are becoming the method of choice for leaders throughout the medical device industry and regulatory officials evaluating the safety and performance of emerging medical technologies [73–79]. In summary, this chapter has discussed the different steps related to preparing, initializing, and implementing CFD methods for modeling rotary blood pumps. We have explored the significance and process of modeling the rheological properties of blood, pump impeller motion, and the inclusion of turbulence models. Different methods to characterize hemolytic and thrombotic risks for blood pumps were presented along with techniques used to validate CFD results through in vitro experimentation. The application of CFD for developing and optimizing medical devices has improved our quantitative understanding of hemodynamic phenomena. These methods offer continued promise in improving the durability and therapeutic longevity of medical devices. CFD methods will remain an essential set of tools in the conceptualization, development, and optimization of medical technologies that serve to reduce the burden of cardiovascular disease worldwide.

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Microfluidic-Integrated Biosensors Fatemeh Shahbazi, Masoud Jabbari, Mohammad Nasr Esfahani, and Amir Keshmiri

Abstract Reducing the number of deaths is a worldwide challenge, and microfluidic-integrated biosensors are one of the most promising solutions. The current leading causes of death are cancer and heart disease; their early and precise diagnosis is essential for patients’ successful prognosis and survival. A biosensor is a device to transfer a biological response into a signal by measuring the interaction between the analyte and transducer. It is on-site user-friendly, cost-effective, and with a fast response time. There is a high demand for biosensors with point-of-care (POC) applications. Technological developments are the key driving factors for the biosensor market. With the advent of microfluidics, the development of microfluidicintegrated biosensors has been intensive because the biological targets are nearly always transported by carrier fluid. Microfluidics also improves the biosensors’ sensitivity, accuracy, and controllability while reducing the sensing region. Despite the current development in this field, the design and fabrication of biosensors are still challenging; accurate numerical simulation could be a game-changer, as it improves the understanding of the detection process and significantly reduces the design time. Hence, the importance of developing a numerical framework for this aim is evident. This chapter introduces microfluidic-integrated biosensors, their applications, challenges, and current numerical models for their design and analysis. In the first section, different biosensors and their applications are studied with this aim. The second part of this chapter introduces the importance of the integration of microfluidics in biosensors, their applications, and recent challenges in their design and fabrication. The F. Shahbazi · A. Keshmiri Department of Mechanical, Aerospace and Civil Engineering, University of Manchester, Manchester M13 9PL, UK e-mail: [email protected] M. N. Esfahani Department of Electrical Engineering, University of York, York YO10 5DD, UK A. Keshmiri Manchester University NHS Foundation Trust, Manchester Academic Health Science Centre, Southmoor Road, Wythenshawe, Manchester M13 9PL, UK M. Jabbari (B) School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_2

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last section studies the current development in numerical methods for designing and analysing microfluidic-integrated biosensors and discusses their advantages and disadvantages.

1 Biosensors The primary purpose of biosensors is the analysis and recognition of bio-molecules (e.g., DNA, proteins, viruses, and cancer cells), which leads to an evident role in saving lives by detecting diseases. These systems decrease the reagent consumption (sample), shorten the experiments’ time, and reduce the overall costs of applications (key features). These lab-on-a-chip devices do not need any laboratory facilities and can be used frequently. Hence, they are the most favourable and effective real-time detection systems. Biosensors consist of four main parts: analyte, receptor, transducer, and electronic display (Fig. 1). A reaction between the immobilised agent and the target molecule is transduced into an electronic signal [1, 2]. This section provides information regarding the science behind the detection process in biosensors, different types of biosensors, and their applications.

1.1 Basic Principals How do we detect diseases? A cell is the fundamental biological unit that carries out all tasks to maintain the well-being of living organisms. Proteins play a significant role in all cellular functions in the cells. Proteins are biological macromolecules consisting of a long chain of amino acid residues. They maintain functional and structural integrity without cells. Proteins play important roles such as regulating enzymes and hormones in the metabolic pathway, providing defensive functions against foreign invaders, signals transduction, and gene expression. For any biological process, proteins naturally associate and dissociate with other molecules. Protein–protein interactions are responsible for transmitting the extracellular signalling stimuli to the cell interior, activating intracellular receptors, and terminating the signal transduction. Any disruption of the signalling process can lead to uncontrollable or inappropriate cell growth and eventually stimulate cancers or other disease states, such as hypertension, heart disease, and diabetes. A variety of weak non-covalent interactions regulates protein–protein/ligand interactions since the biological reaction requires reversibility, VanderWaals interaction, hydrogen bonds, electrostatic forces, and hydrophobic interactions. Protein binding has a high specificity as the simultaneous formation of these weak bonds makes the ligand fit precisely into the protein binding sites. These binding sites are cavities located on the protein surface.

Microfluidic-Integrated Biosensors

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Fig. 1 Four main parts of a biosensor; sample, biological element, transducer, and signal processor [3, 4]

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Antibodies or immunoglobulins are one class of proteins responsible in the immune system for neutralising or tagging foreign objects. It is “Y” shaped and composed of two identical binding sites on each arm of the “Y” for specific recognition of its targets (antigen), linked together by disulfide bonds. Antibodies are secreted by the Lymphocyte (B cells) in response to the foreign molecules. Lymphocyte is a small leukocyte (white blood cell) formed in the lymphatic system. Lymphatic system is a vessel similar to a blood vein that conveys lymph in the body. Lymph consists of 95% water and 5% plasma, proteins, and other chemical substances. They are collected from tissues around the body and returned to the blood. Bio-molecular interactions are dynamic processes that engage association and dissociation rates in the complex biological processes responsible for hereditary and amyloidosis-related diseases. Therefore, information on the interaction dynamics or kinetics of protein–protein interactions is vital for detecting diseases and developing potential treatments. For instance, in drug optimisation, studying kinetic profiling leads to designing effective drug release, which increases patients’ compliance by long dissociation time while reducing the side effects. Scientists employed various bio-physical methodologies to quantify the protein– protein interactions. Homogeneous (solution-based) and heterogeneous (surfacebased) are the main two categories of these methods. The first measures the protein binding in the solution while the analyte and receptor are freely diffused. In the latter category, the analyte reacts with the immobilised receptor on the surface (Fig. 1). The solution-based method provides a homogeneous micro-environment and three-dimensional protein structure, including the rotational diffusion properties. Although, it can only represent a few in-vivo biological interactions. On the other hand, the surface-based method has broader applicability, as it is simple and easily parallelisable. The surface immobilisation could be challenging and cause measurement artefacts, although if it is performed with care, the results match the solution-based methods. End point and real time are two detection categories. End point carries out the detection after the reaction reaches the equilibrium and provides qualitative analysis (unknown concentrations or presence or absence of a target molecule). Real-time detection offers more details regarding the kinetic parameters and dynamics behind protein binding. Current tools for real-time monitoring of protein binding kinetics consist of two categories: classical and alternative technologies. The classical methods require a few hundred microlitres of the sample volume to fill the observation cell. Hence, the affinity-based heterogeneous biosensor is the dominant tool for characterising the protein–protein interactions and quantifying their kinetic parameters in real time. It works based on affinity recognition, which relates the concentration of the target analyte to a quantitative signal [5]. The strength of interaction between proteins is called affinity, translated into the physio-chemical property as an equilibrium dissociation constant (kd ). To calculate it, a reversible reaction with the assumption of simple 1:1 stoichiometry (Eq. 1) is considered.

Microfluidic-Integrated Biosensors

25 kd ,k a

C0 + B ←→ Cs

(1)

where C0 , B, and Cs are the concentration of proteins in the sample (analyte), free surface immobilised proteins (receptor), and bound complex (product), respectively. The terms ka and kd present the forward (association) and reverse reaction (dissociation) rate constants, respectively. The first one determines how fast the complex forms, and the latter defines the stability of the complex. The equilibrium dissociation constant can be determined by the equilibrium analysis (measuring the degree of reaction) or kinetic approaches (how fast the reaction occurs), which leads to the ratio between the reactants and bound complex (Eq. 2). Lower k D (1n M − 1 pM) presents stronger interaction and increase in the products (e.g., small molecules, inhibitors (drug) binding to their target protein). kD =

ka C0 B = kd Cs

(2)

Equation 2 can be transformed into the Langmuir binding equation (Eq. 3). Fraction Saturation =

Cs = C0 + k D bm

(3)

where bm denotes the maximum concentration of the bound complex (total concentration of immobilised receptors), as there is a fixed amount of receptors on the surface, Langmuir binding isotherm is a plot between the fraction saturation and the analyte concentration at a constant temperature, which shows distinctive regions: • when 0 < C 0 < 0.1k D , less than 9% of the receptors are bound to the analyte molecules at equilibrium. • when C0 = k D , half of the receptors are saturated. • when C0 ≥ 10k D , more than 91% of the receptors are occupied. Kinetic analysis results in dynamic characteristics of the reversible interactions, which are associated with two individual parameters (ka and kd ). The basic concept of it is to follow the rates of association and dissociation by varying the concentration of the analyte while keeping the amount of receptors low. Thus, there is a change in the concentration of target molecules in the solution, and the association reaction proceeds with pseudo-first order kinetics. For simple bio-molecular interaction, the rate of complex formation is the difference between the association and dissociation rates (Eq. 4). dC s = ka C 0 B − kd C s = ka C 0 (bm − Cs ) − kd C s dt

(4)

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1.2 Main Categories of Biosensors Biological recognition elements and transducers are two essential parts that distinguish biosensors. Figure 2 presents these two categories. The first category of biosensors (based on the biological element) includes Enzyme-based, Aptamer-based, and Antibody-based (Abs). Enzyme-based biosensors have high selectivity. They produce ions, protons, light, heat, and electrons, monitored with electrochemical methods. They are famous for their reusability, as the enzymes are not consumable. In this group, glucose and urea biosensors are the most popular ones [6]. Aptamer-based biosensors use nucleic acids (e.g., DNA and RNA) to detect proteins and target molecules. The most crucial part of their fabrication is the immobilisation of the receptors. They have high specificity, stability, and affinity. This method minimises the loss of biological elements during the reaction [7–11]. Antibody-based biosensors (immuno-sensors) do not have any need for the purification of immunogens. They are prevalent in environmental detections due to their highly specific detections [12, 13]. The detection target plays an essential role in

Fig. 2 The main categories of biosensors; are based on biological elements and transducer

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adequately selecting the biological element; antibodies and aptamers are more suitable for detecting bacteria or pathogens. For catalytic reactions, enzymes are more convenient. In the second category (based on the transducers), biosensors are grouped into electrochemical, optical, colorimetric, mass-based, and magnetic. The transducer converts a quantity of energy from one form into another. In biosensors, it transforms the bio-molecule-analyte interaction into a measurable optical or electrical signal. Electrochemical biosensors measure the variation of potential during the adsorption of the analyte on the functionalised surface. They have high sensitivity, low cost, low power requirement, and simple instrumentation [14–16]. Based on the measurement parameter of the electrochemical biosensors, they can be grouped into [17]: • • • •

Conductimetric sensor (detects conductivity) Amperometric sensor (detects electroactive species) Potentiometric sensor (detects changes in PH) Impedimetric sensor (detects changes in impedance)

Optical biosensors are easily customizable as there are a significant number of fluorescent molecules available. However, it needs sample preparation and training, and it is costly [18, 19]. Colorimetric is also label-free. Colorimetric and lateral flow tests are simple to use, portable and rapid. Pregnancy, COVID, and HIV POC diagnosis are examples of it in today’s market [20]. Mass-based biosensors have an ultra-sensitive detection and can measure tiny changes on the crystal surface [21]. Magnetic biosensors are highly customizable by providing the ability to manipulate the nanoparticles with the magnetic field. Although, due to the lack of portability and high costs, they are not favoured, which might not be an issue in the future [21, 22]. There are different methods for the immobilisation of biological recognition elements, such as adsorption, covalent binding, entrapment, and membrane confine ment. Among them, the covalent binding has better stability and irreversibility, which prevents leakage of the biological element from the support surface, so commonly this method is used [11, 23]. Another type of category belongs to the mechanical biosensors; Cantilever [24, 25] and quartz crystal microbalance (QCM) [26, 27]. Cantilever sensors are suitable for real-time detection, label-free, have low fabrication cost, high sensitivity, and can detect more than one analyte. This category includes surface stress, mass, heat, photothermal, electrostatic, and magnetic sensor. The QMC works based on mass detection. It is suitable for real-time detection but has a high level of simplicity and compatibility with POC devices.

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Fig. 3 Current challenges in design and fabrication of mobile health systems

1.3 Applications and Challenges Biosensors play an essential role in the majority of industries: • Medical: diagnosis and prognosis of cancer, heart disease, other non- communicable diseases, infectious diseases, mental health and well-being, Geo-health, personalised treatments, and medicine [28]. • Environmental: detection of pollution, toxic elements, chemical compounds, and health hazards [29]. • Food: quality control and assurance purposes from production to processing and distribution [30]. • Education and entertainment: controlling games with the mind, evaluating the level of learning during the class, health tech for workout, and body motion [31]. Technological development is a significant factor in their market’s growth. Despite the current development, there are still s a lot of challenges in design, fabrication, and use. For instance, Figs. 3 and 4 present the current challenges in the design and fabrication of mobile health systems and bio-sensing technologies, respectively [32].

1.4 Summary Biosensors are practical tools for tackling worldwide challenges with comparably low cost, high accuracy, and ease of use. However, their design and fabrication are

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Fig. 4 Current challenges in biosensor technology

still challenging. Microfluidic-integrated biosensors have shown promising results in solving these challenges, which is the topic of the next section (Sect. 2).

2 Microfluidic-Integrated Biosensors Microfluidics can tackle the challenges in bio-sensing technology (Sect. 1.3). Since the advent of microfluidics in 1990, the development of microfluidic-integrated biosensors has been intensive as the biological targets are nearly always transported by carrier fluid. Microfluidics also improves sensitivity, accuracy, and controllability while reducing the sensing region. Figure 5 presents the advantages of integrating biosensors with microfluidics. These biosensors are specifically suitable for DNA and RNA analysis, which is more accurate for detecting diseases [33, 34]. This section provides more information about microfluidics, integration of microfluidics with biosensors, and their benefits and challenges.

2.1 Microfluidics Microfluidics is the science of manipulating and controlling fluids, usually in the range of microlitre to picolitre. Almost all biological microsystems require the use of microfluidics [35]. Microfluidics studies the transport phenomena in devices, where the characteristic length scale is in the order of a micron. Figure 6 visualises the scale of microfluidic-integrated biosensors. The surface effect becomes more important in such devices as the surface-to-volume ratio is higher. Characterisation of the system dimensions by the length scale is a good source for comparison with the macroscale (Table 1). Reducing orders to micro will reduce the amount of required samples. For instance, reducing the scale by a fraction of 103 will reduce the volume by a factor of 109 . This reduction speeds up analysis,

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Fig. 5 Benefits of integrating biosensors with microfluidics

Fig. 6 Visualisation of the scale of microfluidic-integrated biosensors

leads to cheap production, and provides an efficient detection scheme. As Length (L) decreases in microscale, surface forces are likely dominant over volume forces. In miniaturised systems (L → 0), gravity and inertia forces, Weber, Reynolds number, Péclet number (relates the rate of advection to rate of diffusion), Bond

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Table 1 Effect of length scale on the typical physical quantities in microfluidics scale Property (length scale)

Details in microfluidic devices

Volume (L 3 )

Smaller system size, less sample, and fast process

Diffusion time scale, viscous and inertial forces (L 2 )

Creeping (Re 0.015, the droplet generation regime shifts to the other regimes in which a thread of dispersed phase penetrates the continuous phase. In the next three modes, the viscous force becomes an influential factor, and they are generally stable.

2.4.2

Dripping

At higher Ca than squeezing, the viscous shear forces dominate over interfacial forces and they become large enough to break up the interface before channel obstruction. Consequently, smaller droplets than channel dimensions will be formed. In this mode, as the Ca increases (i.e., the Qc increases), the droplet size decreases. For many applications, dripping is favourable over other regimes as the formed droplets are highly uniform, which is attained over a wide range of flow rates. Droplet size and frequency for different geometries at the dripping regime have been studied by researchers [35, 56, 83, 107] as reviewed by Zhu et al. [111].

2.4.3

Jetting

With a further increase in fluids flow rates, the droplet formation regime would be switched to jetting, in which the penetrating thread is decayed into very small droplets with diameters as small as 1 µm as a result of Rayleigh–Plateau instabilities [71]. 

The diameter of the dispersed phase penetrating thread depends on QQdc . In the two regimes previously discussed, monodisperse droplets are generated; but in the jetting regime, capillary perturbations cause the size of the droplets to be polydisperse. In this mode, the Ca in which the droplets start to form, Cacri , is between 1 and 2 and depends on the viscosity ratio [39]. Jets could be either narrowing or widening; in the former viscous and capillary forces and in the latter, inertia and capillary forces predominate.

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One aspect that needs to be further considered in numerical research is that in grid-independent studies usually cases with moderate Ca are tested with different grids. But, if the authors emphasise their findings in the jetting regime, then much finer meshes are required to have grid-independent results. That is due to the fact that the droplets generated in the jetting, are much smaller than the ones formed in the squeezing or dripping regimes. Roughly, each droplet needs at least 6 mesh cells along its diameter. To give a comparison, when a droplet in a squeezing regime in a channel with 200 µm in width is generated, the droplet would have at least 200 µm in length, meaning that even a grid size of 25 µm would give acceptable results. But, if a droplet generation in jetting regime is simulated, it could have a 10 µm diameter, and then the mesh size lower than 2 µm in the droplet generating regions (usually the centre of the channel) would be required, which is much smaller than the usual reported grid size in numerical works. Aside from the droplet size, the very thin thread of dispersed phase formed inside the continuous phase might even need a smaller grid size to be accurately simulated. Yet, unless the hydrodynamics of the thread is a focus of study, our experience shows that satisfying the droplet mesh requirements would give accurate results for the droplet size, composition, and frequency.

2.4.4

Tip-Streaming

This regime was first observed by Suryo and Basaran [86] and they reported that in a co-flow channel, at high Ca, a very thin, cone-shaped jet formed and produced very small droplets that are surprisingly, highly monodisperse, as can be seen in Fig. 2d. When droplets are subjected to intense shear stress, a pointed tip is shaped from which small droplets are emitted. There is not much numerical work reported in this regime, because first, it would be too computationally costly; second, applicationwise, usually larger droplets than the ones generated in the tip-streaming are needed.

2.4.5

Tip-Multi-breaking

Same as the previous mode, due to Rayleigh–Plateau instability [71], small droplets are formed from the tip of the cone-shaped thread. But here, the cone is unstable and periodically polydisperse droplets are formed. This too has gained limited attention because aside from the factors mentioned in the tip-streaming, the formation of nonuniform droplets is another prohibitor as in most practices, uniformity is crucial.

2.4.6

Parallel Flow

There is one more regime called parallel flow in which no droplets are being formed and the two phases move parallel to each other in the channel. This is caused by the high flow rates of both phases. Figure 3 depicts such a regime in T-junction and cross

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Fig. 3 Parallel flow at a T-junction; b cross junction

junction. All around, surface features and flow rate ratios should be designed in a way to avoid parallel flow. Since in this regime, droplets are not formed, it has not been studied much. Guillot et al. [41] initially provided insight into parallel flow and showed that the dispersed phase flow rate is the determining factor and not the Ca. As for CFD approaches, Li et al. used the VOF model in FLUENT to numerically and analytically investigate the mechanisms of droplet formation and discussed parallel flow [58]. They used the pressure profile along the streamwise direction to illustrate the pressure distribution when shifting from jetting to parallel flow. Malekzadeh et al. [62] used OpenFOAM and numerically showed that by changing the wall contact angle and flow rates, the parallel flow could be controlled. They showed that in an air–water system, at very low Ca, changing the contact angle from 180° to 140°, would form a long stream of the dispersed phase parallel to the continuous phase. They have concluded that in lower contact angles, the adhesion between the droplet and channel walls would become stronger, making it more difficult to form the droplets [62].

3 Effect of Operating Parameters As shown in the previous sections, many factors affect droplet formation. Ca, We, and flowrate ratios defined the droplet generation mode which determines the droplet size and frequency. Another factor is surface tension which can be manipulated by adding a surfactant to the fluids. There are two other important factors, surface wettability, and fluid viscosity, which are further explained in the following sections.

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3.1 Surface Wettability Another parameter with a high impact on droplet formation is surface wettability. When aqueous droplets are being formed, the channel walls must be hydrophobic to make sure the continuous phase efficiently wets the walls and helps the dispersed phase to have minimum contact with the wall so that droplets could be formed. It should be noted that in cases where oil droplets need to be formed in the aqueous phase, then the microchannel must have a hydrophilic surface [104]. In experimental studies, it is rather difficult to change the surface wettability, one reason being polymers used to create microfluidic channels mostly possess transient wettability between hydrophilicity and hydrophobicity. Hence, in controlled droplet microfluidics, the channel surface must undergo chemical treatment; for instance, polymeric walls can be coated with a coating agent to guarantee super hydrophobicity or super hydrophilicity as needed [50]. On the other hand, numerical investigation of surface wettability is more feasible as only the value of surface properties needs to be changed in the simulation setup. Several researchers have studied the effect of wall hydrophobicity by changing the wall contact angle boundary condition which quantifies the solid surface wettability in contact with a specific liquid. Bashir et al. [15] have published a remarkable work and used the LS method to numerically examine all the impacts of contact angle on droplet break-up time, generation frequency, and length. They reported that by increasing the contact angle from 120° to 180°, break-up time decreases approximately linearly and droplet frequency increases. Moreover, they stated that at Ca < 0.02, the contact angle did not impose a considerable effect on droplet size, while at Ca > 0.02 its effect became significant [15]. Dang et al. [29] used CLSVOF to investigate the effect of contact angle on the dispersed fluid volume in a 3D converging shape mixing junction and reported that in general, they have a negative correlation with each other. Yet, at high liquid viscosity, the bubble volume does not change much with increasing the contact angle [29]. In a more recent paper, Wang et al. [94] investigated the effect of wall contact angle on a specific geometry called Terrace-based microfluidic devices. This geometry is particularly interesting for step emulsification. They clarified that in that geometry too increasing the wall contact angle causes a definite reduction in droplet size using a VOF model and analysed the results by monitoring the pressure at the breaking point [94].

3.2 Viscosity 3.2.1

Newtonian Fluids

Viscosity and viscosity ratio are other parameters that hugely affect droplet formation. The viscosity effect has been a focus of study since the emergence of droplet microfluidics due to the importance of viscous forces on the interface.

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In the squeezing regime, viscous forces are not dominating, and consequently, the viscosity ratio is not prominent. In 2008, De Menech et al. [31] numerically simulated the flow of two immiscible fluids in a T-junction using a phase-field model. They plotted pressure changes versus Ca for different viscosity ratios, λ = 18 to 1, and showed that in the squeezing regime, the pressure profile was weakly affected by the viscosity ratio and droplet volume was only a function of Q, obeying Vd = 1 + 2Q. Later on, Liu et al. [60] investigated the role of λ during the droplet break-up at a T-junction using LBM and stated that the value of Ca in which the transition from squeezing to dripping occurred, was independent of the λ. At higher Ca corresponding to dripping and jetting, viscose force influences the droplet formation as it causes the interface to rupture. In general, increasing the viscosity of the continuous phase would impose a higher viscous force, causing the detachment to happen sooner. Thus, smaller droplets would be formed. In Zhu et al.’s [111] review paper, the reported equations by different authors are gathered to show the effect of viscosity on droplets’ diameter. Wang et al. [95] used LBM to examine the effect of λ on the droplet detachment point and concluded that at high λ and Q, the location of droplet detachment was shifted further downstream. In more recent work, Li et al. [56] studied the impact of viscosity ratios in a range of slightly different geometries. It was reported that by changing the viscosity ratio, the droplet generation regime could be shifted from dripping to jetting.

3.2.2

Non-Newtonian Fluids

In Newtonian fluids, the viscosity is constant and independent of stress. In nonNewtonian fluids, as the relationship between shear stress and shear rate is non-linear, viscosity can change as a function of strain tensor to either lower or higher values [49]. μ = f (γ˙ ) =

shear stress tensor [Pa] τ = γ˙ shear rate tensor [1/s]

(6)

Newtonian fluids in droplet microfluidics have been extensively studied in both experimental and analytical approaches. Although these studies provide a good understanding of channel hydrodynamics, a wide range of solutions in real-life analysis are non-Newtonian, including proteins and polymers with large and complicated structures. Another example is bacterial studies. The droplet is a microreactor that should have nutrients and enzymes in it so that bacteria can grow. The common reaction environment, i.e., the droplet, is an agarose solution, a shear-thinning (Pseudoplastic) fluid [48]. As shown in Sect. 3.2.1, viscosity could highly influence droplet formation. Therefore, it is vital to examine non-Newtonian flows, too, and compare them with Newtonian cases. A limited number of studies have examined the non-Newtonian behaviour of fluids in droplet microfluidics. In those studies, one of the phases could be nonNewtonian; most of them investigated the effect of the non-Newtonian continuous phase on droplet generation.

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Sang et al. [75] numerically inspected the influence of two types of non-Newtonian fluids, power-law and Bingham, as the continuous phase and found that in the former, increasing the power-law index (n) and the consistency coefficient (k), causes the droplet size to reduce. In the latter, the yield stress dominated over droplet size. Sontti et al. [84, 85] used VOF to explore the Newtonian droplet formation in a non-Newtonian continuous phase for a T-junction [84] and a cross junction [85]. They have concluded that rheological parameters severely controlled the droplet size, frequency, and formation regime. As they increased the apparent viscosity, droplet formation frequency increased, and droplet size decreased. Later on, Chen et al. [23] studied the impact of power-law model indexes on droplet formation and showed that n higher influence on droplet formation than k. More Recently, Besanjideh et al. [17] numerically explored the influence of the non-Newtonian continuous phase which was created by the addition of nanoparticles to it. In nano-fluid solutions, enhancing the shear rate reduces the apparent viscosity. This allowed them to achieve very high viscosity ratios up to 70. They mentioned that changing the Newtonian continuous phase to a non-Newtonian one, i.e., adding Fe2 O3 , shifted the formation regime from squeezing to a transition regime. Moreover, if another type of particle, Al2 O3 , was used instead of Fe2 O3 , dripping and jetting regimes could be observed [17]. So far, all the mentioned works had a non-Newtonian continuous phase. Some researchers investigated the effect of the non-Newtonian dispersed phase on droplet generation. One of the first works on non-Newtonian droplets was conducted by Arratia et al. [11], who experimentally observed a slower decay rate in nonNewtonian polymeric solutions. Wong et al. [97, 98] chose a Carreau-Yasuda polymeric dispersed phase and used the LS method in a series of polymer concentrations and reported that droplet pinch-off time and frequency are influenced by the polymer concentration. They demonstrated that the enhancement of viscous forces delayed the droplet pinch-off time. Taassob et al. [87] numerically explored the effect of nonNewtonian polymeric dispersed phase with moderate concentrations. They stated that in the dripping and jetting regime, monodispersity still can be attained. They also suggested that at any velocity ratio values higher than a critical ratio, droplet volume becomes almost independent of the viscosity [87]. Fatehifar et al. [35] examined the droplet size and detachment time to quantify the impact of the non-Newtonian behaviour of the dispersed phase on droplet generation in different regimes using VOF. They showed that in the squeezing regime, such effect was not much, yet in higher Ca the effect became more apparent, and then in high concentrations of polymer, the regime altered from jetting to tip-streaming. Furthermore, they illustrated that the extensional viscosity of the fluid resisted the pinch-off at the tip of the thread, causing a delay in droplet formation [35]. Even though the works mentioned in this section give a good picture of the effect of non-Newtonian fluids on droplet microfluidics, more research is needed to cover other dynamical aspects of this field. Higher ranges of viscosity, different models for non-Newtonian fluids, time-dependent fluids, and visco-elastic fluids could be further

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studied. The non-Newtonian models and their Constitutive equations are listed by Koh et al. [52]. Moreover, owing to the complex rheological characteristics, achieving monodispersed droplets is not as feasible as in Newtonian cases. Implementation of these models in different CFD software also is not very straightforward and needs more documentation. These eventually provide better, and perhaps full control over the performance of droplet microfluidics so that it becomes suitable for industrial applications.

4 Complex Droplet Formation So far, we have discussed single emulsions where a liquid is dispersed into another liquid. Yet, there are more complex flows in droplet microfluidics of which two of them are introduced and discussed here, double emulsion and encapsulating droplet. In the former, three liquid phases, and in the latter, two liquid phases and one solid phase are in the domain as illustrated in Table 1. Valuable experimental works on both subcategories are available in the literature [50, 78]. Due to the physics added, numerical models would become even more complex and computational costs would rise. As the focus of this book is on modelling and CFD of complex flows, the limited numerical works in the mentioned fields are explored in the following sections.

4.1 Double Emulsions In droplet microfluidics, double emulsions also called multiple emulsions, are structures in which one droplet gets trapped inside another droplet. If the outer droplet (shell) is thin, it is called coating; and if it is thick, it is called droplet encapsulation. The Shell layer serves as a protective layer for the core drops. Table 1 Complex droplet formation in droplet microfluidics Liquid–liquid systems Thin Shell (D/h > 1000))

Liquid–solid systems Thick shell (D/h < 10)

Particle in droplet

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The concept was initially started with water-in-oil-in-water (W/O/W) and oil-inwater-in-oil (O/W/O). It was then extended to three or more different phases. To form a double emulsification, one instability or two instabilities can be used. If the inner and outer droplets are generated at the same time, then the process has one instability. If the first droplet is formed in the first junction and then introduced to another junction to be encapsulated in a different liquid, then it is a two-instabilities process [1] as depicted in Fig. 4a–d and e–f, respectively. For each step, different geometries could be used, for instance, two T-junctions or two co-flow or one of each can be employed to reach double emulsions. In two-step methods, the surface wettability should be manipulated so that the first part of the channel is hydrophobic, and the second part is hydrophilic. In the first part of the channel, the shell phase must wet the channel, and then in the second part, the continuous phase must wet the channel. An advantage of one-step formation over two-step formation is that the surface features do not need to be as strict and precise. As shown in Fig. 4e and f, the penetrating inner jet is surrounded by the middle phase and does not become in contact with the walls. As for that, one-step formation is more feasible in practice. Controlling the surface wettability in a way that changes through the channel, which is particularly difficult in experimental approaches, the surface treatment methods are reviewed by Sattari et al. [76]. In CFD approaches, it is rather to change the values of surface features. Thus, obviously using CFD could help to save time and cost. Some authors used the same inner and outer fluid to make the simulation simpler because this way, models for two-phase flows would be practical. Nabavi et al. [64, 65] harnessed an incompressible three-phase VOF-Continuum Surface Force (VOF-CSF) numerical model and first validated their results against

Fig. 4 Producing double emulsions in a one or two-step process for Cain = 0.011 and Caout = 0.035, a Wein = 0.4, b Wein = 0.6, c Wein = 0.8, d Wein = 0.9, e Wein = 1.1, f Wein = 1.2. The ‘in’ subscript represents the inner phase, shown by the colour white [13]–Reprinted from [13] with permission from AIP Publishing

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experimental work. Then, they conducted a parametric study and investigated the impacts of channel geometry, fluid properties, and phase flow rates on the droplets’ size and frequency. They mentioned that the volume of both inner and outer droplets was reduced as either the outer fluid flow rates or the viscosity increased [65]. Furthermore, they simulated the break-up mechanisms in dripping and jetting regimes and discussed the effect of vortices and velocity gradients at the tip of the thread in each regime. They have further extended their work on simulating multi-core double emulsion droplets [66] and concluded that a maximum of six stable inner droplets could be encapsulated in each compound droplet. As the regimes of interest in double emulsions are dripping and jetting, Chen et al. [25], simulated the behaviour of flow in a flow-focusing device on those regimes and mentioned that the outer fluid flow rate dictates the droplet formation regime and consequently, the droplet size. Moreover, the volume of the formed droplets increases as the viscosity ratio of the core fluid to the shell fluid rises [25]. Adaptive Mesh Refinement (AMR) is a technique for tracking the interfaces and it allows one to control the grid size with respect to a scalar field and can potentially reduce the computational costs and increase the accuracy as the mesh in the interface region is constantly refined by the solver. Thus, a very small grid size is not needed over the entire channel. One of the early works which used (AMR) in a finite element algorithm to simulate the formation of double emulsion in microfluidics was done by Zhou et al. [110], who first validated their work by reproducing the [90] results and then examined the impact of flow rates, viscosity ratio, and viscoelasticity on double emulsion formation. Another code capable of doing so is the Gerris, a freely available adaptive 3D PLIC-VOF code. Azarmanesh et al. [13] used it to adjust the thickness of the shell (the middle phase) in one-step and two-step double emulsion formation; see Fig. 4. LBM is proved to be a valuable tool to simulate such flows as well and many authors have validated their LBM results against experimental works [12, 37, 67, 99, 106]. The work of double emulsion could be further extended to multi-core emulsion which is applicable in multi-component reaction and mixing, multiple cells coculture, and encapsulation of incompatible active materials to avoid cross-contamination [76]. It has been experimentally investigated to some extent [66, 102, 109], but lacks numerical investigations.

4.2 Encapsulating Droplets Droplet microfluidics enables researchers to encapsulate cells or solid particles inside small droplets which are known as a breakthrough specifically in the field of biology. Compared to conventional detection methods with a long process time, microfluidics can shorten this time to minutes [52] as well as provide high accuracy. It is not

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only applicable in real-time detection, bead-based PCR, and tissue printing, but also in single-cell studies, the most fundamental level of biology. Working with single cell is known to be difficult because it is almost impossible to isolate a single cell. Yet, microfluidics made it possible to work with single cells. In 2008, Edd et al. [33], successfully encapsulated single cells into monodisperse picolitre drops in their experiments using a cross junction. One of the drawbacks of single-cell studies is that the loading of cells into droplets is purely random, and the probability of a droplet having exactly 1 cell in it is determined by Poisson statistics. In the best case scenario, the probability is around 30%, i.e., 2 empty droplets and 1 loaded droplet, with most of the works having an output smaller than that. To minimise the number of drops containing more than 1 cell, cell concentration at the inlet should be very low, and consequently, a large number of droplets will be empty and useless. Edd et al. [33] stated that, based on Poisson statistics, to avoid having more than 1 cell in 90% of the loaded droplets, only 15.6% of all the generated droplets would have exactly one cell encapsulated. To address that issue, microfluidics can come to help. Particle/cell sorting in microfluidics has been a point of focus since the emergence of microfluidics and is well studied and developed. A long microfluidic channel at the entry of the droplet generating channel could give control over cell sorting, making it possible to organise the cells to be evenly spaced before they reach the droplet generating junction. There are some experimental works in the literature reporting successful encapsulation of particles/cells in droplets [14, 16, 33]. Beneyton et al. encapsulated single fungal spores in droplets and then incubated them for growth and further analysis [16]. Banerjee et al. [14] encapsulated microparticles in aqueous ferrofluid droplets and then sorted empty and loaded droplets using a magnetic field and concluded that for the aim of encapsulation and sorting, the jetting regime is the most appropriate. In a recent review, Amirifar et al. [6] discussed recent advances in this field. They have mentioned a significant challenge of encapsulating droplets is the precise control of cell counts in droplets. On the other hand, numerical works in encapsulating droplets are scarce. This is due to the presence of particles and cells along with a multiphase flow. Moreover, solid particles and cells could be soft, making it even more complicated. To model multiphase flow with cells, two approaches are possible. In the first one, the Eulerian–Eulerian approach, a multiphase VOF algorithm is used; thus all phases are governed by momentum, mass, and energy conservation equations. The problem with this approach is that the model ignores the discrete nature of cells, meaning that particle–particle and particle–fluid interaction, as well as interphase momentum transfer, are all neglected. The second approach is Eulerian–Lagrangian. Here, the interaction between 2 liquids is solved by the Eulerian part and then the particle motion is modelled by the Lagrangian part. Coupling the Lagrangian solver with the Eulerian solver allows one to capture the trajectories of particles and solid–liquid interactions. The possible approaches for modelling fluid–solid multiphase flow have been reviewed by Ariyaratne et al. [10]. Heinrich et al. [44] used OpenFOAM and an

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adaptive mesh refinement tool to simulate the spray atomisation process employing the two mentioned methods and discussed the superiority of Eulerian–Lagrangian models. To capture the motion of particles in a fluid, there are two transient solvers capable of conveying the effect of solid phase volume fraction to the continuous phase and vice versa. They are Discrete Particle Modelling (DPM) and Multi-Phase Particle In Cell (MPPIC). Originally, the DPM solver was established to model high loads of solid particles and it is designed to directly calculate the particle–particle interaction and involved forces. The solver has been explored and used by many authors [2, 34, 36]. Adeniyi et al. [2] have explained the coupling of DPM with CLSVoF using ANSYS Fluent to model droplet behaviour. Fernandes et al. [36] fully explained the governing equations and involved forces and then validated the DPM solver in OpenFOAM. Still, the direct calculation of particle–particle interactions could be heavy. Hence, to reduce the computational expenses, the MPPIC model was developed by Andrews and O’Rourke [7] which substitute’s the direct calculation of particle collision with some models and the famous Liouville equations. It has been harnessed to model particles in one or two fluids since [51, 54, 59]. Recently, Caliskan et al. [20] employed MPPIC to model large particles in a bubbling fluidized bed. The first effort to simulate this cell encapsulation process was conducted by Yang et al. [105]. They simulated the behaviour of soft cell encapsulation in droplets, generated in a T-junction in the dripping regime, and used the LBM combined with the Immersed Boundary Method (IBM). They stated that the bending total volume constraint and stiffness of the cells must be considered to have physically accurate results. They found that (a) the presence of the cell does not impose any significant changes on the flow streamlines; (b) the cell only deformed in the main channel, where the velocity is higher than in side channels [105]. Yaghoobi et al. harnessed the finite element method—COMSOL Multiphysics software—and level set approach to evaluate the effects of droplet interface and walls on cells—assumed as rigid spherical particles—using static surface tension. They also mentioned that the distance between the particles at the entrance is the main factor affecting the encapsulation [103]. Furthermore, Outokesh et al. [69] also used COMSOL Multiphysics software to track the movement of very small magnetic beads in a droplet-generating microfluidic channel and investigated the effect of magnet distance on the number of particles in droplets during droplet splitting. The lack of numerical work in encapsulating droplets can be addressed using the mentioned models above. CFD approaches can help control the operating parameters in particular the particle concentration at the inlet, i.e., the space between particles, flow rates, and viscosity of the continuous phase so that exactly one particle/cell in each droplet, with a specific size, would be encapsulated.

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5 Summary Droplet microfluidics brings about novel high-through applications along with the known benefits of microfluidic techniques (i.e., Reduced sample usage, shorter process time, higher control). This chapter tried to introduce droplet microfluidics from an engineering point of view and highlighted the applications of CFD approaches in the complex process of droplet generation. Numerical works on the geometry design of the channel are mentioned and it seems that this aspect of droplet generation is almost mature. Yet, more accurate numerical studies using powerful meshing tools are needed, so that near-wall regions and liquid-liquid interface can be modelled with high precision (specifically in narrow jets and droplet splitting). Furthermore, multiphase systems, including 3 liquids, and solid particles in a 2 liquid system, are in need of more investigation. Numerical studies of particle encapsulation in droplets are scarce in the literature and more work is required to understand the physics involved, such as interface and particle forces, which are not very straightforward to assess in experimental works. Gaining better control over the system output can pave the way to accurately design high-throughput droplet microfluidics for real-life assays.

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Subject Specific Modelling of Aortic Flows Amin Deyranlou, Alistair Revell, and Amir Keshmiri

Abstract Cardiovascular diseases (CVDs) are the principal cause of morbidity worldwide. According to the World Health Organisation (WHO), around 17.9 million deaths were reported in 2016 due to CVDs, representing 31% of the global death, while this number is expected to reach over 23.6 million by 2030. Arterial disease, stroke, transient ischaemic attack, and rheumatic heart disease are the most prevalent CVDs. British Heart Foundation (BHF) has reported that around 7.4 million people in the UK are living with CVDs, which imposes a £9 billion annual cost on the healthcare system. In recent years, advances in vascular biology, biomechanics, medical imaging, and computational techniques including Computational Fluid Dynamics (CFD) have provided the research community with a unique opportunity to simulate and analyse blood flow from a new angle and to develop new strategies for intervention. The increasing power-to-cost ratio of computers and the advent of methods for subject-specific modelling of cardiovascular flow have made CFD-based modelling sometimes even more reliable than methods based solely on in-vivo or in-vitro measurement. This chapter explains a workflow for subject-specific modelling of blood flow in the aorta as an exemplar of digitalisation in healthcare. The workflow comprises multi-modal clinical images, a multiscale numerical pipeline, and haemodynamic metrics. Subject-specific modelling primarily relies on clinical data, which is reachable through different clinical imaging modalities to get the anatomy and flow A. Deyranlou (B) Wellcome/EPSRC Centre for Interventional and Surgical Sciences (WEISS), Department of Medical Physics and Biomedical Engineering, University College London, 43-45 Foley Street, London W1W 7TS, UK e-mail: [email protected] A. Deyranlou · A. Revell · A. Keshmiri Department of Mechanical, Aerospace and Civil Engineering (MACE), The University of Manchester, Manchester M13 9PL, UK e-mail: [email protected] A. Keshmiri e-mail: [email protected] A. Keshmiri Manchester University NHS Foundation Trust, Manchester Academic Health Science Centre, Southmoor Road, Wythenshawe, Manchester M13 9PL, UK © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_4

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data of the Region of Interest (ROI). At the next stage, the computational pipeline should be set through appropriate boundary conditions. The latter requires a multiscale approach to couple the three-dimensional CFD model to zero/one-dimensional circuits. These circuits normally mimic upstream and downstream regions, which are not included in the three-dimensional CFD domain, however, they affect crucially and are to be considered for an accurate personalised medicine. Once the pipeline has been set, it can suggest complex blood flow behaviour because of different pathological conditions that might emerge throughout the vascular network. At this stage invoking accurate and reliable haemodynamic metrics can translate the simulated data into interpretable clinical output, which is the main goal of the workflow.

1 Introduction Despite significant developments in clinical diagnosis and treatment of cardio and cerebrovascular diseases, the population suffering remains remarkably high worldwide. World Health Organization (WHO) report shows that the annual death due to Cardiovascular Diseases (CVDs) are increasing. In the UK alone, around 7.4 million people live with a type of cardiovascular disease, which imposes a £9 million annual cost on the healthcare system [1]. Vascular defects incorporate a wide range of different abnormalities in blood vessels such as stenosis, coarctation, dissection, aneurysm, ectasia, malformation, etc. Clinical management of vascular diseases has always been convoluted and multifacet. A broad spectrum of factors, including age, gender, ethnicity, lifestyle, dietary habits, and other co-morbidities, can massively affect disease aetiology and pathophysiological conditions. One very critical aspect which has a significant contribution is blood haemodynamics. Generally, clinical routines for diagnosis and prognosis take place through highly developed imaging modalities. However, the images still lack temporal and spatial resolutions, which are unable to fully resolve flow information. Furthermore, treatment planning and subsequent follow-ups mainly rely on clinical management. In fact, clinical management is based on long-term measurements and previous experiences, while blood haemodynamic impacts are neglected. Within the last decade, endeavours have been triggered to utilise state-of-the-art imaging technology along with high-fidelity computational models to introduce a reliable in-silico platform for precision medicine. The vascular network comprises a huge network of arteries, arterioles, capillaries, venules, and veins. Given the structural and functional discrepancies in various vascular segments, different levels of complexity are included in the model. Nevertheless, a similar computational workflow needs to run for the vascular modelling. In this chapter, the vascular workflow is introduced through the most complex vessel, ‘Aorta’. Aorta is the largest artery, which originates from the left ventricle and branches off towards the upper and lower limbs, to deliver oxygenated blood to the entire body. There is a wide spectrum of aortic diseases, which are aortic aneurysms,

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acute aortic syndromes (including aortic dissection, intramural haematoma, penetrating atherosclerotic ulcer, and traumatic aortic injury), pseudoaneurysm, aortic rupture, atherosclerotic and inflammatory diseases, genetic diseases (e.g., Marfan syndrome, Turner syndrome, etc.), and congenital abnormalities (including the coarctation of the aorta, bicuspid aortic valve, etc.) [2]. Therefore, in-depth knowledge about the aorta including but not limited to its morphology, structure, and blood haemodynamics would help physicians in efficient decision-making and therapy planning. Figure 1 displays a schematic of the aorta with all subsidiary branches at ascending aorta, aortic arch, descending thoracic aorta, and abdominal aorta. The figure distinguishes between the single and paired branches as depicted. Aorta is one of the very diverse arteries, in sense of its huge differences in its morphology amongst different groups of people. In addition to different geometrical sizes, the supra-aortic branches can have different configurations [3, 4]. In this chapter, the entire workflow (Fig. 2) for modelling vascular haemodynamics is illustrated, using an aorta as an exemplar. Therefore, the chapter initially describes the clinical data acquisition and post-processing, then expands on the governing equations, boundary conditions, and main haemodynamic metrics, and finally explains the haemodynamics in the aorta.

2 Clinical Imaging Reconstructing the geometry of the Region of Interest (ROI) is the first stage in course of in-silico modelling of vascular flow. The geometry can be an idealised model [5, 6] or a realistic one [7, 8] of the vessel. For the idealised model, geometrical specifications are derived from the anatomical measurements of different ages, genders, and ethnicities [9–11]. Then, they are combined with appropriate mathematical models to create the geometry. With the development of cutting-edge clinical imaging technology, subjectspecific geometry reconstruction has become a resilient procedure for personalised in-silico modelling of vascular haemodynamics. The aforementioned advancement has been a breakthrough towards the idea of digital twins in the medical sector. In the following section, a list of the most common imaging modalities is provided.

2.1 Imaging Modalities Anatomy of the vascular system can be obtained through different imaging modalities; however, each has been designed to operate better for certain clinical interventions. The most widely used imaging modalities are echocardiography, Magnetic Resonance Imaging (MRI), and Computed Tomography (CT).

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Fig. 1 Schematic of an aorta, including the main aortic conduit and its primary branches

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Pressure Velocity Vortex WSS OSI, …

Fig. 2 A schematic of subject-specific modelling of cardiovascular fluid mechanics workflow

Echocardiography: this modality works based on ultrasound and there are two types of echocardiography, namely, transthoracic echocardiogram (TTE) and transoesophageal echocardiogram (TEE). TTE is a non-invasive method in which a probe is placed on the chest, while TEE is a minimally invasive method in which the transducer passes through the oesophagus to provide more clear images of the ROI. Echocardiography can provide data in 2D planes or 3D volumes. 2D echocardiography has a high temporal resolution, between 30 and 100 frames per second, which makes it a great technique to track rapid valve motion. In 3D echocardiography, the ultrasound sequence gets approximately 20 volumes per second, which is expected to rise with new technologies. Although echo has a good temporal resolution, the images are accompanied by some artefacts and noises, which undermines the overall quality of the images. Cardiac CT: another imaging modality is cardiac CT, which works based on Xray radiation. The modality provides isotropic spatial resolution with a higher quality of around 0.5 mm and an acceptable temporal resolution in a range between 50 and 175 ms. Moreover, 4D cardiac CT provides phasic images, which enables to track of the motion of the vascular wall and myocardium. The main drawback of CT is the ionising radiation hazard, which safety is still under debate [12, 13]. Cardiac MRI: MRI modality is free from the radiation hazard, and it works based on a high magnetic field. It provides nearly a high in-plane spatial resolution of around 1.5 mm but low through-plane spatial resolution, typically above 2.5 mm. Furthermore, MRI scan time is long, so it is accompanied by some artefacts; though, to avoid artefacts, multi-beat protocol and phase-averaging by ECG synchronisation is a common practice. Generally, the MRI data has a lower quality compared to CT, so, geometry reconstruction using the MRI is more cumbersome. In addition to the anatomy, some imaging modalities are designed to provide a dynamic picture of the cardiovascular system, and field data such as intraluminal velocity and pressure. Cine-MRI and Phase Contrast (PC) MRI are two subsidiaries of MRI, which inform physicians about vascular movements and flow haemodynamics in a cardiac cycle, respectively. The images are normally taken for a few phases in one heartbeat, while they are gated by electrocardiogram.

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2.2 Multi-modal Imaging Routine Morphological, structural, and physiological differences of different individuals require a high level of personalisation in subject-specific computational modelling. Therefore, a robust computational workflow for personalised medicine needs to consider all these discrepancies, which are currently taking place through multi-modal imaging techniques. Now the main question to be answered is how multi-modal imaging procedures would help in effective modelling. To clarify the above question, vascular function and its structure should be explained. Generally, vascular walls are made from a multilayer structure, which is mainly tunica intima, media, and adventitia. Each layer comprises different materials, which overall induce heterogeneous, anisotropic properties to the vessel wall. Furthermore, the vascular morphology and flow inside are impacted by different factors, where amongst the most important ones can refer to age, gender, ethnicity, height, weight, and lifestyle. In conclusion, due to the high level of complexity, subject-specific modelling requires an accurate geometry reconstruction and wellimposed boundary conditions. Reaching that point, the practical approach is to use the multi-modal imaging routine, and it means benefiting from different modalities to acquire morphology, wall movement, and flow information inside the lumen and at extremities.

2.3 Segmentation, Geometry Reconstruction, and Flow Data Acquisition After data acquisition, the raw images need to be interpreted to be used for computational modelling. There are different algorithms have been developed over the past few decades. Seminal algorithms majorly require manual segmentation; however, more developed techniques have reduced manual manipulation and are semiautomated. The main algorithms, which have been developed for image segmentation are thresholding-based, region-based, edge-based/boundary-based, clustering method, level-set method, artificial neural networks, atlas-guided approach, and genetic algorithm [14]. Geometry reconstruction starts from image registration and ends up with a CFDready domain. Figure 3 displays the entire pipeline. Utilising some of the abovementioned algorithms, there is a long list of different open-source and commercial software. The most famous available codes, are Mimics [15–18], Simplware ScanIP, VMTK [19–21], VTK [22], GIMIAS [23, 24], 3D-slicer [25], ITK-Snap [26–28], MITK [29], Seg-3D, Medis, Segment [30], CVi42, and GTFlow. These software packages work based on different segmentation and mapping algorithms, which are beyond the scope of this chapter.

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Image registration

Preliminary segmentation

Proofing/finalising segmentation

importing raw data and classification

general morphology of the region of interest (ROI)

verifying the accuracy of the segmented region

CFD ready geometry

CAD treatment

Smoothing

exporting a right format to be ready for meshing

further geometry modification if needed

smoothing irregularities, ridges, steep edges, etc.

Fig. 3 Image processing and geometry reconstruction workflow

Fig. 4 Different velocity profile assumptions as inlet boundary condition for a laminar flow: (a) Plug profile, (b) parabolic profile, (c) Womersley profile, and (d) planar velocity encoded subject-specific profile

3 Multiscale Computational Workflow In this section, the computational method for modelling vascular flow will be explained. Creating an appropriate mesh, selecting well-posed boundary conditions, the governing equations, and the numerical methods, are the main areas that will be further explored. In the beginning, it is aimed to clarify the concept of “multiscale modelling”. The vascular system is constructed from a massive, interconnected network of arteries, arterioles, capillaries, veins, and venules. Given the current development and computational power, it is not feasible and reasonable to entirely simulate

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the 3D model of a vascular bed. Therefore, for each ROI, the rest of the vascular network at flow upstream and downstream are put aside. This point is where the concept of “multiscale” comes into play. As will be explained later in this section, employing a 0D circuit at the inflow and outflow boundaries can physiologically mimic the absent parts. In fact, the technique tries to consider upstream and downstream resistances and compliances, which impose special conditions at extremities. Therefore, they substantially improve the reliability of the computational model by increasing the accuracy of estimation of the field variables, such as velocity vector and pressure. This section firstly elucidates the common types of boundary conditions in aortic flow and then elaborates on governing equations and numerical techniques to establish a robust mathematical model.

3.1 Boundary Conditions In modelling vascular flow, a subtle selection of boundary conditions has paramount importance. In particular, modelling flow in an aorta artery requires special treatment for each boundary type. Subject-specific modelling of the aortic artery requires three types of boundary conditions. Flow at the aortic inlet, just after the aortic valve, flows at the outlets of arteries, which are branched off from the main body of the aorta, and the aortic wall. In this section, each boundary type will be explained, and different possibilities will be discussed. Flow inlet: the blood is pumped out by the heart as a pulsatile waveform. The shape of the flow waveform illustrates important information, which belongs to various instants of a cardiac cycle. Therefore, the waveform of blood at the inlet of an aorta artery should be personalised. The pulses are usually obtained through one of the relevant imaging modalities, such as PC-MRI, Doppler ultrasound, Echocardiogram, etc. Then, the data can be translated into velocity or flow rate, which will be the usable format for the computational simulation. In the course of mathematical modelling of the aorta, if the velocity inlet is chosen, the profile would be taken in one of the following forms, i.e., plug, parabolic, Womersley, or patient-specific [31]. In a plug profile, the boundary layer effect is neglected, and the velocity profile is considered uniform. In a parabolic velocity profile, it is assumed that the flow is fully developed. Therefore, for a circular cross-section and in laminar flow, the mathematical description of the profile is   r 2  V˙ (t) 1− U(r, t) = A R

(1)

where V˙ (t) is the flow rate, A represents the inlet cross-section, r denotes the radial coordinate system, and R is the inlet radius. The third type of velocity profile is the Womersley type [32], which tries to mimic a realistic waveform that is produced

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in pulsatile flows. The analytical solution of the Womersley comes along with the following assumptions: (i) (ii) (iii) (iv) (v) (vi)

Homogenous, incompressible, and Newtonian fluid The flow regime is laminar Pressure is defined as a periodic function to drive the flow Gravitational acceleration has no effect The wall is assumed to be rigid, and the inlet has a circular cross-section At the inlet, the boundary is an axisymmetric and no-slip boundary at the wall.

All together leads to the Womersley velocity profile as defined by Eq. (2) ⎛ ⎞ J0 (βn ( Rr ))  N  r 2   V˙n ⎝ 1 − J0 (βn ) ⎠ inωt 2 V˙0 1− + e U(r, t) = n) R π R2 π R2 1 − β2JJ1 (β n=1 (β ) n 0

(2)

n

where V˙0 is the first harmonic coefficient of the flow rate and it can be decomposed and rearranged based on the Fourier series as follows: V˙ (t) =

N 

V˙n einωt

(3)

n=0

where N is the number of harmonics, and ω is the angular frequency, which is derived from the period of the cardiac cycle. Furthermore, J 0 and J 1 are the Bessel functions of order zero and one, respectively, and βn is defined as follows: βn = i 3/2 αn

(4)

in which αn is the Womersley number of nth harmonics, which is defined as follows:  αn = R

nωρ μ

(5)

As shown above, the second power of the Womersley number defines the ratio of transient inertial force (ρU ω) to viscous force (U μ/R2 ). Also, in this equation, ρ is density, μ is dynamic viscosity, and R as defined earlier is the inlet radius, which is a length characteristic. The fourth type of velocity profile is subject-specific, where the flow information is acquired through clinical images. Then the information is mapped onto the computational mesh, and it can be used as a subject-specific velocity profile [33, 34]. In addition to the velocity inlet, pressure can be applied at the inlet of the aorta [35]. The applied pressure can be the intravascular measurement waveform at the inlet of the aorta, or it can be implicitly obtained from the flow waveform measurement. Then the pressure can be applied perpendicularly to the inlet surface.

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Fig. 5 An example of the left heart model circuit. The circuit comprises the left atrium and ventricle, the left valves, aortic circulation, and systemic circulation

Flow inlet for parametric study: some studies aim to virtually explore the effects of different cardiac functions on aortic circulation. For example, the impact of cardiac outputs, like during rest or in exercise, on aortic circulation in the healthy or diseased aorta. Another example is the effects of abnormal cardiac rhythm, like atrial fibrillation in aortic circulation. [36, 37]. For such studies, instead of modelling the entire heart, which is of course extremely complicated and computationally expensive, one very useful approach is to create a lumped model of the heart. Some lumped models just focus on the left heart [38], and some include the entire heart [39, 40]. Figure 5 displays a model of the left heart, which is initially introduced by Simaan et al. [38], and modified by Deyranlou et al. [37]. This simple model includes a few elements, that can predict the flow and pressure inside the left chamber and across the valves. Applying Kirchhoff’s voltage and current laws, the governing equations for the assumed circuit are presented as follows: C˙ L V 1 1 dp L V =− pL V + ( pL A − pL V ) − ( p L V − p Aor ) dt CLV C L V RL A C L V RL V  ˙ CL A 1 1 1 dp L A p Aor + =− + psys − ( pL A − pL V ) dt CL A Rsys C L A Rsys C L A RL AC L A dpsys 1 1 V˙ Aor =− psys + pL A + dt Rsys Csys Rsys Csys Csys V˙ Aor dp Aor 1 =− + ( p L V − p Aor ) dt C Aor C Aor R L V

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psys d V˙ Aor p Aor R Aor ˙ − V Aor + =− dt L Aor L Aor L Aor

(6)

where p denotes the pressure, V˙ (t) is the flow rate, R refers to the resistance, C is the compliance, which is inversely proportional to the elastance. The dot sign above the compliance denotes the temporal derivative of compliance. The subscripts LA, LV, Aor, and sys refer to the left atrium, left ventricle, aorta, and systemic, respectively. The elastance function, E(t) correlates the intracardiac pressures to the corresponding volume changes. Therefore, the elastance functions for the left atrium and left ventricle are defined as follows: E L A (t) =

1 p L A (t) = VL A (t) − V0,L A C L A (t)

(7)

E L V (t) =

1 p L V (t) = VL V (t) − V0,L V C L V (t)

(8)

where VL A (t) and VL V (t) are associated with the volumes of the left atrium and left ventricle, respectively. Also, V0,L A and V0,L V are theoretical volumes of the left atrium and ventricle at zero pressure, respectively. Moreover, the elastances of the left atrium and left ventricle are estimated mathematically. First, for the left atrium, the elastance is: E L A (t) =

(E L Amax − E L Amin ) e L A (t) + E L Amin 2

(9)

E L Amax and E L Amin are the maximal and minimal elastances associated with the endsystolic and end-diastolic pressure–volume relations. Also, e L A (t) will be described as follows [41]: e L A (t) =

1 − cos

0

t−Tac 2π tcc −Tac

 if 0 ≤ t ≤ Tac , T = 0.80t ac cc if Tac < tcc

(10)

For the left ventricle, the elastance is defined as follows: E L V (t) = (E L V max − E L V min )E n (tn ) + E L V min

(11)

In which E n (tn ) is the double-hill function, which is initially proposed by Stergiopulos et al. [42] and is represented as follows: ⎡  β1 ⎢ E n (tn ) = α1 ⎣

⎤⎡

⎤

⎥⎢  β1 ⎦⎣

1 ⎥  β2 ⎦

tn α2

1+

tn α2

1+

tn α3

(12)

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where αi and β j are the model parameters that are defined as α1 = 1.55, α2 = 0.7, α3 = 1.17, β1 = 1.9, and β2 = 21.9. Moreover, tn = t/Tmax , in which Tmax = 0.2 + 0.15tcc and tcc is the cardiac cycle period. E L V max and E L V min are the maximal and minimal elastances associated with the end-systolic and end-diastolic pressure– volume relations. Flow outlet: the common types of outlet boundary conditions are constant pressure or pressure waveform, traction free, Murray’s law, and the Windkessel model. In a constant pressure outlet, a constant value is assigned to the outlet boundary. If subject-specific pressure is available—which rarely occurs since it’s not part of the routine clinical practice—a waveform will be applied. In the absence of pressure outlet information, the traction-free boundary is the simplest one to mimic the free stream condition at the outlet. In vascular modelling, the latter method ignores the resistance effects of flow downstream and thus is not an appropriate boundary choice. Another method is based on Murray’s Law [43], which works based on energy minimisation of the flow. The theory hypothesises that the flow power to overcome frictional losses and provide the metabolic needs should be minimised. Therefore, employing Poiseuille’s law and correlating the metabolic power to the blood volume, for a laminar flow, results in the following relation between the volumetric flow rate and the outlet radius: V˙ ∝ r 3

(13)

For the turbulent flow inside the vasculature, Williams et al. [44] extended Murray’s law, which culminates in the following relation: 7 V˙ ∝ r 3

(14)

Although Murray’s law can have a good estimation regarding the flow distribution, one must be cautious about is the model would not lead to the violation of mass conservation, which is also accompanied by numerical instabilities. In the absence of a subject-specific pressure waveform, the Windkessel model is a reliable approach that can predict the pressure and flow distributions at the outflows. The model was proposed in the late 1800s by Otto Frank to describe systemic circulation (heart and arteries). The Windkessel model expresses a correlation between the volumetric flow rate, arterial compliance, and peripheral resistance. In Windkessel models, the compliance determines the elasticity of the large-sized vascular wall, such as the aorta, carotid, and subclavian arteries that can store more blood due to their volumetric expansion. Whereas the peripheral resistance is majorly imposed by small vessels. Therefore, the model creates a pulsatile pressure waveform, which can be applied to the outflows. There are three main Windkessel models, which are called two/three/four-element Windkessel models. Figure 6 displays the electric analogy of various types of the Windkessel model. The simplest proposed Windkessel model has two elements, which are one capacitor (arterial compliance) and one resistor (peripheral resistance). Using the electric

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Fig. 6 Windkessel models: (a) two-element, (b) three-element, and (c) four-element Windkessel models

analogy, the theoretical model can be formulated as follows: p(t) d p(t) V˙ (t) = +C Rd dt

(15)

where V˙ (t) is the volumetric flow rate, p(t) is the pressure, Rd denotes peripheral resistance, and C represents the wall compliance. The main weakness of the twoelement Windkessel model is it cannot predict the pressure waveform at high frequencies. Therefore, by introducing a characteristic impedance, the latter drawback will be modified. The impedance considers the wave travelling effects in proximal regions, and it is defined as pulse wave velocity times density, divided by cross-sectional area. Subsequently, it concludes with a three-element Windkessel model [45], which induces small errors in the low-frequency region. Introducing R p , as the proximal resistance, the theoretical model of the three-element Windkessel model is defined as follows:   Rp ˙ p(t) d V˙ (t) d p(t) = (16) 1+ +C V (t) + R p C Rd dt Rd dt Finally, the four-element Windkessel model can compensate for the small errors in low-frequency regions by including the inertial effect through an inductance (L). Thus, the following differential equation defines the model:     Rp ˙ d2 V˙ (t) L d V˙ (t) p(t) d p(t) 1+ + LC = +C V (t) + R p C + 2 Rd Rd dt Rd dt dt

(17)

The selected Windkessel model is then connected to the outlets of the 3D CFD domain. For ample functioning, the Windkessel parameters need to be calibrated. Although the four-element Windkessel model is the most complete one amongst the introduced models, the inductance adds significant complexity to the tuning process. Therefore, the three-element Windkessel model has been recognised as the most efficient technique. In modelling aortic flow, amongst different outlets, the boundaries belonging to coronary arteries require special treatment. Since the coronaries are located in the heart, therefore, the inflow is affected by the heart movement. It means, that in the systolic phase, less flow goes to the coronary arteries, and the main portion goes

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Fig. 7 Lumped model for the coronary outlet

inside during the diastolic phase. Thus, for more accurate modelling, this phase lag is considered through another lumped model as shown in Fig. 7. Thus, using Kirchhoff’s law for the shown electrical circuit, the differential equation for coronary outlets is defined as follows: q0 V˙ (t) + q1

d V˙ (t) d 2 V˙ (t) dp(t) d 2 p(t) dpv (t) + q2 + p2 = p0 p(t) + p1 −b 2 2 dt dt dt dt dt (18)

where q0 , q1 , q2 , b, p0 , p1 , and p2 are defined as follows: q0 = Rv + Ra + Ram q1 = Ra Rv (Ca + Cv ) + Ram (Rv Cv + Ra Ca ) q2 = Ra Ram Rv Ca Cv b = Rv C v p0 = 1 p1 = Rv (Ca + Cv ) + Ca Ram p2 = Ram Rv Ca Cv Vessel wall: the aortic wall is compliant, and it has heterogeneous anisotropic properties. Therefore, subject-specific modelling of the aortic wall is drastically complicated. The simplest approach is to use a rigid wall assumption, where the aortic wall deformation and movement are neglected. Thus, considering an impervious wall, for a viscous flow, no slip and no infiltration boundary condition is selected. Introducing n and t as the normal and tangential vectors to the wall, respectively, the

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mathematical description of the boundary conditions can be defined as follows: (u − v) · t = 0, no slip

(19)

(u − v) · n = 0, no infiltration

(20)

To consider the wall movement, there are two approaches, one is the Fluid–Structure Interaction (FSI), and another one is the geometry-prescribed moving boundary method. The heterogeneous structure of the aortic wall and its different thicknesses at different locations, make the aortic wall extremely subject-specific. Given the current development, there have not been sufficient resources to personalise wall thickness and wall constitute model. Thus, the FSI method seems less practical for subject-specific modelling. The second technique is the geometry-prescribed moving boundary method, where clinical images provide motion information of the endocardium, valve, and luminal surface [7, 46–49]. In this method, although the wall effects on the fluid domain are neglected, the method has still more potential to be employed for personalised medicine. In the geometry-prescribed method, ideally 4D images—three in space and time—are taken in a cardiac cycle. Then, a 4D segmentation is employed based on deformable models [50–53] and statistical shape modelling [54–56]. This approach requires appropriate mesh manipulation at different phases to have a smooth mesh transition and to avoid any negative volume. Given the required level of time refinement, a certain number of grid networks are created between every two consecutive phases [7, 57]. Initial work in this area was carried out by Saber et al. [46, 58], which were later developed [7, 57, 59, 60]. Another technique has been recently developed by Mirko et al. [61], which reduces the complexity of the moving boundary technique. In this method, the regional distensibility is calculated using Cine-MRI data, then the distensibility is distributed throughout the domain by interpolating between different regions.

3.2 Governing Equations This section explains the governing equations for modelling blood haemodynamics in an aorta artery. Flow regime: at this stage, the first question to be answered is what should be the flow regime in vascular flow dynamics? The flow is usually considered laminar, however, some studies suggest the existence of turbulent flow at some points in a cardiac cycle or specific locations like stenosis, aneurysm and the regions the mixing occurs [62–64]. To define the onset of turbulent flow in physiological flows, Peacock et al. [65] proposed an empirical correlation, based on the critical Reynolds number, which is determined as follows:

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Recr = 169α 0.83 St −0.27

(21)

where α is the Womersley number and St is the Strouhal number, and both are dimensionless numbers. The Womersley number expresses the ratio of pulsatile flow frequency to viscous effects and is defined as follows: α=

 1 Din ωρ 2 2 μc

(22)

Also, the Strouhal number is the ratio of inertial forces due to the flow unsteadiness or local acceleration to the inertial forces due to the velocity difference between points in the flow field. The Strouhal number is defined as follows: St =

f Din 2 U peak − Umean

(23)

To evaluate whether the physiological flow should be considered laminar or turbulent, initially, the peak Reynolds number must be calculated as follows: Re p =

U peak Din ρ μc

(24)

√ where Din = 2 Ain /π , in which Ain is the inlet area. Moreover, in the above equations, ω = 2π f , in which f is cardiac cycle frequency, ρ is blood density, which is taken equal to 1060 kg/m3 , and μc is the blood characteristic viscosity. To define the characteristic viscosity of blood, its viscosity must be determined first. The blood is composed of plasma and cellular elements, i.e., leukocytes (white blood cells), erythrocytes (red blood cells), and thrombocytes (platelets), which prompts shear-thinning effects and reveals blood non-Newtonian effects, especially for the shear rates below 100 s−1 [66, 67]. To consider blood non-Newtonian properties, a few models for blood viscosity are more common, which are Carreau, CarreauYasuda, modified Casson, power-law, and generalized power-law [68]. Amongst the aforementioned models, one of the popular ones is the Carreau-Yasuda model [69], which is defined as follows: a  nCY −1   μa = μ∞ + (μ0 − μ∞ ) 1 + λCY γ˙i j CY aCY

(25)

where μa is the apparent viscosity, μ0 denotes the low shear viscosity and is normally set to 0.16 Pa s, μ∞ is the high shear viscosity and is equal to 0.0035 Pa s, λCY denotes the time constant, which is equal to 8.2 s, n CY is the power-law index that is equal to 0.2128, and aCY is the Yasuda exponent, which is equal to 0.64 [70]. Furthermore, γ˙ is the shear rate, which is defined based on the second invariant of the rate of deformation tensor as follows:

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∂ ui ∂x j

85



∂uj ∂ ui + ∂x j ∂ xi

 (26)

u and x are velocity and displacement vectors. Now the blood viscosity is defined, the characteristic viscosity for defining Womersley and Reynolds numbers can be calculated. For these calculations, μc is estimated from the Carreau-Yasuda model, by substituting U peak /Din for the shear rate, γ˙i j , as suggested by Cagney and Balabani [71]. The final note in this section is about flow regime; in cases the turbulent flow is considered, the Large Eddy Simulation (LES) is normally invoked, as it can successfully model the transitional flow between the laminar and turbulent flows [72]. Also, k − ω Shear Stress Transport (SST) Reynolds Averaged formulation of the Navier– Stokes equations is another approach taken for modelling turbulent flow in the aorta [73]. Governing equations inside the fluid domain: in the laminar flow regime, the mass conservation and momentum balance for an incompressible, non-Newtonian fluid in Arbitrary Lagrangian–Eulerian (ALE) formulation can be written as follows: ∂ ∂ρ + (ρ(ui − vi )) = 0 ∂t ∂ xi

(27)

 ∂σ f i j ∂ρu j ∂  + + ρfj ρ(uk − vk )u j = ∂t ∂ xk ∂ xi

(28)

where ρ is the density, u denotes the fluid velocity vector, v is the mesh velocity vector, which is zero for a rigid stationary model and in the case of FSI, it is equal to the concomitant motion of fluid and solid domains at the interfaces. Also, x is the coordinate system, f j denotes the body force per unit of volume, and σi j is the surface stress tensor as determined below: σ f i j = − pδi j + τi j

(29)

where p is the pressure, δi j is the Kronecker delta, and τi j denotes the shear stress tensor. For a Newtonian fluid the stress tensor in the equation is defined as follows: σ f i j = − pδi j + λδi j

  ∂uj ∂ uk ∂ ui +μ + ∂ xk ∂x j ∂ xi

(30)

in which μ is the first coefficient of viscosity (dynamic viscosity), and λ is the second coefficient of viscosity (volume viscosity). For incompressible flows, volume viscosity is equal to zero. Since the blood is non-Newtonian, an appropriate viscosity model—like the Carreau-Yasuda model defined in Eq. (25)—is chosen and replaced by the apparent viscosity. Governing equations inside the solid domain: for the solid domain, the conservation of momentum can be described as follows:

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ρs

∂σs i j ∂2 Xi = + ρs f js ∂t 2 ∂ Xi

(31)

where ρs is the density of the solid part—here it would be an arterial wall, σs i j is the solid stress tensor, and f js is the body force per unit of volume. In the above equation, the solid stress tensor, σs i j , can be defined as follows:   σs i j = λs tr i j + 2μs i j

(32)

In solid stress tensor, i j is called Piola Kirchhoff stress tensor and is defined as follows:   ∂Xj 1 ∂ Xi (33) +

i j = 2 ∂X j ∂ Xi Furthermore, the Lame coefficients, λs and μs , are characterised based on Young’s modulus and Poisson coefficient [74]. In the case of FSI modelling, the interface condition between the fluid and solid domains is set to satisfy the displacement and momentum continuity: xi = X i

(34)

σ f i j · n f = σs i j · ns

(35)

3.3 Numerical Method Like many other physical phenomena, computational modelling of aortic flow requires a robust implementation of numerical methods, which are accurate and computationally meaningful. Generally, there are two broad classifications for numerical modelling of vascular flow in computational domains. The first one is the continuum approach, which comprises two main techniques, i.e., the Finite Element Method (FEM) and Finite Volume Method (FVM). The second approach is the molecular technique, which is mostly recognised by Lattice Boltzmann Method (LBM) and Smoothed Particle Hydrodynamics (SPH). Furthermore, in the case of FSI or moving boundary method, two general approaches are employed, which are body conformal technique such as ALE formulation, and fixed grid techniques such as Immersed Boundary Method (IBM). Grid network: continuum methods require splitting the computational domain into small-sized computational meshes. There are different meshing codes using different algorithms. Amongst the most popular ones are Ansys meshing packages (ICEM, Ansys-Meshing, Fluent-Meshing, and T-Grid), STAR-CD+, Cadence (previously

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known as Pointwise), and snappyHexMesh (OpenFOAM mesh). Generally, created meshes can be divided into structured and unstructured, with different cell types. For the vascular modelling and in the aorta artery, given the intricate morphology, the unstructured mesh is more efficient. Additionally, special treatment needs to be performed for the boundary layer to appropriately capture the velocity gradient in the vicinity of the vessel wall. In turbulent flow, the fluid flow is divided into three layers, i.e., a viscous sublayer (transport occurs by diffusion, velocity is linear, and flow is laminar), a buffer layer (transport occurs both by diffusion and turbulent mixing), and a turbulent layer (transport occurs by turbulent mixing). Generally, to have a good resolution at the boundary, the height of the first grid cell should be in a range corresponding to y + ∼ 1. Therefore, to calculate the height of the first cell for a sufficient boundary layer mesh, the steps explained in continue should be taken. In step one, using the definition of y + , the height of the first layer can be defined as follows: y+ =

y+ν yu τ →y= ν uτ

(36)

where u τ is the friction velocity, y is the distance from the wall, and υ is kinematic viscosity. In step two, the friction velocity can be expressed as follows: uτ =



τw /ρ

(37)

where wall shear stress, τw , can be calculated as follows in step three: τw =

1 2 C f ρU∞ 2

(38)

In step four, the friction coefficient, C f , should be calculated, for which, the following formula can be employed that belongs to flow over a flat plate: Cf =

0.026 1

(39)

Red 7

where, Red = U∞ D/ν is the Reynolds number with a characteristic length of D that can be the inlet diameter. Additionally, Pier and Schmid [75] suggested that the total thickness of the pulsatile boundary layer can be estimated as follows: δ=



ν/ f

(40)

in which f is the pulsation frequency. Therefore, that approach helps to choose a fine enough cell size near the wall region to resolve the boundary layer flow effectively. Solver codes: nowadays, in addition to in-house developed codes, there are several well-known open-source and commercial codes, which are widely used for numerical modelling in general. Ansys-Fluent, Ansys-CFX, COMSOL, OpenFOAM,

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Autodesk CFD, Altair Hyperworks, SimulationX, PowerFLOW, FLOW-3D, SimVascular, and CRIMSON are some of the widely used codes. Noteworthy to mention that the last two codes are specifically designed to model cardiovascular flows. Furthermore, the above codes provide more additional features by extending their user-defined capabilities. Developing and coupling user-defined functions are performed through programming languages, such as FORTRAN, C++, and Python. This section is not allocated to equation discretisation methods and associated techniques, thus, to get familiar with the computational methods and underlying numerical schemes, the readers are encouraged to refer to the codes’ theory guides and relevant textbooks in numerical methods and CFD [76, 77]. Windkessel parameters calibration, outlet boundaries: as explained in the previous section, the three-element Windkessel model needs to be calibrated based on the specific characteristics of each individual. There are different approaches to tuning the Windkessel parameters [78–80]. Whichever technique is used for the calibration, the main objective is to tune the Windkessel parameters to fall within the physiological range. In this section, one of the most widely employed calibration techniques will be explained, which is the basis for more developed methods. The Windkessel calibration pipeline can be explained through the following stages: Stage 1: Finding the total vascular compliance, using a single block three-element Windkessel model as proposed by Les et al. [62]. In this approach, the extracted subject-specific waveform at the inlet of the aorta is fed as input. Then the RCR values are found through an iterative process to satisfy the target aortic pressure (Fig. 8). The aortic pressure can be either obtained through invasive intravascular measurement, or through non-invasive brachial cuff pressure, which is more common in clinical routine. Studies have shown that the diastolic pressure does not change significantly throughout the vascular tree, however, the systolic pressure changes [81]. Therefore, aortic diastolic pressure is approximately equal to brachial diastolic pressure. For the systolic aortic pressure, due to its compliance, the pressure slightly reduces, and it can be estimated within the brachial pressure [81] as follows: psa = 0.83 psb + 0.15 pdb Fig. 8 A single block three-element Windkessel model

(41)

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In Eq. (41) subscripts sa, sb, and db refer to the systolic aorta, systolic brachial, and diastolic brachial, respectively. Xiao et al. [79] proposed an iterative approach to modify total resistance and compliance through first-order Taylor expansion of the following expression for Rtot and Ctot : pmean V˙in mean

(42)

V˙max − V˙min t psys − pdia

(43)

Rtot = Ctot =

where V˙in mean , V˙max , and V˙min are mean, maximum, and minimum values of the measured waveform, V˙ (t), at the inlet of the aorta, respectively. Also, t is the time difference between V˙max , and V˙min . Moreover, pmean , psys , and pdia are the mean, systolic, and diastolic pressures resulting by using V˙ (t) at the inlet. Then at every iteration, total resistance and compliance are modified as defined below: k+1 k = Rtot + Rtot

k+1 k Ctot = Ctot −

k pdia mean V˙ikmean

V˙max − V˙min 2

( p kpulse )

tp kpulse

(44)

(45)

k k = pda − pdia , which is the difference where k is the iteration number, and pdia mean between diastolic target pressure (aortic pressure) and diastolic pressure at iteration k k − pdia ), which is the differnumber k. Furthermore, p kpulse = ( psa − pda ) − ( psys ence between target pulse pressure and pulse pressure at iteration number k. The final point at this stage is to set the ratio of proximal resistance to total resistance, Rtot p = 0.056Rtot [79]. Noteworthy to mention that this ratio can vary between 0.05 and 0.1 depending on target pressures. Stage 2: after finding the total compliance of the vascular system, the compliance and resistance of each terminal branch must be found. This stage needs another series of iterations to reach the correct values. To expedite the iteration, the 0D (lumped) model of the 3D CFD model should be made, with the aim to find the aortic compliance and the resistances and compliances of all branches. To make the lumped model of the entire aorta, each section is modelled with a block of R, RL, or RLC based on the importance of friction (reflects in R), inertial effects (reflects in L), and compliance (reflects in C), which are shown in red blocks of RLC in Fig. 9 as an example. To estimate RLC for the lumped model, there are different methods, which are presented in continue. The resistance R can be approximated using Poiseuille’s law, which is developed for a steady, laminar flow in a straight tube with length l and diameter D:

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Fig. 9 An example of a 3D model and corresponding lumped model (the geometry of the aorta was adapted from Deyranlou et al. [37])

R=

128μl π D4

(46)

To have a better estimation, another method is to obtain the resistance of each segment using a steady rigid simulation of the 3D model of an aorta by applying the pressure at the inlet and target flow rate at the outlets. The applied pressure is defined based on the aortic target pressure and obtained pressure from the single block three-element Windkessel model: pin mean = FF( psa − pda ) + pda

(47)

where FF represents the pressure waveform factor suggested by Chemla et al. [82] and is obtained as follows:

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FF =

91

pmean − pdia psys − pdia

(48)

Therefore, the resistance of the lumped model for each segment of the aorta can be approximated as the ratio of pressure difference and flow rate, as defined below: Rtot,i =

p V˙i

(49)

In order to obtain the segmental inertial effects, L, Westerhof et al. [81] suggested the following relation, which is obtained from the momentum balance for a plug flow at the inlet of a rigid straight tube with length l and diameter D: L=

4ρl π D2

(50)

Also, Westerhof et al. [81] for the straight tube offered the following relation to approximate the compliance based on geometrical specification: C=

3π D 3l 16Eh

(51)

where E denotes Young’s modulus and h is the wall thickness. Even though the above formula gives a reasonable estimation for the compliance, if local distensibility information is available for the aortic geometry, that would give a more accurate estimation. So, the segmental compliance, Cv , can be defined as follows: Cv = Distensibilit y × V

(52)

where distensibility is Distensibilit y =

A 1 Amin p pulse

(53)

in which A is the maximum area change in a cardiac cycle, Amin is the diastolic area, and p pulse is the pulse pressure. Having phasic anatomy information, such as CineMRI, the segmental distensibility can be determined as defined above. Furthermore, for branches, an empirical relation based on Pulse Wave Velocity (PWV) by Reymond et al. [83] can be employed whenever cross-sectional variations are not available: Distensibilit ybranch =

1 1 ρ PW V 2

(54)

Stage 3: once the lumped model of the entire aorta is created, then the goal is to find the aortic compliance and Windkessel parameters for each branch. Figure 10 demonstrates the iteration process to reach the target values. The target values are

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Fig. 10 The iteration process to find the final values of aortic compliance and Windkessel parameters

target aortic pressure at the inlet and target flow rates at the outlets of the aorta. At this stage, initially, the aortic compliance and Windkessel parameters are changed in lumped model to reach the target values, then a few more iterations may be required in the 3D aortic model to acquire the correct values. The main goal of running the first few iterations in the lumped model is to reduce the number of iterations in the 3D CFD model to decrease the calibration time. One additional note to be considered is in a rigid simulation of the aorta, to obtain realistic compliances (total, peripheral, and aortic compliances), and to reach more conveniently to target pressure and flow rates, it is recommended to introduce aortic compliance at the inlet of the aorta [78]. Therefore, the boundary at the inlet can be modified as follows: dp(t) V˙inlet,3D (t) = V˙wave f or m (t) − Caor ta dt

(55)

Interpolation on inlet waveforms: the flow rate or velocity at the inlet of an aorta is normally acquired for certain time points in a cardiac cycle. Since the data is not continuous, and the temporal resolution needs to be small enough, therefore, interpolation should be performed. Although there are different functions that can be fit, one of the most widely employed and efficient ones is the Fourier series. Assume that there is a set of data, (xi , yi ), through which a Fourier series, Fr (x), is to be fitted: Fr (x) = a0 +

N  2π n (an cos ωn x + bn sin ωn x), ωn = tcc n=1

(56)

where x is an independent variable, and a0 , an , and bn are the Fourier coefficients. By employing the least square method and using the assumed data set (xi , yi ; i = 1, m), initially, the error function, Er , is to be formed as below:

Subject Specific Modelling of Aortic Flows

Er =

m  i=1

(yi − Fr (xi )) = 2

m 



93

N  (an cos ωn xi + bn sin ωn xi )) yi − (a0 +

2

n=1

i=1

(57) To find a perfect fit, a0 , an , and bn should be found to minimise the error function. Therefore, differentiating both sides the following expression will be obtained: d Er =

∂ Er ∂ Er ∂ Er ∂ Er ∂ Er ∂ Er da0 + dan + dbn = 0 → = 0, = 0, =0 ∂a0 ∂an ∂bn ∂a0 ∂an ∂bn (58)

Solving and rearranging the above equations, the following matrices of coefficient, variables, and constant values are obtained, and the final solution determines the Fourier series coefficients. ⎡

   sin ω1 xi 1 cos ω1 xi   ⎢ 2 cos ω1 xi sin ω1 xi cos ω1 xi ⎢ cos ω1 xi ⎢   2 ⎢ sin ω1 xi cos ω1 xi sin ω1 xi sin ω1 xi ⎢ ⎢ . . . ⎢ ⎢ ⎢ . . . ⎢ ⎢ . . . ⎢   ⎢ ⎣ cos ωn xi cos ω1 xi cos ωn xi sin ω1 xi cos ωn xi    sin ωn xi cos ω1 xi sin ωn xi sin ω1 xi sin ωn xi ⎤ ⎡  yi ⎥ ⎢ y cos ω1 xi ⎥ ⎢ ⎥ ⎢ i ⎥ ⎢ sin ω x y 1 i ⎥ i ⎢ ⎥ ⎢ . ⎥ ⎢ =⎢ ⎥ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎣ yi cos ωn xi ⎦  yi sin ωn xi

. . . . . . . .

. . . . . . . .

  . cos ωn xi sin ωn xi   . cos ωn xi cos ω1 xi sin ωn xi cos ω1 xi   . cos ωn xi sin ω1 xi sin ωn xi sin ω1 xi . . . . . . . . .  2  . cos ωn xi sin ωn xi cos ωn xi   2 . cos ωn xi sin ωn xi sin ωn xi

⎤⎡

a0 ⎥⎢ ⎥⎢ a 1 ⎥⎢ ⎥⎢ b1 ⎥⎢ ⎥⎢ ⎥⎢ . ⎥⎢ ⎥⎢ . ⎥⎢ ⎥⎢ . ⎥⎢ ⎥⎢ ⎦⎣ a n bn

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(59)

In the matrices above, the summation bound is between 1 and m (i = 1, …, m), where m is the number of (xi , yi ) data points. Normally a Fourier series with eight harmonics provides an accurate fit to the data. Also, the matrix can be easily solved by employing the Gauss–Seidel method.

4 Blood Haemodynamics in the Aortic Artery Understanding blood haemodynamics in an aorta artery helps in better recognition of the possible link between vascular pathogenesis and certain flow patterns. Studies in this area occur through in-vivo/vitro measurements and computational simulations. Furthermore, in the diseased aorta, the intervention can be generally classified into

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endovascular treatment and surgery. In both cases, evaluating the important haemodynamic metrics provides important information for treatment planning, optimisation, and post-intervention follow-ups. In the following section, the most popular haemodynamic metrics are summarised. Thereafter, some well-known attributes of aortic flow will be discussed through the concepts of velocity, pressure, WSS, and helicity.

4.1 Haemodynamic Metrics Wall Shear Stress (WSS) and related metrics: WSS is a well-recognised haemodynamic metric in biofluid mechanics, which indicates the locations are prone to accumulation of fatty deposits and atherogenesis [84]. Peiffer et al. [85] demonstrated that low WSS and high oscillatory flow are two factors, which promote the possibility of atherosclerosis. WSS is defined as follows:   ∂u WSS = τwall = −μ ∂n

(60)

in which μ is the dynamic viscosity, u represents the velocity vector parallel to the wall, and n is the normal vector to the wall. In pulsatile physiological flows, TimeAveraged WSS (TAWSS) is a metric, which describes mean WSS behaviour in a cardiac interval, RR, as presented in Eq. (61). Another subsidiary of the WSS vector is the TAWSS gradient (TAWSSG), which is originally introduced by Lei et al. [86]. The metric describes the mean magnitude of the WSS gradient within a cardiac cycle and is considered a marker for tension in the endothelial layer. 1 TAWSS = RR

RR |τwall |dt

(61)

0

1 TAWSSG = RR

 RR  0

∂τx ∂x

2

 +

∂τ y ∂y

2

 +

∂τz ∂z

2 dt

(62)

Vascular flow is pulsatile, and it is not unidirectional, therefore, it alters the instantaneous direction of WSS vectors. The metric that evaluates the directional changes is the Oscillatory Shear Index (OSI), which was introduced by Ku et al. [87]. OSI varies between 0 and 0.5, where 0 refers to the unidirectional WSS vectors, while 0.5 refers to the fully reversed WSS vectors. ⎛ OSI = 0.5⎝1 −

⎞



  1  RR τ dt RR  0 wall  1 RR

 RR 0

|τwall |dt

⎛

⎠ = 0.5⎝1 −

⎞



  1  RR τ dt RR  0 wall  TAWSS

⎠

(63)

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Himburg et al. [88] found that there is a relation between the residence time of the near the wall particles, and WSS and OSI. This relation was formulated in Relative Residence Time (RRT) as follows: RRT =

1 1  =  1  RR (1 − 2 × OSI) × TAWSS τ dt RR  0 wall 

(64)

Another metric that assimilates the concomitant effects of WSS and OSI is Endothelial Cell Activation Potential (ECAP). It was shown that in a flow regime with high WSS, the endothelial cells tend to align in the flow direction, therefore, intercellular distance decreases and it leads to a lower permeability, which in turn blocks the infiltration of micron-sized elements. However, once the WSS decreases and the OSI increases, the endothelium cells reconfigure to form a more rounded shape and thus the permeability of the layer increases, which allows infiltration of fatty materials into the wall [89]. Therefore, the ECAP, which was introduced by Achille et al. [90], is defined as the ratio of OSI to TAWSS: ECAP =

OSI TAWSS

(65)

Q-criterion: In cardiovascular dynamics, there is some evidence, which suggests some relations between the vortex creation and its structure and pathological phenotypes [91]. Consequently, quantifying the vortex intensity and its structure in the haemodynamics of vascular flow can provide beneficial information. To quantify the vortex structure in cardiovascular mechanics, there are different criteria [92], amongst them, the Q-criterion [93] is the most employed one [8, 15, 94]. This criterion is defined based on the second invariant of the velocity gradient tensor presented in Eq. (71). Velocity gradient tensor can be shown as follows: ⎡ ⎤ d11 d12 d13 ∂ ui = Si j + i j = ⎣ d21 d22 d23 ⎦ Di j = ∂x j d31 d32 d33

(66)

In Eq. (66), Si j describes the symmetric part, which is called the rate of strain tensor and determines the shearing rate. Furthermore, i j refers to the asymmetric part, which defines the rotation rate and is called the vorticity.   ∂uj 1 ∂ ui + Si j = 2 ∂x j ∂ xi   ∂uj 1 ∂ ui i j = − 2 ∂x j ∂ xi

(67) (68)

In order to define the invariants of the velocity gradient tensor, the characteristic equation for Dij is presented as follows:

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ζ 3 + I1 ζ 2 + I2 ζ + I3 = 0

(69)

where I1 , I2 and I3 , are invariants of the velocity gradient tensor and are defined through the following equations:

I2 =

  I1 = −trace D = −[d11 + d22 + d33 ]

(70)

 1    1  1 2 I1 − trace DD = i j i j − Si j Si j = ||2 − |S|2 2 2 2

(71)

   1  I3 = −det D = −I1 3 + 3I1 I2 − trace DDD 3

(72)

Helicity: helicity is another important metric in the haemodynamics of the aorta, which describes the spiral motion of blood flow, from the aortic root towards the descending aorta. The concept of helicity was originally introduced by Moffat in 1969 [95, 96]. Therefore, the flow helicity in the aorta can be defined as follows: H (t) = u(x, t) · ω(x, t)

(73)

where, u(x, t) and ω(x, t) are the velocity and vorticity vectors, respectively. Furthermore, employing the localised normalised helicity (LNH), the direction of flow rotation is expressed as follows: LNH =

u(x, t) · ω(x, t) |u(x, t)||ω(x, t)|

(74)

In order to analyse the helicity and its regional magnitude, the ‘time-averaged helicity’ and ‘helicity intensity’ inside a 3D domain with a volume of V and cycle period of tcc are introduced as follows [31]: 1 H1 = tcc V

tcc ˚ H (t)d V dt

(75)

|H (t)|d V dt

(76)

0

1 H2 = tcc V

tcc ˚ 0

Intraluminal pressure and flow distributions: pressure and flow distribution are two metrics that have paramount importance for physicians, and they are regarded as important biomarkers. Evaluating pressure distribution and flow perfusion helps in decision-making and treatment optimisation. These two metrics are field variables, and they are automatically obtained by solving Navier–Stokes equations.

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Cardiac Metrics: there are a few cardiac metrics, which are useful for the workflow of personalised medicine: Stroke Volume (SV) = End Diastolic Volume (EDV) − End Systolic Volume (ESV) (77) Ejection Fraction (EF) =

SV × 100 EDV

Cardiac Output (CO) = SV × HR

(78) (79)

where SV, EF, and CO are normally defined in a unit of litre, percentage, and litre per minute, respectively.

4.2 General Characteristics of Flow in Aorta Arteries In a normal aorta, there are some general flow features, which exist irrespective of age, gender, and ethnicity. The importance of recognition of normal flow in the aorta would help to discern between healthy and diseased aorta. Furthermore, it helps to evaluate the efficiency of the treatment and the likelihood of reintervention for diseased aortas. This section is devoted to briefly highlighting the main widely seen aortic flow characteristics. Velocity: in the main aortic conduit, which is similar to candy-cane geometry, the highest velocity takes place in central areas of each cross-section normal to the centreline, and lower velocity at the wall vicinity. Furthermore, the velocity is higher in ascending and descending aortas compared with the transverse aorta (the arch segment). Pressure: in a clinical setting, intravascular pressure is an important and popular metric amongst physicians. Studies have shown that, in healthy aortas, the pressure distribution is almost uniform, and differences mainly stem from the transient and pulsatile nature of blood flow, particularly in systole, which is around 15 mmHg. However, in the case of aortic disease, the distribution is not uniform and local pressure rise occurs. An example of pressure distribution in healthy and diseased aortas is shown in Fig. 11. WSS: in a normal aorta, the aortic valve is tricuspid, therefore, the jet flow that exits the left ventricular outflow tract is mostly laminar with streamlines aligned with the axial axis. Therefore, the flow friction reduces, with a uniform WSS distribution, where the axial vector of WSS is the dominant component [97]. While, where the flow is helical, the WSS circumferential component becomes dominant. Also, the magnitude of the WSS varies in a cardiac cycle due to the velocity changes, nevertheless, in a normal healthy aorta the TAWSS—mean WSS in a cardiac cycle—in most of the locations is less than 4 Pa.

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Different frames in a cardiac cycle

a

b

c

d Fig. 11 An example of convective pressure distribution in healthy and diseased aortas: (a) healthy aorta, (b) complex aortic disease including bicuspid aortic valve, ascending aorta dilation, pseudocoarctation, (c) aortic dissection, and (d) Marfan syndrome. (The figure is adapted from Pitcher et al. [98], and the figure labels have been updated)

Helicity: spiral flow in the aorta is named helical flow. The location it occurs, the number of discrete helices, the strength, the extension, and the duration they last are some main features that have been paid particular attention to. Most studies have confirmed that the blood that enters the aorta is helical. Recently, Ebel et al. [99, 100] demonstrated that the helical flow triggers at the outlet of the aortic valve and predominately exists in ascending aorta. Additionally, as the helix evolves in ascending aorta, it forms two opposite helices, where some retrograde flow is observed that may contribute to flow towards coronary arteries [101]. Furthermore, the helical flow begins in the systolic phase and lasts at least for the duration of systole (Fig. 12).

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Fig. 12 An example of helical flow in the aorta in six timepoints in a cardiac cycle (the results were taken from Deyranlou et al. [36] and the figure was edited)

4.3 Limitations Over the past few decades, subject-specific modelling of vascular flow has been developed significantly. Despite all the advancements, there are still some areas that need to be improved. In this section, the main limitations of the in-silico modelling are explained. (i)

A successful segmentation of vasculature requires high-quality imaging data. Furthermore, the segmentation and geometry reconstruction are not fully automated. Thus, it is a time-consuming process, and the result depends on the expertise of the user. Furthermore, some slight differences may be seen if the segmentation is performed by different users or on different occasions. (ii) For modelling different vascular networks, current imaging technology sometimes is not adequate. Therefore, the user is not able to resolve the anatomy easily or completely. (iii) Personalisation of boundary conditions is complicated, and it still requires some user-dependent manipulation for a successful implementation. Additionally, a thorough personalisation requires obtaining enough flow data for the inflow and outflow boundaries and structural data for the vessel wall. Since the acquisition of all data is not always part of the clinical routine, the lack of enough data is another challenge. (iv) Current workflow from the beginning of image segmentation to the postprocessing stage requires a week or a few weeks and it depends on the level of expertise of the users. Although the duration can meet most surgical planning timelines, for some urgent interventions it is not able to accomplish the job within the required timescale.

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References 1. British Heart Foundation (2020) UK Factsheet. British Heart Foundation 1–21 2. Erbel R, Aboyans V, Boileau C et al (2014) 2014 ESC guidelines on the diagnosis and treatment of aortic diseases. Eur Heart J 35:2873–2926. https://doi.org/10.1093/eurheartj/ehu281 3. Popieluszko P, Henry BM, Sanna B et al (2018) A systematic review and meta-analysis of variations in branching patterns of the adult aortic arch. J Vasc Surg 68:298-306.e10. https:// doi.org/10.1016/j.jvs.2017.06.097 4. Aboulhoda BE, Ahmed RK, Awad AS (2019) Clinically-relevant morphometric parameters and anatomical variations of the aortic arch branching pattern. Surg Radiol Anat 41:731–744. https://doi.org/10.1007/s00276-019-02215-w 5. Nakamura M, Wada S, Mikami T et al (2003) Computational study on the evolution of an intraventricular vortical flow during early diastole for the interpretation of color M-mode Doppler echocardiograms. Biomech Model Mechanobiol 2:59–72. https://doi.org/10.1007/ s10237-003-0028-1 6. Zhang LT, Gay M (2008) Characterizing left atrial appendage functions in sinus rhythm and atrial fibrillation using computational models. J Biomech 41:2515–2523. https://doi.org/10. 1016/j.jbiomech.2008.05.012 7. Schenkel T, Malve M, Reik M et al (2009) MRI-Based CFD analysis of flow in a human left ventricle: methodology and application to a healthy heart. Ann Biomed Eng 37:503–515. https://doi.org/10.1007/s10439-008-9627-4 8. Otani T, Al-Issa A, Pourmorteza A et al (2016) A computational framework for personalized blood flow analysis in the human left atrium. Ann Biomed Eng 44:3284–3294. https://doi. org/10.1007/s10439-016-1590-x 9. Lang RM, Bierig M, Devereux RB et al (2006) Recommendations for chamber quantification. Eur J Echocardiogr 7:79–108. https://doi.org/10.1016/j.euje.2005.12.014 10. Lang RM, Badano LP, Mor-Avi V et al (2015) Recommendations for cardiac chamber quantification by echocardiography in adults: an update from the american society of echocardiography and the European association of cardiovascular imaging. J Am Soc Echocardiogr 28:1-39.e14. https://doi.org/10.1016/j.echo.2014.10.003 11. Bommer W, Weinert L, Neumann A et al (1979) Determination of right atrial and right ventricular size by two-dimensional echocardiography. Circulation 60:91–100. https://doi. org/10.1161/01.CIR.60.1.91 12. Engla NEW (2010) Is computed tomography safe? Perspective 363:1–3. https://doi.org/10. 1056/NEJMp1002530 13. Smith-Bindman R, Miglioretti DL, Johnson E et al (2012) Use of diagnostic imaging studies and associated radiation exposure for patients enrolled in large integrated health systems. JAMA, J Am Med Assoc 307:2400–2409. https://doi.org/10.1001/jama.2012.5960 14. Dar AS, Padha D (2019) Medical image segmentation a review of recent techniques, advancements and a comprehensive comparison. Int J Comput Sci Eng 7:114–124. https://doi.org/10. 26438/ijcse/v7i7.114124 15. Vedula V, George R, Younes L, Mittal R (2015) Hemodynamics in the left atrium and its effect on ventricular flow patterns. J Biomech Eng 137:1–8. https://doi.org/10.1115/1.4031487 16. Vedula V, Seo JH, Lardo AC, Mittal R (2016) Effect of trabeculae and papillary muscles on the hemodynamics of the left ventricle. Theoret Comput Fluid Dyn 30:3–21. https://doi.org/ 10.1007/s00162-015-0349-6 17. Koizumi R, Funamoto K, Hayase T et al (2015) Numerical analysis of hemodynamic changes in the left atrium due to atrial fibrillation. J Biomech 48:472–478. https://doi.org/10.1016/j. jbiomech.2014.12.025 18. van Ooij P, Markl M, Collins JD et al (2017) Aortic valve stenosis alters expression of regional aortic wall shear stress: new insights from a 4-dimensional flow magnetic resonance imaging study of 571 subjects. J Am Heart Assoc 6:1–14. https://doi.org/10.1161/JAHA.117.005959

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3D Printing of Polymer Composites Hamid Narei and Masoud Jabbari

Abstract Material Extrusion Additive Manufacturing (MEAM) has recently been acknowledged as a time-efficient and reliable process of manufacturing polymer composite for prototypes and finished products. As properties of resins used in MEAM, including rheology and the mechanical characteristics, extrudate swell, the orientation of fibres, and the processing conditions, including strand-to-strand distance, layer thickness, infill density and build orientation, significantly affect the mechanical and thermal properties and surface roughness of MEAM-fabricated parts, fabrication time, and the manufacturing resolution, they should be thoroughly investigated. However, the interplay of various parameters hinders the proper understanding of the influences each parameter exerts on the performance of the fabricated part. Considering the capabilities of numerical modelling using Computational Fluid Dynamics (CFD) to simulate a process with completely tailored specifications, it can be used to address the aforementioned barrier. Hence, this chapter aims to provide a brief overview of the numerical modelling of MEAM technology for the production of polymer composites.

1 Introduction Additive Manufacturing (AM), commonly known as three-dimensional (3D) printing, is a process whereby 3D parts are directly created from computer-aided design data without the need for fabrication of expensive part-specific tooling. Major advantages of AM over conventional subtractive (e.g., turning, milling, electrical discharge machining, drilling, etc.) or formative (e.g. casting, forging, bending, powder metallurgy, injection moulding, etc.) manufacturing methods include lowering production costs, shortening the design manufacturing cycle, and H. Narei Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M13 9PL, UK M. Jabbari (B) School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_5

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increasing the degree of automation. Hence, AM has recently drawn substantial worldwide attention and secured different applications in the automotive, aerospace, electrochemical, electronics and biomedical sectors [9, 41]. Among different AM technologies available in the market, including material extrusion, binder jetting, material jetting, vat polymerization, sheet lamination, powder bed fusion, and directed energy deposition [13], Material Extrusion AM (MEAM) has gradually been gaining more acceptance owing to its comparatively low cost, wide availability, relatively minor safety concerns regarding the process, and ease of use [15]. In this manufacturing method, filaments are first melted in the heated nozzle and the molten material is extruded from a moving head along a predefined toolpath. then, the controlled deposition of successive layers of this extruded material in a build platform forms the 3D parts [8, 11] (Fig. 1). However, on the one hand, parts fabricated by MEAM have inferior mechanical properties, including low elastic behaviour, possible delamination, and poor mechanical integrity, due to the layer-by-layer nature of the deposition and the presence of numerous voids [3, 5]. Moreover, layer-based manufacturing methods produce rough surfaces, which are more difficult to postprocess than metals [40, 43]. On the other hand, the thermoplastic polymer matrix, commonly used in the MEAM process for manufacturing various parts, in essence, suffers from inferior mechanical properties, exacerbating the position of fabricated parts as fully functional and load-bearing components [8]. The existence of such imperfections makes the improvement of materials used in the MEAM necessary to make sure that the structural functionalities of fabricated parts meet the functional requirements of different applications. Polymer composites reinforced with different forms of additives such as particles, short and continuous fibres, and nanomaterials for MEAM feedstock have been developed to address some of the shortcomings associated with manufacturing. Polymer

Fig. 1 Schematic diagram of the MEAM process [14]

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composites are characterised by high performance, enhanced mechanical properties, and extra functionality including biocompatibility, electroconductivity, or higher heat conductivity [2, 5, 8, 33]. First endeavours focused on the insertion of various types or sizes of particles into a polymer matrix. Although parts fabricated using composites containing particles demonstrate some improvements in terms of the mechanical, electrical, and thermal properties, the level of performance obtained was low compared to composites fabricated by conventional composite manufacturing methods [36, 37]. Indeed, in terms of improved mechanical properties, composites reinforced with short or continuous fibres are more credible [26]. Nonetheless, the prohibitively expensive facilities and equipment required by the conventional manufacturing processes of fibre-reinforced polymer composites, including autoclaves or complicated moulds used in out-of-autoclave procedures hamper the wide application of this type of composites [8]. MEAM can be a promising replacement for the costly conventional methods such as filament winding, vacuum forming, and pultrusion for manufacturing fibre-reinforced polymer composites. As a predictive tool, numerical simulation has widely been used to analyse a variety of manufacturing processes, including MEAM [20, 42]. MEAM simulation, however, is extremely challenging due to the complex multiphysics phenomena occurring on different temporal and spatial scales [10]. Hence, different aspects of the polymer composite extrusion flow in the MEAM, including the internal flow of the reinforced molten material inside the extruder, extrudate swell, the fibre orientation of the fibre-reinforced polymer composites inside the nozzle and in the extrudate during the deposition process, and the thermo-mechanical behaviours of the fabricated parts, are separately simulated by different numerical techniques, including Finite Difference (FD), Finite Element (FE) and Smoothed-Particle Hydrodynamics (SPH) methods. In this chapter, we provide a brief overview of the studies focused on numerical analysis of the polymer composites manufactured via the MEAM process. Unfortunately, a few studies have focused on the numerical simulation of the MEAMmanufactured two-phase polymer composites. First, we focus on fibre fillers and investigate the effect of the filler on the fibre orientation and resulting mechanical and thermal properties. Then, we summarise the limited studies examining the MEAM-manufactured polymer composites reinforced with non-fibre fillers. Next, we briefly explain core–shell polymer composites and a study focusing on the numerical simulation of a core–shell strand deposition. In the end, a summary is provided to properly summarize the key points of the discussion and the future outlook of numerical simulation of the polymer composites manufactured via the MEAM is presented.

2 Fibre Fillers Fibre-reinforced polymer composites have gained considerable attention for application in MEAM as parts manufactured from fibre-reinforced feedstock have higher

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stiffness, strength and thermal stability. However, they exhibit higher anisotropy than their virgin counterparts, mainly stemming from the shear alignment of fibres during the printing process, where the fibre suspension is oriented by the flow field within the polymer. Composite materials reinforced with fibres show properties more similar to those of the fibres in the direction in which the majority of fibres are aligned and to those of the matrix in directions with less fibre alignment [19]. Hence, it is of utmost importance to quantify the fibre orientation, which is mostly dependent on the flow field velocity gradients inside the nozzle and in the extrudate during the deposition process, as it directly affects the thermal and mechanical properties of the manufactured parts [39]. The fibre orientation in beads deposited by MEAM is extremely complex and significantly varies depending on the location within the deposited beads. Numerical simulation may offer detailed insight into the complex physics of manufacturing fibre reinforced polymer composited with MEAM and help to predict the fibre orientation pattern in a printed composite. This knowledge may finally be used to determine the thermal and mechanical properties of deposited beads and, eventually, manufactured parts.

2.1 Fibre Orientation and Mechanical Properties Nixon et al. [30] were one of the first groups to focus on the fibre orientation state within an MEAM nozzle in 2014. They employed Moldflow and the modified FolgarTucker diffusion model to numerically evaluate the effects of three different nozzles (divergent, straight and convergent), analysed as axisymmetric 2D cross-section, on the resulting fibre orientation state. They reported the highest fibre alignment was observed in the convergent geometry, while the divergent nozzle brought about the lowest. They also suggested increasing the nozzle divergence would decrease the first principal tensile modulus. In a similar study, Garcia [16] also investigated the effects of the nozzle geometries on the fibre orientation state; however, he considered a 3D domain and calculated the 3D random fibre orientation. These studies ignored the extrudate swell, the sudden variation in the flow boundary at the nozzle exit due to sudden pressure change, which greatly affects the fibre orientation state and, consequently, the mechanical properties of the produced prints. Numerical simulations performed by a team led by Douglas E. Smith of Baylor University have provided valuable insight into the nozzle extrusion flow and polymer deposition flow in short fibre-reinforced polymer composites manufactured via the MEAM process [17, 39, 46–49]. They covered different areas including the extrudate swell, elastic constants and CTE of a deposited Carbon Fibre (CF) reinforced Acrylonitrile Butadiene Styrene (ABS) polymer composite. In 2016, Heller et al. [17] numerically investigated the effects of nozzle geometry and extrudate swell of CF-reinforced ABS composite on the fibre orientation for a conventional MEAM. The COMSOL Multiphysics integrated with a MATLAB interface was used to develop a Finite Element Method (FEM) model, where an incompressible fluid flow with a Newtonian fluid melt was considered. To explain the

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fibre orientation, they used the Advani and Tucker model and Folgar-Tucker model for isotropic rotary diffusion and orientation, respectively. Their results showed that extrudate swell had a significant effect on the fibre orientation state, and thereby, mechanical properties of short CF reinforced polymer composites. They reported the free surface expansion, as a result of the melt swell, markedly decreased the fibre orientation, leading to an approximately 20% decrease in the axial modulus of elasticity and a 6.62% increase in the radial modulus. In addition, they conducted a parametric analysis to further investigate the effects of nozzle geometry on the fibre alignment state and mechanical properties. In another study, Heller et al. [18] extended their previous work to simulate the deposition flow of a large-scale MEAM. They evaluated the fibre orientation in a CF/ABS deposited bead and its effects on the thermal and mechanical properties. Their results demonstrated the fibre alignment was high in the flow direction, increased in the convergence zone and slightly decreased in the straight section of the nozzle. Wang and Smith [46] numerically studied the effects of melt rheology on the die swell, fibre orientation and elastic properties of a fibre-reinforced polymer composite extruded in a large-scale MEAM with the finite element method using ANSYS-Polyflow. They compared several rheology models including Newtonian, Carreau–Yasuda, Power law, Simplified Viscoelastic (SV) and multi-mode Phan– Thien–Tanner (PTT) material models. They reported the PTT model predicted a higher die swell than those of the Generalised Newtonian Fluid (GNU) models (i.e., Carreau–Yasuda and Power law models) and the Newtonian model. Their simulations also showed that the PPT melt model resulted in the lowest fibre principal alignment, while the Power law model yielded the highest in the polymer extrusion direction. Furthermore, Young’s modulus along the principal direction computed through the GNF models was higher, while the PPT model yielded a lower Young’s modulus but higher shear moduli. Finally, they recommended the SV model as the melt rheology model for the modelling of large-scale MEAM over conventional viscoelastic fluid models like PPT as the fibre orientation state and elastic properties computed through the SV model were relatively similar to those yielded from the PPT model, while its computational cost was lower. Wang and Smith [47] also examined the effects of the single screw swirling motion of a large-scale pellet-based MEAM on the resulting fibre orientation state in an extrudate of CF-reinforced polymer composite as it was reported the single screw rotation uniquely aligned fibres within the deposited composite bead. Using the FEM, they simulated an axisymmetric non-Newtonian viscoelastic flow, and their model included the flow near the screw tip and within the nozzle, as well as a short section of the free extrudate. To solve the fibre orientation state, they applied the Wang-O’Gara-Tucker RSC fibre orientation model. Their findings indicated that the swirling motion of the flow resulting from the screw rotation yielded rotational streamlines and elongational flows, causing fibres to travel longer pathlines through and orientate within the flow domain which are notably different as compared to nonswirling flows. Moreover, they suggested that the principal tensile modulus yielded by the swirling flow model was approximately 20% higher than that of the nonswirling flow. Comparing their results with related experimental data for a similar

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system, they also reported the data obtained from the swirling model using the RSC closure showed more favourable agreement with the experiments than the results predicted by the non-swirling flow model. In the aforementioned studies focused on fibre orientation, the weakly coupled flow-fibre formulation was used to predict the fibre alignment within the polymer composite extrude. In this approach, to solve the polymer melt flow kinematics, it is first assumed that the effect of fibres is negligible in the flow computation and the fibre orientation state is then predicted from the computed flow fields [49]. However, researchers observed that the reinforcement of polymer feedstock with fibres significantly affected the flow behaviour of the resulting composite in an MEAM process [6]. Hence, to better understand the related physical phenomena, it is of crucial importance to consider the mutually dependent effects between the reinforced fibres and polymer flow. In this regard, Mezi et al. [27] developed a fully coupled numerical model to study the fibre orientation state within the die swell of an MEAM process. They tested different values for the fibre–fibre interaction coefficient and the coupling parameter to examine their effects on the final fibre orientation state and free surface shape. They observed when fibres were randomly distributed at the flow inlet, increasing the fibre coupling parameter resulted in a rise in the die swell ratio. On the other hand, they noticed the increase of the fibre coupling parameter decreased the die swell ratio for perfectly aligned fibres at the entrance with low fibre interaction coefficients. Wang and Smith [49] also developed a FEM-based algorithm to numerically quantify the mutually dependent relations between non-Newtonian polymer melt flow and fibre orientation state of fibre reinforced polymer composites manufactured via the MEAM process. They applied the computation approach to analyse an axisymmetric flow with a free surface defining the melt extrudate. They used the non-Newtonian power-law to model the shear-thinning rheological behaviour of the polymer melt. They also simulated the fibre orientation state within the flow domain through the Advani-Tucker orientation tensor approach. To compute the die swell of the free extrudate, they employed a streamline-based remeshing technique. Comparing the results obtained from the proposed fully coupled flow model with those obtained from the weakly-coupled flow solution, they indicated that flow field magnitude along the extrusion direction and the fibre alignment in the flow direction increased when the fully coupled model was used. Furthermore, they reported the extrudate swell ratio computed by the fully coupled flow model showed a 2-time reduction as compared to the weakly-coupled flow solution. Moreover, using the fully coupled approach resulted in a difference of approximately 8% in the bead-average tensile modulus along the direction of material deposition. It should be mentioned that, in this study, the researchers only investigate the full coupling of reinforced fibres and polymer flow in an axisymmetric flow with a free surface without considering any deposition feature. In 2021, Wang and Smith [48] extended the method they developed in their previous work [49] to investigate the effect of flow–fibre interaction for the extrusion and deposition processes of short CF/ABS composites produced via a large-scale MEAM. In this study, they modelled the molten composite flow inside the nozzle

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and a strand deposited on the build plate using a 2D planar Newtonian creeping flow. They employed Advani–Tucker fibre orientation tensors to simulate the fibre orientation, where the orthotropic fitted closure and isotropic rotary diffusion are used. Comparing the results from the fully coupled model with those of the weakly coupled, they reported significant differences between the solutions of uncoupled and fully coupled models near the nozzle convergence and extrusion-deposition transition zones. It should be noted that in these zones intense flow-fibre interaction occurs. Moreover, the fully coupled model was very sensitive to the value of the fibre–fibre interaction coefficient. Finally, they indicated that their material properties predicted via the fully coupled flow/fibre orientation solution were in good agreement with related experimental studies. It should be mentioned that, in this study, researchers characterised the extrusion deposition flow only using a Newtonian flow model partly because, in complex domains, when the flow fields are coupled with second-order fibre orientation tensors, anisotropic constitutive behaviour, which is based on tensors, exhibit convergence issues. Furthermore, the researchers only employed averaged fibre aspect ratio up to 15 in their fully coupled simulations also because of the difficulty in computational convergences [45, 48]. Wang [45] simplified the FEM-based fully coupled flow/orientation model proposed by Wang and Smith [49] using an optimised scalar representation of the viscosity where the fourth-order fibre orientation tensor was substituted for the fourth-order viscosity tensor. He used the simplified fully coupled model to investigate the effects of averaged fibre aspect ratio and material deposition rate on the fibre orientation state for a large-scale MEAM process. For this purpose, he applied the power law rheology model for the flow fields computation. He also characterised the extrudate swell of the planar deposition flow using a 1D remeshing technique. His findings demonstrated that increasing the material deposition rate would reduce the extrudate swell of the deposited bead, while a rise in the averaged fibre aspect ratio would increase the principal fibre alignment, particularly at the bottom of the deposition flow. Furthermore, he reported that changes in the material deposition rate resulted in a marked difference in the local orientation state, mainly at the bottom of the flow in contact with the build plate and the core of the flow. They also mentioned that a high averaged fibre aspect ratio would cause increased difficulty in the numerical convergence of the solution. In most of the previous literature, the researcher used the Advani-Tucker fibre orientation tensor approach [1] for the modelling of the fibre orientation state within the composite flow. The Advani-Tucker model introduces orientation tensors representing fibre orientation states to the classic FolgarTucker model to extend its usefulness [21]. Although it has been indicated that the application of the Advani-Tucker method to simulate the fibre orientation state was computationally cost-effective, it has also been demonstrated that the classic FolgarTucker model resulted in faster overprediction of the orientation tenor results [44, 48]. To address this overprediction, Wang [44] used the principle Anisotropic Rotary Diffusion Reduced Strain Closure (pARD-RSC) model for the estimation of the fibre alignment in the newly developed fully coupled flow/fibre orientation model [49]. The researcher also investigated the effects of the strain slip factor and the fibre–fibre interaction coefficient on the fibre orientation state in deposited beads

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of a large-scale MEAM flow. He reported that predicted results of the pARD-RSC model were mainly sensitive to the strain slip factor. It was demonstrated that higher values of the fibre–fibre interaction coefficient would decrease the principal fibre alignment, while an increase in the strain slip factor would notably increase it in the deposited bead. Furthermore, the findings of simulations, confirmed by the experimentally reported data, showed the fibre orientation state in the deposited beads was significantly affected by the initial fibre alignment pattern of the flow inlet. Lewicki et al. [23] numerically investigated the flow field and fibre orientation of CF-reinforced epoxy composites during the extrusion process. In this study, short fibres were modelled as discrete particles and their interactions with matrix, other fibres and walls were considered. They initially assumed randomly oriented fibres inside the filament and the simulations predicted higher fibre orientation near the walls, attributed to a wall-dominated shear alignment (Fig. 2). Few researchers focused on the application of numerical particle methods, including SPH, to simulate extrusion and deposition processes of both continuous and discontinuous (short) fibre-reinforced polymer composites in MEAM. As a meshfree Lagrangian method, SPH enjoys several advantages, including being inherently mass conservative, well-suited in dealing with large deformations and free surfaces and moving interfaces and well-characterised numerical stability [24]. Owing to the mentioned advantages, there have been some efforts to model the MEAM process of fibre-reinforced polymer composites. In one of these efforts, Bertevas et al. [4] numerically studied the fibre orientation state in fibre-reinforced composite beads deposited by MEAM using SPH. For this purpose, they implemented a microstructural fibre suspension model in an SPH framework. They investigated the effects of certain parameters, such as fluid viscosity, initial fibre orientation, printing head internal angle, the substrate-to-extrusion velocity ratio, and the ratio between extrusion and substrate velocities, on the orientation state in the printed bead. They reported that the orientation state of the deposit was slightly affected by fluid viscosity, initial fibre orientation and printing head internal angle. Their main finding was the prediction of a skin/core structure in the deposited bead, where the skin regions exhibited a higher fibre alignment parallel to the substrate compared to one detected in the core region. They indicated that, for a constant aspect ratio, increasing the fibre volume fraction would increase this skin/core fibre effect, which was also sensitive to the ratio between extrusion and substrate velocities. The skin/core effect was also enhanced when the substrate velocity was lower than the extrusion velocity. Yang et al. [50] coupled SPH with the discrete element method (DEM) to model the MEAM process of fibre-reinforced polymer composites. They simulated both continuous and discontinuous (short) fibres, represented by discrete DEM particles bonded together. They modelled the polymer matrix with discrete SPH particles. Using their coupled method SPH-DEM model, they provided limited predictions of fibre orientation and deformation. Recently, Zang et al. [51] developed a coupled multiphase model based on CFD and DEM to numerically study the nozzle clogging in the MEAM process of fibre-reinforced composites. They investigate the effects of polymer viscosity, fibre volume fraction and fibre length on the MEAM process. They treated the polymer matrix as an incompressible Newtonian flow in CFD, while

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Fig. 2 Fibre orientation evolution with time in 3D within the computational domain under simulated conditions of printing [23]

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they modelled short CFs as rigid bodies through clumping discrete spheres in DEM. Applying the Hertz-Mindlin contact law, they considered the Collisions between fibres in DEM. They demonstrated that the nozzle might clog when the fibre volume fraction and/or the fibre length increase. They also suggested when manufacturing composites with fairly short fibres, using a polymer matrix with a lower viscosity could be beneficial to avoiding nozzle clogging issues, while, in the case of long fibres, the nozzle clogging issue was almost independent of the polymer viscosity.

2.2 Thermal Properties Owing to the relatively high Coefficient of Thermal Expansion (CTEs) of conventional polymers used in MEAM, the process is particularly prone to adverse warping and delamination between layers due to thermal stresses [12]. When it comes to largescale MEAM, these issues are dramatically intensified as even a small thermal strain may lead to several millimetres of deformation. Recent studies [25, 38] have shown adding CFs to polymer feedstock could extremely dwindle warping in manufactured parts because of the near-zero CTE of CFs along their axis, thereby, the reduced CTE of the resulting polymer/CF composites. Moreover, it has been demonstrated that the higher thermal conductivity of printed parts parallel to the printing direction could reduce thermal gradients and, therefore, distortions throughout the parts. The combination of these factors can profoundly improve the geometric accuracy of the parts manufactured with CF/polymer composites compared to those printed with unreinforced polymers [25]. Furthermore, the low CTE parallel to the printing direction of CF/polymer composites enables tailoring the in-plane expansion behaviour of printed tools to specific needs [6]. Brenken et al. [7] developed a physics-based simulation tool Additive3D in Abaqus© 2017 to model the MEAM process for fibre-reinforced thermoplastic composites. They determined the temperature history of the deposition process for the 50 wt.% CF/Polyphenylene Sulphide and modelled the anisotropic thermal conductivity of CF-reinforced polymer composite. Compton et al. [12] studied the thermal evolution in thin walls of CF/ABS composites manufactured with Big Area Additive Manufacturing (BAAM). For this purpose, they developed a simple 1D transient thermal model to simulate the build process and performed experiments using a BAAM 3D printer. The predictions of layer temperatures obtained from their finite element model were in good agreement with temperatures of corresponding physical experiments measured using infrared imaging. Russel et al. [39] also developed a computational approach for predicting the spatially varying fibre orientation within the flow of a large-scale MEAM. They eventually predicted the CTE tensors for the bead of a fibre-reinforced polymer composite. In a recent study, Hoskins et al. [19] simulated the residual thermal stress of a cuboidal part produced from CFreinforced ABS feedstock using. For this purpose, the CTE variations across the

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cross-sections of beads, experimentally characterised using thermomechanical analysis, are used as a non-homogenized input for a finite element model. Then, predictions obtained from this model were compared with strain maps of large-scale printed parts measured using 2-D digital image correlation. They reported that the CTE results of the finite element model and digital image correlation were not extremely similar. They attributed this discrepancy to the fact that porosity in the measured cubes had not been accounted for in the finite element model. In the study explained above, Wang and Smith [48], using the fully coupled flow/orientation algorithm based on the FEM approach, computed the CTE for short CF-reinforced ABS polymer composite manufactured via a large-scale MEAM process. Their simulations suggested that the transverse CTE exhibited some skewness, which might affect the inter-bead void formation and result in uneven distortion during the manufacturing process. Using SPH, Ouyang et al. [31] explored the thermal behaviour of non-isothermal fibre-reinforced polymer composite produced via a 3D printing process. They implemented a classic, microstructure constitutive model for fibre suspensions combined with the temperature-dependent power law model for the viscosity in the SPH framework. In the simulations, the fibre evolution equation, the suspension momentum conservation equation and the temperature transport equation were fully coupled. Their findings suggested that increasing the temperature dependency coefficient would reduce the fibre alignment along the printing direction in the top half of the deposited layer and enhance it in the bottom half. On the contrary, A rise in the Peclet number, defined as the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid, would reduce the fibre alignment in the bottom half and enhance it in the top half. Furthermore, the researcher indicated that increasing the Peclet number decreases the thickness of the printed layer. It should be mentioned that, in this study, they assumed the thermal conductivity of fibre-reinforced polymer composites was constant (isotropic). Ouyang et al. [32] extended their previous work [31] by considering an anisotropic thermal conductivity (fibre orientation-dependent thermal conductivity tensor) for the composite. They also investigated the effect of the temperature gradient in the nozzle on the fibre alignment for the one-layer deposition and thermal interactions between two deposited beads. They reported that the anisotropic thermal conductivity enhanced the thermal conduction along the printing direction. Furthermore, at relatively high-volume fractions, the anisotropic thermal conductivity reduced the fibre alignment along the printing direction in the lower half and increased it in the upper half. Regarding the nozzle temperature, they suggested that its increase would enhance the fibre alignment in the lower half of the layer when one layer of composite was deposited, and this alignment became more with the rise in both the fire aspect ratio and volume fraction. The deposition of the second bead affected the fibre orientation of the first bead because of the squeezing from the second layer; the fibre alignment of the upper half in the first layer increased, while the lower half experienced a reduction in its fibre alignment along the printing direction. A rise in both the fibre aspect ratio and volume fraction also amplified this variation in the fibre alignment of the first layer.

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3 Non-fibre Fillers The number of studies on the numerical simulation of the MEAM manufactured composites filled with non-fibre fillers is extremely limited. Nikzad et al. [29] studied the melt flow behaviour in an MEAM process for iron particle-reinforced ABS composite. Using 2D and 3D simulations respectively performed in ANSYS FLOTRAN and ANSYS CFX CFD tools, they analysed the velocity, pressure drop during the deposition and temperature distribution along the melt flow. By comparing the results of simulations, they indicated that two analyses exhibited a strong correlation in predicting the flow behaviour. Kim et al. [22] also investigated the effects of two different nozzle geometries (flat and circular) on the orientation of silver nanowires (AgNWs) inside the extruded photo-curable thermoset polymer composite. They reported changing the nozzle shape led to different fluidic behaviours and the circular nozzle generated an aligned distribution of AgNWs, while the flat one induced a random distribution.

4 Core–Shell Polymer Composites To improve the properties of materials commonly used in the MEAM process, core– shell structured filament has recently been introduced. In this novel material design approach, a polymer matrix with a high glass-transition temperature, like Polycarbonate or a blend of ABS and Polycarbonate, is used as the core material to create a stiff skeleton. Another polymer enjoying a low glass-transition temperature, like high/low-density polyethene or Surlyn, acts as the shell to make the improved interdiffusion of polymers between adjacent layers possible. 3D printed parts manufactured from the core–shell filaments demonstrated good dimensional accuracy, enhanced elongation at break and unprecedented impact resistance [34, 35]. Narei et al. [28] recommended using the one-stage manufacturing process where the core polymer is embedded into the shell inside the printing head and the resulting composite is immediately printed onto the build plate, to reduce the energy consumption of the conventional two-phase manufacturing process, as well as to accelerate the overall manufacturing process. They employed numerical simulations within the CFD framework to identify the best combinations of processing parameters for the MEAM process of a core–shell polymer composite (Fig. 3a). Their objectives were to find strands with the highest possible volume fraction of core polymer to maximise the dimensional accuracy of the resulting strand and, at the same time, with the full encapsulation of the core matrix inside the shell to improve the interdiffusion of polymers between adjacent layers, leading to the improved interfaces. In this study, they controlled the deposition flow by three dimensionless parameters, namely, the diameter ratio of core material to the nozzle, the normalised gap between the extruder and the build plate, and the velocity ratio of the moving build plate to the average velocity inside the nozzle. They used the Volume-of-fluid method coupled

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Fig. 3 a Geometry of the computational domain, and b cross-sections of core–shell strand for t/D = 1, d/D = 0.7 and V /U = 1 [28]

with the level-set to capture the molten core and shell interface and free surface of the extruded materials. They reported the shape, size and cross-sections of deposited strands, demonstrating the effects of operating parameters on the encapsulation of the core matrix inside the shell. Moreover, they suggested that increasing the diameter ratio would result in a high-volume fraction of core material, and, thereby, better dimensional accuracy. Furthermore, they indicated that the diameter ratio of 0.7 (d/D = 0.7), the normalised gap of 1 (t/D = 1), and the velocity ratio of 1 (V /U = 1) would result in a full encapsulation of the core inside the shell with the highest volume fraction (Fig. 3b).

5 Summary and Future Outlook MEAM is a promising manufacturing technique for the production of polymer composites parts. This chapter briefly reviewed the literature on the numerical simulation of polymer composites manufactured via MEAM. We tried to highlight the applications of CFD methods in the modelling of the complex process of MEAM. Unfortunately, few researchers have numerically studied the numerical simulation of the MEAM manufactured two-phase polymer composites. The effects of fibre fillers on the fibre orientation and resulting mechanical and thermal properties were investigated. Furthermore, few studies focusing on the MEAM-manufactured polymer composites reinforced with non-fibre fillers were also summarised. In the end, a novel polymer composite, namely core–shell polymer composite, was also investigated. As the number of studies in this area is extremely limited, we believe the future research needs in this area are as follows: • Advanced process modelling and optimization of process parameters should be conducted to improve the reliability of the MEAM. The effects of various process variables in a multi-scale modelling paradigm should be thoroughly investigated

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to ensure that an MEAM process with a repeatable outcome is achievable for the reliability and commercial viability of the manufactured composites parts. • More research efforts should be focused on modelling the continuous fibre reinforced composites manufactured via MEAM. Although continuous fibre reinforced composites offer better thermo-mechanical properties compared to those of short fibre-reinforced composites, the number of studies that investigated the numerical simulation of continuous fibre-reinforced composites produced via MEAM is extremely low. • More studies should be focused on the numerical simulation of polymer composites reinforced with nanomaterials, including different nanoparticles, nanotubes and nanowires, as they have demonstrated exceptional mechanical, thermal, electrical and chemical properties. Although structural characteristics of nanotubes and nanowires somehow resemble fibres, they enjoy unique structural characteristics and chemical and physical properties. • More studies are needed to thoroughly model core–shell polymer composites manufactured via MEAM. For instance, the successive deposition of strands should be numerically simulated to examine the effect of various processing parameters on the properties of the formed mesostructure.

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Magnetorheological Fluids Hesam Khajehsaeid, Ehsan Akbari, and Masoud Jabbari

Abstract A Magnetorheological fluid (MRF) is a suspension of micron-sized ferromagnetic particles in a nonmagnetic carrier fluid, including some additives mainly added to prevent agglomeration and sedimentation of the particles. In contrast to ordinary fluids, MRFs can tolerate shear stress until a threshold value. This threshold stress which strongly depends on strength of the applied magnetic field, is called yield stress of the MRF. The rheological properties of MR fluids depend on the applied field in addition to the shear rate. The major issue in the preparation of MRFs is the selection and amount of constituents. Stability is an important factor which determines the extent of applications of an MRF. Several factors must be considered to achieve a stable MRF. Resistance against movement in both off-state and on-state is another important characteristic. Thanks to their interesting magneto-mechanical properties, MRFs are utilized in a variety of applications that require rapid, continuous, and reversible changes in the state of the system. In general, MRF devices use one of the main modes of operation based on the type of deformation applied to the fluid. This includes valve mode, shear mode and squeeze mode. Common MR devices are brakes, clutches, dampers, and mounts. A study on the rheology, magnetic behavior, and working modes of MRFs is necessary prior to designing any MR device. To design magnetorheological devices and to understand how they work, one should first realize the relationship between the shear stress and shear rate in the incorporated magnetorheological fluid. Several phenomenological models have been proposed to predict the magneto-mechanical behaviour of MRFs as well as performance of MR devices. This chapter will review the current state-of-the-art for development, applications, and modelling of magnetorheological fluids. H. Khajehsaeid (B) WMG, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected] E. Akbari The Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran e-mail: [email protected] M. Jabbari School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_6

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1 Introduction Magnetorheological (MR) materials are a category of smart materials that exhibit magneto-mechanical coupling. Mechanical properties of these materials are changed in the presence of a magnetic field. Based on the type of carrier matrix and material’s physical state in the absence of magnetic field, MR materials can be generally classified into MR fluids (MRF), MR elastomers (MREs), MR powders (MRPs) and MR gels (MRGs) [1]. A magnetorheological fluid is a suspension of micron-sized ferromagnetic particles in a nonmagnetic carrier fluid, often including some additives mainly added to prevent agglomeration and sedimentation of the particles. In contrast to ordinary fluids, which cannot often sustain shear stress, however small, MRFs can tolerate shear stress until it reaches a threshold value. This threshold stress is called the yield stress of the MRF and strongly depends on strength of the applied magnetic field. For Newtonian fluids, the shear rate is almost proportional to the applied stress with a proportionality coefficient known as viscosity of the fluid. Viscosity is one of the important characteristics that industrial applications of a fluid may be limited due to its magnitude. MRFs exhibit a reversible sudden change from a Newtonian-like fluid to a semisolid state upon application of an external magnetic field [2, 3]. It means that the rheological properties of MR fluids, e.g., viscosity depend on the magnitude, (and sometimes direction) of the applied field in addition to the shear rate. In the presence of a magnetic field, randomly dispersed magnetic particles are mutually attracted and rearranged to form chain-like microstructures aligned in the direction of the applied field as shown in Fig. 1. These chains restrict the fluid motion and convert the fluid entirely/partially to a semi-solid state (see Fig. 2) which results in a yield stress [4]. Beyond yield, the chains may collapse, however, there might still be particle agglomerations that significantly increase the fluid viscosity compared with the off-state (zero magnetic field) condition [5]. Yield stress and off-state viscosity are very important in characterization of MRFs. Yield stress is the maximum tolerable shear stress without considerable deformation, and off-state viscosity corresponds to the fluid’s viscosity in the absence of a magnetic field. A negligible off-state viscosity and a high yield stress are favorable for MRFs in most applications though almost any attempt to improve one of these, negatively affects the other one. These properties depend on viscosity of carrier fluid, size,

Fig. 1 Distribution of magnetic particles in an MRF: (left) off-state and (right) on-state

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Fig. 2 An MRF in the on-state

shape and concentration of particles as well as additives [6, 7]. Viscosity change of MRFs is reversible and can be toggled in a few milliseconds which is desirable in applications requiring rapid changes in the state of a system. Both yield stress and viscosity can easily be regulated by adjusting the intensity of the applied field. This provides the opportunity to adjust stiffness and damping properties of MR systems to adapt to different conditions. MR suspensions are prepared in a rather simple procedure which is mixing all the constituents. However, the major issue is the type and amount of constituents. An MRF includes three main components: carrier fluid, magnetic particles and additives. Several factors must be considered in the selection of constituents to achieve a stable and promising MRF. Stability is an important factor which determines extent of the applications of an MRF. Resistance against movement in both off-state and on-state is another important characteristic. The carrier fluid (or base fluid) is a medium which the magnetizable particles are suspended in. Viscosity of carrier fluid is one of the important factors that affects magnetorheological properties of an MRF. An appropriate MRF should contain a carrier fluid with low viscosity in order to achieve considerable MR effect [8]. Another important feature of carrier fluid is its vapor pressure. If carrier fluid is not vaporized easily, it can be used in a wide range of temperatures. Petroleumbased oils, mineral oils, industrial hydrocarbon oils, paraffin oils, water, and silicon oil are some of the common carrier fluids used in the MRF technology. Carrier fluid should have a good temperature stability (i.e., its viscosity should not be affected significantly over the intended temperature range) and a high boiling point. It should also be noncorrosive, and nonreactive with magnetic and abrasive particles. For any application, an appropriate carrier fluid can be chosen based on the required properties, stability and extent of the required magnetorheological effect. For example, for magnetorheological polishing processes, considering cost, environmental issues and cooling efficiency, water is often the best candidate [9]. For damper, clutch and brake applications, noncorrosive silicone or oil-based carrier fluids are often preferred.

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Magnetizable particles play a crucial role in the MR effect. To choose a proper magnetizable particle for an MRF, some characteristics should be considered including magnetic behavior (magnetization and hysteresis), size, shape, and density of particles. The amount of the particles in an MRF is also very important. A variety of metals, alloys and ceramics composites have been used as magnetic particles for MRFs. Some examples are ferrite-polymer, iron–cobalt alloy, carbonyl iron, nickel–zinc ferrites [10], iron and its compounds [8, 11]. Among these, carbonyl iron and its alloys have the highest saturation magnetization [12, 13]. These particles are magnetically multidomain and demonstrate a low magnetic coercivity, that is why they are called magnetically soft [3]. Soft materials are temporary magnets, a feature which is very important for reversibility of the magnetorheological effect. Since soft magnetic materials are easily magnetized and demagnetized, it is possible to control the rheological characteristics of MRFs reversibly [14]. Though in the past Fe3 O4 was extensively used in making magnetorheological suspensions due to its nontoxic nature and extensive availability, however, its application has been limited due to its low magnetic saturation. On the other hand, using Fe3 O4 in an MRF often incorporates some problems including sedimentation and aggregation of particles [15]. Nowadays, the most common magnetic particle in MRF technology is carbonyl iron (with approximate density 7.91 g/cm3 ) which is chosen because of its high magnetic saturation (approximately 2.1 T) and proper particle size (average particle size: 4.25 µm) [3, 12, 16]. Nevertheless, high density of carbonyl iron and its sedimentation has somehow limited its use in commercial applications. A considerable difference between densities of carrier liquid and magnetic particles often leads to rapid sedimentation, which reduces the efficiency of MRFs [17, 18]. To prepare an efficient and stable MRF, the selection of magnetizable particles must be done carefully. Coercivity and remnant magnetization of particles must be low for reversibility of the MR effect. For a promising MR effect, magnetic saturation must be as high as possible. A typical value of magnetic saturation around 2 T is considered to be appropriate. Other influencing factors are size, shape and density of particles. Researches show that the proper particle size range is 0.1–10 µm for a good MR effect. Low-density particles are preferable for the preparation of a stable MRF. Other parameters like compatibility with the carrier fluid and also chemical stability of particles must also be considered. As mentioned earlier, one of the main concerns in development of MRFs is the sedimentation of magnetic particles through operation time. The most commonly used methods to solve this problem is to add surfactants and use a dispersed solid colloidal stabilizer to give plastic, thixotropic properties to the carrying fluid [19]. Various types of additives (stabilizers and surfactants) are added to MRFs to prevent sedimentation, produce a stable suspension, enhance lubrication, and change the initial viscosity of the MRF [20]. Surfactants can also reduce the surface tension between magnetic particles and carrier liquid. Additives may include suspending as well as thixotropic agents, anti-friction and antiabrasion/erosion compounds. Highviscosity materials such as grease or other thixotropic materials might be added to improve the stability of particles against sedimentation. Iron Naphthenates or Iron Oleate may be added to MRFs as dispersants, while metallic soaps like lithium

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stearate and/or sodium stearate are added as thixotropic additives. Additives are necessary to control the viscosity of the fluid, sedimentation of particles and interparticle friction, in addition to prevent thickening of the fluid over time, which is called In-Use Thickening (IUT) [8, 13]. Methods used for measuring sedimentation ratio include visual observation, measurement of the magnetic permeability of suspension, and some other sophisticated methods. Though the former is still the most extensively used one, it is difficult to be applied in opaque media. Using the optical methods, it is also difficult to measure stability of slowly sedimenting suspensions. Centrifuges are mostly used to quicken the stability measurements which normally take long (a few days or weeks). Since permeability of MRFs strongly depends on volume fraction of magnetizable particles, sedimentation of these particles cause the upper layer of MR fluids to have lower particle concentration and, as a result, lower magnetic permeability. Thus, it is possible to measure the particles sedimentation by means of measuring the changes in the MRF’s magnetic permeability as well.

2 Operation Modes Depending on the type of flow, MR devices are designed to operate in either valve mode, or shear mode, or squeeze mode [8, 21]. Figure 3 shows these operation modes schematically.

Fig. 3 Operation modes of MRF devices: a valve mode, b shear mode, c squeeze mode

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2.1 Valve Mode In the valve mode, a pressure drop causes MRF between two stationary plates flow (Fig. 3a). There is also a magnetic field applied perpendicular to the flow direction to control MRF’s viscosity and, consequently the flow rate. Furthermore, a pressure drop would resist an output force attacking the device. It is normally derived by two independent viscous components in a valve-mode device, i.e., a magnetorheological component that depends on the magnetic field [19, 22], and a pure rheological component This operation mode is widely used in applications such as dampers, and shock absorbers particularly in the automotive industry.

2.2 Shear Mode In the shear mode, surfaces have relative motion. This means that, one surface is stationary while the other one moves, giving a velocity gradient to the incorporated MRF. In special case, both surfaces might be moving, however, with different velocities to maintain a shear on the MRF. In this mode, the magnetic field is applied perpendicular to the direction of flow, as shown in Fig. 3b. Some examples of applications of this mode are brakes, clutches, and dampers.

2.3 Squeeze Mode The squeeze mode is a recent development compared to the other two operation modes discussed in the above. As shown in Fig. 3c, in this mode, the MRF is located between two parallel plates while a mechanical force is applied to the plates in the same direction as the magnetic field to decrease or increase the distance between the plates, which results in a squeeze flow. The magnetic field is applied similar to the valve and shear modes in order to control the fluid’s viscosity, so the flow rate. Since in this operation mode, the acting force is in the direction of the magnetic particle chains, the MRF will have little or no flow. Thus, this mode is usually used for low-motion and high-force applications [19, 23] like small-amplitude vibration and impact dampers [24]. Due to the existence of a mechanical force inline with the particle chains, the yield stress developed in this mode is signifiantly higher than the valve and shear modes.

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3 MRF Applications Thanks to their interesting magneto-mechanical properties, MRFs can be utilized in a variety of applications that require rapid, continuous, and reversible changes in the state of system. These merits have enabled MRFs to be used in a broad range of devices. In general, MRF devices use one or two of the operation modes explained in the previous section. This includes valve mode, shear mode and squeeze mode or any combination of these, which show different performances in various practical applications [8, 25, 26]. Common MR devices include brakes, clutches, dampers, and mounts. A study of the rheology, magnetic behavior, and working modes of MRFs is necessary prior to designing any MR device. Nowadays, MRFs are being used in a variety of fields, and perform exceptionally in engineering and medical applications [27]. This is because MRFs exhibit a fast, stable and reliable magnetorheological effect with excellent controllability [28, 29]. This offers highly regulable devices with a low failure rate. MR devices can be split to those that utilize translational motion and those that use MRFs to convey a rotational motion. Translational MR devices are commonly used in semi-active dampers and controllable shock absorbers. Translational devices can act as linear force transmission units or temporarily lock an object in a defined position. These devices absorb vibrations by creating a variable damping force which is controlled by the applied magnetic field in the real time. Furthermore, the mechanical resistance of MRFs in the presence of a magnetic field can absorb energy of moving bodies. In rotational MR devices, an MRF is typically utilized for controlled transmission of torque from an input shaft/disc to an output shaft/disc. This mechanism is utilized in clutches to transmit torque. It can also be used as a brake in order to control a rotational motion by means of a magnetic field-induced braking torque. Moreover, such rotating devices can also be utilized as dampers to eliminate rotational vibrations. In the following, the principles, characteristics, and engineering applications of MRF devices are briefly described.

3.1 MRF Brakes A brake is a mechanical device that inhibits motion of a moving part. Brakes are widely used in transportation and construction equipment to slow, stop, or maintain the stopped state of moving machinery or moving/rotating parts. Most common brakes use friction to convert kinetic energy into heat, but other methods of energy conversion may also be used [30]. Magnetorheological fluids are currently being used in development of variableresistance brakes. The main aim of developing new MRF brakes is to increase brake torque, optimize device volume and weight, and reduce energy consumption [31]. Braking torque of an MR rotary brake can change rapidly in the presence of an externally applied magnetic field as viscosity of the fluid is altered. In this case, MRF

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is sandwiched between two cylinders, a rotating inner one and a static outer one. In the absence of an external field, the rotating cylinder would rotate freely, however, magnetic field causes MRF solidify, which means that the MRF phase changes from liquid to semi-solid (plastic), providing resistive torque to the rotating cylinder. This action occurs in milliseconds once the magnetic field is applied. The braking torque can be controlled by the intensity of the applied magnetic field in the real time [32]. MRF brakes come in a variety of designs including drum, disk, T-shaped rotor and multiple disks [33]. In general, basic MRF brake systems used in automotive applications consist of a rotating disk enclosed by an MRF gap and a static casing. Conventional disk brake systems have disadvantages such as high energy consumption, delayed response in the effect of pressure build up, noise caused by metal-onmetal friction, periodic maintenance of pad and hydraulic line, and need for auxiliary components. In contrast, MRF brakes benefit from quick response, minimum periodic maintenance need, simple design, and good controllability. These advantages have prompted industries and technology developers to consider developing MRF brakes to replace the conventional ones. In 2018, Wu et al. proposed a high-torque radial multi-pole and multi-layer MRF brake [34]. In this MRF brake, inner and outer coils with twelve magnetic poles generate two superpositioned magnetic fields, where four MRF layers were located in these coils. The device’s braking performance was great, but the additional coils increased the weight and energy consumption. Later, an MRF brake was designed with a multi-drum architecture that was compact and light-weight, generating four gaps of MRF, which enhances the shear surface area but also calls for more precise manufacture [35]. Nguyen et al. developed an MRF brake with three coils on each side of the brake housing in order to increase the braking torque in addition to the aforementioned structural improvements [36]. The proposed MRF brake provided better braking performance than a traditional single side-coil MRF brake. However, disadvantages such as high weight, high energy consumption, and temperature interference could not be avoided. In another MRF brake design, permanent magnets were used to attract MRF into a neighboring gap to reduce loss. As a result of the revised design, power loss due to the zero-field viscous torque was reduced by removing contact between MRF and rotor, but in contrast, in the braking state of the device, the effective shear area of MRF was smaller which limited the braking torque [38]. In 2019, a micro-brake based on MRF and MR grease was developed to control a miniature turbine generator [39, 40]. An electromagnetic coil filled internal space in the MR device, and low-permeability materials were employed to maximize magnetic field density. When the turbine generator is operating at high wind speeds, the MR brake can provide adequate resistance. A novel hollowed multi-drum MRF brake was proposed to deal with the magnetic hysteresis [41]. The device had a multi-drum mode, and a hollow casing filled with an actuator to obtain a compact structure, however, these compact constructions tend to raise the MRF temperature which consequently reduces the braking torque. Later, to reduce the influence of temperature on MRF, a squeezing MR brake was introduced [42]. Several flumes were incorporated in the device to disperse heat of the MRF and

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the brake disc where the water-cooling system enhanced the working duration and maintains high braking torque. Furthermore, an MRF brake in squeeze-shear mode was proposed with a squeezing bolt presenting compressive force for transmission performance and a magnetic vane set to ensure that the magnetic flux traverses both sides of the rotor [43]. The MRF device produced higher torque in the combined mode compared to those without compression.

3.2 MRF Clutches Clutches are mechanical components that engage/disengage power transfer from a driving shaft/disc to a driven one. A brake is actually a clutch with a fixed output shaft or disc. Hydraulic clutches use a fluid as a mechanical coupler to transmit torque, however, an MRF clutch transfers torque via the shear stress induced in the embedded MRF. MR clutches benefit from controllable and smooth torque transmission considered as the main advantages over conventional hydraulic types. Compared to conventional hydraulic clutches, MR clutches respond more quickly to a control input. Moreover, by adjustment of the applied magnetic field, torque transmission can be controlled easily and precisely. Some designers have proposed multi-disk and multilayered MR clutches to increase the torque/power transmission capability [44–46]. MR clutches are often designed in disk or cylindrical forms. In the cylindrical type, MRF fills the gap between two coaxial cylinders, whereas in the disc type, MRF is contained between two parallel circular plates as shown in Fig. 4. These clutches typically operate in the shear mode where the MRF layer works as a connecting medium between the input and output shafts/discs. The first MR clutch was developed by Rabinow who introduced advantages of MRFs in controlled torque transmission [2]. Kikuchi et al. developed a multi-layered MR clutch with very small gaps between disks to reduce magnetic resistance as well as the clutch size [47]. Heavy magnetic particles are centrifuged at high rotational speeds in MR clutches, thus MRFs often lose their homogeneity while in operation. To overcome this limitation, a sponge-type cylindrical clutch was proposed by Neelakantan and Washington to mitigate the centrifuging of particles at high rotational speeds [48]. Fernandez et al. designed an MR clutch using permanent magnets. They developed a new mechanism to alter the intensity of the magnetic Fig. 4 Cylindrical type (left) and disk type MRF clutch (right)

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field around MRF in order to adjust the transmitted torque. This system allows the covering cylinder to move in and out to alter the field intensity [49]. In most MR clutches, magnetic field is generated by a coil. Magnitude of the electric current in the coil determines intensity of the magnetic field applied to the MRF, thus the transmitted torque. With no current, clutch works in a disengaged state, where only a drag torque due to the base viscosity of the MRF. In the engaged state, the transmitted torque can only be maintained by continuous supply of the electric energy to the coil. The transmittable torque is an important parameter in the design of MR clutches. This is usually adjusted by the design parameters regarding the intended application. For example, the required torque in robotic applications is usually in the range of 0.1–1 Nm [49, 50], and often below 10 Nm in rehabilitation applications [51]. However, there are high-power applications needing torques in the range of a few thousands Nm [46, 52]. Maximum torque transmitted by an MR clutch can be increased by altering the size, input speed, number of layers, thickness of the plates/discs, and finally making use of MRFs with higher yield stress and viscosity. Thus, in order to rationally design an MR power transmission device, torque transmission capability needs to be evaluated as a design requirement.

3.3 MRF Dampers Damper is a device which restricts movement or reduces adverse vibration. MRF dampers are able to provide variable damping force by means of a controlled magnetic field. These dampers may operate in the valve, shear, or squeeze modes. Development of high-performance MRF dampers for use in a variety of engineering applications has always been a research hotspot. Adjusting the size of structures, enhancing internal magnetic field, and modernizing designs are the subject of the current researches. Dampers in the valve mode are typically built as a cylinder-piston system. Once the magnetic field is applied, the orifice in the piston or specific bypass in the cylinder acts as a valve to control the MRF flow. In terms of structure, they are classified as single-ended piston-rod and double-ended piston-rod systems. Lord® Corporation developed a linear MRF damper, which uses a single-ended pistonrod system. This is to be used in semi-active vibration control systems for heavy-duty vehicle seat suspensions [53, 54]. Knee prosthesis, active engine mounts, vibration dampers, propshaft mounts, and seismic dampers for civil applications are some of the other applications for MRF dampers. Shtarkman [56, 57] developed a rotary MRF shock absorber which can be used in active suspension systems for passenger cars. Marathe et al. [58] proposed replacing typical passive dampers with novel MRF dampers to improve lag mod damping in the helicopter rotor system due to the controllable nature of damping force. A locking device is an example of the dampers in the squeeze mode application. It works by moving a small steel disk into a chamber filled with an MRF. Lord® have developed a damper that operates in the squeeze mode which is widely used in the real-time, active damping control of industrial applications [59].

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MRF dampers with inner and outer chamber damping units have also been proposed based on the combined operation modes [60, 61]. The combined operation modes significantly widen the damping force control range and the stroke range of the piston-rod systems. However, as the damping force is regulated jointly by the spring and MRF, the accurate output of the device is more complicated. In 2020, a damper was designed with the intention of minimizing MRF sedimentation [62]. A permanent magnet was incorporated into the piston to move particles back and forth, and a conductive strip was placed to monitor the particle chains to measure the sedimentation ratio. The permanent magnet, on the other hand, made the piston more resistant to normal movement. Furthermore, a self-powered MRF damper was developed for washing machines to reuse the vibration energy [63]. The device incorporates energy harvesting technology, in which induced power from induction coils is directly transmitted to damper excitation coils, generating a commensurate damping force.

3.4 MRF Valves and Seals In MRF valves, a magnetic field causes an increase in the viscosity of an MRF as it flows through the valves. This change in viscosity causes a resistance to the flow through the valve, which raises the inlet pressure and, as a result, slows or stops the flow. Nguyen and Choi presented an annular MR relief valve and an MR valve with both annular and radial flow paths, which are two structural variants of the common MR valves [65]. The annular path design includes valve coils, covers, and cores, where the MRF flows through the annular ducts between cores. The magnetic field induced by the coil causes the MRF to transform from a liquid to a semi-solid state, which increases viscosity of the fluid and eventually stops the flow. When the supply pressure is high enough to overcome the MRF’s yield stress, the fluid will start to flow through the valve. Unlike the typical hydraulic proportional valves which are mostly uncontrollable, several controllable MR valves have been developed [64, 66]. MRF seals are used in a variety of applications, including vacuum equipment and dust prevention tools. Advanced materials, like semiconductors, must be fabricated in high-vacuum environments, and even a small amount of vaporized active gas must be avoided. For example, Yamamoto et al. achieved an ultra-low vapor pressure (7.3 × 10−11 Pa) at room temperature using a magnetic field strength of 31 mT to activate a perfluoropolyether oil-based MRF [67]. However, multistage seals are required to maintain high vacuum in magnetic seals. An MRF seal can create highpressure seal of 3 × 105 Pa due to high magnetization of the suspension [68]. The seal width and magnetic flux density vary depending on the gap size. The main advantages of MRF seals are structural simplicity, high sealing provision, lack of wear in mating parts, and ease of maintenance. Consequently, MRF seals perform well in the static regime or at low rotation speeds of the shaft to be sealed. Fujita et al. investigated MRF seal and rotary shaft seal characteristics by dispersing micronsized iron particles in a low vapor pressure solvent [69]. The results showed that

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the burst pressure of the MR seal is affected by particle size and volume fraction, solvent viscosity and surface tension, and seal gap. In other words, once iron particles larger than a few micrometers were used, the solvent surface tension had an effect on the burst pressure. Furthermore, in a large gap, the burst pressure in a stationary condition raises the pressure via the MRF yield stress. The rotation condition of the burst pressure is similar to the stationary condition in the case of a small gap. This is because the arranged iron particles are unable to reconstruct in a small gap. Furthermore, the burst pressure would increase as the particle volume fraction and solvent viscosity increase due to an increase in the MRF’s plastic viscosity and yield stress.

3.5 MRF Polishing Devices The MR polishing, also known as Magnetorheological Finishing, is a new magneticassisted hydrodynamic polishing method [70]. This technique is used for optical glasses, plastics, ceramics, and some nonmagnetic materials. Development of this technology has removed many conventional limitations of finishing techniques in particular for sophisticated shapes like spheres. In other words, traditional finishing methods are comparatively inferior in finishing complex freeform surfaces, due to the lack of control on finishing forces and limitations of polishing tool movement over the complex freeform contour of the components [71]. MR finishing methods include magnetorheological finishing, magnetorheological abrasive flow finishing, rotational magnetorheological abrasive flow finishing, and ball end magnetorheological finishing. Additionally, the MR finishing may eradicate figure mistakes as well as smooth small-scale microroughness without causing surface or subsurface damage [72].

3.6 Medical Applications MRFs become increasingly important in medical applications as remote surgery and robot-assisted technologies grow. Tactile feedback devices and medical wearable rehabilitation devices are two areas where MRFs are commonly being used. Current research focus on using MRFs to simulate feedback from various robotic surgery applications to provide operators a real tactile experience. A force generator module with MRF was developed in 2018 for surgical robotic applications to provide forcefeedback data [73]. This device can measure a wide range of force and is capable to quickly reproduce forces generated during tele-robotic bone biopsy procedures. An MR spherical actuator with haptic feedback was proposed for medical joystick applications in 2019 [74]. The actuator has a unique stator that replaces the traditional single coil with eight separate coils and magnetic circuits, allowing for force control in multiple interaction directions. In addition, an endovascular catheterization

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system with a master and slave device has been proposed. The catheter in the slave device moves through the blood vessel and transmits the real resistance measured by the sensor to the master device. The master device simulates resistance using MRF, providing the remote physician with a realistic sense which improves surgical safety [75]. Similarly, to simulate the stiffness and damping of human tissues, a controllable tactile device was made by immersing an MRF in porous polyurethane foam [76]. The device can capture several repulsive forces generated by human organs and can improve remote real tactile sensing. Furthermore, MRFs have numerous new applications in medical rehabilitation equipment. A prosthetic knee with an MRF brake was proposed in 2018, assisting humans to achieve normal gait movement in where an MRF brake accommodates the need for flexible braking torque variation. A multi-freedom MRF damper with a ball-and-socket structure was proposed to improve the rehabilitation of the human shoulder and upper limb [77, 78]. MRF dampers provide a setting for rehabilitation training, which can effectively simulate the motion of human joints. Furthermore, some MRF microneedles have been developed using a drawing lithography technique for minimally invasive surgery, and smart wearable equipment [79–81]. In a gradient magnetic field, MRFs are magnetized, so can generate fusiform patterns, which results in different forms of microneedle arrays after heating and solidifying. MRF microneedles are rather cheap and easier to manufacture than traditional precision machined ones. A semi-active ankle–foot orthosis with an MRF link-mechanism was designed to prevent paralysis and gait abnormalities affecting human ankle joints [82]. This MRF device, in conjunction with a compression spring, can reduce foot slap and toe drag during various phases of movement. Similarly, an MRF variable-stiffness leg was designed for movement recovery to improve energy efficiency and gait stability [83]. Furthermore, a novel soft exoskeleton with an MRF damper was proposed to suppress pathological tremor based on assistive technologies [84]. The MRF damper in conjunction with a flexible elliptic spring can help suppress tremor of the wrist joint in 3 degrees of freedom. This device prevents wrist tremor with a less constrained area on the hand surface, which reduces the suppression effect of a small amplitude tremor.

4 Constitutive Models To design magnetorheological devices and to better understand how they work, one should first know the relationship between the shear stress, magnetic field and the shear rate in the magnetorheological fluid. Several constitutive models which are mostly phenomenological, have been proposed by researchers for MRFs to predict the performance of MR devices. MR device modeling is often done by one of the following approaches: (a) Applying a specific MRF constitutive model to the device geometry in question. The model is chosen based on a representation of the fluid behavior. The model

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parameters are adjusted so that the predicted performance of the device matches experimental data acquired from it. (b) The device is viewed as a whole and a model is fit to the behavior of the device linking its input and output parameters. The dynamic viscosity η of a Newtonian fluid is defined as the ratio between the shear stress τ and the shear rate γ˙ . This relationship is expressed as: τ =η

∂u = ηγ˙ ∂y

(5)

where u is the velocity of the fluid and y is a spatial coordinate perpendicular to the flow direction. These quantities can be seen in the schematic diagram in Fig. 5, which shows the velocity profile of a fluid adjacent to a stationary wall. For a nonNewtonian fluid, the viscosity may depend on the shear rate. It may increase by increasing the shear rate (shear-thickening fluids) or decrease by the increase of shear rate (shear-thinning fluids). In the case of MRFs, in addition to the shear rate, the viscosity depends on intesity of the applied magnetic field, too. On the other hand, as MRFs exhibit semi-solid, liquid, and mixed phases, it is not always easy to express the shear stress in terms of the parameters that play role. In general, a reliable constitutive model should be able to represent the yield stress of MRFs in terms of the magnetic field intensity (as well as magnetic particles shape, size, volume fraction, etc. in advanced models), and capture the nonlinear flow behaviour of MRFs by as small number of parameteres as possible. Rheological characteristics of MR fluids can be measured by means of viscometers or rheometers. Such devices generally measure the rheological properties of fluids by measuring the shear stress at given shear rates or vice versa. Rotational rheometers measure the required torque for a given angular velocity. As rhemeters offer more measurement modes, they are often preffered for MRFs. Another advantage of rotational rheometers compared to other types is their ability to operate continuously over a given shear rate range so that steady-state measurements can be conveniently performed. The measured data are then used to calculate the rheological properties like viscosity and yield stress. Accurate assessment of the yield stress of MR fluids is inevitably required for successful development of a new MRF because Fig. 5 Velocity profile of a flow adjacent to a stationary wall

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the yield stress is the most important controllable parameter that governs the performance of MR devices. However, there is no unianimous interpretation for the yield point of MRFs as the phase change occurs gradually over a shear rate range rather than in a single point. As this might be different for different MRFs, it is important to study the flow curves (shear stress versus shear rate) of MR fluids. The shear rate range at which MRF’s viscosity drops by several decades is often assumed as the yield point of the MRF, so the corresponding stress as the yield stress. Micromechanical studies are being carried on to investigate the behaviour of MRFs at yield to provide more accurate insight into phase change of these materials. There are two basic methods of measuring viscosity; using a Couette cell (or torsional flows), and using a Poiseulle flow. In the case of the Couette cell, fluid is sheared between two coaxial cylinders, one rotating while the other one is stationary, yielding a linear velocity profile, as shown in Fig. 6a. The rotational coaxial cylinder viscometer consists of two coaxial cylinders, where fluid is placed in the annulus between the cylinders. The outer cylinder is fixed. The torque and rotational velocity of the inner cylinder are measured to determine the shear stress and shear rate, which are needed for viscosity calculation. For the rotational coaxial cylinder viscometer, shear stress can be calculated directly from the measured torque. In the case of the Poiseulle flow, fluid is made to flow through a passage, yielding a velocity profile that is parabolic, as shown in Fig. 6b. A third method is to shear the fluid between a rotating disk above another fixed parallel disk, forming a cylindrical cavity, as shown in Fig. 6c. In this method, shear rate depends on the radial distance from the axis of rotation as well as the gap between the two disks. The shear rate and shear stress can be obtained directly from the acquired torque and angular velocity data. This technique is similar to the Couette cell, however, the shear rate is not constant across the area of the plates. There are a variety of constitutive models for MR fluids. An important part of the MRF models is the method they use for calculation of the yield stress. Researchers have followed both micromechanical and phenomenological approaches

Fig. 6 Methods of viscosity measurement

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to express MRFs’ yield stress in terms of the involved parameters. Most micromechanical models use the concept of magnetization and the Maxwell stress tensor to do so. These models took the concept of mutual attractions of magnetic particles in a chain structure. The simple dipole model (SDM) which assumes that the particles are much smaller than the inter-particle distances is probably the most common approach of this kind. Such models often need a statistical approach to analyze aggregation of magnetic particles (as dipoles forming chains) and its contribution to resistance against flow. Since these models are based on physical assumptions, they often have complicated mathematical formulations, and as they need so many simplifications, they are not still as accurate as phenomenological models. That is why phenomenological models are often used for almost all practical applications. Some common phenomenological models are discussed in the following.

4.1 Bingham-Plastic Model Fluids, depending on their response to shear, are considered as Newtonian fluids (where the shear stress is proportional to the shear rate), or non-Newtonian fluids (where viscosity decreases or increases as shear rate increases). In the absence of magnetic field, most MRFs behave almost like Newtoninan fluids, so their mechanical behavior can be fairly well described by the famous Newton’s law of viscosity: τ = ηγ˙

(7)

In (7), τ is the shear stress, η is the viscosity of the fluid and γ˙ is the shear rate. However, in the presence of a magnetic field, MRFs exhibit a semi-solid behavior with a yield stress which depends on intensity of the applied field. The Binghamplastic model is an idealized model to study the rheology of non-Newtonian fluids. At stress levels higher than the yield stress, the fluid behaves like a Newtonian fluid with a constant post-yield viscosity. However, below the yield stress there is no considerable flow [86]: τ > τ y : τ = τ y + η p γ˙

(8)

τ ≤ τ y : γ˙ = 0 where shear stress τ consists of a yield stress τ y and a Newtonian term which is actually the multiplication of the high-shear rate viscosity η p and the shear rate γ˙ . In other words, when the applied shear stress is larger than the yield stress (τ > τ y ), the material flows with a post-yield viscosity, η p . When the shear stress is less than the yield stress, (τ ≤ τ y ), the material behaves like a solid with no remarkable flow. This model assumes that the shear rate does not affect the post-yield viscosity. However,

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experiments show that MRFs’ viscosity decreases with the increase of shear rate at high shear rates, known as the shear-thinning phenomenon [87].

4.2 Herschel-Bulkley Model The Herschel-Bulkley model is another common model for non-Newtonian fluids which focuses on capturing the shear-thinning behavior by adopting a shear ratedependent viscosity in the post-yield regime. This is achieved by representing the shear stress with a power law of the shear rate, which can be expressed as [88]: τ = τ H + k γ˙ n

(9)

In (9) τ H is the Herschel-Bulkley yield stress, k and n are the model parameters which depend on the magnetic field intensity for MRFs. The exponent ‘n’ defines the behaviour of the fluid in the post-yield region, and is called the flow behavior index. In the case of n > 1, fluid is said to exhibit shear-thickening behaviour, and when n < 1, the fluid exhibits shear-thinning behaviour. Note that the Herschel-Bulkley model reduces to the Bingham-plastic model for n = 1.

4.3 Casson Model The Casson model is a structure-based model, which can also be used to describe the shear-thinning behavior of MR fluids as follows [89]: √ √ √  τ = τc + ηc γ˙

(10)

In (10), τc and ηc are the Casson yield stress and the Casson viscosity, respectively. Both Casson and Herschel-Bulkley models provide better fits to MRFs’ experimental rheology data compared to the Bingham-plastic model. Neither Herschel-Bulkley nor Casson models do not explicitely include the intensity of applied magnetic field. This means that these models need to be calibrated at each field intensity they are being used.

4.4 Khajehsaeid et al. Model A nonlinear constitutive model has been proposed for MRFs which possesses remarkable features in predicting the magneto-mechanical behavior of these smart materials.

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The model is defined based on the magnetic flux density rather than the field intensity so rheometric measurements can be directly analyzed by this model without a need for hysteresis curve of the material under study. There exists a systematic method for the identification of the model parameters so that once the model is calibrated at an arbitrary magnetic flux density, it can be used at different fluxes with a reasonable accuracy without a need for further calibration. It has been shown that predictions of the model are in good correlation with the experimental data [90]. The model is represented as follows: τ = a1 γ˙ + a2 B + a3 B.ln(1 + γ˙ )

(11)

where a1 , a2 and a3 are the model parameters and B is the magnetic flux density in Tesla. a1 is the off-state viscosity of the MRF, i.e., the viscosity of the fluid in absence of magnetic field. The second term indicates the static yield stress which is acheived just before onset of flow in the MRF. This linear term can be substituted by a micromechanics-motivated term, a higher-order polynomial or a nonlinear term to increase the accuracy of the model [91]. It is also possible to include effects of magnetic field, and particles volume fraction on the yield stress of MRFs. A good choice for the second term of the above constitutive model is expected to be able to account for the magnetic saturation of MRFs, too. The third term corresponds to the shear-thinning of MRFs in the presence of external magnetic field [90]. As the model parameters are related to MRFs’ physical properties, it is easy to identify a unique set of model parameters for any given experimental flow curves (i.e., shear stress vs. shear strain curves). None of the above-mentioned constitutive models include the effect of temperature on the magneto-mechanical behaviour of MRFs as different MRFs exhibit somehow different temperature responses. However, it is still possible to include temperaturedependent terms/parameters in the above constitutive models once enough experimental flow curves are available at different temperatures for the MRF under study.

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Ceramic Manufacturing for Green Energy Applications Masoud Jabbari, Hesam Khajehsaeid, Mohammad Souri, and Mohammad Nasr Esfahani

Abstract (Multi-)functional ceramics are being used in many different applications, e.g. thermal barrier coatings, piezoactuators, capacitors, solid oxide fuel cells and electrolysis cells, batteries, membranes, and filters. Quite often, the performance of a ceramic component is varied/controlled by altering its composition and/or microstructure. Recent studies show that the final properties of (multi-)functional ceramics and (multi-)materials can be controlled and optimised by correctly adjusting the manufacturing process parameters. However, there are several hurdles to be overcome for the successful fabrication of (complex ceramic) structures. Rheological properties of ceramic pastes play an extremely important role in the processing of (multi-material) ceramics. Only by matching the rheological properties of the different pastes, a reproducible and well-defined gradient composite will be formed. Tape casting (classed as an extrusion process) has been used extensively in manufacturing of (mutli-)functional ceramics. Main challenge with tape casting is, as mentioned, to control the rheological properties of the slurries/pastes as they strongly affect the process and the quality of the final product. This chapter will review the current state-of-the-art for modelling and simulation of tape casting process with the main focus on the rheology of ceramic pastes during the shaping process as well as the downstream drying stage.

M. Jabbari (B) School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] H. Khajehsaeid WMG, University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected] M. Souri Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Allston, MA 02134, USA e-mail: [email protected] M. N. Esfahani Department of Electrical Engineering, University of York, York YO10 5DD, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_7

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1 Introduction Tape Casting has been used for decades to produce thin layers of ceramics to be used as single layers or multilayered structures [15, 41]. Today, tape casting is the basic fabrication process that provides multilayered capacitors and multilayered ceramic packages. It has been used extensively for producing thin ceramic layers [38], electric substrates [1], multilayer ceramics (MLCs) [37], and most recently in lithium-ion batteries for energy storage purposes [53]. The process itself starts with mixing different ingredients and homogenising all into a slurry. The amount and combination of the ingredients normally vary depending on the application and desired properties. For example, the role of each additive in the process and the properties of the final tapes has been investigated by Moreno [43, 44], with additional statistical design of experiments on the parameters like the weight fraction of ceramic powder, dispersant, plasticizer, binder and solvent. As an example, typical ingredients used in the La0.85 Sr0.15 MnO3 (LSM) ceramic slurry with an indication of their function is summarised in Table 1. The fluidised slurry is then put into the tape casting machine that comes with a doctor blade configuration. The slurry undergoes the narrow region of the doctor blade and spreads over the moving surface forming a thin layer—ranging between 100–3000 µm [18]. The produced so-called “wet tapes” are then go through down stream processes like drying and sintering for consolidation and improving solidstate mechanical properties. A schematic of the tape casting process from when the slurry is inputted into machine to the drying stage is illustrated in Fig. 1. In principle, the factors influencing the tape casting process are either related to the slurry itself (additive contents) or the machine configuration like doctor-blade height and casting speed. The first set of parameters will play a role by changing the “Rheological” behaviour of the slurry [3], which is very important when it comes to numerically analysing and/or digitally twinning the process. The latter is also impact the rheology to a certain extent, but rather controls the forming of the tapes as well as their geometrical sizes [34]. To that end, the rheology of the ceramics slurry is a key factor in producing tapes and/or analysing the process numerically.

Table 1 Material content and their function for LSM slurry Material Function La0.85 Sr0.15 MnO3 (LSM) Methyl ethyl ketone Ethanol Polyvinyl pyrrolidone (PVP) Polyvinyl butyral (PVB) Polyethylene glycol (PEG) Dibutyl phthalate (DBP) Additol

Ceramic powder Solvent Solvent Dispersant Binder Plasticizer Plasticizer Deflocculant

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Evaporation

Free Flow Reservoir

Porous Medium

Doctor Ceramic Slurry

y

Blade x

h

Inlet 0

Drying

Shaping

Fig. 1 Schematic illustration of the tape casting process

Hence, in this chapter a brief overview of rheological constitutive models used in analysing ceramics slurry with focus on analytical and numerical modelling will be given—focused on the “Shaping” stage. Later the downstream “Drying” process will be reviewed once in complete isolation from the shaping stage (decoupled), and one with weak coupling (sequential) between the shaping and the drying process.

2 Shaping Stage 2.1 Rheological Behaviour The way fluids deform in response to an applied force characterises them from solids, where they show a “plastic flow” rather than elastic deformation. However, it is variation in the plastic flow that changes fluids’ behaviour and leads to different rheological characteristics. When it come to analysing fluids in tape casting, generally the coupled mass and momentum conservation equations are solved ∂ρ + ∇ · [ρu] = 0, ∂t  ρ

 ∂u + u · ∇u = −∇ P + ∇ · T + F ∂t

(1)

(2)

where ρ is density, u is velocity, P is pressure, T is stress tensor and F is tensor of the contribution from external forces. It is the description of the T that results

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in different rheological behaviour. In the following all common rheological models used in the tape casting process will be reviewed briefly.

2.1.1

Newtonian Fluids

One of the first and mostly used models in terms of analysing slurry flow in the tape casting process is the Newtonian assumption as follows τ = μ∇ · u = μγ˙

(3)

where τ is the shear stress tensor, ∇ · u is the velocity gradient vector, γ˙ is the shear strain vector, and μ is the Newtonian dynamic viscosity—see Fig. 2. Under the assumption of incompressible flow, the Newtonian model has been used successively in analysing flow behaviour in the tape casting process [10, 12, 23, 34].

2.1.2

Non-Newtonian Fluids

For an incompressible flow the Navier–Stokes equations—Eqs. (1) and (2)—will reduce to ∂u 1 + u · ∇u = − ∇ p + ν∇u, (4) ∂t ρ ∇ · u,

(5)

where ν is kinematic viscosity. For non-Newtonian fluids the kinematic viscosity become a function of velocity gradient, and one way to resolve this non-linearity is to use generalised apparent viscosity, based on Taylor’s series of expansion as      ν (|D (u)|) D (u) = ν D uold  D uold  d [ν (|D (u)|) D (u)]  old   u + u − uold d (u)   + O u − uold , where D (u) =

1 2



(6)

 u + u , and

   1   d [ν (|D (u)|) D (u)]  old  u I + I = ν D uold  d (u) 2 (7) d [ν (|D (u)|)]  old   old  u + D u . d (u)

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  Neglecting the higher order terms, O u − uold , and inputting Eq. (7) into Eq. (6) will lead to the Newton type iteration. By neglecting the second term in the right-hand side of Eq. (7), the viscous term becomes linearised and fixed-point iterations will be achieved as    ν (|D (u)|) D (u) ≈ ν D uold  D (u) .

2.1.3

(8)

Power Law Fluid

Generally, fluids that do not obey the Newtonian behaviour are categorised as nonNewtonian fluids. Rearranging Eq. (3) for the viscosity, one could get a more generic form for the “apparent viscosity” as follows μa =

τ ≡ f (γ˙ ) , γ˙

(9)

and it is the function f that defines different types of non-Newtonian fluids. The most simple form of such “generalised Newtonian” flows is the “Power-law” fluid (also known as Ostwald-de-Waele model) in which the apparent viscosity is μa = k γ˙ n−1 ⇐⇒ τ = μa γ˙

(10)

where n is the power-law index and k is called the consistency factor—see Fig. 2. This model is the most common used rheological behaviour of the slurry in tape casting [19, 20, 26, 48, 51]. The limits of this model is due to the unboundedness of the viscosity function and the lack of a non-zero viscosity at zero shear rate—which does not match experimental results for real fluids.

Shear stress [Pa ]

Fig. 2 Schematics of different rheological models Newtonian Power Law Bingham Herschel-Bulkley Casson

Shear rate [1/ s]

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Bingham Fluid

There are some fluids that behave like solids in the sense that they appear not to flow until the magnitude of the applied force (shear stress) shoots over a fixed value a.k.a. the yield point (stress). Such fluid is collectively known as “Bingham Fluid” with a yield stress, τ y , in the constitutive behaviour as   τy γ˙ , τ = μB + |γ˙ |

|τ | > τ y

(11)

where μ B is the Bingham viscosity—also classed as a viscoplastic fluids—see Fig. 2. This model has been used extensively in numerical/analytical flow analysis of the tape casting process mostly in the shaping stage [16, 24, 32, 35, 49, 52]. Using the Bingham fluid in the tape casting process will result in an important parameter called critical velocity, vcr , which is the minimum velocity required to overcome the yield point. The critical velocity will change by altering machine configuration, e.g. the doctor blade height, h, and ultimately influence the thickness of the produced tapes (δ)—c.f. Fig. 3. 2.1.5

Herschel-Bulkley Model

Another flow model with a yield point, τ y , with viscoplastic shear-thinning behaviour is called “Herschel-Bulkley Fluid”—see Fig. 2—that is written as follows   τy γ˙ , τ = k |γ˙ |n−1 + |γ˙ |

|τ | > τ y

(12)

where k and n are material constants.

Fig. 3 Influence of substrate velocity on the tape thickness for different doctor blade height of a h = 4 mm, and b h = 5 mm—Reprinted from [21] with permission from Elsevier

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2.1.6

155

Casson Model

The “Casson Model” [47] is another class of viscoplastic fluids—see Fig. 2—that was originally used for studying blood flows, however, later was used successfully for tape casting of ceramics [7, 13]. The model is written as  τ=

√ μC +



τy |γ˙ |

2 γ˙ ,

|τ | > τ y

(13)

Generally, there are other rheological models available in the open literature, i.e. Prandtl-Eyring, Powell-Eyring, and Modified (bi-viscosity) Viscoplastic Model [21], however, they are not used often for studying ceramic slurries in tape casting.

2.2 Tracking Interface When the ceramic slurry leaves the doctor blade region, it opens up to the atmospheric environment creating a free surface. Tracking this interface has significant influence on the manufacturing and the final quality of the tapes. Two very well-known methods of tracking interfaces when computationally simulating such complex flows are the volume of fluid (VOF) and level set. Both methods are used in literature for capturing the free surface of tapes in computational fluid dynamics (CFD) simulations using finite element method (FEM) and finite volume method (FVM) [19, 20, 35, 36]. An example of such simulation is shown in Fig. 4.

2.3 Tracking Particles As discussed earlier, the viscous stress tensor, τ , and the strain rate tensor, γ˙ , for constitutive behaviour of a generalised Newtonian fluid is τ = μa γ˙ . For a nonNewtonian fluid with the Ostwald-de Waele power-law model the apparent viscosity, thence, becomes

Fig. 4 An example of interface tracking for CFD simulation of the tape casting process—Reprinted from [21] with permission from Elsevier

Symmetry plane

Inlet pe Ta

t cas

Moving substrate

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  η−1 μa = k γ˙ n−1 = k γ˙ 2 2

(14)

For a power-law suspension including particles with fraction of θ we get  2 γ˙ m μa = (1 − θ ) μm γ˙ a2

(15)

in which γ˙ 2m is the average of square of local shear

rate experienced by the matrix fluid, and μm is the matrix viscosity evaluated at γ˙m2 . Based on Eqs. (14) and (15): γ˙ a2 (θ ) = γ˙ 2m (0)



1−θ μm

2 η+1

(16)

 Based on Eq. (16), for any value of average shear rate in the matrix fluid, γ˙ 2m , and matrix viscosity, μm , the macroscopic apparent shear rate can be calculated for any θ [29]. Employing such approach allows to track the particle migration in the ceramic manufacturing and to predict the locally packed areas as for example illustrated in Fig. 5.

Fig. 5 Particle distribution in the ceramic tape casting process a without, b with particle tracking method—Reprinted from [29] with permission from Elsevier

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3 Drying Stage In tape casting aqueous-based ceramic slurries are mostly used as they are more environmental friendly and they are cheap to produce [5, 14]. However, the downfall of using aqueous-based slurries are longer times for “drying” and higher probability of shrinkage and delamination [9, 33, 39]. A tape layer can be considered as a porous medium containing powders (particles) and a liquid phase, where the evaporation takes place by one-sided heat and mass transport [5]. As a downstream process after shaping, thence, the drying stage becomes important in terms of influencing the final properties of the tape layers. The drying stage and the characterisation of it in the form of final shrinkage is often measured experimentally, simply by the weight difference of the green and dried tapes as well as “in situ” weight-loss measurement. Drying often contains different kinetics, i.e. evaporation, viscous deformation, flow in porous media, and diffusive transport [8, 25], whose coupled behaviour influences the final properties of the thin layers. It is fair to say that a tape layer can be considered as a porous medium which contains powders and liquid phases. Modelling of drying in this respect can be approached in two different ways; one where the drying stage and modelling (DM) of it is analysed in complete isolation of the shaping stage—in particular ignoring particle migration models (PMM)—using prescribed initial conditions, Fig. 6a, and the one in which the drying model reads some upstream information from the shaping stage, Fig. 6b. The former is known as “decoupled” modelling, where as the latter represent a weakly (sequential) “coupled” analysis for the entire shaping and drying processes. The model concept for simulating drying in general combines a two-phase flow (gas and liquid) in the porous medium, and a single-phase flow (gas phase) in the free-flow region. The following processes are to be described: transfer of heat and vapour across the interface, evaporation and condensation at the interface, evaporation/condensation, and dissolution/degassing inside the porous medium. The 2dimensional problem setting is illustrated in Fig. 7 in which two domains ff and pm are separated by the interface = ∂ff ∩ ∂pm with the outward unit normal vectors nff and npm . It is assumed that the flow velocities in the porous medium are slow (Re 1) allowing an application of the multi-phase Darcy’s law for the phase velocities. The model, moreover, allows the transfer of components from one phase into another and thus describes evaporation and condensation as well as dissolution and degassing within the porous medium [30, 31, 45]. The mass balance for the water component w and the air component a in the porous-medium sub-domain is

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DM Coupled free flow (ff)

Porosity,

and porous media (pm)

Permeability, K

Stokes flow in ff Multiphase Darcy in pm

(b) Coupled (shaping + drying) Inputs

PMM

DM

Process parameters:

Navier–Stokes

Coupled free flow (ff)

casting speed

Non-Newtonian flow

and porous media (pm)

height in reservoir

VOF

Stokes flow in ff

dimmensions

Particle motion

Multiphase Darcy in pm

Fig. 6 Overall overview of model concepts for a decoupled, and b coupled analysis Transient, Stokes

npm

δ ff = 0.005 m Ωff

y Γ = 0.005 m

Γ

Ωpm δ pm

nff

= 0.005 m

y

Transient, Darcy x L = 0.01 m

Fig. 7 Model set-up with a single phase in the free-flow region (ff ) that interacts with two-phase flow in the porous medium (pm ) via the interface ( )

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⎧  ∂ (α X κ Sα )  κ α ⎪ φ + ∇ · Fκ − qα = 0, ⎪ ∂t ⎪ ⎪ α∈{l,g} α∈{l,g} ⎪ ⎪ ⎪ ⎨    κ Fκ = α vα X ακ − Dα,pm α ∇ X ακ ⎪ ⎪ α∈{l,g} ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ κ qα : source or sink term

159

∀ κ ∈ {w, a} (a) (b)

(17)

(c)

κ where the macroscopic diffusion coefficient of component κ in phase α, Dα,pm , is κ = σ φ Sα Dακ Dα,pm

(18)

which is a function of porosity, phase saturation, and tortuosity—that is calculated by [40] (φ Sα )7/3 (19) σ = φ2 The multi-phase Darcy’s law is employed for calculating the phase velocities vα = −

kr α K (∇ pα − α g) , α ∈ {l, g} μα

(20)

The energy balance is formulated as below ⎧  s cs T ) ⎪ φ ∂(α∂tu α Sα ) + (1 − φ) ∂(∂t + ∇ · FT − qT = 0 (a) ⎪ ⎪ ⎨ α∈{l,g}  ⎪ ⎪ α h α vα − λpm ∇T ⎪ ⎩ FT = α∈{l,g}

(21) (b)

A single-phase flow (gas phase: g) and two components in the gas phase (water and air) is considered in the free-flow region. The mass balance for each component is given by ⎧ ∂ (g X gκ ) ⎪ ⎪ + ∇ · Fκ − qgκ = 0 (a) ⎪ ∂t ⎪ ⎪ ⎨ (22) Fκ = g vg X gκ − Dgκ g ∇ X gκ (b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ q κ : source or sink term (c) g The mass conservation equation for the gas phase is given by

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⎧ ∂g + ∇ · Fm − qg = 0 (a) ⎪ ⎪ ∂t ⎪ ⎪ ⎨ Fm = g vg (b) ⎪ ⎪ ⎪ ⎪ ⎩ (c) qg = vg g X gκ

(23)

Neglecting the non-linear inertia term of the Navier–Stokes equation, the momentum balance using the Stokes equation is given by ⎧∂ v ( g g) ⎪ ⎨ ∂t + ∇ · Fv − g g = 0 (a)   ⎪ ⎩ F = p I − μ ∇v + ∇v (b) v g g g g

(24)

The energy balance equation for the free-flow sub-domain is given as ⎧∂ u ( g g ) + ∇ · F − q = 0 (a) ⎪ T T ⎪ ∂t ⎪ ⎪ ⎨ (b) FT = g h g vg − λg ∇T ⎪ ⎪ ⎪ ⎪ ⎩ (c) qT = vg g h g T

(25)

The coupling conditions are used at the free-flow-porous-medium interface, they allow exchange of different physics between the subdomains and are valid on the REV scale [45]. Both the thermodynamic equilibrium and continuity of fluxes are assumed to be ensured across the interface. The continuity of the normal stresses resulting in a possible jump in the gas-phase pressure is given by n·



  ff   pm pg I − μg ∇vg + ∇v n = pg g

(26)

The continuity of the normal mass fluxes is conserved by 

 pm ff  g vg · n = − g vg + l vl · n

(27)

For the tangential component of the traction in the free-flow, the Beavers-JosephSaffman condition is used [6, 50] as the Dirichlet boundary for the tangential freeflow velocity. √

 vg +

 ff ki   ∇vg + ∇vg n · ti = 0, i ∈ {1, . . . , d − 1}

αB J

(28)

where ki = t · (Kt) is the tangential component of the permeability tensor. Baber et al. [4] showed that the Beavers-Joseph coefficient does not affect the drying curve, and hence in this study α B J = 1 is used.

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The thermal equilibrium for the free-flow-porous-medium interface is similar to the condition proposed by [2], showing the continuity of temperature [T ]ff = [T ]pm

(29)

as well as the continuity of heat fluxes across the interface  ff   pm  g h g vg − λg ∇T · n = − g h g vg + l h l vl − λpm ∇T · n

(30)

For the free-flow-porous-media interface the chemical equilibrium is given by the continuity of mass fractions  ff  pm (31) X gκ = X gκ and the continuity of the component fluxes across the interface 

 ff   pm g h g X gκ − Dgκ g ∇ X gκ · n = − g h g vg + l h l vl − λpm ∇T · n

(32)

The control-volume finite-element method (CVFEM or box method) is used for the discretisation of both subdomains ff and pm together with an implicit Euler time discretisation. The coupled problem (free-flow domain, porous-media domain and interface) is written in the following operator form ∂M (u) − ∇ · F (u) = Q (u) (33) ∂t   pm pm w,pm where u = pgff , X gw,ff , vffg , T ff , pg , Sl or X g , T pm is the solution vector, M (u) is the storage, F (u) the flux and Q (u) the source/sink term. The non-linear algebraic system at each time step is treated via a standard Newton solver, which is a stable and robust scheme, as follows:      n+1,i ∂R (34) − un+1,i−1 = −R un+1,i u ∂u n+1,i       u J(un+1,i )   where J un+1,i is the Jacobi matrix calculated by numerical differentiation, u is  the correction to the primary variables u, and R un+1,i is the residual at time level n + 1 and iteration i. The linear problem at each Newton iteration step is solved using the direct linear solver SuperLU [11], and for the time integration, a fully implicit Euler scheme with a heuristic time-step control based on the convergence rate of the Newton solver is used.

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3.1 Decoupled Drying For the decoupled systems, the only information used in the drying stage is the prescribed porosity and/or permeability. The porosity (φ) and particle diameter (d p ) values are assumed based on the material load (ceramic and polymer particles), and Ergun’s equation is used for calculating the intrinsic permeability of the porous ceramics [17], φ 3 d 2p , (35) K= 150 (1 − φ)2 and it is inputted into Eq. (20). This way the tape layer can be studied with a constant homogenous porosity (and permeability), or can be prescribed as a layered structure—c.f. Fig. 8. Numerical results of spatial and temporal evolution for the saturation of water in the porous-medium (Sw ) as well as mass fraction of vapour in the gas phase (X gw,ff ) for a single layer are shown in Fig. 9. As seen from Fig. 9, during the evaporation process, the saturation of the water in the porous medium is decreasing with time, and thence, the mass fraction of vapour in the free-flow (especially in the interface region) is increased. In the early stages when evaporation occurs, the temperature at the surface of the layer drops because of a loss of heat due to the latent heat of vaporization of the water—see Fig. 10b. On the other hand, heat flows to the surface from the atmosphere, thus quickly establishing thermal equilibrium, and with decreasing degree of evaporation the porous medium starts to reach the thermal equilibrium—see Fig. 10d. These models will allow to study impact of different influential factors like the initial temperature of the free flow region (T ff ) and the porous medium (T pm ), the velocity of the air in the free flow region, particle size (d p ), and the porosity in the porous medium (φ) [28]. Such studies will help in analysing and better controlling of key parameters in the drying of ceramic tapes, namely the maximum drying rate, R˙ max , and the final time for drying, t f . Moreover, using statistical analysis, one can look into combined effects of the above-mentioned factors on R˙ max and t f —c.f. Fig. 11. (a)

(b)

(c)

free flow

free flow

free flow

φ 2 , K2 φ, K

φ 1 , K1

φ 2 , K2 φ 1 , K1

Fig. 8 Prescribed initial values of porosity and permeability for possible configuration a single layer, b graded layers, c bi-layers

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Fig. 9 Evolution of water saturation in the porous-media and mass fraction of vapour in the freeflow region after a 0 [min], b 100 [min], c 350 [min], and d 480 [min]—Reprinted from [27] with permission from Elsevier

Fig. 10 Evolution of temperature in the porous-media and the free-flow region after a 0 [min], b 100 [min], c 350 [min], and d 480 [min]—Reprinted from [27] with permission from Elsevier

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1200

17.5 17

1000

1000

16.5

16

800 800

600 400

600

200 6 10 -6

17

16

15

15.5

14 15

13 4 2

302

300

298

296

294

400

14.5

0.15 0.1

302

300

298

296

294

14

Fig. 11 a Influence of combined T pm and d p on t f at φ = 0.4, and b impact of combined T ff and vmax on R˙ max at T pm = 298.15 K and d p = 3.5 µm—Reprinted from [28] with permission from Elsevier

3.2 Coupled Approach As illustrated in Fig. 6, for the coupled approach, initially the shaping stage is analysed with the purpose of capturing the realistic patterns for particle distribution (and hence the porosity). The apparent viscosity of the ceramic slurry, μa = m γ˙ η−1 , will play a big role in the resultant particle distribution. An example of such study is illustrated in Fig. 12, where the particle distribution θ along the height of the produced tapes are simulated. Existence of these models allow controlling the particle distribution patterns during the manufacturing by fine tuning the process parameters and/or rheological behaviour of the fluid. The evolution of field functions for three different cases are compared: (I) a decoupled approach; (II) coupled approach with m = 3 and η = 0.2; and (III) coupled

Fig. 12 a Impact of changing consistency factor (m) on particle distribution, and b the apparent viscosity of a power-law fluid with consistency factor value. The magnetic dashed-lines show the minimum, average, and maximum shear rate seen in the simulation of each particular fluid— Reprinted from [22] with permission from Elsevier

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Fig. 13 Water saturation in the porous-media (bottom half of each sub-plot) and mass fraction of vapour in the free-flow region (upper half of each sub-plot) for three different cases of I, II, and III. The plots are shown at dimensionless time of a t ∗ = 0, b t ∗ = 0.25, c t ∗ = 0.75, and d t ∗ = 0.98—Reprinted from [22] with permission from Elsevier

approach with m = 12 and η = 0.4. The results are depicted in Fig. 13. The drying in a ceramic tape layer is generally controlled by two major mechanisms: (M1) the rate of water evaporation from the surface of the cast layer, and (M2) the rate of water diffusion through the tape (from the bottom of the tape)to the drying front. When the water reaches the top surface of the cast layer, it takes energy from the air and from the rest of the porous medium and evaporates into the free-flow region. In drying the limiting factor is normally the migration of the water to the top surface of the tape layer. Generally, M1 is faster than M2, and this leads to form a dried crust across the surface of the tape. Such behaviour can be easily seen in Fig. 13 for the three cases, where a zero saturated region is formed in the top region of the tape layer. However, the shape and size of the zero saturated region is different for each case that is mainly driven by the particle distribution pattern in porous medium. Presence of such heterogeneity in drying pattern of the cast layers promotes distortion in porous structures in later processes like sintering [42, 46]. The results presented by Jabbari and Esfahani [22] show that heterogeneity in the particle distribution coming from PMM will alter the drying behaviour of the tapes. Ceramic tapes with low consistency and low power-law index values will reduce the drying rate (slightly) as well as the final drying time, that favours the manufacturing of tapes by reducing the risk of crack initiation/growth in ceramics.

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4 Summary Tape Casting has been used for producing thin ceramics layers for different green energy applications. Although the process is well-established and has be extensively used, there is still room for improving the process. These improvements could either focus to avoid any manufacturing induced deficiencies, or mostly in terms controlling and tailoring the final properties. One of the key factors that impacts the shapes (and uniformity) as well as the final properties of the produced tapes is the rheology of the ceramic slurry. The current chapter has briefly reviewed the common (and existent) rheological models and their use in modelling and simulation of tape casting (with especial focus on flow analysis). Choosing a right material content and thence rheology for the ceramic slurry will lead to a proper dimensional control (and quality). This, moreover, will play a significant role in the downstream processes like drying and sintering. The drying process of green tapes were, moreover, reviewed in this chapter with emphasis on better controlling and/or avoiding process-induced issues like shrinkage and crack. Optimising process parameters—i.e. the free-flow velocity and temperature—and the tape cast parameters—i.e. ceramic particle size and porosity of the tapes—will help reduce drying time as well as drying rate (both detrimental for quality of dried tapes). Finally, numerical simulation and modelling of the tape casting process (at different stages of shaping and drying), and digitally twinning the whole process will help manufacturers to control and improve the quality of the parts. This will, furthermore, reduce waste and promotes a sustainable manufacturing process. It is the firm believe of the authors that presence of such modelling frameworks and with further improvements (and emerging areas like machine learning) will help to shift the paradigm from “manufacturing-property” to “property-manufacturing”.

References 1. Abhilash P, Roshni SB, Mohanan P, Surendran KP (2018) A facile development of homemade substrate using ‘quench free’ glass-ceramic composite and printing microstrip patch antenna on it. Mater Des 137:38–46 2. Alazmi B, Vafai K (2001) Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int J Heat Mass Transf 44:1735–1749 3. Albano MP, Garrido LB (2005) Influence of the slip composition on the properties of tape-cast alumina substrates. Ceram Int 31:57–66 4. Baber K, Mosthaf K, Flemisch B, Helmig R, Müthing S, Wohlmuth B (2012) Numerical scheme for coupling two-phase compositional porous-media flow and one-phase compositional free flow. IMA J Appl Math 77:887–909 5. Bauer C, Cima M, Dellert A, Roosen A (2009) Stress development during drying of aqueous zirconia based tape casting slurries measured by transparent substrate deflection method. J Am Ceram Soc 92:1178–1185 6. Beavers GS, Joseph DD (1967) Boundary conditions at a naturally permeable wall. J Fluid Mech 30:197–207

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7. Bitterlich B, Lutz C, Roosen A (2002) Rheological characterization of water-based slurries for the tape casting process. Ceram Int 28:675–683 8. Brinker C, Scherer G (2013) Sol-gel science: the physics and chemistry of sol-gel processing. Academic Press 9. Briscoe B, Biundo GL, Özkan N (1998) Drying kinetics of water-based ceramic suspensions for tape casting. Ceram Int 24:347–357 10. Chou YT, Ko YT, Yan MF (1987) Fluid flow model for ceramic tape casting. J Am Ceram Soc 70:C–280 11. Demmel J, Eisenstat S, Gilbert J, Li X, Liu J (1999) A supernodal approach to sparse partial pivoting. SIAM J Matrix Anal Appl 20:720–755 12. Gaskell P, Rand B, Summers J, Thompson H (1997) The effect of reservoir geometry on the flow within ceramic tape casters. J Eur Ceram Soc 17:1185–1192 13. Gurauskis J, Baudín C, Sánchez-Herencia AJ (2007) Tape casting of Y-TZP with low binder content. Ceram Int 33:1099–1103 14. Hotza D, Greil P (1995) Aqueous tape casting of ceramic powders. Mater Sci Eng, A 202: 206–217 15. Howatt G, Breckenridge R, Brownlow JM (1947) Fabrication of thin ceramic sheets for capacitors. J Am Ceram Soc 33:237–242 16. Huang X, Liu C, Gong H (1997) A viscoplastic flow modeling of ceramic tape casting. Mater Manuf Process 12:935–943 17. Innocentini MD, Sepulveda P, Salvini VR, Pandolfelli VC, Coury JR (1998) Permeability and structure of cellular ceramics: a comparison between two preparation techniques. J Am Ceram Soc 81:3349–3352 18. Jabbari M (2014) Modelling of tape casting for ceramic applications 19. Jabbari M, Bulatova R, Hattel J, Bahl C (2013) Quasi-steady state power law model for flow of (La0.85Sr0.15)0.9MnO3 ceramic slurry in tape casting. Mater Sci Technol 29:1080–1087 20. Jabbari M, Bulatova R, Hattel JH, Bahl CR (2014) An evaluation of interface capturing methods in a VOF based model for multiphase flow of a non-Newtonian ceramic in tape casting. Appl Math Model 38:3222–3232 21. Jabbari M, Bulatova R, Tok A, Bahl C, Mitsoulis E, Hattel JH (2016) Ceramic tape casting: a review of current methods and trends with emphasis on rheological behaviour and flow analysis. Mater Sci Eng, B 212:39–61 22. Jabbari M, Esfahani MN (2019) The role of rheological parameters on drying behaviour of a water-based cast tape. Chem Eng Res Des 152:269–277 23. Jabbari M, Hattel J (2011) Numerical modeling of fluid flow in the tape casting process. In: AIP conference proceedings. American Institute of Physics, pp 143–146 24. Jabbari M, Hattel J (2014) Bingham plastic fluid flow model in tape casting of ceramics using two doctor blades-analytical approach. Mater Sci Technol 30:283–288 25. Jabbari M, Hattel J (2016) Modelling coupled heat and mass transfer during drying in tape casting with a simple ceramics-water system. Drying Technol 2:244–253 26. Jabbari M, Hattel JH (2013) Numerical modeling of the side flow in tape casting of a nonNewtonian fluid. J Am Ceram Soc 96:1414–1420 27. Jabbari M, Jambhekar V, Hattel JH, Helmig R (2016) Drying of a tape-cast layer: numerical modelling of the evaporation process in a graded/layered material. Int J Heat Mass Transf 103:1144–1154 28. Jabbari M, Nasirabadi PS, Jambhekar V, Hattel JH, Helmig R (2017) Drying of a tape-cast layer: numerical investigation of influencing parameters. Int J Heat Mass Transf 108:2229–2238 29. Jabbari M, Spangenberg J, Hattel JH (2016) Particle migration using local variation of the viscosity (LVOV) model in flow of a non-Newtonian fluid for ceramic tape casting. Chem Eng Res Des 109:226–233 30. Jambhekar V, Helmig R, Schröder N, Shokri N (2015) Free-flow-porous-media coupling for evaporation-driven transport and precipitation of salt in soil. Transp Porous Media 110:251–280 31. Jambhekar V, Mejri E, Schröder N, Helmig R, Shokri N (2016) Kinetic approach to model reactive transport and mixed salt precipitation in a coupled free-flow-porous-media system. Transp Porous Media 114:341–369

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Rheology and Cure Kinetics of Modified and Non-modified Resin Systems Hatim Alotaibi, Constantinos Soutis, and Masoud Jabbari

Abstract Resins, in particular thermosets, are defined as a polymer material used with synthetic/natural fibres by reinforcement during liquid composite moulding (LCM) processes to produce composite products. The rheological behaviour and cure kinetics of resins are crucial and can be influenced by modification factors that include non-isothermal conditions and the addition of nanofillers. The thermoset flow will behave as a non-Newtonian fluid in which viscosity is not constant. This complex flow problem has been an area of interest over decades to meet industry demands for more developed composites, and to quantify relevant issues by applying innovative optimisation methods. In the present chapter, a thorough investigation of the current approaches and future potential of rheological behaviour and curing kinetics of thermosets is presented. Keywords Cure kinetics · Rheolgy · Nanotechnology-based · Thermosetting resin · Porous media · LCM · Modelling

1 Introduction Thermosetting resins have an extensive use within aerospace and automotive industries as adhesives, coating, wet lay-up lamination, insulation, and moulding. They exhibit a diverse range of physical, mechanical, and chemical characteristics that can contribute to the thermal stability, mechanical integrity, and solvent resistance of H. Alotaibi Department of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester M13 9PL, UK C. Soutis Northwest Composites Centre, Department of Materials, The University of Manchester, Manchester M13 9PL, UK Aerospace Research Institute, The University of Manchester, Manchester M13 9PL, UK M. Jabbari (B) School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Azizi (ed.), Applied Complex Flow, Emerging Trends in Mechatronics, https://doi.org/10.1007/978-981-19-7746-6_8

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the final products. In liquid moulding processes, thermosets (liquid resins) are fabricated by compounding with fibres to form high-quality, complex-shaped components. During the fabrication process, resins are injected or infused to permeate throughout fibre preforms to reach a complete saturation—voids free. The permeation is affected by fabric types or classifications (c.f., Fig. 1) in addition to the porosity nature of the medium. This shows, for instance, mats are dissimilar to those made with weaves in which the latter resist progressing (impregnating) flows more due to two length-scales of pores, i.e. intra-tow and inter-tow porosities (see Sect. 3.2). Apart from design and process parameters, heat or mould-wall temperature influences thermosetting resins with respect to cure kinetics and rheology. This can be explained by a chemical conversion of the liquid resin which induces a cross-linking activity (connection) of individual polymer chains to form a solid (glassy) three-dimensional (3D) polymer network. It becomes more critical in event of modified thermosets, in other words nano-filled thermosets, owing to the presence of high thermal properties (e.g., thermal conductivity) of the nano-filler. The prominently applied nanomaterials would include graphene (Gr), carbon nanotubes (CNTs), and nanoclays (NCs) as a means to tailor thermosetting resins. A new potential for manufacturing composites in an effort to improve physical, thermal, and mechanical properties. This chapter will cover material types of thermosets used in LCM processes along with nanotechnology-based thermosets to review and discuss effects of processing parameters, cure kinetics, and chemo-rheology.

2 Material and Methods 2.1 Thermosetting Polymers (Resins) Several types of thermosetting polymers used in aeronautical and space industries would include epoxy, polyester, vinyl ester, phenolic, polyethylene, and benzoxazines [2]. These offer outstanding mechanical and thermochemical integrity of the reinforced and bonded fibres (filaments). Furthermore, thermosets can be processed at room temperature and are being easy to shape allowing the advantage of large-scale production. When hardening (completely cured) such viscous liquids, they prove to be resistant to melting and can be treated under various temperature conditions. Epoxy resins are one of the widely used high-performance thermosets in LCM processes for aerospace applications. This includes coatings, adhesives, and reinforcements (moulding compounds) because of a low shrinkage rate they possess during curing [3]. Curing temperatures range from 18 to 160 °C, and therefore this conveys that the lower the temperature, the longer the cycle time, and vice versa [3]. The use of epoxy resins has still been popular in aerospace structures, in particular primary (e.g., fuselage, and wing skin) and secondary (e.g., inspection panels, and spoilers) [2]. Phenolic resins are other crucial liquid resins that take place in tailoring structural components within the aerospace industry for their superior thermal and mechanical

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Fig. 1 Classifications of fibre preforms that include mats, unidirectional (UD), weaves, braids, and knits [1]

properties. They are usually used for ablation in order to provide protective surfaces resisting undesirable environmental conditions [4, 5]. Processing or moulding such resins can be manufactured by LCM processes (e.g., RTM) for compounding liquid resins with fibrous reinforcements such as carbon fibres, whereas curing can be achieved with heating capable of (up to) 180 °C mould temperature [4, 5]. Polyester

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Table 1 Thermosetting resins and their relevant applications and LCM processes [3–10] Resin system

Application

LCM process

Epoxy

Adhesives, coatings, and moulding compounds

RRIM, SRIM, RTM, VARTM, and SCRIMP

Phenolic

Adhesives, coatings, laminating (wet lay-up) resins, and moulding compounds

IM, RRIM, and RTM

Polyester

Coatings, moulding compounds, and laminating resins

IM, RRIM, SRIM, RTM, VARTM, and SCRIMP

Vinyl ester

Coatings, moulding compounds, and laminating resins

RTM, VARTM, SCRIMP, and RRIM

Polyurethane

Adhesives, coatings, and moulding compounds

RRIM, SRIM

RRIM: Reinforced reaction injection moulding SRIM: Structural reaction injection moulding RTM: Resin transfer moulding VARTM: Vacuum assisted resin transfer moulding SCRIMP: Seemann composite resin infusion moulding process IM: Injection moulding

resins are applied in a broad range of fields including automotive and naval apart from aerospace. This class of resins are economical (inexpensive) and can be produced in distinct outputs of properties, i.e. ductile or rigid (brittle) [6, 7]. It is unlike the epoxy family of resins in terms of shrinking since polyester resins experience higher volumetric shrinkage (5−12%) and that is usually quantified via integrating thermoplastic constituents into the polyester matrix [6, 7]. Because of high reactivity (polymerisation), the recommended curing temperatures operate form ambient temperature till 70 °C [7]. Table 1 presents popular resins that are commonly manufactured utilising liquid moulding processes of composites within aerospace and automotive industries.

2.2 Nanotechnology-Based Thermosetting Polymers Thermoset nanocomposites (modified resin systems) are attractive alternatives to pristine (neat) resins by virtue of high specific strength, thermal stability, and optimised activation energy in reaction kinetics [11–13]. Progress on modification of resins has been a subject of research and makes them competitive to conventional macrocomposites. A resin system is modified at a nano-scale level by the addition of nano-fillers prepared using consecutive experimental methods and tooling (see Sect. 2.3). The increasingly nano-fillers being used in aeronautical, automotive, and marine applications include graphene (Gr), carbon nanotubes (CNTs), nanoclays (NCs), and nano-silica (NS) for nanotechnology-based thermoset composite structures [11–13]. Modifying thermosetting resins with the aforementioned nanofillers are found to be useful, as they tend to toughen a resin system to strengthen

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Fig. 2 Different crystal structures of carbon. From lift to right: 3D diamond and graphite; 2D graphene; 1D nanotubes; and 0D buckyballs [15]

its resistance to crack propagations in the produced composite parts [11]. Grapheneenhanced polymers become a popular choice in tailoring resins for manufacturing and producing advanced fibre-reinforced polymer composites. Graphene, a single layer atoms composed of sp2 carbon  in a hexagonal array, proves unprecedent surface to  weight ratios 2000 m2 /g , and thermal conductivity of 5000 [W/m K] aside from the possession of superior mechanical properties—a Young’s modulus of 1 [TPa], and an ultimate strength of 130 [GPa] [11, 14]. Graphene is incorporated into resins with different forms such as graphene oxide (GrO) and reduced graphene oxide (rGrO). It is initially fabricated—prior to mixing—adopting one of the popular available approaches comprising mechanical exfoliation, liquid-phase exfoliation (LPE), and chemical vapour deposition (CVD) [11, 14]. As to cure kinetics and rheology, graphene can slightly accelerate the exothermic reaction of resins during manufacturing processes for advanced composites owing to the reduced amount of oxidation groups in the nanomaterial [11, 14] (Figs. 2 and 3).

Fig. 3 Production methods of graphene [14]

174 Table 2 A summary of peak cure temperatures and total heat of reactions for modified and non-modified resin systems at a non-isothermal heating rate of 10 [◦ C/min]

H. Alotaibi et al. System

T p [◦ C]

  Htot J/g

Ref [18]

DGEBA-DDM

163

450

DGEBA-DDM + 1 wt.% GNP

166

440

DGEBA-DDM

169.7

460.3

DGEBA-DDM + 5 wt.% GrO

122.7

455.1

Epoxy

116.4

510.1

0.2 CNTs + Epoxy

115.5

493.1

0.2 NC + Epoxy

116.7

490.6

DGEBA/DQPB

177

247

DGEBA/DQPB + 10 wt.% NS

171

151

[19] [20]

[21]

DGEBA-DDM: diglycidyl ether of bisphenol A/4-4 diaminodiphenylmethane DGEBA-DQPB: diglycidyl ether of bisphenol A/3,5-diamino-N(4-(quinolin-8-yloxy) phenyl) benzamide Epoxy: Araldite LY 5052 Resin/Aradur 5052 Hardener

Carbon nanotubes, nano-scale hollow tubes, are made of rolled up graphene sheets or graphite layers of bonded carbons in the form of cylindrical shape. Three methods are available for the fabrication of CNTs, i.e., chemical vapour decomposition, arc discharge, and laser ablation (vaporisation) [11, 16]. The produced CNTs can be single-walled (SWCNT) or multi-walled (MWCNT) in which the latter consists of multiple nested single-wall carbon nanotubes inside each other, a concentric interconnected nanotubes [11, 20]. Adding CNTs to resins during the modification process can be with a direct dispersion of SWCNTs/MWCNTs or with functionalised CNTs (e.g., amino-functionalised MWCNTs). Both methods give rise to enhanced mechanical and thermal properties of the produced composites. Mainly, improving fracture toughness, flexural strength and flexural modulus from the mechanical perspective, while enhancing glass transition temperature from the thermal prospective [11, 16, 17]. Similar to graphene, curing and rheological behaviours are affected by nanofillers (CNTs) with incorporation causing an acceleration in cure and an increase in the viscosity of the resin system at the early stage of polymerisation. Table 2 explains the influence of nano-fillers on the exothermic reaction and temperature of the resin system. The peak exothermic temperature decreases in the event of thermoset-based nanocomposites because of the enhanced heat transfer properties (e.g., thermal conductivity).

2.3 Preparation The preparation of the mixture, i.e., nano-fillers and thermosets will generally include the use of magnetic stirrer, ultrasonic homogeniser, and vacuum heat oven (e.g., degasification) [11, 17]. This process technique is known as an in-situ polymerisation

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in which the nanofiller is dispersed into a solution followed by mechanical mixing or magnetic stirring to homogenise the mixture. A resin system is then added, heated, stirred for a certain amount of time and at a recommended manufacturing temperature. Degasification is usually required afterwards to vacuum air bubbles out of the modified resin system (c.f., Fig. 4). Other processing techniques of thermoset-based nanocomposites include solution casting (mixing), melt compounding, spinning, and the recently introduced additive manufacturing (3D printing) [17]. A thermal analysis method, a differential scanning calorimetry (DSC), is used to characterise the nanotechnology-based thermosetting resin by obtaining the heat flow data or the socalled exothermic peaks, and also reaction rates profiles for the different isothermal or non-isothermal experiments. From the DSC data, the degree of cure, and viscosity evolution can be calculated using the analysis results of cure kinetics and chemorheology parameters. Moreover, the effect of the nanofiller on viscosity of resin flow and permeability of the fibre preforms during the infusion process can be studied. This is to examine the permeability variation during the resin impregnation because of the usual agglomeration formed by nano-platelets (e.g., graphene), which may occur at the beginning of the liquid moulding process.

Fig. 4 Schematic diagram for the development nano-technology-based thermosetting resins for cure and rheology characterisation, liquid moulding of composites

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3 Processing 3.1 Resin Liquid Moulding Processes of Composites Composites, a fibre reinforcement and a matrix, are still playing a crucial role within aerospace, automotive, marine, and construction industries in producing lightweight, durable components with outstanding mechanical properties. Fibre reinforcements are classified as continuous (long) or discontinuous (short) fibres. Such a reinforcement can contribute to prevent failure of the composite part as loads being endured by polymers will be transferred to fibres, for instance, microcracking propagation [8– 10]. Fabric topologies or architectures are imperative in composites manufacturing in view of the fact that a permeability (ability of transmitting fluids) within (intra) and in-between (inter) fibre bundles can tailor the quality of the final product. The flow parameter (permeability) is relevant to curing and rheology for the reason that a delay (lag) of flow advancement within fibre bundles (yarn/tow) could cause an undesirable potential early cure. This, however can be addressed using numerical methodologies to predict intra- and inter-tow permeabilities customising accurately and efficiently liquid moulding of composites for an industrial use (see Sect. 3.2). composite manufacturing processes are categorised as open and closed mould processes, wherein the LCM processes—broadly known for manufacturing thermoset-based composites— fall under closed mould category (c.f., Fig. 5). In LCM, simple and complex shapes can be handled and manufactured by a diverse range of techniques (see Table 3) that involves the well-known resin transfer moulding (RTM). These processes are preferable in aerospace and automotive industries for their low cost in terms of equipment and maintenance, and high-quality (well-finished) composite parts. Processing techniques in LCM incorporate operating and design parameters, in particular pressure injection, flow rates (infusion), positions of inlet and vent ports, permeability of the fibre preform, and heating (e.g., mould temperature). A temperature control is vital during resin impregnation of fibrous reinforcements since it is necessary to initiate curing after a complete saturation of porous media and mould filling.

3.2 Resin Impregnation of Dual-Scale Fabrics Resin Impregnation of porous media in LCM is typically described by Darcy’s law (see Eq. (1)), nonetheless this description is restricted to single-scale transport phenomena. It stems from the fact the hypothesis does not account for porosity (pores) of fabrics, in lieu it relies on the volume averaged fluid velocity together with the injection parameter (pressure) and viscousness of the resin (viscosity). As mentioned earlier, flows within fibre bundles, intra-tow (micro) flows, are significant and required attention when moulding fabric topologies like weaves and braids. Such fabric architectures will normally include two pore regions, i.e., open (inter-tow) and porous (intra-tow). Researchers [22–24] attempted to characterise the dual-scale

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Fig. 5 Composite manufacturing processes and classifications [8–10]

flow applying high-tech techniques (e.g., ultrasonic and dielectric sensors) during the filling stage. This includes an experimental work by Pierpaolo et al. [24] that utilises a dielectric monitoring system and an optical microscope to track preform saturation and to build a microstructure image of the manufactured fibre laminated matrix composite, respectively. Thereby, this provided information on micro-meso regions allowing the calculation of the dual-scale permeability of textiles as shown below by Fig. 6. An illustration of resin impregnating fibrous reinforcements through two length-scale of pores (intra- and inter-tow) is given by Fig. 7 for a unit cell of a plain weave. Trial-and-error experiments can be expensive and time-consuming, on the other hand, numerical tools prove to be cost-effective and efficient in monitoring resinfront, predicting dual-scale permeability, and characterising cure and rheological behaviour. In a numerical perspective, Navier–Stokes (N–S) equations are employed

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Table 3 Description of the different processing techniques in LCM [8–10] LCM process

Description

Injection moulding (IM)

Thermoset/thermoplastic None, short fibres resins are injected into the mould after being molten

Reinforcements

Wide range of thermoplastics, while polyester, phenolic for thermosetting resins

Reaction injection moulding (RIM)

Polymer mixture is injected at low pressure into a mould to expand and cure

Thermoset resins, polyurethane (PU-RIM), polyamides 6 (NyRIM)

None

Structural reaction SRIM is a part of RIM in Fibre mats, meshes injection moulding which a fabric structure and preforms (SRIM) is place in the mould prior to the injection

Polymers

Thermoset resins (e.g., polyurethane, polyester)

Resin transfer moulding (RTM)

Resin flow is injected by low pressure into closed mould to impregnate the preplaced preform

Mats, foam cores, Thermoset resins (e.g., unidirectional, 2D and epoxy, polyester, 3D woven fabrics, etc. phenolic and vinylester)

Vacuum assisted resin transfer moulding (VARTM)

Resin flow is vacuumed Mats, foam cores, Thermoset resins (e.g., and drawn from the resin unidirectional, 2D and epoxy, polyester, and supply in a one-sided 3D woven fabrics, etc. vinylester) mould with a vacuum bag to impregnate the fibre preform until being fully saturated and gelled

Seemann composite resin infusion moulding process (SCRIMP)

Resin flow is pulled by a vacuum to be distributed through the media (flow channels) into the preplaced fibre lay-up

Fabrics, core materials Thermoset resins (e.g., epoxy, polyester, and vinylester)

Fig. 6 The micro-meso structure of a 2D woven fabric by an optical microscopy: a 2D woven RVE, b meso-scale region, and c micro-scale region [24]

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Fig. 7 Resin impregnation a inside the intra-tow region (microscopic), b between the yarns (mesoscopic) [30]

with an additional source term accounting micro-flows (i.e., a microscopic permeability) to obtain flow modelling through open and porous regions. The added microscopic permeability (intra-tow permeability) can be any of the available analytical models [25–29] in literature, e.g., Kozeny-Carman (KC) semi-empirical equation (see Eq. (3)). These theoretical models require the information of the fibre (filament) radius and fibre bundle volume fraction or porosity (i.e., gaps between fibres). Resin liquid moulding of fibre preforms usually experiences a low Reynold number (Re  1) because of the low velocity of the resin flow, and with no compression during the filling process. Therefore, adopting the assumption of incompressible creeping (viscous) Newtonian (no viscosity changes during resin impregnation) flow regimes will be valid. This would reduce the flow governing equations (N–S) to the so-called Stokes-Brinkman equations (see Eq. (5)) allowing the characterisation of dual-scale resin impregnation of unidirectional, woven, and braided fabrics. u=−

Ko ∇p μ

(1)

⎡

⎤ ⎤ ⎡ ⎡ ⎤ ux ∂ p/∂ x Kxx 0 0 ⎣ u y ⎦ = − 1 ⎣ 0 K yy 0 ⎦ × ⎣ ∂ p/∂ y ⎦ μ uz 0 0 K zz ∂ p/∂z Kt =

r 2f

(φ)3 4kc (1 − φ)2

(2)

(3)

∂ (ρu) + ∇ · (ρuu) = −∇ p + μ∇ 2 u + ρg + S ∂t

(4)

∂ (ρu) + μ∇ 2 u + ∇ p = S ∂t

(5)

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∂ρ + ∇ · (ρu) = 0 ∂t S=−

μ u Kt

(6) (7)

  where Ko m2 is a permeability tensor (overall permeability), u[m/s] is the volumeaveraged   velocity, ∇ p is the pressure gradient, μ[Pa · s] is the viscosity of the liquid, Kt m2 is the intra-tow permeability, r f [m] is radius of the fibre, φ[−] is the porosity of the medium, kc [−] is the Kozeny–Carman constant. The source term, in which a theoretical equation can be applied, is denoted by S from Eqs. (4) and (5), while ρg, μ∇ 2 u, and ∇ · (ρuu) stand for the body force term, the diffusion term, and the convection term, respectively. Modelling of the aforementioned topologies is applied on a representative volume element (RVE) or a unit cell that will yield a value representing the global level. Figure 7 depicts a two-scale flow level throughout a unit cell and how voids could form during the impregnation process. Various numerical discretisation schemes are used to divide the domain into elements or control volumes (CV) and to convert partial deferential equations to an algebraic form that can be numerically solved. These schemes include finite volume method (FVM), finite difference method (FDM), and finite element method (FEM). Table 4 illustrates numerical works that successfully investigated dual-scale resin flow in porous media. Furthermore, Table 5 shows part of numerical contributions that evaluated simulation tools.

4 Cure Kinetics and Rheology 4.1 Curing Kinetics Characterising polymerisations of a resin system is required, and it becomes essential when a liquid composite moulding process is subject to variable heating rates during fill and cure sages. The polymerisation of a polymeric material transforms a resin system from liquid (prepolymer) to a glassy rigid material (polymer) by crosslinking individual polymer chains to form a 3D network structure (i.e., a thermoset polymer) [43–46]. This complex chemical transformation is primarily affected by heat that is to say the temperature parameter. Thus, a resin liquid would undergo critical transformations that include gelation, and vitrification stages. In the gelation stage, the liquid resin becomes more viscous (viscosity evolved), and also can explain the maximum pressure needed for such a property. The vitrification point provides details about the ultimate chemical reaction that a thermosetting resin could achieve and the transition to a glassy (solid) material. The cross-linking reactions can be attained by applying any of the differential thermal analysis techniques that comprise DSC, and Fourier Trans-form Infrared Spectroscopy (FTIR). Thermal analysis methods can be dynamic (non-isothermal) or static (a series of isothermal experiments) to bring forth

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Table 4 Numerical modelling considering dual-scale porous media [30] Reference

Ko /Ks [−]

φt [%]

φo [%]

Packing arrangement

Sadiq et al. [31]

1.0406 1.198

25 31

61.6 70.9

Array of solid and 1D Darcy’s equation porous circular fibre bundles

Ranganathan et al. [32]

1 Ko /Ks > 1

30 If φt > 30

45.0

Hexagonal arraignments in elliptic tows

Stoke flow equations for open region and brinkman equation for porous region

Nedanov and Advani [33]

1.003

29

39.0

Hexagonal packing of fibres in woven fabric

CFD package FIDAP, numerically solves stokes flow and brinkman’s equations

Belov et al. [34]

1.25

42

66.0

A plain-woven fabric

Lattice Boltzmann method, WiseTex software

Tahir et al. [35]

1.03

25

62.0

Hexagonal arrangements of fibres in circular tows within unit cell

ANSYS-Fluent, Navier–stokes equations for dual scale

Syerko et al. [36]

4.6

36

62.0

Quadradic Packing

Applied brinkman equations and mass conservation

Dual scale approach

Ko : Overall or dual-scale permeability Ks : Single-scale permeability φ t : Tow porosity permeability   φ o : Overall tow porosity φ o = φ s + φ t − φ s φ t Table 5 Part of numerical contibtuions that evaluated simulation tools for flow-front modelling References

Fabric architecture

Injection method

Flow modelling

Computational approach

Tan et al. [37]

bi-axial

Unidirectional

Dual-scale

FE/CV-PoreFlow

Simacek et al. [38]

UD

Unidirectional

Dual-scale

FE/CV-LIMS

Rodrigue et al. [39]

Fibre-mats

Unidirectional

–

Darcy-based PAM-RTM

Grössing et. al. [40]

UD/Triaxial NCF

Radial

Dual-scale

FVM-VOF OpenFoam

Sas et al. [41]

UD

Unidirectional

–

FEM-LSM COMSOL

Wei et al. [42]

Fibre-mats

Unidirectional

–

FVM-VOF Moldex3D

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data of cure kinetics together with glass transition and crystallisation temperatures. In cases such as thermoset-based nanocomposites, a nano-filled (modified) resin system, the modified polymeric material will exhibit different trends in the context of cure kinetics and chemo-rheological behaviours. The nano-fillers (e.g., CNTs, Gr, etc.) are found to be effective in accelerating the cross-linking reaction by dint of extraordinary heat transfer characteristics that a nano-filler contains [18–21]. Small portions 0.05−0.2 wt.% of such nano-reinforcements are able to tailor mechanical, thermal, and physical characteristics of thermosets [11]. An experimental work by Umer et al. [47] investigated the impact of graphene (graphene oxide) incorporation into epoxy resins using DSC for curing analysis VARTM for permeability characterisation. The authors [47] stressed that an early cure was observed with resins modified with 0.2 wt.% of graphene oxide content, as a result, this was validated during the VARTM experiment wherein the graphene-enhanced resin system progressed slower compared to the non-modified system. Further modifications, for example, functionalising the nano-filler can also enhance the exothermic reactions of the mixture (a functionalised nano-filler added to a resin system). Neda et al. [48] showed that a modified graphene oxide, functionalised with the silane agent, improved the activation energy in contrast to the unmodified graphene oxide for an unsaturated polyester resin (c.f., Fig. 8). Generally, the extent (degree) of cure (α) is determined by the total and residual heat of the reaction of the modified and non-modified resin samples in DSC during the exothermic reaction as shown below in Eqs. (8) and (9).

α=

Ht Htot

(8)

Fig. 8 A comparative analysis of activation energy and degree of cure with temperature for neat unsaturated polyester (UP), UPGrO, and UP modified-GrO [48]

Rheology and Cure Kinetics of Modified and Non-modified Resin Systems

1 α= Htot

t 

d Ht dt dt

183

(9)

0

    Here, Ht J/g is the residual heat of the reaction and Htot J/g is the ultimate heat of the reaction at the curing time (t[s]). The rate of the reaction can be calculated by rearranging the above-mentioned equations as illustrated below.  d Ht 1 dα = dt Htot dt

(10)

Over the last decades, empirical models have been developed to determine the curing parameters (e.g., degree- and rate-of-cure) for a variety of thermosetting polymers, for example, epoxy [49–52], and polyesters [53–56]. This is to reduce expensive experimental equipment and trials by employing empirical models into sophisticated numerical software to enable characterising convection–diffusion-dominated transport phenomena in LCM. Kinetics models can be divided into two types that are mechanistic and phenomenological. Sections 4.1.1 and 4.1.2 give a background of cure kinetics formulations and discuss them thoroughly.

4.1.1

Mechanistic Model

The mechanistic model was first introduced by Stevenson [57] on the basis of the free radical polymerisation approach. The radical polymerisation method considers the termination term along with other chemical transformational stages of the polymeric material including initiation, inhibition, and propagation. This is nonetheless modified and simplified by ignoring the termination term owing to the complexity of identifying reaction terminations empirically as discussed by a considerable amount of research [52, 54, 55, 58]. The well-known modified mechanistic kinetics model was developed by Kamal and Sourour [52, 59, 60], in which autocatalytic reactions can be described with such a model. dα = (k1 + k2 α)(1 − α)(B − α) dt

(11)

where B is the amine to epoxides (functional group) initial ratio, k1 [1/s] and k2 [1/s] are the rate constants. Kamal and Sourour [52, 59, 60] explained the curing reactions of epoxy/diamine systems by the mechanistic kinetics model for isothermal curing conditions. It was found that in later stages (diffusion) of the curing process, this model wouldn’t be valid. It was concluded that results matched well (±5%) with the DSC experimental data. However, the phenomenological models (see Sect. 4.1.2) are found to be simple, accurate, and suitable for a broad range of thermosets, as such types neglect the details of polymerisation and assume one equation can describe the entire chemical reaction.

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Phenomenological Model

Phenomenological empirical models, nth order and autocatalytic, incorporate rate constants k1 and k2 that can be calculated adopting an Arrhenius form. These constants require a knowledge of kinetic data, in particular, heat flow profiles to allow modelling the exothermic reaction rate and the degree of cure or conversion. As previously stated, the kinetic data is obtained either by dynamic (non-isothermal) DSC or a series of isothermal DSC experiments. Consequently, a non-linear regression technique is used to solve the rate constants as well as the reaction orders m and n. Obeying the Arrhenius law, the activation energy (forward or peak reaction) of the liquid resin will be populated yielding values of the rate constants, and hence the result of the chemical transformation rate. The proposed autocatalytic model by Kamal and Sourour [52, 59, 60] proves its applicability for a broad spectrum of resin systems (e.g., unsaturated polyester, and epoxy resins), and furthermore including ones that reach an ultimate polymerisation rate at the midpoint (intermediate stage of the reaction) of conversion (degree of cure). This is unlike the nth order kinetics model that ignores the activation energy during the reaction progress simplifying the formula with merely reactants as the influential factors [46, 61]. Therefore, the well-known Kamal model (Eq. (12)) is increasingly being used for its accuracy and sufficiency in computing rate of chemical reactions of the resin systems.  dα  = k1 + k2 α m (1 − α)n dt  −E a 1 k1 = A1 exp RT  −E a 2 k2 = A2 exp RT

(12) (13) (14)

where A1 and A2 are the pre-exponential constants [1/s], E a i is the reaction activation energy [J/mol], R is the universal gas constant [J/mol · K], and T is the isothermal temperature [K]. The above form has been simplified further to include only one kinetic rate constant (k2 = k), this due to the observations found for the rate of the reaction (dα/dt[1/s]) being approximately zero for all isothermal curing experiments [59, 62]. The simplified form of Eq. (12) would eliminate k1 and give the following: dα = kα m (1 − α)n dt  −E a k = A exp RT

(15) (16)

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185

The phenomenological models have been modified further to account for the diffusion-controlled reaction that may occur at the vitrification stage [63–65]. This modification could provide complete conversion/extent of cure and curing kinetics characterisation.

4.2 Rheological Behaviour A resin flow during the impregnation process may, in certain circumstances, experience increase in viscosity inducing gelation ahead of time affecting the moulding process of the liquid resin. This is explained by factors dominating the chemical formulation of the system—time and temperature [46, 61]. In LCM (e.g., RTM), low viscosity thermosets are usually processed, which means they are vulnerable to high temperatures (greater than manufacturer recommended temperature). In such situations, the resin system encounters rapid reactions leading to non-linear viscosity variations. Castro and Macosko [66] developed a semi-empirical equation that is capable of predicting viscosity evolution (dynamic viscosity) during the cure of a resin system. The chemo-rheological model [66] requires the knowledge of rate constants and a viscosity-dependent activation energy, and that is acquired utilising a rheometer instrument—an instrument for measuring the viscosity of a substance—for cure experiments. This model, moreover, can be employed via numerical solvers (e.g., ANSYS-Fluent, OpenFoem, etc.), computational fluid dynamics tools, to address changes in the rheology of resins throughout the chemical conversion. The two main rheological models that are commonly used in literature, are developed by Castro and Macosko [66] and Roller [67]. The Castro-Macosko model calculates the viscosity as a function of the degree of cure and temperature as given below in Eq. (17). 

Eμ μ(α, T ) = μ0 exp RT



αgel αgel − α

a+bα (17)

where μ[Pa · s] is the dynamic viscosity, μ0 [Pa · s] is a pre-exponential factor, a[−], and b[−] are exponents, E μ [J/mol] is the viscosity activation energy, and αgel [−] is the conversion at the gel point. In addition, α holds if it reaches the αgel value, which is a limitation that should be considered when applying this semi-empirical solution. Roller [67] proposed a simple approach that assumes no control over the chemical conversion and the early curing of a resin system shows a Newtonian behaviour of the viscosity. The model’s kinetic analogues kk and E k , and the initial viscosity μ0 follow an Arrhenius form. This rheological solution (c.f., Eq. (18)) has sufficiently been applied to a wide range of resin systems, and model’s viscosity variations as a function of time at a constant (isothermal) temperature. ln(μ(t, T )) = ln(μ0 ) +

 Ek Eμ + tkk exp RT RT

(18)

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Table 6 Numerical studies of chemo-rheology and cure kinetics of resin systems during impregnation process Reference

Flow simulation

Numerical method

Lee et al. [69]

Two-dimensional

(CV/FE)

Cheung et al. [72]

Three-dimensional

Galerkin finite element method

Abbassi et al. [70]

Two-dimensional

Implicit-FDM

Leistner et al. [73]

Three-dimensional

FEM-based implicit Runge–Kutta with Multilevel-Newton (DIRK/MLNA)

Poodts et al. [74]

Three-dimensional

PAM-RTM

Deléglise et al. [75]

Three-dimensional

LIMS

Sandberg et al. [76]

Three-dimensional

COMSOL

4.3 Heat Transfer and Cure Modelling A numerical analysis of resin transfer (liquid moulding) solves a moving boundary transport problem that is dominated by convection and polymerisation. Two common approaches are adopted to track free-surface flows, namely the Lagrangian moving mesh (grids) and Eulerian fixed mesh. Eulerian receives more attention for its broad applicability, simplicity, and effectiveness, since the method treats a resin movement within a domain as a volume faction of a fluid phase. This is different form the Lagrangian or the so-called Arbitrary Lagrangian–Eulerian (ALE) method which moves and deforms domains creating new meshes (regeneration of mesh), and hence a large computational time [68]. Coupling heat transfer (energy) equations with polymerisation (species) and impregnation (flow) equations has been a popular approach by many [68–71] to analyse and optimise convection–diffusion-dominated flows during liquid moulding of porous media. Table 6 summarises some of the numerical findings available in literature relating to the use of distinct numerical methods to design mould filling and processing parameters and to characterise the inevitable polymerisation of the resin system. The heat balance equation (see Eq. (19)) incorporates a source term accounting the generation of heat of a resin system caused by the exothermic reaction. Local thermal equilibrium (resin and fibre shares the same temperature) is usually assumed by reason of a creeping (slow) flow motion which is normally the case for LCM processes. Properties (e.g., density) of macro-materials as well as nano-materials can be calculated following the rule of mixture. Equation (22) is involved and used as provision for thermoset-based nanocomposites. ρC p

∂T dα + ρr C pr (u · ∇T ) = ∇(k · ∇T ) + φH tot ∂t dt

(19)

Rheology and Cure Kinetics of Modified and Non-modified Resin Systems

⎧ ⎪ ⎪ ⎨

(ρ f ρr ) (ρ f wr +ρr w f ) k k k = k w( r+kf )w ⎪ ( r f f r) ⎪ ⎩ C p = wr C pr + w f C p f ⎧ r)  ⎨ wr =  φ(φ/ρ1−φ ρ=

ρf

+

ρr

⎩ w =1−w f r

⎧ ⎪ ⎪ ⎨

187

ρfiller ρr = ψ ρ ρ+resin1−ψ  filler resin ( filler )ρfiller  +2kresin +2ψfiller (kfiller −kresin ) kr = kresin kkfiller filler +2kr esin −ψfiller (kfiller −kresin ) ⎪ ⎪ ⎩ C = 1 − ψ C + ψfiller Cfiller pr filler resin

(20)

(21)

(22)

    where ρ kg/m3 , Cp J/(kg · K) , k[W/(m · K)], w[−], and ψfiller [−] are the density, specific heat, thermal conductivity, weight fraction, and filler volume fraction, respectively. The fibre and resin subscripts are f , and r , respectively. Htot is the total heat of reaction, and dα/dt is the rate of polymerisation. On the left-hand side of Eq. (19), the two terms denote the transient and convection. On the right-hand side, the diffusion and heat generation terms are represented, respectively. The chemical complexity (transformation) of the polymerised liquid resin is expressed by species (transport) conservation equations (see Eq. (23)). This formulation describes the conversion of monomers to polymers using convective fluxes (fluid transporting species), a scaler quantity (degree of cure), and rate of reaction (polymerisation) at the fluid phase. Equation (23) presents transient (first) and convection (second) terms at the left-hand side, while a source term is added at the right-hand side denoting the rate of reaction. φ

dα ∂α + (u · ∇α) = φ ∂t dt

(23)

5 Conclusion and Future Trends Polymeric materials, modification, processing, cure kinetics, and rheology are thoroughly investigated in the present chapter. The incorporation of nano-fillers such as graphene, carbon nanotubes, nano-silica, and nano-clays proves their ability to tailor thermosetting resins in the context of accelerating the exothermic reaction (cure) during liquid moulding processes of fibrous reinforcements (e.g., RTM). This, furthermore, enhances the mechanical, thermal, and chemical characteristics of the final product (manufactured composites). The chapter introduces relevant theoretical and numerical means and reviews with the aim of setting a benchmark allowing the characterisation of convection–diffusion-dominated transport problems. It can be argued that the functionalisation (modification) of nano-fillers is a present (existing)

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and a future potential for improving the functionality of thermosetting resins. Moreover, hybridisation of nano-fillers (e.g., Gr/CNTs) is an area of interest and can be a promising potential. Development of numerical solutions—cost-effective and efficient—is imperative, in view of the fact that it will allow controlling and optimising resin impregnation and polymerisation in LCM processes.

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