Computational Analysis of Heat Transfer in Fluids and Solids (Defect and Diffusion Forum) 3035714118, 9783035714111

Table of contents :
Computational Analysis of Heat Transfer in Fluids and Solids II
Preface
Table of Contents
A Study of Transient Heat Transfer through a Moving Fin with Temperature Dependent Thermal Properties
Determination of Proper Fin Length of a Convective-Radiative Moving Fin of Functionally Graded Material Subjected to Lorentz Force
A Note on the Similar and Non-Similar Solutions of Powell-Eyring Fluid Flow Model and Heat Transfer over a Horizontal Stretchable Surface
Biomechanics of Surface Runoff and Soil Water Percolation
Finite Element Numerical Investigation into Unsteady MHD Radiating and Reacting Mixed Convection Past an Impulsively Started Oscillating Plate
Analytical and Numerical Study on Cross Diffusion Effects on Magneto-Convection of a Chemically Reacting Fluid with Suction/Injection and Convective Boundary Condition
Physical Aspects on MHD Micropolar Fluid Flow Past an Exponentially Stretching Curved Surface
MHD Flow of Non-Newtonian Molybdenum Disulfide Nanofluid in a Converging/Diverging Channel with Rosseland Radiation
Buoyancy Effects on Human Skin Tissue Thermoregulation due to Environmental Influence
Turbulent Heat Transfer Characteristics of a W-Baffled Channel Flow - Heat Transfer Aspect
MHD Boundary Layer Flow over a Cone Embedded in Porous Media with Joule Heating and Viscous Dissipation
Squeeze Film Lubrication on a Rigid Sphere and a Flat Porous Plate with Piezo-Viscosity and Couple Stress Fluid
Influence of Homogeneous and Heterogeneous Chemical Reactions and Variable Thermal Conductivity on the MHD Maxwell Fluid Flow due to a Surface of Variable Thickness
Heat Transfer Analysis of Three-Dimensional Mixed Convective Flow of an Oldroyd-B Nanoliquid over a Slippery Stretching Surface
Effects of Variable Fluid Properties on Oblique Stagnation Point Flow of a Casson Nanofluid with Convective Boundary Conditions
Keyword Index
Author Index

Citation preview

Computational Analysis of Heat Transfer in Fluids and Solids II

Edited by Prof. Oluwole Daniel Makinde

Computational Analysis of Heat Transfer in Fluids and Solids II

Special topic volume with invited peer-reviewed papers only

Edited by

Prof. Oluwole Daniel Makinde

Copyright  2020 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of the contents of this publication may be reproduced or transmitted in any form or by any means without the written permission of the publisher. Trans Tech Publications Ltd Kapellweg 8 CH-8806 Baech Switzerland http://www.scientific.net

Volume 401 of Defect and Diffusion Forum ISSN print 1012-0386 ISSN cd 1662-9515 ISSN web 1662-9507 (Pt. A of Diffusion and Defect Data – Solid State Data ISSN 0377-6883)

Full text available online at http://www.scientific.net

Distributed worldwide by Trans Tech Publications Ltd Kapellweg 8 CH-8806 Baech Switzerland Phone: +41 (44) 922 10 22 Fax: +41 (44) 922 10 33 e-mail: [email protected]

Preface Heat transfer and fluid flows seems to pervade all aspects of our life, since almost everything experiences heating or cooling of some kind and the entire life depends on fluid flows. Finding appropriate solutions to heat transfer problems in fluid and solid materials will enhance functional success of the materials and facilitate new product development in industries and engineering. This special issue on “Computational Analysis of Heat Transfer in Fluids and Solids II” in the journal “Defect and Diffusion Forum” addresses various nonlinear models involving heat transfer phenomenon in fluids and solids. Computational techniques are employed to analysis the problems and numerical results are discussed quantitatively in order to demonstrate the salient features of practical engineering and industrial applications. The topics covered by excellence research papers in this issue include: tribology, extended surfaces fins, reactive flow problem, Newtonian and nonNewtonian flow, nanofluids dynamics, boundary layer flow, natural convection, hydrodynamic stability, biomechanics, plasma physics, physics of dusty plasma, forced convection, mixed convection, magnetohydrodynamics, thermal radiation, porous media flow, and irreversibility analysis. The pertinent results obtained are very staple in understanding the complex interaction of heat transfer with fluids flow and solids mechanics. Wide range of applications of the work in this special issue can be found in the area of materials development, hydrodynamic lubrication, thermal storage, biomedical, solar heating, nuclear system cooling, micromixing technologies, military equipment storage, cooling of electrical and electronics components, product management, power production, pollution control and safety assessment. It is our hope that this special issue will inspire and help a wide audience of researchers, scientists, engineers and educators from various fields of human activity. Our appreciation goes to all the participants for their excellent contribution toward the success of this special issue. The outstanding work of the reviewers and their constructive comments are highly appreciated. Professor Oluwole Daniel Makinde Editor

Table of Contents Preface A Study of Transient Heat Transfer through a Moving Fin with Temperature Dependent Thermal Properties L.P. Ndlovu and R.J. Moitsheki Determination of Proper Fin Length of a Convective-Radiative Moving Fin of Functionally Graded Material Subjected to Lorentz Force G.A. Oguntala, G. Sobamowo, Y. Ahmed and R.A. Abd-Alhameed A Note on the Similar and Non-Similar Solutions of Powell-Eyring Fluid Flow Model and Heat Transfer over a Horizontal Stretchable Surface R. Khan, M. Zaydan, A. Wakif, B. Ahmed, R.L. Monaledi, I.L. Animasaun and A. Ahmad Biomechanics of Surface Runoff and Soil Water Percolation J.M.M. Deng and O.D. Makinde Finite Element Numerical Investigation into Unsteady MHD Radiating and Reacting Mixed Convection Past an Impulsively Started Oscillating Plate B.P. Reddy, P.M. Matao and J.M. Sunzu Analytical and Numerical Study on Cross Diffusion Effects on Magneto-Convection of a Chemically Reacting Fluid with Suction/Injection and Convective Boundary Condition S. Eswaramoorthi, M. Bhuvaneswari, S. Sivasankaran and O.D. Makinde Physical Aspects on MHD Micropolar Fluid Flow Past an Exponentially Stretching Curved Surface K.A. Kumar, V. Sugunamma, N. Sandeep and S. Sivaiah MHD Flow of Non-Newtonian Molybdenum Disulfide Nanofluid in a Converging/Diverging Channel with Rosseland Radiation J. Raza, F. Mebarek-Oudina, P. Ram and S. Sharma Buoyancy Effects on Human Skin Tissue Thermoregulation due to Environmental Influence L.H. Adeola and O.D. Makinde Turbulent Heat Transfer Characteristics of a W-Baffled Channel Flow - Heat Transfer Aspect Y. Menni, A.J. Chamkha and O.D. Makinde MHD Boundary Layer Flow over a Cone Embedded in Porous Media with Joule Heating and Viscous Dissipation S. Devi and M.K. Sharma Squeeze Film Lubrication on a Rigid Sphere and a Flat Porous Plate with Piezo-Viscosity and Couple Stress Fluid B.N. Hanumagowda, C.K. Sreekala, Noorjahan and O.D. Makinde Influence of Homogeneous and Heterogeneous Chemical Reactions and Variable Thermal Conductivity on the MHD Maxwell Fluid Flow due to a Surface of Variable Thickness G. Sarojamma, K. Sreelakshmi, P.K. Jyothi and P.V.S. Narayana Heat Transfer Analysis of Three-Dimensional Mixed Convective Flow of an Oldroyd-B Nanoliquid over a Slippery Stretching Surface K.V. Prasad, H. Vaidya, K. Vajravelu, G. Manjunatha, M. Rahimi-Gorji and H. Basha Effects of Variable Fluid Properties on Oblique Stagnation Point Flow of a Casson Nanofluid with Convective Boundary Conditions H. Vaidya, K.V. Prasad, K. Vajravelu, A. Wakif, N.Z. Basha, G. Manjunatha and U.B. Vishwanatha

1 14 25 36 47 63 79 92 107 117 131 140 148 164

183

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 1-13 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-03-29 Revised: 2019-06-14 Accepted: 2019-09-18 Online: 2020-05-28

A Study of Transient Heat Transfer Through a Moving Fin with Temperature Dependent Thermal Properties Partner Luyanda Ndlovu1,2,a and Raseelo Joel Moitsheki1,b,∗ 1 School

of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, South Africa

2 Standard

Bank of South Africa, 30 Baker Street, Rosebank, Johannesburg, 2196, South Africa a

[email protected], b [email protected]

Keywords: Heat transfer, analytical solutions, DTM, extended surface

Abstract. In this article, heat transfer through a moving fin with convective and radiative heat dissipa­ tion is studied. The analytical solutions are generated using the two­dimensional Differential Trans­ form Method (2D DTM) which is an analytical solution technique that can be applied to various types of differential equations. The accuracy of the analytical solution is validated by benchmark­ ing it against the numerical solution obtained by applying the inbuilt numerical solver in MATLAB (pdepe). A good agreement is observed between the analytical and numerical solutions. The effects of thermo­physical parameters, such as the Peclet number, surface emissivity coefficient, power in­ dex of heat transfer coefficient, convective­conductive parameter, radiative­conductive parameter and non­dimensional ambient temperature on non­dimensional temperature is studied and explained. Since numerous parameters are studied, the results could be useful in industrial and engineering applications. Introduction An increasing number of engineering applications are concerned with energy transport requiring rapid heat dissipation. To increase the heat transfer rate from a heated surface, fin (or extended surface) assembly is commonly used. The use of fins is seen in various industrial applications such as oil carrying pipelines, space nuclear reactor power systems and many more. The heat transfer mechanism of a fin is to conduct heat from a heat source by thermal conduction, and then dissipate heat to an ambient fluid by the effect of thermal convection and radiation. An extensive review of extended surface heat transfer is given by Kern and Kraus [1] and Kraus et al. [2]. A brief review of published work that is of relevance to this article is presented next. A detailed study of temperature distribution in a moving fin is provided for various embedding parameters in a triangular fin [3], rectangular fin [4], an exponential fin [5] and a trapezoidal cross section [6]. Singla and Das [7] studied an inverse problem to estimate the speed of a moving fin using the binary­coded generic algorithm. Sun et al. [8] applied the Spectral Collocation Method (SCM) to derive the solution for a convective­radiative heat transfer through a moving rod with variable thermal conductivity. Sun and Xu [9] further applied the SCM to predict the temperature distribution in moving fins of complex cross­sections. Recently, Ma et al. [10] simulated a combined conductive, convective and radiative heat transfer in irregular moving porous fins using Spectral Element Method (SEM). In­ vestigation of mixed convection heat transfer along a continuously moving heated vertical plate with suction and blowing was conducted by Al­Sanea [11]. The radiation effects are quite significant in var­ ious engineering and industrial processes especially in the design of reliable equipments, nuclear plants and gas turbines. Razelos and Kakatsios [12] investigated the optimum dimensions of convecting­ radiating rectangular fins. They studied the influence of all dependent parameters on optimization and performance of rectangular fins. They also investigated the effect of the temperature­dependent thermal conductivity and emissivity on the temperature profile. An approximate analytical solution for convection­radiation heat transfer from a continuously moving fin with temperature­dependent thermal conductivity was developed by Aziz and Khani [13]. Torabi and Zhang [14] investigated ther­ mal performance of a convective­radiative straight fin by considering different profiles containing

2

Computational Analysis of Heat Transfer in Fluids and Solids II

nonlinearities and further studied the fin efficiency. Sun et al. [15] studied the different temperature­ dependent thermal properties for predication of heat transfer in a convective­radiative fin. Kundu and Wongwises [16] presented a decomposition analysis of a convective­radiating fin with one side of the primary surface being heated by a fluid with high temperature. Aziz and Makinde [17] applied a two­dimensional heat conduction model to obtain the thermal performance and entropy generation in an orthotropic convection pin fin used in advanced light weight heat sinks. The study illustrated the simultaneous realization of the least material and minimum entropy generation in pin fin designs. Mhlongo et al. [18] applied the local and nonlocal symmetry techniques to study transient response of rectangular fins to step change in base temperature and base heat flux. The authors showed that the governing equations can be reduced to the tractable Ermakov­Pinney equation. The DTM is a semi­numerical­analytical method based on the Taylor series expansion and was first proposed by Zhou [19] in 1986 for the solution of linear and nonlinear initial value problems that appear in the analysis of electrical circuits. This method was subsequently used to obtain analytical solutions of various types of higher­order differential equations. Torabi et al. [20] applied the DTM to derive approximate explicit analytical expressions for the temperature distribution in a moving fin with temperature­dependent thermal conductivity and experiencing simultaneous convective­radiative surface heat loss. Moradi and Rafiee [21] extended the work of Torabi et al. to study a similar model for various fin profiles. Ndlovu and Moitsheki [22] successfully applied the 2D DTM to a transient heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins. Some interesting results were obtained and the effects of the parameters appearing in the model on the tem­ perature distribution were illustrated and explained. Mosayebidorcheh et al. [23] studied transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles. These authors ap­ plied the Hybrid Differential Transform Method­Finite Difference Method (DTM­FDM) to generate the numerical solutions to the problem. Fallo et al. [24] applied the 3D DTM for the first time to study heat transfer in a cylindrical spine fin with variable thermal properties. Hatami et al. [25] applied the Differential Quadrature Method (DQM) together with the DTM to study magnetohydrodynamic (MHD) two­phase Couette flow analysis for fluid­particle suspension between moving parallel plates. To the best our knowledge, no research to date has been performed to study transient heat transfer in a moving fins. Numerous studies have been devoted to steady state analysis with the assumption that the transient response dies out quickly [26, 27]. In this article, 2D DTM is applied to study simultaneous conductive, convective and radiative heat transfer in a moving fin of rectangular profile. Thermal conductivity, heat transfer coefficient and surface emissivity are all temperature dependent. The problem formulation is presented in Section 2. A brief discussion on the fundamentals of the 2D DTM is provided in Section 3. The validation of the analytical solutions together with analytical results are given in Section 4. Lastly, we provide some discussions based on the results obtained in Section 5 and the conclusions summarized are Section 6. Problem Formulation We consider a one dimensional rectangular moving fin of length L, with cross­sectional area Ac , thickness δ and perimeter P while it moves horizontally with a constant velocity U as depicted in Figure 1. The fin surface is exposed to a convective and radiative environment of temperature Ta and the base temperature of the fin is Tb > Ta . The fluid surrounding the moving fin is heated by the convective­radiative fin and induces a horizontal flow field (advection). The radiative component would be more prominent if fin undergoes natural convection or if the forced convection is rather weak or absent. The energy balance equation for the moving fin losing heat by simultaneous convection and radiation can be expressed as   ∂T ∂ ∂T P H(T ) P σε(T ) 4 ∂T ρc = K(T ) − (T − Ta ) − (T − Ta4 ) − ρcU , 0 ≤ X ≤ L. (1) ∂t ∂X ∂X Ac Ac ∂X

Defect and Diffusion Forum Vol.401

3

For most materials, the thermal conductivity varies linearly with temperature, that is, K(T ) = ka [1 + γ(T − Ta )],

(2)

the surface emissivity is also assumed to vary linearly with temperature, ε(T ) = εa [1 + η(T − Ta )], and heat transfer coefficient may be given by a power law function of temperature,  n T − Ta h(T ) = hb . Tb − Ta

(3)

(4)

Fig. 1: Schematic representation of a rectangular moving fin. Here, ka is the thermal conductivity of the fin at ambient temperature, γ is a measure of thermal conductivity variation with temperature, εa is the surface emissivity of the fin at ambient temperature, η is a measure of surface emissivity variation with temperature, ρ is the density of fluid; c is the specific heat of fluid; ε and σ are the emissivity and Boltzman constant respectively, hb is the heat transfer coefficient at the fin base and n is a constant. The constant n may vary between ­6.6 and 5. However, in most practical applications it lies between ­3 and 3 [28]. The exponent n represents laminar film boiling or condensation when n = −1/4, laminar natural convection when n = 1/4, turbulent natural convection when n = 1/3, nucleate boiling when n = 2, radiation when n = 3 and n = 0 implies a constant heat transfer coefficient. Assuming that the fin tip is adiabatic (insulated) and the base temperature is kept constant, then the boundary conditions are given by [2] et al., T (t, L) = Tb and ∂T = 0. ∂X X=0

(5) (6)

Initially the fin is kept at the ambient temperature, T (0, X) = Ta .

(7)

4

Computational Analysis of Heat Transfer in Fluids and Solids II

Introducing the following dimensionless variables, X ka t T Ta , τ= , θ = , θa = , ζ = ηTb , β = γTb , 2 L ρcL Tb Tb n 2 2 3 P hb Tb L P εa σL Tb ka UL Nc = , Nr = , α = , and P e = , n ka Ac (Tb − Ta ) ka Ac ρc α x=

reduces the energy equation (1) to   ∂θ ∂ ∂θ = {1 + β(θ − θa )} − Nc (θ − θa )n+1 − Nr [1 + ζ(θ − θa )](θ4 − θa4 ) ∂τ ∂x ∂x ∂θ − P e , 0 ≤ x ≤ 1. ∂x

(8)

(9)

The prescribed boundary conditions are given by, θ(τ, 1) = 1 and ∂θ = 0. ∂x x=0

(10)

θ(0, x) = 0.

(12)

(11)

and the initial condition becomes, The dimensionless variable θ represents the temperature, θa represents the dimensionless ambient tem­ perature, x is the dimensionless space variable, τ is the dimensionless time variable, β is the thermal conductivity gradient, ζ is the surface emissivity gradient, Nc is the convection­conduction parameter, Nr is the radiation­conduction parameter, P e is the Peclet number which represent the dimensionless speed of the moving fin, α is the thermal diffusivity of the fin. Fundamentals of the Two­Dimensional Differential Transform Method In this section, the basic idea underlying the two­dimensional DTM is briefly introduced. If function θ(t, x) is analytic and differentiated continuously with respect to time and the spatial variable x in the domain of interest, then we let   1 ∂ κ+s ϕ(t, x) Φ(κ, s) = , (13) κ!s! ∂tκ ∂xs (0,0) where the spectrum Φ(κ, s) is the transformed function, which is also called the T­function (see [29, 30]). The differential inverse transform of Φ(κ, s) is defined as ϕ(t, x) =

∞ X ∞ X

Φ(κ, s)tκ xs ,

(14)

κ=0 s=0

and from equations (13) and (14) it can be concluded that   ∞ X ∞ X 1 ∂ κ+s ϕ(t, x) ϕ(t, x) = t κ xs . κ ∂xs κ!s! ∂t (0,0) κ=0 s=0

(15)

Defect and Diffusion Forum Vol.401

5

In real applications, the function ϕ(t, x) is expressed by a finite series, and equation (14) can be written as: ϕ(t, x) =

n m X X

Φ(κ, s)tκ xs ,

(16)

κ=0 s=0

Equation (16) implies that

∞ X

ϕ(t, x) =

∞ X

Φ(κ, s)tκ xs ,

(17)

κ=m+1 s=n+1

is negligibly small. Some of the useful mathematical operations performed by the differential transform method are given in Table 1 . Table 1: Fundamental operations of the differential transform method Original function ϕ(t, x) = f1 (t, x) ± f2 (t, x) ϕ(t, x) = αf (t, x) (t,x) ϕ(t, x) = ∂f∂t (t,x) ϕ(t, x) = ∂f∂x 2 ϕ(t, x) = ∂ f∂t(t,x) 2 ∂ 2 f (t,x) ϕ(t, x) = ∂x2 ϕ(t, x) = tm xn ϕ(t, x) = xm eat

Transformed function Φ(κ, s) = F1 (κ, s) ± F2 (κ, s) Φ(k, s) = αF (κ, s) Φ(κ, s) = (κ + 1)F (κ + 1, s) Φ(κ, s) = (s + 1)F (κ, s + 1) Φ(κ, s) = (κ + 1)(κ + 2)F (κ + 2, s) Φ(κ, s) = (s + 1)(s + 2)F (κ, s + 2) Φ(κ, s) = δ(κ − m)δ(s − n) κ Φ(κ, s) = aκ! δ(s − m)

The Kronecker delta function δ(κ − s) is given ( by 1 if κ = s δ(κ − s) = 0 if κ ̸= s. Derivation of Analytical Solutions In this section we provide an analytical solution to the problem under consideration. Using 2D­DTM to solve PDEs consists of three main steps. The steps are: transforming the PDE into algebraic equations, solving the equations, and inverting the solution of algebraic equations to obtain an infinite series solution or an approximate solution. Taking the two­dimensional differential transform of equation (9), given n = 2, we obtain the following recurrence relation, (κ + 1)Θ(κ + 1, h) = (1 − βθa )(h + 1)(h + 2)Θ(k, h + 2) +β

κ X h X

(h + 1 − j)(h + 2 − j)Θ(k − i, j)Θ(i, h + 2 − j)

i=0 j=0

+β

κ X h X

(j + 1)Θ(k − i, j + 1)(h + 1 − j)Θ(i, h + 1 − j)

i=0 j=0

− Nc

h−j κ X κ−i X h X X i=0 l=0 j=0 p=0

Θ(i, h − j − p)Θ(t, j)Θ(k − i − t, p)

6

Computational Analysis of Heat Transfer in Fluids and Solids II

+ 3Nc θa

κ X h X

Θ(i, h − j)Θ(k − i, j) + Nc θa3

i=0 j=0

− Nr ζ

h−j h−j−p h−j−p−l κ X κ−i κ−i−t h X X X κ−i−t−z X X X X i=0 t=0

z=0

q=0

j=0 p=0

l=0

w=0

Θ(i, h − j − p − l − w)Θ(t, j)Θ(z, p)Θ(q, l)Θ(k − i − t − z − q, w) − Nr (1 − θa ζ)

h−j h−j−p κ X κ−i κ−i−t h X X X X X i=0 t=0

z=0 j=0 p=0

l=0

Θ(i, h − j − p − l)Θ(t, j)Θ(z, p)Θ(k − i − t − z, l) + Nr (ζθa4 Θ(k, h) − θa4 (1 + ζθa )) − P e(h + 1)Θ(k, h + 1), where Θ(κ, h) is the differential transform of θ(τ, x). Taking the two­dimensional differential transform of the initial condition (12) and boundary con­ dition (11) we obtain the following transformations respectively, Θ(0, h) = 0, h = 0, 1, 2, . . .

(18)

Θ(κ, 1) = 0, κ = 0, 1, 2, . . . .

(19)

We consider the other boundary condition as follows, Θ(κ, 0) = a, a ∈ R, κ = 1, 2, 3, . . .

(20)

where the constant a can be determined from the boundary condition (10) at each time step after obtaining the series solution. Substituting equations (18)­(20) into (??) we obtain the following, Θ(1, 2) = θa3 (θa (1 + θa ζ)Nr + Nc ), 1 Θ(2, 2) = θa3 (θa (1 + θa ζ)Nr + Nc )(13 − 12θa β − 3P e + θa4 ζNr − 3θa2 Nc ), 2 Θ(1, 3) = θa3 (θa (1 + θa ζ)Nr + Nc ), 1 Θ(2, 3) = θa3 (θa (1 + θa ζ)Nr + Nc )(21 − 20θa β − 4P e + θa4 ζNr − 3θa2 Nc ) 2 .. .

(21) (22) (23) (24)

Substituting equations (18)­(24) into (16) we obtain the following closed form of the solution, θ(τ, x) = aτ + aτ 2 + (θa3 (θa (1 + θa ζ)Nr + Nc ))τ x2 + (θa3 (θa (1 + θa ζ)Nr + Nc ))τ x3 1 + θa3 (θa (1 + θa ζ)Nr + Nc )(13 − 12θa β − 3P e + θa4 ζNr − 3θa2 Nc )τ 2 x2 + aτ 3 2 1 + θa3 (θa (1 + θa ζ)Nr + Nc )(21 − 20θa β − 4P e + θa4 ζNr − 3θa2 Nc )τ 2 x3 + . . . 2

(25)

The constant a can be determined from the boundary condition (10). To obtain the value of a, we substitute the boundary condition (10) into (25) at the point x = 1. Thus, we have, θ(τ, 1) = aτ + aτ 2 + (θa3 (θa (1 + θa ζ)Nr + Nc ))τ + (θa3 (θa (1 + θa ζ)Nr + Nc ))τ 1 + θa3 (θa (1 + θa ζ)Nr + Nc )(13 − 12θa β − 3P e + θa4 ζNr − 3θa2 Nc )τ 2 + aτ 3 2 1 + θa3 (θa (1 + θa ζ)Nr + Nc )(21 − 20θa β − 4P e + θa4 ζNr − 3θa2 Nc )τ 2 + . . . = 1 2

(26)

Defect and Diffusion Forum Vol.401

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We then obtain the expression for θ(τ, x) upon substituting the obtained value of a into equation (25). Using the first 25 terms of the power series solution, we benchmark the series solutions to the numerical solution obtained through employing the inbuilt numerical solver in MATLAB (pdepe). As seen from the benchmark analysis in Figure 2, the series solution is in agreement with the numerical solution. 1 0.9 0.8 0.7 0.6 0.5 0.4 2D DTM: n = 0 2D DTM: n = 2 NS: n = 0 NS: n = 2

0.3 0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 2: Temperature distribution in a moving fin for varying n. With the confidence obtained from the benchmark analysis, we plot the solution (25) for various parameters as shown in Figures below. For all analytical results depicted in the Figures in this article, the following values of parameters are used unless stated otherwise. n = 2, β = 1, Nc = 4, Nr = 0.25, θa = 0.10, P e = 3, τ = 0.30

(27)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 = 0.1 = 0.2 = 0.3 = 0.5

0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 3: Temperature distribution in a moving fin for varying time, τ .

Results and Discussion The results have been analyzed graphically and this is useful in terms of discussing the thermo­physical parameters presented by the solutions. This section describes the impact of the embedding parameters

8

Computational Analysis of Heat Transfer in Fluids and Solids II

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

= 0.25 = 0.75 = 1.00 = 1.25

0.2 0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 4: Effect of the thermal conductivity gradient, β , on temperature distribution. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

Nc = 1 Nc = 25

0.2

Nc = 50

0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 5: Effect of the thermo­geometric parameter, Nc , on temperature distribution. 1 0.9 0.8 0.7 0.6 0.5 0.4 Nr = 0.10

0.3

Nr = 10

0.2

Nr = 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 6: Effect of the thermo­geometric parameter, Nr , on temperature distribution.

Defect and Diffusion Forum Vol.401

9

1 0.9 0.8 0.7 0.6 0.5 0.4 a

0.3

a a

= 0.65 = 0.70 = 0.75

0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 7: Effect of the thermo­geometric parameter, θa , on temperature distribution. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

= 0.10 = 50 = 150 = 250

0.2 0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 8: Effect of the surface emissivity conductivity, ζ , on temperature distribution. 1 0.9 0.8 0.7 0.6 0.5 0.4 Pe Pe Pe Pe

0.3

=0 =1 =2 =3

0.2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 9: Effect of varying speeds, P e, on temperature distribution..

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Computational Analysis of Heat Transfer in Fluids and Solids II

0.7

0.6

0.5

0.4

0.3

0.2 Fin Tip (x = 0) x = 0.25 x = 0.50

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 10: Temperature distribution along the fin.

1 0.8 0.6 0.4 0.2 0 0.8 0.6

1 0.8

0.4

0.6 0.4

0.2 0

0.2 0

x

Fig. 11: Transient temperature distribution in a moving fin. on the temperature distribution of a moving fin of rectangular profile. The governing equations were solved analytically using the 2D DTM and validated numerically using the the inbuilt numerical solver in MATLAB (pdepe). Figure 2 shows the effect of the nature of heat transfer on transient temperature distribution and also gives a comparison of the 2D DTM and the numerical results. It is observed that the approximate analytical solutions are in good agreement with the numerical results. We also note that the temperature distribution of the fin is an increasing function of the exponent n. Figure 3 shows that the fin temperature increases with time. It has been shown that the transient temperature profile approaches the steady state solution as time evolves [22]. Figure 4 shows the impact of the change in thermal conductivity gradient. We observe that as the values of β decreases, the temperature in the fin decreases signifying an increased heat loss to the ambient fluid. Increasing the thermal conductivity gradient enhances heat conduction from the fin base and leads to increased temperatures inside the fin. Figure 5 illustrates the temperature distribution for varying values of the convective­conductive parameter Nc . For Nc = 0, this indicates the absence of convection while Nc = 1 indicates that con­ vection and conduction are at equal rates [3]. An increase in the values of the thermo­geometric fin parameter leads to a decrease in temperature in the fin. A similar result is observed in Figure 6 which represents the impact of the radiation parameter Nr on temperature distributions of a moving fin. Fig­

Defect and Diffusion Forum Vol.401

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ure 7 shows the variation in temperature distribution for varying values of the convective­radiative sink parameter θa . The ambient temperature plays an important role in transient heat dissipation. Higher ambient temperature lead to a reduced heat transfer rate as indicated by high values of fin temperature for higher values of the convective­radiative sink temperature. The effect of varying surface emissivity parameter ζ is depicted in Figure 8. As expected, we observe that as the surface emissivity increases, the radiative heat loss decreases resulting in lower fin temperatures. Figure 9 illustrates the effect of the Peclet number on the temperature distribution in the fin. As P e increases, (fin moves faster), the temperature inside the fin decreases rapidly due to increased advective effect on the surface of the fin. We also observe that for P e = 0, i.e., stationery fin, the cooling of the fin takes more time as shown by higher temperatures when compared to a moving fin. Figure 10 shows the temperature variation along the fin profile. The fin temperature gradually increases as the axial parameter, x, increases and this is in line with the imposed boundary condition at the fin base. Lastly, Figure 11 we observe that the temperature distribution along the three dimensional spaces corresponds to the boundary conditions and to all the results described above. Concluding Remarks The 2D DTM was successfully applied to study transient heat transfer in a longitudinal moving fin subject to convective and radiative heat losses. The closed­form solution of the governing highly nonlinear equation was obtained and the embedding parameters were illustrated and explained. The accuracy of the analytical solution was validated using the numerical solution. Acknowledgments PLN, thanks Standard Bank of South Africa for funding the doctoral studies. RJM thanks the National Research Foundation South Africa for financial support. References [1] D.Q Kern, A.D. Kraus, Extended Surface Heat Transfer, McGraw­Hill, New York, 1972. [2] A.D. Kraus, A. Aziz, J. Welty, Extended Surface Heat Transfer, John Wiley and Sons, New York, 2001. [3] P. Kanti Roy, A. Mallick, H. Mondal, P. Sibanda, A Modified Decomposition Solution of Trian­ gular Moving Fin with Multiple Variable Thermal Properties, Arabian J Science and Engineering, 43(30): 1485­1497, 2017. [4] A.S. Dongonchi, D.D. Ganji, Convection­radiation heat transfer study of moving fin with temperature­dependent thermal conductivity, heat transfer coefficient and heat generation, Ap­ plied Thermal Eng. 103: 705­712, 2016. [5] M. Turkyilmazoglu, Heat transfer from moving exponential fins exposed to heat generation, Int J Heat and Mass Transfer, 116: 346­351, 2018. [6] M. Turkyilmazoglu, Efficiency of the longitudinal fins of trapezoidal profile in motion, Heat Transfer, 139(9): 4 pages, 2017. [7] R.K. Singla, R. Das, Application of decomposition solution and inverse prediction of parameters in a moving fin, Energy Conversion and Management, 84: 268­281, 2014.

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Computational Analysis of Heat Transfer in Fluids and Solids II

[8] Y.S. Sun, J. Ma, B.W. Li, Spectral collocation method for convective­radiative transfer of a mov­ ing rod with variable thermal conductivity, Int J Thermal Sciences 90: 187­196, 2015. [9] Y.S. Sun, J.L. Xu, Thermal performance of continuously moving radiative­convective fin of com­ plex cross­section with multiple nonlinearities, Int Comm Heat and Mass Transfer, 63: 23­34, 2015. [10] J. Ma, Y. Sun, B. Li, Simulation of combined conductive, convective and radiative heat transfer in moving irregular porous fins by spectral element method, Int J Thermal Sciences, 118: 475­487, 2017. [11] S.A. Al­Sanea, Mixed convection heat transfer along a continuously moving heated vertical plate with suction or injection, Heat Mass Transfer, 47(6­7): 1445­1465, 2004. [12] P. Razelos, X. Kakatsios, Optimum dimensions of convecting­radiating fins: Part I ­ longitudinal fins, Applied Thermal Engineering 20: 1161­1192, 2000. [13] A. Aziz, F. Khani, Convection­radiation from a continuously moving fin of a variable thermal conductivity, Franklin Institute, 34: 640­651, 2011. [14] AM. Torabi, Q. Zhang, Analytical solution for evaluating the thermal performance and efficiency of convective­radiative straight fins with various profiles and considering all nonlinearities, En­ ergy Conversion and Management 66: 199­210, 2013. [15] Y. Sun, J. Ma, B. Li, Z. Guo, Predication of nonlinear heat transfer in a convective­radiative fin with temperature­dependent properties by the collocation spectral method, Numerical Heat Transfer Part B, 69(1): 68­83, 2016. [16] B. Kundu, S. Wongwises, A decomposition analysis on convective­radiating rectangular plate fins for variable thermal conductivity and heat transfer coefficient, Franklin Institute, 349: 966­ 984, 2012. [17] A. Aziz, O.D. Makinde, Heat transfer and entropy generation in a two­dimensional orthotropic convection pin fin, International J. of Exergy, 7(5), 579­592, 2010. [18] M.G. Mhlongo, R.J. Moitsheki, O.D. Makinde, Transient response of longitudinal rectangular fins to step change in base temperature and in base heat flow conditions, International J. of Heat and Mass Transfer, 57, 117­125, 2013. [19] J.K Zhou, Differential transformation and its application for electrical circuits, Huazhong Uni­ versity Press, Wuhan China, 1986. [20] M. Torabi, H. Yaghoobi, A. Aziz, Analytical solution for convective­radiative continuously mov­ ing fin with temperature­dependent thermal conductivity, Int J Thermophysics, 33: 924­941, 2012. [21] A. Moradi, R. Rafiee, Analytical solution to convection­radiation of a continuously moving fin with temperature­dependent thermal conductivity, Thermal Science, 17(4): 1049­1060, 2013. [22] P.L. Ndlovu, R.J. Moitsheki, Application of the two­dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins, Comm Nonlinear Science and Numerical Simulation, 18: 2689­2698, 2013.

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[23] S. Mosayebidorcheh, M. Rahimi­Gorji, D.D Ganji, T. Moayebidorcheh, O. Pourmehran, M. Biglarian, Transient thermal behavior of radial fins of rectangular, triangular and hyperbolic pro­ files with temperature­dependent properties using DTM­FDM, Central South University, 24(3): 675­682, 2017. [24] A. Fallo, R.J. Moitsheki, O.D. Makinde, Analysis of heat transfer in a cylindrical spine fin with variable thermal properties, Defect and Diffusion Forum, 387, 10­22, 2018. [25] M. Hatami, K. Hosseinzadeh, D.D. Ganji, M.T. Behnamfar, Numerical study of MHD two­phase Couette flow analysis for fluid­particle suspension between moving parallel plates, Taiwan In­ stitute of Chemical Engineers, 45(5): 2238­2245, 2014. [26] B. Kundu, K. Lee, A proper analytical analysis of annular step porous fins for determining max­ imum heat transfer, Energy Conversion and Management, 110: 469­480, 2016. [27] P.L. Ndlovu, R.J. Moitsheki, Predicting the Temperature Distribution in Longitudinal Fins of Various Profiles with Power Law Thermal Properties Using the Variational Iteration Method, Defect and Diffusion Forum, 387: 403­416, 2018. [28] H.C. Ünal, An analytical study of boiling heat transfer from a fin, Int J Heat and Mass Transfer 31: 1483­1496, 1988. [29] F. Kangalgil, F. Ayaz, Solitary wave solutions for the KdV and mKdV equations by differential transform method, Chaos, Solitons and Fractals, 41(1): 464­472, 2009. [30] F. Ayaz, On the two­dimensional differential transform method, Computational and Applied Mathematics, 143: 361­374, 2003.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 14-24 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-03-25 Revised: 2019-10-21 Accepted: 2019-10-21 Online: 2020-05-28

Determination of Proper Fin Length of a Convective-Radiative Moving Fin of Functionally Graded Material Subjected to Lorentz Force G. OGUNTALA1,a*, G. SOBAMOWO2,b, Y. AHMED2,c, R. ABD-ALHAMEED1,d Department of Biomedical and Electronics Engineering, Faculty of Engineering and Informatics, University of Bradford, West Yorkshire, United Kingdom

1

Department of Mechanical Engineering, Faculty of Engineering, University of Lagos, Akoka, Lagos, Nigeria

2

[email protected], [email protected], [email protected], d [email protected]

a

Keywords: analytical solution, convective-radiative fin, effective length, functionally graded material, Kummel function, moving fin

Abstract. Excess fin length results in material waste and additional weight leading to increased cost with no benefit in return. Moreover, extra fin length affects the overall performance of the fin as fluid motion is suppressed, resulting in reduced convective heat transfer coefficient. To achieve a miniaturised system with effective cooling, the determination of appropriate length of extended surfaces becomes a key performance and fabrication process factor. Therefore, the present work aims at determining the proper or effective length of a convective-radiative moving fin of functionally graded material under the influence of a magnetic field. The developed governing equation of the analysis is solved analytically with the aid of Kummer’s function. The analytical solutions are used to investigate the effects of non-homogeneity, convective, radiative and magnetic parameter on the thermal performance and the proper fin length. The present study is hoped to assist in making costeffective decisions on designing cooling approaches for different consumer electronics and highpower systems under various operating conditions. Introduction Fins are extended surfaces useful for cooling various electronic systems and heavy machinery. The diverse area of applications of fins has necessitated the intense attention it is receiving in the literature. Over the past few decades, different researches have been conducted on the thermal characteristics of fins under various operating conditions. In [1], Aziz and Enamul-Huq analytically studied the thermal behaviour of a convecting fin of variable thermal conductivity, whilst Aziz extends the investigation on the fin behaviour under the effect of uniform internal heat generation [2]. However, with the aid of the method of successive approximation, Campo and Spaulding studied the thermal behaviour of uniform circumferential fins [3]. These notable works influence the proliferation in research on the subject of effective cooling of various thermal and electronic systems via fins. Recent times have witnessed the development and use of different exciting methodologies including the analytical, semi-analytical, numerical and experimental approach to analyse and enhance the thermal performance of fin of temperature-dependent thermal conductivity under a convective-radiative environment [4-24]. Most of these existing works investigate the thermal performance of homogenous fins under various conditions. However, the use of materials with changing composition, microstructure across its volume have been identified as an effective way to improve the thermal performance of extended surfaces. Functionally graded materials (FGM) are materials with varying electrical, chemical, mechanical, magnetic, thermal properties over the volume of the bulk material. The continuous variations in the properties of FGM along a specific axis are based on different factors including porosity and pore size, chemical and microstructural gradientstructures. Moreover, due to the varying physical properties of FGMs, it is used for a wide range of applications including nuclear, automobile structure, aerospace, and optoelectronics. Furthermore, owing to the significant capabilities and enhanced heat transfer characteristics of FGM, several

Defect and Diffusion Forum Vol.401

15

studies have been conducted on effective cooling of various systems using FGM fin heat sink [2529]. The applications of heat transfer in a moving continuous surface such as hot rolling, extrusion, casting, wire, sheets and glass fibre drawing, the flow of liquids where the material transfer heat to the surroundings while moving through a channel have gained research interests in recent times. Such applications can be modelled as one-dimensional heat conduction under a continuous motion. In view of this, several investigations have been conducted on the thermal behaviour of fins subjected to motion [30-35]. With the increasing depletion of natural materials used for heat transfer enhancement, the need to determine technical parameters such as spacing, thickness, and length of extended surfaces in order to avoid waste and achieve desired thermal performance is very important [36-38]. Therefore, it becomes imperative to determine the effects of material parameters such as effective length on the fin temperature distributions and overall thermal performance. Nevertheless, there is an idealization that the longer the fin, the larger the surface area and the higher the rate of heat transfer. The idealisation suggests that maximum heat transfer is achieved, as the fin is infinitely long. However, such an idealization is not consistent in practice because temperature drops along the fin exponentially and reaches the ambient temperature at certain length. Consequently, the remaining fin length beyond the length where ambient temperature is reached does not contribute to heat transfer as fin temperature is at ambient temperature as illustrated in Fig. 1.

Fig. 1 Temperature drops along the fin length. Source: [36] From Fig. 1, it is obvious that designing such an extra-long fin is not cost effective as it does not affect the overall thermal performance of the fin. Such condition results in material waste, extra weight, and size with no benefit in return. In practice, the extra fin length affects the overall thermal performance since fluid motion is suppressed, leading to reduced convective heat transfer coefficient [36]. Furthermore, the extra fin length is ineffective since the minimal increase in heat transfer at the tip region does not justify the disproportionate increase in weight and cost. Therefore, in the fin design and fabrication process, after the selection of fin material and cross-section, the determination of the appropriate length of the extended surfaces becomes a key performance requisite. This implies that the determination of effective fin length via theoretical and experimental investigation to elucidate the fundamental physical mechanisms is important. The present work aims at the determination of the proper or effective length of a convective-radiative moving fin of FGM under the influence of Lorentz force. The developed governing equation of the thermal analysis is solved analytically via the aid of Kummer’s function. The analytical solutions are used to investigate the effects of inhomogeneity, convective, radiative and magnetic parameters on the thermal performance and examine the effective fin length. Problem Formulation Fig 2 considers a rectangular solid fin of FGM with length b and thickness t and exposed on both faces to a convective-radiative environment at temperature T∞ . The fin moves at variable speed U(x)

16

Computational Analysis of Heat Transfer in Fluids and Solids II

in the axial coordinate x which is measured from the base of the moving fin. Assume the medium is homogeneous, isotropic and saturated with a single-phase fluid. The physical properties of solid as well as fluid are constant, the fin temperature variation is assumed one-dimensional and steady. Moreover, there is no thermal contact resistance at the fin base and the fin tip is at atmospheric temperature.

Fig. 2. Schematic diagram of the fin problem under investigation Following the above assumptions, the governing differential equation is developed as d  dT  k ( x) dx  dx

J ×J ∂T σε P 4  hP T − Ta4 − c c + q ( x ) = ρ c pU ( x ) (T − Ta ) − − A A A ∂x σ 

(

)

(1)

where the Lorentz force is given as: Jc × Jc

σ

(2)

= σ Bo2u 2

substituting Eq. (2) in (1) and taking the magnetic term as a linear function of temperature gives: ∂  ∂T  k ( x) ∂x  ∂x

σ Bo u σε P 4  hP T − Ta4 − (T − Ta ) − − A A  A

(

)

2

2

ρ c pU ( x ) (T − Ta ) + q ( x ) =

∂T ∂x

(3)

To determine the proper length of the fin, the fin is taken as an infinitely long fin. Therefore, the boundary conditions are

= x 0,= T Tb

= x L= T∞ ∞, T

(4)

dT = x L= , 0 dX

(5)

However, it is worth noting that

Considering a case where the temperature difference within the fin material during the heat flow is small, then the expression T4 is expressed as: (6) T 4 ≅ 4T∞3T − 3T∞4 It is interesting to note that Eq. (6) can be linearized without any loss of generality. Therefore, Eq. (3) becomes σ B2u 2 4σε PTa3 ∂  ∂T  hP ∂T ρ c pU ( x ) (T − T∞ ) − o (T − T∞ ) + q ( x ) =  k ( x )  − (T − T∞ ) − A A ∂x  ∂x  A ∂x

(7)

Using the following dimensionless parameter, = X

σε PL2Tb3 L∞ T − T∞ Ph x k = = = , L∞ , θ , K , Nc 2 , Nr , = = L L Tb − T∞ ko Ako Ak0

σ Bo2 u 2 L2 UL ρ c qA , Pe = ,Q Ha = = hP (Tb − T∞ ) Ako ko

(8)

Defect and Diffusion Forum Vol.401

17

The governing equation is expressed as: d dX

dθ  K (X ) dX 

dθ  − ( Nc 2 + Nr + Ha ) θ + Q ( X ) = 0  − Pe ( X ) dX 

(9)

and the dimensionless boundary conditions are: = X 0,= θ ( 0) 1

(10)

= X L= θ ( L∞ ) 0 ∞,

In the present work, the thermal conductivity of the FGM and the internal heat generation are taken as: (11) k (X ) =γ X2 (12)

Q (= X ) λ1 X + λ2 X 2

Also, it is worth stating that the fin is allowed to both moves and stretch/shrink horizontally at speed Usx2. Therefore, the dimensionless speed of the fin is taken as: (13)

Pe ( X ) = α X 2

on substitution, we obtain γ X2

d 2θ dθ dθ + 2γ X −α X 2 − ( Nc 2 + Nr + Ha ) θ + Q ( X ) = 0 dX dX dX 2

(14)

by dividing Eq. (14) through by γ X X

d 2θ dX 2

2  α  dθ ( Nc + Nr + Ha ) 1 +2− X  − θ+ Q(X ) = 0 γX γX  γ  dX

(15)

Eq. (15) can be transformed into Kummer’s equation or standard confluent hypergeometric equation. The solution of Eq. (15) subject to the boundary conditions in Eq. (10) is:

θ (X )

     γ 

               k +1 ∞     λ1 k!X  2    γ + + X X      ∑  ( Nc2 + Nr + Ha )     k =2 k  ( n + 1)2 + ( n + 1) − ( Nc2 + Nr + Ha )      1 − a −  ∏ n=2        γ γ            

                        1 −   a +b  a + b     +X 2 M , b + 1, X      γ  2                       

      λ1  − λ2  2  ( Nc + Nr + Ha )    1 − a −    γ      2 ( Nc + Nr + Ha )  6−  γ  

              k +1  ∞       k ! X   L2∞ +  ∑      2   ( Nc + Nr + Ha )        k =2 k 2    ∏ n = 2  ( n + 1) + ( n + 1) −       γ                      λ1        + L       ∞ 2     λ + + Nc Nr Ha ( )       1  − λ2  1 − a −         Nc 2 + Nr + Ha )    γ (   γ 1 − a −          ×       γ                  Nc 2 + Nr + Ha ) (  6−       γ                 a+b 2 a + b   , b + 1, L∞  L∞ M    2   

The temperature gradient of the fin is expressed in the form:

(16)

18

Computational Analysis of Heat Transfer in Fluids and Solids II

dθ dθ1 dθ 2 = + dX dX dX

(17)

where

dθ1 dX

          λ1       − λ2           2  + Nr + Ha )   Nc  ( k   ∞        γ 1 − a − ( k + 1) k ! X λ1      1 + 2 X +  ∑      γ 2 2    Nc + Nr + Ha )    Nc + Nr + Ha )        k =2 k  ( ( 2       2  γ 1 − a −  ∏ n = 2  ( n + 1) + ( n + 1) − + + Nc Nr Ha ( )      γ γ    6−          γ                          ∞      k + 1) k ! X k ( 2     L∞ +  ∑      2      Nc + Nr + Ha )      k =2 k ( 2           ∏ n = 2  ( n + 1) + ( n + 1) −          γ                              λ1        L +     1 −   ∞      λ1 Nc 2 + Nr + Ha )     a +b−2 (          − λ   a+b a + b     − − γ 1 a 2 2        + M , b + 1, X     X   Nc 2 + Nr + Ha )   γ (  2 2    γ 1 − a −               ×        γ                     Nc 2 + Nr + Ha )   (       6−    γ                         a+b   2 a + b  L∞ M  , b + 1, L∞     2     

(18)

                     ∞      k + 1) k ! X k ( 2     L∞ +  ∑      2      Nc + Nr + Ha )      k =2 ( k 2           ∏ n = 2  ( n + 1) + ( n + 1) −          γ                              λ1        + L  a +b  1 −       ∞    λ1 Nc 2 + Nr + Ha )     a +b   (            dθ 2   2   2 − λ   a + b + 2     − − a 1 γ 2        = X M , b + 2, X      Nc 2 + Nr + Ha )    γ (  2 dX  b + 1     γ 1 − a −             ×        γ     2                2        ( Nc + Nr + Ha )       6−    γ                          a+b   2 a + b  + , 1, L M b L ∞ ∞    2                               ∞      ( k + 1) k ! X k      L2 +       2     ∞ ∑  Nc + Nr + Ha )       k =2 ( k 2            ∏ n = 2  ( n + 1) + ( n + 1) −            γ                                − +           λ a +b       1   − λ2  a + b   2 2    M , b + 1, X           + + Nc Nr Ha ( )     2        γ 1 − a −   ×       γ                 2    Nc + Nr + Ha )     (     6−      γ                       a+b   2 a + b  , b + 1, L∞  L∞ M     2     

and

(19)

Defect and Diffusion Forum Vol.401

a+b   a + b   2 n n M X , b + 1, X  = 1+ ∑  2  n =1 ( b + 1) n n ! ∞

19

(20)

is the Kummer’s function, where a + b  a + b  a + b a+b  a + b  a+b  + 1  + 2 + 3  ...  + n − 1 =   2  2  2 n  2  2   2 

( b + 1)n =( b + 1)( b + 2 )( b + 3) ... ( b + n )

(21)

It is worth noting that the above solution satisfies the condition α = γ . For a homogenous fin moving at a constant speed, the solution becomes: θ (X )

  χ 2 + χ1 ( cosh µ2 + sinh µ2 )   ( cosh µ1 X + sinh µ1 X )  ( cosh µ2 + sinh µ2 ) − ( cosh µ1 + sinh µ1 )    χ 2 + χ1 ( cosh µ1 + sinh µ1 ) −   ( cosh µ2 X + sinh µ2 X )  ( cosh µ2 + sinh µ2 ) − ( cosh µ1 + sinh µ1 )    λ2 λ2 1  λ1 − + X2 + Pe  X 2 2 2 ( Nc + Nr + Ha ) ( Nc + Nr + Ha )  ( Nc + Nr + Ha )  +

( Nc

1

+ Nr + Ha )

2

2

(22)

    λ2  2λ2 −  λ1 −  Pe 2    ( Nc 2 + Nr + Ha )    

where

(

1 Pe + Pe 2 + 4 M 2 2

µ1 =

χ1 =− 1

χ2 =

( Nc

( Nc

+

1

2

+ Nr + Ha )

λ2

+ Nr + Ha )

2

( Nc

+ Nr + Ha )

2

µ2 =

(

1 Pe − Pe 2 + 4 M 2 2

)

    λ2  2λ2 −  λ1 −  Pe 2    ( Nc 2 + Nr + Ha )    

  λ2  λ1 − Pe  L∞ 2 2 ( Nc + Nr + Ha )  ( Nc + Nr + Ha )      λ2  2λ2 −  λ1 −  Pe 2  2   ( Nc + Nr + Ha )    

L2∞ +

1

2

2

)

1

(23)

Therefore, the rate of heat transfer from the fin is obtained from Eq. (18) and (19) whilst the ratio of heat transfer is expressed as:

Qratio =

Q finite lenght fin Qinf inite lenght fin

(24)

Results and Discussion The results of the present method using the Kummer’s function for the determination of the effective length of the solid moving fin is presented in this section. For parametric study, the aluminium fin with the following thermogeometric parameters including length = 0.1m, width = 0.05m, thickness = 0.03m, ambient temperature = 28oC, base temperature = 200oC, convective heat transfer coefficient

20

Computational Analysis of Heat Transfer in Fluids and Solids II

= 25 W/m2 oC is used. From Figures 3-6, the effects of convective, radiative, magnetic and moving parameters on the effective length of the fin are presented. 1

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0

0.2

0.4

0.6

0.8

1 X

1.2

1.4

1.6

1.8

1

0

2

Fig. 3. Effects of convective parameter on the proper length of the fin

0.8

0

θ(X)

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.6

0.8

1 X

1.2

1.4

1.6

1.8

1 X

1.2

1.4

1.6

0

2

Fig. 5. Effects of magnetic field parameter on the proper length of the fin

1.8

2

Pe = 0.05 Pe = 0.10 Pe = 0.15

0.5

0.4

0.4

0.8

0.8

0.6

0.2

0.6

1

0.7

0

0.4

0.9

0.7

0

0.2

Fig. 4. Effects of radiative parameter on the proper length of the fin

Ha = 0.2 Ha = 0.4 Ha = 0.6

0.9

θ(X)

0.5

0.4

0

Nr = 0.1 Nr = 0.2 Nr = 0.3

0.9

θ(X)

θ(X)

1

Nc = 0.2 Nc = 0.4 Nc = 0.5

0.9

0

0.2

0.4

0.6

0.8

1 X

1.2

1.4

1.6

1.8

2

Fig. 6. Effects of moving parameter on the proper length of the fin

From Figure 7, the effect of fin thickness (based on the number of fins) on the proper length of the fins is illustrated. It can be seen from the results that the proper length of fins is directly proportional to the thickness of the combined number of fins. This finding helps in decision making so as to avoid extra costs and work at high efficiency. In practice, the less thick fin is good for effective heat transfer rate in the extended surfaces. However, the decision on the magnitude of the low thick fin that corresponds to short fin as shown in Figure 7 is to be made based on the amount of heat dissipated for effective thermal performance of the fin. Using the fins for effective heat dissipation, the number of low thick fins can be increased and arranged on the prime surface. Consequently, the increased number of fins on the prime surface causes a high degree of heat dissipation as shown in Figure 8. 0.04

300

250

0.03

Heat transfer rate (W)

Proper lenght of fin (m)

0.035

0.025 0.02 0.015

150

100

50

0.01 0.005

200

0

0.005

0.02 0.015 0.01 Thickness of the fin (m)

0.025

0.03

Fig. 7. Effects of fin thickness on the proper length of the fin

0

0

0.005

0.01 0.015 0.02 Thickness of the fin (m)

0.025

0.03

Fig. 8. Effects of fin thickness on the heat transfer rate

1

1

0.8

0.8

Heat transfer ratio

Heat transfer ratio

Defect and Diffusion Forum Vol.401

0.6

0.4

0.6

0.4

0.2

0.2

0

21

0

0

0.5

1

3

2.5

2

1.5 Nc

Fig. 9. Effects of convective parameter on the heat transfer ratio

0

0.05

0.1

0.15 Nr

0.2

0.25

0.3

Fig. 10. Effects of radiative parameter on the heat transfer ratio

1

Heat transfer ratio

0.8

0.6

0.4

0.2

0

0

0.05

0.1

0.15

0.2

0.25 Ha

0.3

0.35

0.4

0.45

0.5

Fig. 11. Effects of the magnetic parameter on the ratio of heat transfer To establish the heat transfer characteristics of a long fin as a means to determine the proper or effective fin length, Figures 9-11 shows the effects of convective, radiative and magnetic parameters on the ratio of heat transfer of fin of finite length to heat transfer from an infinitely long fin under the same conditions. From Figures 9-11, it is shown that the rate of heat transfer from the fin increases as the convective, radiative and magnetic parameter increases. However, at certain values of these parameters, each curve reaches a plateau, after which the curves reaches some value for the fin of infinite length. It is worth stating that since the thermogeometric parameters are directly proportional to fin length, the fin temperature drops along the fin exponentially and reaches the ambient temperature at a certain length. Consequently, beyond the fin length at which the ambient temperature is reached, the remaining length does not enhance heat transfer. Therefore, designing such an extralong fin is not cost effective as it does not affect the overall thermal performance of the fin. Furthermore, the extra fin length constitutes material waste, extra weight, and size with no benefit in return. Hence, to achieve the desired fin performance with reasonably low cost, there must reasonable compromise between heat transfer performance and fin size. Conclusion In this work, we carried out an investigation on the determination of the effective length of a functionally graded convective-radiative moving fin under the influence of magnetic field. The developed governing equation of the thermal analysis is solved analytically with the aid of Kummer’s function. The analytical solutions are used to analyse the impacts of convective, radiative and magnetic parameters on the thermal performance and the proper fin length. The findings of the present study will assist in making decisions on the appropriate length of fin for effective thermal performance of fins under various operating conditions.

22

Computational Analysis of Heat Transfer in Fluids and Solids II

Nomenclature A h H k P Q qb t Tb T Ta x X w M Pe Ha Nr Nc

Cross-sectional area of the fins, m2 Heat transfer coefficient over the fin surface, W/m2K Dimensionless heat transfer coefficient at the base of the fin, Wm-2k-1 Thermal conductivity of the fin material, Wm-1k-1 Perimeter of the fin, m Dimensionless heat transfer rate per unit area Heat transfer rate per unit area at the base, W/m2 Thickness of the number of fins, m Base temperature, K Fin temperature, K Ambient temperature, K Axial length measured from fin base, m Dimensionless length of the fin Width of the fin, m Thermogeometric parameter Peclet number Hartmann number Radiative term Convective number

Greek Symbols β δ θ

inhomogeneity index thickness of single fin, m dimensionless temperature

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Computational Analysis of Heat Transfer in Fluids and Solids II

[24] G.A. Oguntala, G.M. Sobamowo, N.N. Eya, R.A. Abd-Alhameed, Investigation of simultaneous effects of surface roughness, porosity, and magnetic field of rough porous microfin under a convective–radiative heat transfer for improved microprocessor cooling of consumer electronics, IEEE Transactions on Components, Packaging and Manufacturing Technology, 9 (2019) 235-246. [25] R. Kandasamy, X.-Q. Wang, A. S. Mujumdar, Transient cooling of electronics using phase change material (PCM)-based heat sinks, Applied Thermal Engineering, 28 (2008) 1047-1057. [26] G. Oguntala, G. Sobamowo, Y. Ahmed, R. Abd-Alhameed, Thermal prediction of convectiveradiative porous fin heat sink of functionally graded material using Adomian decomposition method, Computation, 7 (2019) 19. [27] M. G. Sobamowo, G. A. Oguntala, A. A. Yinusa, Nonlinear transient thermal modeling and analysis of a convective-radiative fin with functionally graded material in a magnetic environment, Modelling and Simulation in Engineering, 2019 (2019) 16pages. [23] S. Mosayebidorcheh, M. Rahimi-Gorji, D.D Ganji, T. Moayebidorcheh, O. Pourmehran, M. Biglarian,Transient thermal behavior of radial fins of rectangular, triangular and hyperbolic profiles with temperature-dependent properties using DTM-FDM, Central South University, 24(3) (2017) 675-682. [29] R. Hassanzadeh, M. Bilgili, Improvement of thermal efficiency in computer heat sink using functionally graded materials, Communications on Advanced Computational Science with Applications, 2014 (2014) 13 Pages. [30] M. V. Karwe, Y. Jaluria, Numerical simulation of thermal transport associated with a continuously moving flat sheet in materials processing, Journal of Heat Transfer, 113 (1991) 612-619. [31] S. Roy Choudhury, Y. Jaluria, Forced convective heat transfer from a continuously moving heated cylindrical rod in materials processing, Journal of Heat Transfer, 116, (1994) 724-734. [32] A. Aziz, R. J. Lopez, Convection-radiation from a continuously moving, variable thermal conductivity sheet or rod undergoing thermal processing, International Journal of Thermal Sciences, 50 (2011) 1523-1531. [33] M. Torabi, H. Yaghoobi, A. Aziz, Analytical solution for convective–radiative continuously moving fin with temperature-dependent thermal conductivity, International Journal of Thermophysics, 33 (2012) 924-941. [34] R. K. Singla, R. Das, Application of decomposition method and inverse prediction of parameters in a moving fin, Energy Conversion and Management, 84 (2014) 268-281. [35] D. Bhanja, B. Kundu, A. Aziz, Enhancement of heat transfer from a continuously moving porous fin exposed in convective–radiative environment, Energy Conversion and Management, 88 (2014) 842-853. [36] Y. A. Cengel, A. J. Ghajar, Heat and Mass Transfer. Fundaments & Applications, 5th ed. USA: McGraw-Hill Education, 2013. [37] F. P. Incropera, D. P. Dewitt, Introduction to Heat Transfer. USA: John Wiley & Sons, Inc, 2011. [38] G. Sidebotham, Heat Transfer Modeling, in Energy, 1st Ed. ed. Heidelberg, Netherlands Springer, 2006.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 25-35 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-02-22 Revised: 2019-10-02 Accepted: 2019-10-02 Online: 2020-05-28

A Note on the Similar and Non-Similar Solutions of Powell-Eyring Fluid Flow Model and Heat Transfer over a Horizontal Stretchable Surface R. Khan1,a, M. Zaydan2,b, A. Wakif 2,c*, B. Ahmed1,d, R.L. Monaledi3,e, I.L. Animasaun4,f, A. Ahmad1,g 1Department 2Hassan 3Faculty

of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, 44000 Islamabad, Pakistan. II University, Faculty of Sciences Aïn Chock, Laboratory of Mechanics, B.P. 5366 Mâarif, Casablanca, Morocco.

of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa.

4Department

of Mathematical Sciences, Federal University of Technology, Akure, Nigeria.

[email protected], [email protected], c*[email protected], [email protected], [email protected], [email protected], g [email protected]

a d

Keywords: Similar and non-similar analysis, non-Newtonian fluid, flow and heat transfer, PowellEyring viscosity model, numerical solutions, correlation expression

Abstract: Deliberation on the dynamics of non-Newtonian fluids, most especially Powell-Eyring fluid flow can be described as an open question. In this investigation, the flow and heat transfer characteristics are examined numerically by means of similarity analysis for a Powell-Eyring fluid moving over an isothermal stretched surface along the horizontal direction, whose velocity varies nonlinearly as a function of 𝑥𝑥 and follows a specified power-law degree formula. In order to solve the problem under consideration, the resulting system of coupled nonlinear partial differential equations with their corresponding boundary conditions is transformed into a correct similar form by utilizing appropriate similarity transformations, which are exceptionally acceptable for a particular form of the power-law stretching velocity, whose exponent is equal to 1⁄3. From the mathematical point of view, the similar equations of the studied flow cannot be obtained for any form of the powerlaw surface stretching velocity. As a result, it was found that the use of a general power-law stretching velocity results in non-similar equations. Also, appropriate numerical methods for similar and nonsimilar equations are used to discuss the results of engineering significance. Furthermore, correlation expressions for the skin friction and Nusselt number have been derived by applying the linear regression on the data outputted from the used computational methods. On the contrary to the heat transfer rate, it was found that the local skin friction coefficient is a decreasing property of powerlaw stretching. Introduction The field of non-Newtonian fluid dynamics emerges exponentially in the last few decades due to frequent and wide involvement both in natural and technological applications. The engineering phenomena involving the non-Newtonian fluids is far more challenging than that entailing the Newtonian fluids. The Non-Newtonian fluids under appropriate circumstances present shearthinning, shear-thickening, viscoplastic, time-dependent and viscoelastic behaviors. Power-law [1], Powell-Eyring [2], Reiner-Phillipoff [3] and Ellis [4] are some empirical models explaining the stressstrain relationship for different non-Newtonian fluids. Acrivos et al.[5] studied the forced convection flow of a power-law fluid moving over a wedge. Lee and Ames [6] studied the power-law fluids for momentum transfer in general Falkner-Skan flows, Goldstein flows, momentum and energy transfer in forced convection about a right angle wedge. Hansen and Na [7] analyzed the similarity solutions of some empirical models for non-Newtonian fluids. After these pioneering works on non-Newtonian fluids in a semi-infinite domain, researchers focused their attention to this topic followed by bunches of articles on non-Newtonian fluids; see Ahmad and Asghar [8], Halim et al. [9]. Other interesting

26

Computational Analysis of Heat Transfer in Fluids and Solids II

works related to non-Newtonian fluids are reported clearly in Refs. [10–17]. Also, for Newtonian fluid and nanofluid flow problems, the readers can refer to the Refs. [18–27], which are solved in various physical configurations. The analysis of Powell-Eyring fluid flow and heat transfer has been scarcely discussed in the literature due to the complexities associated with the explicit expressions of stress components and velocity. The Powell-Eyring viscosity model is considered to explain the stress-strain relationship of pseudoplastic non-Newtonian fluid. Mathematically, the Powell-Eyring model is complex from the computational point of view. However, this rheological model has a distinct advantage for accommodating the flow, as well as high shear-rate Newtonian flow regions manifested by some nonNewtonian fluids. Also, the Powell-Eyring fluid model is very useful in the formulation of powdered graphite and ethylene glycol. Due to its shear-thinning properties, this fluid model has been proposed by Bessonov et al. [28] as the best possible model for blood in the literature. The significance of quartic autocatalysis on the dynamics of the three-dimensional flow of Eyring-Powell was presented by Koriko et al. [29]. It was found that the concentration of the homogeneous bulk fluid (EyringPowell nanofluid) increases and decreases with the volume fraction and Prandtl number. In a theoretical study on the flow of Eyring-Powell fluid due to the catalytic surface reaction by Abegunrin et al. [30], it was observed that the vertical velocity, horizontal velocity, temperature gradient, and concentration are decreasing properties of the material parameter dependent on viscosity. Increase in the magnitude of the first Eyring-Powell parameter corresponds with a significant decrease in the viscosity of the non-Newtonian fluid. Animasaun et al. [31] illustrated the effects of partial slip on the flow of Eyring-Powell conveying gold nanoparticles and remarked that partial slip is suitable to enhance the transport phenomenon near the wall. Based on Eyring reaction rate theory, the base of Eyring-Powell model gives it a strong thermodynamic foundation. According to Barth [32], the 3parameter Powell-Eyring model is given by −1

 ∂u   ∂u  (1) µ = µ∞ + ( µo − µ∞ )  λ  sinh −1  λ  ,  ∂y   ∂y  where µo shows the dynamic viscosity at zero-strain rate, µ∞ represents the limiting viscosity as

( ∂u

∂y ) → ∞ and λ denotes the characteristic time. In non-dimensional form, Eq. 1 read as −1

∂u  µ µ∞  µ∞   ∂u  −1  = + 1 − µ=   λ  sinh  λ  , µo µo  µo   ∂y   ∂y 

(2)

Keeping in mind the applicability and complexity related to the Powell-Eyring fluid model and the effective role of surface velocity in the flow control and prevention of heat loss, this article examines in detail the flow and heat transfer characteristics of a Powell-Eyring fluid moving over a power-law heated stretching sheet. In this study, a particular form of power-law stretching velocity has been recognized for which the governing equations have been reduced to a correct similar form. Also, the governing equations have been transformed into a non-similar form to study the flow behavior due to the general power-law stretching velocity. Appropriate numerical methods are adopted to obtain the solutions of the transformed equations. Further, the numerical data have been used to obtain the correlation expressions in the form of dimensionless strain and heat transfer rates for various values of the involved parameters. These correlation expressions are obtained by applying the linear regression on the numerical data. Mathematical Model Consider an incompressible two-dimensional laminar flow of a non-Newtonian fluid moving over a heated horizontal sheet with a nonlinearly stretching velocity 𝑢𝑢𝑤𝑤 (𝑥𝑥). The x-axis lies in the direction of the stretching surface, whereas the y-axis is taken normal to this surface as shown in Fig. 1.

Defect and Diffusion Forum Vol.401

27

𝑦𝑦, 𝑣𝑣

𝑥𝑥, 𝑢𝑢

Fig. 1: Sketch of the geometry of the boundary layer flow problem. Using the Powell-Eyring viscosity model described above by Eq. (1), we get the following stress-strain relationship

= τ yx µ∞

write

 ∂u  ∂u µo − µ∞ + sinh −1  λ  ∂y λ  ∂y 

(3)

In the dimensionless form, Eq. (3) alters to λτ yx µ∞  ∂u   µ∞  ∂u  −1  = τ= (4)  sinh  λ  ,  λ  + 1 − µo µo  ∂y   µo   ∂y  By taking the second-order approximation of the function 𝑠𝑠𝑠𝑠𝑠𝑠ℎ−1 (𝑥𝑥) when |𝑥𝑥| ≪ 1, we can 3

 ∂u  ∂u 1  ∂u  − λ  , sinh  λ  ≅ λ ∂y 6  ∂y   ∂y  Therefore, Eq. 3 may be reformulated as follows −1

λ

∂u 1). On the contrary, when the flow timescale is much greater than the fluid time-scale (i. e. , 𝑊𝑊𝑊𝑊 < 1), the elastic effects relax sufficiently and the viscous behavior prevails in the medium. It is worth noting that the Newtonian fluid flow case can be recovered from the system of Eqs. (15)-(18) by setting ε = 0 in Eq. (15). Furthermore, the resulting boundary value problem described mathematically above by Eqs. (15)-(18) has been solved numerically by means of the shooting method (SM) based on the Runge-Kutta-Fehlberg scheme (RKFS).

Defect and Diffusion Forum Vol.401

29

Non-Similarity Solution (Stretching Velocity Axn) For the general case of the power-law stretching velocity 𝑢𝑢𝑤𝑤 (𝑥𝑥) = 𝐴𝐴𝑥𝑥 𝑛𝑛 , it is more physically realistic to introduce the following transformations

= ψ

T − T∞ = ,η To − T∞

uw ( x)ν x f (η , ξ ) = , θ (η , ζ )

uw ( x) x = y , ξ . νx L

(22)

As shown above, the expression of the coordinate η involves both 𝑥𝑥 and 𝑦𝑦. This component is also called the pseudo-similarity variable as reported by Sparrow and Yu [33]. In Eq. (22), the variable ξ is chosen to avoid the explicit appearance of 𝑥𝑥 in the transformed equations. After introducing the above transformations into Eqs. (7)-(11), we get (n + 1) − ff ′′ − nf ′2= ξ f ′h′ − f ′′h , (1 + ε ) f ′′′ − ε δ ξ (3n 1) f ′′2 f ′′′ + (23) 2

(

)

1  n +1  ′ θ ′′ +  ξ ( f ′φ − θ ′h )  fθ = Pr  2 

(24)

′ (η , ξ ) 1 , θ= f= (η , ξ ) 1= at η 0,

(25)

f ′(η , ξ ) → 0 , θ (η , ξ ) → 0 as η → ∞,

(26)

where ℎ = (𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜉𝜉 ) and 𝜙𝜙 = (𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜉𝜉 ). After introducing these transformations with the successful use of some simplifying mathematical operations, the streamwise dependence related to the natural growth of the boundary can be reduced significantly as in the case of similar boundary layers. The existence of the streamwise dependence is due principally to the particularity of the physical constraints arising from the stretching velocity 𝑢𝑢𝑤𝑤 (𝑥𝑥). As illustrated graphically in Fig. 5, it is found that the left-hand sides of Eqs. (23) and (24) reduces to those shown in Eqs. (15) and (16) when 𝑛𝑛 = 1/3. In this case, the right-hand sides of Eqs. (23) and (24) will go to zero. The local non-similarity method (LNSM) has been used as a powerful procedure to solve accurately the resulting boundary value problem described above by Eqs. (23)-(26). For further information about LNSM, the readers can see Refs. [33,34].

Results and Discussion From the engineering point of view, the important physical quantities of interest are the skin friction and heat transfer rate at the surface. In the dimensionless form, these quantities are reduced in terms of skin friction coefficient 𝑆𝑆𝑆𝑆𝑟𝑟 and Nusselt number 𝑁𝑁𝑁𝑁𝑟𝑟 as follows τ Cf = w2 , (27) ρ uw Nu =

qw x . k (Tw − T∞ )

(28)

In Eqs. (27) and (28), the Powell-Eyring fluid shear stress and heat flux are given by ∂u µ − µ∞ τw = µ∞ + o ∂y λ

 ∂u 1  ∂u 3  − λ   , λ  ∂y 6  ∂y   y =o

qw = − k

∂T ∂y

.

(29)

(30)

y =0

After incorporating Eqs. (29) and (30) into Eqs. (27) and (28) and using the suggested similarity transformations, we obtain

30

Computational Analysis of Heat Transfer in Fluids and Solids II

 ε  Sf r = Re x1/2 Cf = (1 + ε ) f ′′ ( 0 ) 1 − δ f ′′2 ( 0 )  .  3 

(31)

Nur = Re x −1/2 Nu = −θ ′ ( 0 ) .

(32)

This section will discuss the special effects of the involved parameters on the velocity distribution, fluid temperature, skin friction, heat transfer and Nusselt number for a Powell-Eyring fluid moving over a stretching sheet. In the case of similar boundary layer flows, �i. e. , 𝑢𝑢𝑤𝑤 (𝑥𝑥) = 𝐴𝐴𝑥𝑥 1⁄3 � the involved parameters are ε and δ only. As the value of ε approaches to zero, the rheological behavior of the fluid tends to become Newtonian. For δ = 0 , Eq. (15) reduces to the form of the Casson fluid [35]. In the case of non-similar flows (i. e. , 𝑢𝑢𝑤𝑤 (𝑥𝑥) = 𝐴𝐴𝑥𝑥 𝑛𝑛 ), 𝜉𝜉 and 𝑛𝑛 are two additional embedded parameters. Statistically, the skin friction can be correlated as follows Sf r = CD δ + CP ε + C.

(33)

This expression has been evaluated for different values of Pr, 𝑛𝑛, and 𝜉𝜉 by applying the linear regression on the data obtained numerically and listed in Table 1. From this tabular illustration, the values of the correlation coefficients 𝐶𝐶𝐷𝐷 , 𝐶𝐶𝑃𝑃 and 𝐶𝐶 are computed for different values of the involved parameters along with the maximum percentage error. In Fig. 2, the velocity of the fluid is plotted for different values of 𝜀𝜀. It is observed that the velocity of the fluid increases with the increasing values of 𝜀𝜀. As mentioned earlier, the parameter ε increases the pseudoplasticity property of the fluid. Hence, with the increase in the values given to 𝜀𝜀, the fluid becomes more pseudoplastic. As a result of this state, the consistency of the fluid decreases with an increase in the shear rate. So, more the fluid is pseudoplastic, greater the velocity it will possess due to the motion of the surface. Fig.3 shows the temperature profile of fluid for different values of 𝜀𝜀. It is found that the temperature of the fluid decreases with the rise in the values of 𝜀𝜀. This result can be explained by the intensification in the pseudoplasticity behavior of the fluid. Fig. 4 exhibits the impact of the power-law index 𝑛𝑛 on the velocity field. As highlighted in Fig.2, the consistency of the Powell-Eyring fluid decreases with the increase in the values given to the shear rate. Therefore, the fluid velocity and the boundary layer thickness decrease with the increase in the values of n characterizing the velocity of the surface. The rate of change in the velocity profile with respect to 𝜉𝜉 (𝑖𝑖. 𝑒𝑒. , 𝑔𝑔′ (𝜉𝜉, 𝜂𝜂) = 𝜕𝜕𝑓𝑓 ′ ⁄𝜕𝜕𝜉𝜉 ) is displayed graphically in Fig. 5 for different values of 𝑛𝑛. It can be seen from this graphical representation that the contribution of this term is zero in the similar case, which corresponds to 𝑛𝑛 = 1⁄3. Thus, the nonsimilar terms can be logically removed from Eqs. (23) and (24). Further, this contribution is positive for values less than 𝑛𝑛 = 1⁄3 and negative for these greater than 𝑛𝑛 = 1⁄3. The non-similar temperature profile is plotted in Fig.6 for different values of 𝑛𝑛. It is clearly noticed that the fluid temperature decreases due to the increase in the values of n. The effects of the involved parameters on the reduced skin friction coefficient 𝑆𝑆𝑆𝑆𝑟𝑟 is depicted in Fig. 7. It is observed that the engineering quantity 𝑆𝑆𝑆𝑆𝑟𝑟 is an increasing function of 𝛿𝛿 and a decreasing function of both 𝑛𝑛 and 𝜀𝜀. As proved above, the parameter 𝜀𝜀 increases the pseudoplasticity of the fluid. So, more the fluid is pseudoplastic, greater the velocity it will possess due to the motion of the surface. Accordingly, the effects of the viscous friction forces decrease significantly at the surface. Further, as the parameter 𝛿𝛿 increases the viscous effect relaxes for the elastic behavior of the fluid to dominate, which results in a decrease in the velocity with a simultaneous increase in the physical quantity 𝑆𝑆𝑆𝑆𝑟𝑟 due to the movement of the surface. These findings are revealed in a tabular form as shown in Table 1 and formulated mathematically as suggested in Eq. (33). Also, it is found from this tabular illustration that the coefficient 𝐶𝐶𝐷𝐷 of 𝛿𝛿 is positive , whereas the coefficient 𝐶𝐶𝑃𝑃 of 𝜀𝜀 has an opposite sign. Hence, the physical results are in good agreement with the mathematical outputs described by Eq. (33). Also, the change in skin friction 𝑆𝑆𝑆𝑆𝑟𝑟 with respect to 𝜀𝜀 is much smaller as compared to the change with respect to 𝛿𝛿. Besides, the skin friction is perceived as a decreasing function of 𝑛𝑛 since the consistency of Powell-Eyring fluid decreases with an increase in the shear rate. For this reason, the skin friction is observed to be a decreasing function as 𝑛𝑛 increases.

Defect and Diffusion Forum Vol.401

31

Table 1: Estimation of the reduced skin friction coefficient Sf r for different values of the involved parameters. Pr

1/3

𝜉𝜉

0.0

𝐶𝐶𝐷𝐷

0.190

𝐶𝐶𝑃𝑃

-0.012

𝐶𝐶

Max. percentage error

0.7

𝑛𝑛

-0.668

2.5756

1.0

1/3

0.0

0.151

-0.015

-0.655

4.606

2.0

1/3

0.0

0.143

-0.015

-0.652

5.181

0.8

0.1

0.1

0.035

-0.020

-0.518

5.352

0.8

0.1

0.8

0.168

-0.004

-0.534

0.9515

0.8

1.0

0.8

0.167

-0.025

-0.915

5.1454

2.0

0.1

0.8

0.127

-0.009

-0.525

2.9215

Based on our results presented in Table 2 for the dimensionless physical quantity 𝑁𝑁𝑁𝑁𝑟𝑟 , the heat transfer rate can be estimated accurately by utilizing the following correlated expression Nur = C D δ + C P ε + C.

(34)

where 𝐶𝐶𝐷𝐷̅ , 𝐶𝐶𝑃𝑃̅ and 𝐶𝐶̅ are the correlation coefficients for 𝑁𝑁𝑁𝑁𝑟𝑟 .

Table 2: Estimation of the reduced Nusselt number Nur for different values of the involved parameters. Pr

1/3

𝜉𝜉

0.0

𝐶𝐶𝐷𝐷̅

0.026

𝐶𝐶𝑃𝑃̅

0.0

𝐶𝐶̅

Max. percentage error

0.7

𝑛𝑛

0.434

0.699

1.0

1/3

0.0

0.027

-0.002

0.519

0.924

2.0

1/3

0.0

0.000

-0.002

0.771

1.076

0.8

0.1

0.1

0.020

-0.001

0.435

0.348

0.8

1.0

0.1

0.027

0.000

0.538

2.157

0.8

1.0

0.8

0.027

-0.003

0.539

2.279

2.0

0.1

0.1

0.026

-0.004

0.709

0.756

2.0

1.0

0.1

0.036

0.000

0.928

1.937

2.0

0.1

0.8

0.032

-0.001

0.709

0.496

2.0

1.0

0.8

0.035

-0.005

0.929

2.028

The values of the coefficients 𝐶𝐶𝐷𝐷̅ , 𝐶𝐶𝑃𝑃̅ and 𝐶𝐶̅ shown above in Table 2 are estimated for different values of 𝑃𝑃𝑃𝑃, 𝑛𝑛, and 𝜉𝜉 along with their corresponding maximum percentage errors by utilizing the linear regression model on the numerical data obtained for the reduced Nusselt number 𝑁𝑁𝑁𝑁𝑟𝑟 . The impacts of the involved parameters on the reduced Nusselt number 𝑁𝑁𝑁𝑁𝑟𝑟 are portrayed graphically in Figs. 8 and 9, in which the engineering quantity 𝑁𝑁𝑁𝑁𝑟𝑟 is plotted against 𝜀𝜀 for various values of δ, Pr and 𝑛𝑛. From these figures, it is observed that the physical quantity 𝑁𝑁𝑁𝑁𝑟𝑟 is a decreasing function of ε due to the increase in the pseudoplasticity of the fluid. Table 2 also depicts that the rate of change in the quantity 𝑁𝑁𝑁𝑁𝑟𝑟 with respect to 𝜀𝜀 is negative (i. e. , 𝐶𝐶𝑃𝑃̅ < 0). In Fig.8 each encircled triplet of curves is plotted for different values of 𝛿𝛿 keeping Pr fixed. It is seen that the Nusselt number

32

Computational Analysis of Heat Transfer in Fluids and Solids II

𝑁𝑁𝑁𝑁𝑟𝑟 is an increasing function of Pr and 𝛿𝛿. In addition, it is noticed that the temperature variation with respect 𝛿𝛿 at the surface is very small. The non-similar Nusselt number 𝑁𝑁𝑁𝑁𝑟𝑟 is plotted against 𝜀𝜀 as pictured in Fig. 9. It is demonstrated graphically that the quantity 𝑁𝑁𝑁𝑁𝑟𝑟 is an increasing function of 𝑛𝑛. The middle triplet presented in Fig.8 for Pr = 2.0 and the middle triplet plotted in Fig. 9 for n =1/3 are exactly the same, which confirms the reliability of the two numerical methods. Similar behavior of 𝑁𝑁𝑁𝑁𝑟𝑟 towards 𝛿𝛿 can be viewed in Table 2, in which the change in Nusselt number with respect to 𝛿𝛿 is positive (i. e. , 𝐶𝐶𝐷𝐷̅ > 0).

Fig.2: Velocity profiles with ε .

Fig.4: Velocity profiles with n.

Fig. 3: Temperature profiles with ε .

Fig. 5: Velocity gradient profiles with n .

Defect and Diffusion Forum Vol.401

Fig. 6: Temperature profile with n .

33

Fig. 7: Skin friction coefficient with n , δ and ε .

Fig. 8: Nusselt number with δ , Pr, ε , n = 1 3 . Fig. 9: Nusselt number with δ , n , ε , Pr = 2.0 .

Conclusions The similarity and non-similarity of the boundary layer flows have been analyzed more clearly in this paper, in order to estimate the flow and heat transfer characteristics of Powell-Eyring fluids, in the case where these non-Newtonian fluids are moving over a heated power-law stretching sheet. In the first step, a particular form of stretching velocity is considered to formulate the governing equations in a similar form. Later on, the physical problem was discussed non-similarly for a general power-law stretching velocity. In the end, the correlation expressions for the skin friction and Nusselt number are developed mathematically with their corresponding maximum percentage errors from the tabular data outputted numerically with the help of the linear regression model. These analytical expressions are in coherence with the numerical results. As main results, we can state the following points: • The effects of the parameters 𝜀𝜀 and 𝛿𝛿 on the skin friction coefficient are completely different. • The skin friction coefficient increases with the increase in 𝛿𝛿 and decreases with the increase in 𝜀𝜀. • The effect of the independent variable 𝜉𝜉 is very less on the skin friction coefficient. • The skin friction coefficient decreases with the increase in 𝑛𝑛. • The heat transfer rate increases locally with the increase in 𝑛𝑛 and Pr.

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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13]

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M. Jalil, S. Asghar, Flow of power-law fluid over a stretching surface: A Lie group analysis, International Journal of Non-Linear Mechanics, 48 (2013) 65–71. R.E. Powell, H. Eyring, Mechanism for relaxation theory of viscosity, Nature, 154 (1944) 427– 428. T.Y. Na, Boundary layer flow of Reiner-Philippoff fluids, International Journal of Non-Linear Mechanics, 29 (1994) 871–877. Y.D. Wadhwa, Generalized Couette flow of an Ellis fluid, AIChE Journal, 12 (1966) 890–893. A. Acrivos, M.J. Shah, E.E. Petersen, Momentum and heat transfer in laminar boundary-layer flows of non-Newtonian fluids past external surfaces, AIChE Journal, 6 (1960) 312–317. S.Y. Lee, W.F. Ames, Similarity solutions for non-Newtonian fluids, AIChE Journal, 12 (1966) 700–708. A.G. Hansen, T.Y. Na, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids, Journal of Basic Engineering, 90 (1968) 71–74. A. Ahmad, S. Asghar, Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field, Applied Mathematics Letters, 24 (2011) 1905– 1909. N.A. Halim, R.U. Haq, N.F.M. Noor, Active and passive controls of nanoparticles in Maxwell stagnation point flow over a slipped stretched surface, Meccanica, 52 (2017) 1527–1539. W. Ibrahim, O.D. Makinde, Magnetohydrodynamic stagnation point flow of a power-law nanofluid towards a convectively heated stretching sheet with slip, Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 230 (2014) 345–354. G. Singh, O.D. Makinde, Mixed convection slip flow with temperature jump along a moving plate in presence of free stream, Thermal Science, 19 (2015) 119–128. W.A. Khan, J.R. Culham, O.D. Makinde, Combined heat and mass transfer of third-grade nanofluids over a convectively-heated stretching permeable surface, The Canadian Journal of Chemical Engineering, 93 (2015) 1880–1888. O.D. Makinde, M.T. Omojola, B. Mahanthesh, F.I. Alao, K.S. Adegbie, I.L. Animasaun, A. Wakif, R. Sivaraj, M.S. Tshehla, Significance of buoyancy, velocity index and thickness of an upper horizontal surface of a paraboloid of revolution: The case of non-Newtonian Carreau fluid, Defect and Diffusion Forum, 387 (2018) 550–561. O.D. Makinde, N. Sandeep, T.M. Ajayi, I.L. Animasaun, Numerical exploration of heat transfer and Lorentz force effects on the flow of MHD Casson fluid over an upper horizontal surface of a thermally stratified melting surface of a paraboloid of revolution. International Journal of Nonlinear Sciences and Numerical Simulation, 19(2/3) (2018) 93–106. K. Gangadhar, K.R. Venkata, O.D. Makinde, B.R. Kumar, MHD flow of a Carreau fluid past a stretching cylinder with Cattaneo-Christov heat flux using spectral relaxation method, Defect and Diffusion Forum, 387 (2018) 91–105. S.G. Kumar, S.V.K. Varma, P.D. Prasad, C.S.K. Raju, O.D. Makinde, R. Sharma, MHD reacting and radiating 3-D flow of Maxwell fluid past a stretching sheet with heat source/sink and Soret effects in a porous medium, Defect and Diffusion Forum, 387 (2018) 145–156. K. Sreelakshmi, G. Sarojamma, O.D. Makinde, Dual stratification on the Darcy-Forchheimer flow of a Maxwell nanofluid over a stretching surface, Defect and Diffusion Forum, 387 (2018) 207–217. A. Wakif, Z. Boulahia, R. Sehaqui, Numerical study of the onset of convection in a Newtonian nanofluid layer with spatially uniform and non- uniform internal heating, Journal of Nanofluids, 6 (2017) 136–148.

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[19] A. Wakif, Z. Boulahia, R. Sehaqui, Numerical analysis of the onset of longitudinal convective rolls in a porous medium saturated by an electrically conducting nanofluid in the presence of an external magnetic field, Results in Physics, 7 (2017) 2134–2152. [20] A. Wakif, Z. Boulahia, F. Ali, M.R. Eid, R. Sehaqui, Numerical analysis of the unsteady natural convection MHD Couette nanofluid flow in the presence of thermal radiation using single and two-phase nanofluid models for Cu–water nanofluids, International Journal of Applied and Computational Mathematics, 81(4) (2018) 1–27. [21] M.I. Afridi, A. Wakif, M. Qasim, A. Hussanan, Irreversibility analysis of dissipative fluid flow over a curved surface stimulated by variable thermal conductivity and uniform magnetic field: Utilization of generalized differential quadrature method, Entropy, 20 (2018) 1–15. [22] A. Wakif, Z. Boulahia, S.R. Mishra, M.M. Rashidi, R. Sehaqui, Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model, European Physical Journal Plus, 133 (2018) 1–16. [23] S. Shaw, S.S. Sen, M.K. Nayak, O.D. Makinde, Boundary layer non-linear convection flow of Sisko-nanofluid with melting heat transfer over an inclined permeable electromagnetic sheet, Journal of Nanofluids, 8 (5) (2019) 917-928. [24] M.K. Nayak, A.K. Hakeem, O.D. Makinde, Time varying chemically reactive magnetohydrodynamic non-linear Falkner-Skan flow over a permeable stretching/shrinking wedge: Buongiorno model, Journal of Nanofluids, 8 (3) (2019) 467-476. [25] M.K. Nayak, S. Shaw, O.D. Makinde, A.J. Chamkha, Investigation of partial slip and viscous dissipation effects on the radiative tangent hyperbolic nanofluid flow past a vertical permeable Riga plate with internal heating: Bungiorno model. Journal of Nanofluids, 8 (1) (2019) 51-62. [26] A.K. Hakeem, M.K. Nayak, O.D. Makinde, Effect of exponentially variable viscosity and permeability on Blasius flow of Carreau nanofluid over an electromagnetic plate through a porous medium, Journal of Applied and Computational Mechanics, 5 (2) (2019) 390-401. [27] K.V. Prasad, H. Vaidya, O.D. Makinde, B.S. Setty, MHD mixed convective flow of Casson nanofluid over a slender rotating disk with source/sink and partial slip effects, Defect and Diffusion Forum, 392 (2019) 92-122. [28] N. Bessonov, A. Sequeira, S. Simakov, Y. Vassilevskii, V. Volpert, Methods of blood flow modelling, Mathematical Modelling of Natural Phenomena, 11 (2016) 1–25. [29] O.K. Koriko, I.L. Animasaun, M.G. Reddy, N. Sandeep, Scrutinization of thermal stratification, nonlinear thermal radiation and quartic autocatalytic chemical reaction effects on the flow of three-dimensional Eyring-Powell alumina-water nanofluid, Multidiscipline Modeling in Materials and Structures, 14 (2017) 261–283. [30] O.A. Abegunrin, I.L. Animasaun, N. Sandeep, Insight into the boundary layer flow of nonNewtonian Eyring-Powell fluid due to catalytic surface reaction on an upper horizontal surface of a paraboloid of revolution, Alexandria Engineering Journal, 57 (2018) 2051–2060. [31] I.L. Animasaun, B. Mahanthesh, O.K. Koriko, On the motion of non-Newtonian Eyring–Powell fluid conveying tiny gold particles due to generalized surface slip velocity and buoyancy, International Journal of Applied and Computational Mathematics, 137(4) (2018) 1–22. [32] W.L.Barth, Simulation of non-Newtonian fluids on workstation clusters, (Doctoral dissertation). The University of Texas, Austin, 2004. [33] E.M. Sparrow, H.S. Yu, Local non-similarity thermal boundary-layer solutions, Journal of Heat Transfer, 93 (1971) 328–334. [34] M. Massoudi, Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge, International Journal of Non-Linear Mechanics. 36 (2001) 961–976. [35] S. Nadeem, R. Ul Haq, C. Lee, MHD flow of a Casson fluid over an exponentially shrinking sheet, Scientia Iranica, 19 (2012) 1550–1553.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 36-46 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-05-22 Revised: 2019-10-02 Accepted: 2020-04-15 Online: 2020-05-28

Biomechanics of Surface Runoff and Soil Water Percolation James Makol Madut Deng1,a, Oluwole Daniel Makinde2,b* Department of Theoretical and Applied Physics, African University of Science and Technology, Abuja, Nigeria

1

Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

2

[email protected], [email protected]

a

Keywords: Surface runoff, soil water percolation, soil erosion rate, soil heat transfer rate, groundwater aquifer

Abstract: In this study, the complex interaction of surface runoff with the biomechanics of soil water transport and heat transfer rate is theoretically investigated using mathematical model that rely on the two phase flows of an incompressible Newtonian fluid (stormwater) within the soil (porous medium) and on the soil surface (runoff). The flow and heat transfer characteristics within the soil are determined numerically based on Darcy-Brinkman-Forchheime rmodel for porous medium coupled with appropriate energy equation while analytical approach is employed to tackle the model for interacting surface runoff stormwater. The effects of various embedded biophysical parameters on the temperature distribution and water transport in soils and across the surface runoff together withsoil-runoff interface skin friction and Nusselt number are display graphically and discussed quantitatively. It is found that an increase in surface runoff over tightly packed soil lessens stormwater percolation rate but enhances both soil erosion and heat transfer rate. Introduction Surface runoff and soil water percolation are closely associated with rainfall and melting of snow, or glaciers. Soil inability to absorb excess stormwater and meltwater due to heavy rainfall, high melt rate of snow and glacier, soil saturation, impervious resulting from surface sealing or pavement, etc., do lead to surface runoff [1].Surface runoff is the major cause of soil erosion and surface water pollution. In urban areas, runoff is the main cause of flooding which may damage properties and infrastructures including loss of life [2]. In order to alleviatethe unpleasant effects of surface runoff,several proactive measures are needed to boost soil absorption of stormwater and meltwater. These measures may include minimizing impervious surfaces in urban areas, adopting soil erosion and flood control programs, etc. Moreover, percolation describes the downward flow rate of the stormwater or meltwater within the soil [3]. Water percolation in the soil contributes to the formation of groundwater aquifers which serves as a freshwater storage that can be utilized during droughts when surface water supplies are reduced.Generally, soil is regarded as a porous media;the soil loose sediments like sand and gravel are porous and permeable. It can hold water and allows water to flow through [4]. While the amount of porosity in a soil depends onits mineral content and structure, the rate of water percolation depends on soil permeability (i.e. the size of the soil porespaces and how the pores are connected). For instance, sandy soils have large well connected pores and higher permeability than the clay soils [5]. The use of mathematical models to tackle the menaceof surface runoff and enhance the soil water percolation for the formation of groundwater aquifers has attracted the attention of several scientists and researchers [6-11]. For soil with high permeability like sandy soil, the relationship between the flow rates and the pressure gradient would be practically linear based on the Brinkman form of Darcy law, while this relationship may be nonlinear for soil with low permeability lay clay soil (Darcy–Forchheimer law) [12-14]. Bristow and Horton [15] theoretically investigated the influence of surface mulch soil water flow and heat transfer. The effects of temperature gradient on the soil water flow were studied by Gurr et al. [16]. Numerical results on soil water flow and heat transfer rate together with soil-atmosphere interaction was reported by Fetzer et al. [17]. In all the

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above studies, it is observed that mathematical model ofsoil-runoff interface at the continuum scale where water and energy fluxes are highly dynamic are often magnified. This may lead to inaccuracy in the result obtained. In this present study, the biomechanics of surface runoff and its interaction with soil water percolation is numerically examined. The Darcy-Brinkman-Forchheimer nonlinear model for porous medium coupled with appropriate energy equation is employed in order to analysis the soil percolation rate while the model representing interacting surface runoff is based on modified Blasius flow with heat transfer characteristics. The groundwater aquifer servers as the lower boundary of the porous medium domain while the soil surface represents the upper boundary to the porous medium domain and is dramatically influenced by changes in velocity and temperature gradients of both runoff and stormwater percolation. In the following section, the model and its mathematical equations are obtained, analysed and solved. Pertinent results are graphically presented and discussed. The paperprovides a mathematical treatment of surface runoff menace and veritable platform to understand the complex interaction between the surface runoff and the soil water percolation rate. Model Problem The soil is regarded as a permeable porous media due to its composition of compactly packed different particle sizes that form pore spaces which permit both infiltration and percolation of water to take place. It is bounded below by the groundwater accumulation or bedrock with temperature T0 and above by the presence of runoff water at varying temperature T2 and velocity u2 caused by excessive rainfall that exceed permeability, soil saturation, snowmelt or other sources. The soil temperature and water velocity due to percolation within the soil is given by T1 and u1 respectively as shown in figures (1a,b). It is assumed that the runoff water is incompressible with constant properties.

Fig. 1a: Problem geometry

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Fig. 1b: Schematic diagram of the problem Under these assumptions, the model momentum and energy equations for both the soil region-I and the runoff water region-II can be written as: Region I – Soil (Water Percolation) du1 cu12 1 ∂P µ1 d 2 u1 µ1u1 −V =− + − − , ρ1 ∂x ρ1 dy 2 ρ1 K ρ1 K dy

dT1 k1 d 2T1 µ −V = + 1 2 dy ρ1c p dy ρ1c p

(1)

2

 du1  cu13 µ1u12   + , + ρ1c p K ρ1c p K  dy 

(2)

Region II –Runoff Water d 2u2 du µ2 + ρ 2V 2 = 0, 2 dy dy

(3) 2

 du  d 2T2 dT2   = 0, (4) k2 + c V + ρ µ pf 2 2 dy dy 2  dy  The appropriate boundary conditions at therunoff water free surface as well as the interface between the soil and the runoff water including the underground soil bound are given as, du1 (a) = µ 2 dy dT T1 (0) = T0 , T1 (a ) = T2 (a ), k1 1 (a ) = k 2 dy u1 (0) = 0, u1 (a ) = u 2 (a ), µ1

du 2 (a ), u 2 → U ∞ , dy dT2 (a ), T2 (∞) → T∞ , dy

(5)

where u1 is the soilwater percolation velocity (m/s), u2 is the runoff water velocity (m/s), T1 is the soil temperature (Kelvin), T2 is the runoff water temperature(Kelvin), T0 is the groundwater or bedrock temperature (Kelvin), T∞ is the runoff water free stream temperature(Kelvin), a is the soil depth (m), k1 is the saturated soil thermal conductivity (W/mKelvin), k2 is the runoff water thermal conductivity (W/mKelvin), K is the soil permeability rate (m2), V is the soil water suction velocity(m/s), µ1 is the soil water dynamic viscosity (kg/ms), µ2 is the runoff water dynamic viscosity(kg/m.s), ρ1 is the soil water density(kg/m3), ρ2 is the runoff water density(kg/m3), cpf is the runoff water specific heat capacity(J/kgKelvin) and cp is the soil specific heat capacity(J/kgKelvin). In order to render the model equations dimensionless, the following variables and parameters are introduce into equations (1)-(5);

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θ1 =

39

T1 − T0 T − T0 u u U y x cVa 2 ,θ 2 = 2 , W1 = 1 , W2 = 2 ,η = , X = , S = ,L = ∞ , T∞ − T0 T∞ − T0 V V a a V µ1 K

µ1 c p µ 2 c pf k µ µ aP K Va , Da = 2 , Re = ,υ1 = 1 ,υ 2 = 2 , Pr1 = , Pr2 = ,m = 1 , k1 k2 k2 µ1V υ1 ρ1 ρ2 a c pf µ υ V2 dP γ = 1 , Ec = ,δ = ,A= − ,n = 1 , c pf (T∞ − T0 ) cp dX µ2 υ2 and we obtain P=

(6)

Region I –Soil (Water Percolation) d 2W1 dW1 W1 + Re − − SW12 + A = 0, 2 dη Da dη

(7)

  dW  2 W 2 dθ 1 d 2θ 1 3 1 1   Re Pr Pr + + − = + SW Ec δ  1 , 1  1  Da dη dη 2   dη 

(8)

Region II –Runoff Water d 2W2 dW2 + n Re = 0, 2 dη dη

(9) 2

d 2θ 2 dθ  dW2   = 0 , + n Pr2 Re 2 + Pr2 Ec 2 dη dη  dη  with the appropriate boundary conditions in dimensionless form as dW1 dW2 (1), W2 → L, (1) = W1 (0) = 0, W1 (1) = W2 (1), γ dη dη dθ dθ θ 1 (0) = 0, θ 1 (1) = θ 2 (1), m 1 (1) = 2 (1), θ 2 (∞) → 1, dη dη

(10)

(11)

where Re is the soil water suction Reynolds number, Da is the soil Darcy number, S is the Forchheimer parameter (i.e. soil nonlinear permeability parameter), Ec2 is the runoff water Eckert number,δ is the surface runoff-soil water specific heat capacity ratio, Ec is the runoff and soil water Eckert number, Pr1 is the soil water Prandtl number, Pr2 (≈6.2) is the runoff Prandtl number, L is the free stream velocity parameter, m is soil water-runoff thermal conductivity ratio, γ is soil waterrunoff dynamic viscosity ratio and n is the soil water-runoff kinematic viscosity ratio. It is important to note that Eckert number Ec represents the effects of internal heat generation due to soil water and surface runoff energy dissipation. The Darcy number Da and Forchhiemer number S show the both linear and nonlinear soil permeability rate due to its composition compactly packed particles which form a porous matrix that permit water percolation. Other parameters of interest are the skin friction coefficients (Cf) that is the leading cause of surface runoff soil erosion and the Nusselt number (Nu) at the soil interface with the runoff water, are given as: a 2τ w dW1 aq w dθ = =− 1 , , Nu = Cf = (12) 2 dη η =1 k1 (T0 − T∞ ) dη η =1 ρ1V where the shear stress τw and the heat flux qw at the soil surface are given as

τ w = µ1

du1 dy

, q w = − k1 y =a

dT1 dy

. y =a

(13)

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Computational Analysis of Heat Transfer in Fluids and Solids II

Solution Procedure The dimensionless model equations (7)-(10) together with its boundary conditions (11)form a three points boundary value problem that seems intractable numerically, however, we reduce the problem to atwo points boundary value problem by analytically determined the solutions for region-II (Surface runoff water) that satisfy the free stream conditions. The velocity and temperature profiles of the surface runoff water are given as:

W2 (η ) = L + A1e − n Reη , θ 2 (η ) = 1 +

Ec Pr2 A12 − 2 n Reη e , 2(Pr2 − 2) )

(14)

where A1 is to be numerical determined base on the prescribed interface conditions between the soil surface and the runoff water in equation (11). The two point’s boundary value problem of region-I (soil water percolation) is numerically tackled using Runge-Kutta-Fehlberg integration scheme coupled with shooting method. From the numerical solution for velocity and temperature profiles, we compute the values for the skin friction (Cf) (soil erosion factor) and the Nusselt number (Nu) as given by equations (12). Results and Discussion For numerical results, the following parameter values A=1, Da=0..0.3, S=10..700, Ec=0..5, Re=0..1, L=0.5..1, Pr1=20, Pr2 = 6.2, δ=0.5, m = 2, n =5, γ = 10 are utilised. Graphically results depicting the velocity and temperature profiles of surface runoff and soil water percolation as well as soil-runoff interface skin friction and Nusselt number are displayed. Surface Runoff and Soil Percolation Velocity Profiles Figures 2-5 illustrate the impact of embedded biophysical parameters on stormwater velocity profiles both within the soil in the region 0 ≤η≤ 1 and the runoff in the region η> 1. The das lines correspond to water percolation rate within the soil while the thick lines indicate surface runoff velocity. It is interesting to note an escalation in the velocities of both soil water percolation and surface runoff with increasing values of soil suction parameter Re and runoff rate parameter L as shown in figures 2 and 3. This may be due to loose texture of soil composition which enhanced stormwater infiltration and percolation into the soil with increasing rate of surface runoff, leading to an increase in groundwater accumulation. Figure 4 show that an increase in soil permeability boosts the stormwater percolation. As Darcy number Da increases, the percolation rates and the soil pressure gradient would be practically linear, consequently, the stormwater infiltration and groundwater accumulation amplify. This scenario is evidence in runoff over a sandy or loosely packed soil composition. Moreover, as Forchheimer parameter S rises, the soil permeability diminishes, leading to a tightly packed soil composition like clay soil. Hence, the percolation rates and the soil pressure gradient becomes nonlinear, consequently, the stormwater percolation into the soil drastically reduced as shown in figure 5.

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Fig.2: Velocity profiles with increasing Re. Fig.3: Velocity profiles with increasing Re.

Fig.4: Velocity profiles with increasing Da. Fig.5: Velocity profiles with increasing S. Surface Runoff and Soil Percolation Temperature Profiles Figures 6-10 show the effects of various biophysical parameters on the stormwater temperature profiles both within the soil in the region 0 ≤η≤1 and the runoff in the region η>1. The das lines correspond to the temperature distribution within the soil water while the thick lines depict the temperature distribution within the surface runoff. Figure 6 revealed that a boost in suction Reynolds number causes the soil temperature to rise while the stormwater runoff temperature decreases. This may be due to exchange of energy between the surface runoff and the soil coupled with increasing rate of percolation. Meanwhile, an increase in the runoff rate parameter L and the Eckert number Ec enhance the soil temperature as shown in figures 7 and 8. These parameters enhance the percolation rate which invariably lead to an elevation in the rate of internal heat generation within the soil. In figure 9 we observed that an increase in soil permeability (with Da increases) decreases the soil temperature. This can be attributed to loose texture of soil composition which enhance heat loss. The trend is opposite with a rise in Forchheimer parameter S due to a

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decrease in the soil permeability as shown in figure 10, the soil composition is tightly packed like clay soil, consequently, the heat loss diminishes and soil temperature augments.

Fig.6: Temperature profiles with increasing Re. Fig.7: Temperature profiles with increasing L.

Fig.8: Temperature profiles with increasing Ec. Fig.9: Temperature profiles with increasing Da.

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Fig.10: Temperature profiles with increasing S. Soil Surface Erosion (Skin Friction) and Heat Transfer Rate (Nusselt Number) Figures 11-14 demonstrate the effects of various biophysical parameters on the coefficient of skin friction which invariably lead to soil erosion and the heat transfer rate at the soil surface (η=1) due to interaction between the runoff and the soil water percolation. Interestingly, the soil surface erosion caused by skin friction escalates with increasing values of suction Reynolds number Re, Forchheimer parameter S and runoff rate parameter L as shown in figures 11 and 12. As these parameters increase, stormwater runoff velocity gradient at the soil surface amplifies and soil permeability diminishes, consequently, the rate of erosion of top soil surface escalates. This extreme scenario may damage tightly packed soil surface and cause surface water pollution. Moreover, a rise in soil permeability with increasing Darcy number Da, causes the skin friction to fall and lessens the effect soil erosion. This may be attributed to an increase in the soil permeability which enhance stormwater percolation and diminish the runoff velocity gradient at the soil surface.The effects of various parameters on heat transfer rate at the soil surface are depicted in figures 13 and 14. The Nusselt number is rises with an amplification in the parameter values of Re, L, Ec and S. As these parameters increase, the soil becomes tightly packed and the surface runoff rate rises, leading to an escalation in the temperature gradient at the soil surface and heat flux. The trend is opposite when the soil is loosely packed with high permeability (Da increasing), consequently, the Nusselt number deceases.

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Fig.11: Skin friction with increasing Re, L, Da. Fig.12: Skin friction with increasing S.

Fig.13: Nusselt number with increasing Re, L, Da. Fig.14: Nusselt number with increasing Ec, S. Conclusions The complex interaction between the surface runoff and the soil water percolation is theoretically investigated using two phase flow model that relies on the continuum mechanics principle of conservation laws of mass, momentum and energy. The model boundary value problem is numerically tackled. The impact of various embedded biophysical parameters on the velocity and temperature distribution of stormwater transport within the soil and across the surface runoff together with soil-runoff interface skin friction and Nusselt number are determined. Our results can be summarised as follows: • Increase in Re, L, and Da boost the soil water percolation rate coupled enhanced rate of runoff while increase in S lessens it. • Increase in Ec, S, L and Re boost the soil temperature while increase in Da lessens it.

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Increase in Re, L and S boost skin friction and promote soil surface erosion while increase in Da lessens it. Increase in Re, L, Ec and S enhances Nusselt number while increase in Da lessens it.

Finally, the biophysical conditions of the soil and its usefulness for agricultural and industrial purpose may be determined by its interaction with surface runoff. Adequate knowledge of complex interaction of surface runoff and stormwater percolation are essential in order to thwart flooding, soil erosion, surface water pollution and augment groundwater accumulation. Our results will no doubt be of agricultural and environmental interest. References [1]

J.F. Zuzel, R.R. Allmaras, R. Greenwalt, Runoff and soil erosion on frozen soils in Northeastern Oregon. J. Soil & Water Conserv. 37, 351-354, 1982.

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F. M. dos Santos, J. A. de Lollo, F. F. Mauad, Estimating the surface runoff from natural environment data, Management of Environmental Quality: An International Journal, 28(4), 515-531,2017.

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M. Danáčová, P. Valent, R. Výleta, Evaluation of surface runoff generation processes using a rainfall simulator: A small scale laboratory experiment. IOP Conf. Series: Earth and Environmental Science 95,022016(pp1-8), 2017.

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D.D. Davis, R. Horton, J. L. Heitman, T. S. Ren, An experimental study of coupled heat and water transfer in wettable and artificially hydrophobized soils, Soil Sci. Soc. Am. J., 78(1), 125–132, 2014.

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S.V. Nerpin, A.F. Chudnovskiy, Soil physics. Science, Moscow, 1967.

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L. Rundora, O. D. Makinde, Effects of Navier slip on unsteady flow of a reactive variable viscosity non-Newtonian fluid through a porous saturated medium with asymmetric convective boundary conditions. Journal of Hydrodynamics,Ser. B, 27(6), 934-944, 2015.

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[10] L. Rundora, O.D. Makinde, Effects of suction/injection on unsteady reactive variable viscosity non-Newtonian fluid flow in a channel filled with porous medium and convective boundary conditions. Journal of Petroleum Science and Engineering, 108, 328-335, 2013. [11] O.D. Makinde, T. Chinyoka, L. Rundora, Unsteady flow of a reactive variable viscosity nonNewtonian fluid through a porous saturated medium with asymmetric convective boundary conditions. Computers and Mathematics with Applications, 62, 3343–3352, 2011. [12] S. Whitaker, Flow in porous media I: A theoretical derivation of Darcy's law, Transport in Porous Media. 1: 3–25, 1986. [13] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles, Applied Scientific Research,1, 27–34, 1949. [14] D.A. Nield, A. Bejan, Convection in porous media, Springer-Verlag, New-York 1992.

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[15] K.L. Bristow, R. Horton, Modeling the impact of partial surface mulch on soil heat and water flow, Theor. Appl. Climatol.,54(1/2), 85–98,1996. [16] C.G. Gurr, T.J. Marshall, J.T. Hutton, Movement of water in soil due to a temperature gradient, Soil Sci., 74, 335–345, 1952. [17] T. Fetzer, J. Vanderborght, K. Mosthaf, K. M. Smits, R. Helmig, Heat and water transport in soils and across the soil-atmosphere interface: 2. Numerical analysis, Water Resour. Res., 53, 1080–1100, 2017.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 47-62 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-06-13 Revised: 2019-10-21 Accepted: 2019-10-21 Online: 2020-05-28

Finite Element Numerical Investigation into unsteady MHD Radiating and Reacting Mixed Convection Past an Impulsively Started Oscillating Plate B. Prabhakar Reddy1,a*, P.M. Matao1,b, J.M. Sunzu1,c Department of Mathematics, CNMS, The University of Dodoma P. Box No. 338, Dodoma, Tanzania

1

[email protected], [email protected], c [email protected],

a

Keywords: MHD, radiation parameter, chemical reaction parameter, magnetic parameter, Hall current

Abstract: In this article, numerical investigation is carried out for the unsteady MHD mixed convection flow of radiating and chemically reacting fluid past an impulsively started oscillating vertical plate with variable temperature and constant mass diffusion. The transport model employed includes the Hall current. A uniform magnetic field is applied transversely to the direction of the fluid flow. The flow consideration is subject to small magnetic Reynolds number. The Rosseland approximation is used to describe the radiation heat flux in the energy equation. The dimensionless governing system of partial differential equations of the flow has been solved numerically by employing the FEM. The influence of pertinent parameters on primary velocity, secondary velocity, temperature and concentration are presented graphically whereas primary skin friction, secondary skin friction, Nusselt number and Sherwood number are presented in tabular form. The findings of the present study are in good agreement with the earlier reported studies. Introduction The study of MHD with heat and mass diffusion in the presence of radiation has attracted the attention of a large number of researchers due to its diverse applications. Many processes in new engineering occur at high temperature and knowledge of radiation heat transfer becomes necessary for the design of the pertinent equipment. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites, and space vehicles, etc are examples of such engineering areas. The radiation effects on MHD flow under the different conditions were investigated [1-10]. Nandkeolyar et. al [11] studied unsteady hydro-magnetic natural convection flow of a dusty fluid past an impulsively moving vertical plate with ramped temperature in the presence of thermal radiation. Magneto-convective boundary layer slip flow of nanofluid past a convectively heated vertical plate was reported by Das et. al [12]. Recently, Muhammad and Makinde [13] presented thermo-dynamic analysis of unsteady MHD mixed convection with slip and thermal radiation over a permeable surface. In all the above cited studies, the effects of chemical reaction on the flow are not taken into account. Investigation of MHD flow with chemical reaction is of much significance due to its wide applications in many branches of science and engineering. In many chemical engineering processes, chemical reaction takes place between a foreign mass and the working fluid in which the plate is moving. These processes take place in numerous industrial applications such as manufacturing of ceramics or glassware, polymer production and food processing. In nature, the presence of pure air or water is not possible. Some foreign mass may be present either naturally or mixed with the air or water. The presence of a foreign mass in air or water causes some kind of chemical reaction. The effects of chemical reaction on the flow with different situation were investigated [14-16]. Makinde [17] studied chemically reacting hydro-magnetic unsteady flow of radiating fluid past a vertical plate with constant heat flux. Recently, Jayakar et al [18] presented thermo-diffusion effects on MHD chemically reacting fluid flow past an inclined porous plate in a slip flow regime.

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Computational Analysis of Heat Transfer in Fluids and Solids II

However, in all these investigations, the effects of Hall current are not taken into account. The effect of Hall current cannot be completely ignored if the strength of the magnetic field is strong and number of density electrons is small as it is responsible for the charge of the flow pattern of an ionized gas. Hall current results in a development of an additional potential difference between opposite surfaces of a conductor for which a current is induced perpendicular to both the electric and magnetic field. This current is known as Hall current. It plays an important role in determining features of the fluid flow problems because it induces secondary flow in the flow field. Several authors presented the applications of MHD with Hall current in different fields [19-21]. The effects of Hall current on MHD flow under the different conditions were investigated [22-25]. Recently, Reddy [26] investigated the effect of Hall current on MHD transient flow past an impulsively started infinite horizontal porous plate in a rotating system. The objective of the present work is to study the unsteady MHD mixed convection flow past an impulsively started oscillating vertical plate with radiation, chemical reaction and Hall current. The FEM has been adopted to solve the governing equations of the flow. The effects of embedded parameters on primary velocity, secondary velocity, temperature, concentration, primary skin friction, secondary skin friction, Nusselt number and Sherwood number are presented through the graphs and tables and then discussed. According to the best of authors’ knowledge this work not yet received the attention of researchers though it has significant applications in the fields of science and engineering. Model Problem Consider the unsteady MHD flow of viscous incompressible electrically conducting, radiating and reacting fluid past an impulsively started oscillating infinite vertical plate. A uniform magnetic →

field B of strength B0 is applied in the direction perpendicular to the fluid flow. In the Cartesian coordinate system, the x '− axis is taken along the plate in the vertically upward direction, the y '− axis perpendicular to the direction of the plate and the z '− axis is normal to the x ' y '− plane. The physical model of the problem is shown in Fig.1. Initially, at time t ' ≤ 0, the temperature of the fluid and the plate is T∞' and the concentration of the fluid is C∞' . Subsequently, at time t ' > 0, the plate starts oscillating in its own plane with frequency ω ', the temperature of the plate and the concentration of the fluid, respectively are raised to Tw' and Cw' . It is assumed that the radiation heat flux in the x '− direction is negligible as compared to that in y '− direction. As the plate is of infinite extent and electrically non-conducting, all the physical quantities, except the pressure, are functions of y ' and t '.

Fig. 1: Physical model and coordinate system.

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49

The generalized Ohm's law on taking Hall current into account Cowling [28] is given by → ωτ → → → → → J + e e  J × B=  σ  B + q× B  , B0    

(1)

→ → → →

where q , B, E , J , σ , ωe and τ e are respectively, velocity vector, magnetic field vector, electric field vector, current density vector, electric conductively, cyclotron frequency and electron collision →

time. The equation of continuity ∇. q = 0 gives v ' = 0 everywhere in the flow since there is no →

variation of the flow in y '− direction, where q = ( u ', v ', w ') and u ', v ', w ' are respectively, velocity components along the coordinate axes. The magnetic Reynolds number is so small that the induced →

magnetic field produced by the fluid motion is neglected. The solenoid relation ∇. B = 0 for the →

magnetic field B = ( Bx ' , By ' , Bz ' ) gives By ' = constant say B0 . i.e., B = (0, B0 , 0) everywhere in the →

→

→

flow. The conservation of electric current ∇. J = 0 yields j y ' = constant, where J = ( jx ' , j y ' , jz ' ) . This constant is zero since j y ' = 0 at the plate which is electrically non- conducting. Hence, j y ' = 0 everywhere in the flow. In view of the above assumption, Eq. (1) yields (2) jx ' − mj y ' = σ ( Ex ' − w ' B0 ) , jz ' + mjx ' = σ ( Ez ' + u ' B0 ) ,

(3)

where m ( = ωeτ e ) is the Hall parameter which represents the ratio of electron-cyclotron frequency and the electron-atom collision frequency. Since the induced magnetic field is neglected, →

→ ∂H ∂Ex ' ∂Ez ' Maxwell equation ∇ × E = − becomes ∇ × E = 0 which gives = 0 and = 0. This ∂t ∂y ' ∂y ' implies that Ex ' = constant and Ez ' = constant everywhere in the flow and choose this constants

→

E= 0, E= 0. Solving for jx ' and jz ' from (2) and (3), on using E= equals to zero, i.e., E= x' z' x' z' σ B0 = jx ' (4) ( mu '− w ') 1 + m2 σ B0 = jz ' (5) ( mw '+ u ') 1 + m2 Taking into consideration the assumptions made above, under the Boussinesq’s approximation, and using (4) and (5), the basic governing equations of the flow are derived as:

σ B02 ∂u ' ∂ 2u ' = v 2− ∂t ' ∂y ' ρ 1 + m 2

(

)

σ B02 ∂w ' ∂2w ' v = + ∂t ' ∂y '2 ρ 1 + m 2

(

∂T ' ∂t '

ρ C= k p

( u '+ mw ') + g β (T '− T∞' ) + g β * ( C '− C∞' )

(6)

( mu '− w ')

(7)

)

∂ 2T ' ∂qr − ∂y '2 ∂y '

∂C ' ∂ 2C ' = D − γ ' C '− C∞' 2 ∂t ' ∂y '

(

(8)

)

The initial and boundary conditions for the problem are:

(9)

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Computational Analysis of Heat Transfer in Fluids and Solids II

t ' ≤ 0; u ' = 0, w ' = 0, T ' = T∞ ' , C ' = C∞ '

(

for all y ' ≥ 0

)

t ' > 0; u ' = u0 cos ω ' t ', w ' = 0, T ' =T∞' + Tw' − T∞' , C ' = Cw' u ' → 0, w ' → 0, T ' → T∞ ' , C ' → C∞ '

at y ' = 0

as y ' → ∞

(10)

where u ', w ', g , β , β ∗ , t ', C ', C∞' , Cw' , D, T ', T∞' , Tw' , k ,υ , ρ , qr and C p are respectively, velocity of the fluid in x '− direction, velocity of the fluid in z '− direction, acceleration due to gravity, volumetric coefficient of thermal expansion, volumetric coefficient of concentration expansion, time, species concentration in the fluid, concentration in the fluid far away from the plate, species concentration at the plate, mass diffusion, temperature of the fluid, temperature of the fluid near the plate, temperature at the plate, thermal conductivity, kinematic viscosity, fluid density, radiation heat flux and specific heat at constant pressure. The radiation heat flux under the Rosseland approximation [17, 27] expressed by qr = −

4σ ∗ ∂T '4 3k ∗ ∂y '

(11)

where σ ∗ is the Stefan-Boltzmann constant and k ∗ is the mean absorption coefficient. It is assumed that temperature difference within the flow are sufficiently small, then Eq. (11) can be linearized by expanding T ′4 into the Taylor series about T∞′ which, after neglecting higher-order terms, takes the form:

(12)

T ′4 ≅ 4T∞′3T ′ − 3T∞′4

In view of Eqs. (11) and (12), Eq. (8) reduces to

 16σ ∗T '3∞ ∂T ' ρ C p = k 1 + ∂t ′ 3kk ∗ 

 ∂ 2T '  2  ∂y '

(13)

Let us introduce the following non-dimensional parameters and quantities:

u =

µC p y ' u0 t ' u0 2 σ B0 2 v u' w' v ω 'v γ 'v ,y ,t ,w = ,ω , , , ,γ , S P M = = = = = = = c r 2 2 u0 v v u0 u0 D k u02 ρ u0

16σ ∗T∞'3 ,θ R = = 3kk ∗

−T ) −C ) (T= (C = ,φ ,G (T − T ) ( C − C ) '

' ∞

'

' ∞

' w

' ∞

'

' ∞

r

(

)

(

g βν Tw' − T∞' g βν C ' − C∞' , Gm = u0 3 u0 3

)

(14)

Using Eq. (14), into Eqs. (6), (7), (9) and (13), the following dimensionless governing equations of the flow are obtained:

∂u ∂ 2u M = 2− ∂t ∂y 1 + m2

(

)

M ∂w ∂ 2 w = + 2 ∂t ∂y 1 + m2

(

( u + mw ) + Grθ + Gmφ

)

( mu − w )

(15)

(16)

∂θ (1 + R ) ∂ 2θ = ∂t Pr ∂y 2

(17)

∂φ 1 ∂ 2φ = − γφ ∂t Sc ∂y 2

(18)

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51

where u , w, θ , φ , Gr , Gm , M , m, Pr , R, Sc and γ are respectively, dimensionless primary fluid velocity, dimensionless secondary fluid velocity, dimensionless fluid temperature, dimensionless fluid concentration, thermal Grashof number, mass Grashof number, magnetic field parameter, Hall parameter, Prandtl number, radiation parameter, Schmidt number and chemical reaction parameter. The initial and boundary conditions (10), in non-dimensional form become: for all y ≥ 0 , t ≤ 0; u = 0, w = 0, θ = 0, C = 0 t > 0; u = cos ωt , w = 0, θ = t , C = 1 u → 0,

w → 0, θ → 0, C → 0

at y = 0 ,

as y → ∞ ,

(19)

The dimensionless primary and secondary skin frictions are given by ∂w ∂u and τ z = , τx = ∂y y =0 ∂y y =0 The dimensionless Nusselt and Sherwood numbers are given by ∂φ ∂θ and Sh = . Nu = ∂y y =0 ∂y y =0 Numerical Procedure The dimensionless governing system of partial differential equations (15) − (18), subject to initial and boundary conditions (19) have been solved numerically for various values of parameters involved in the problem by using the FEM. The FEM is extremely efficient technique to solve ordinary and partial differential equations which occurs in wide range of engineering problems such as fluid mechanics, heat transfer, electrical systems, solid mechanics and chemical engineering, etc. Detailed discussion of the method is given in Zienkiewiez [28], Reddy [29] and Bathe [30]. Variational Formulation The variational form associated with equations (15) − (18), over a typical two nodded linear

element ( ye , ye +1 ) is given by ye+1

∫

ye

ye+1

∫

ye ye+1

∫

ye ye+1

∫

ye

 ∂u   ∂ 2u  M w1   −  2  +  ∂t   ∂y  1 + m 2

(

)

 ∂w   ∂ 2 w  M w2  − 2 +  ∂t   ∂y  1 + m 2

(

 ∂θ w3   ∂t

  u mw G G θ φ 0, + − + ( ) ( r m ) dy = 

(20)



)

0, ( mu − w )dy = 

2  (1 + R )  ∂ θ   − 0   dy =  Pr  ∂y 2    ,

 ∂φ  1  ∂ 2φ   0 w4   −  2  + γφ dy = t S y ∂ ∂     c   ,

(21)

(22)

(23)

where w1 , w2 , w3 and w4 are arbitrary test functions and may be viewed as the variations in u , w, θ and φ , respectively. After reducing the order of integration and non-linearity, the following equations are obtained:

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Computational Analysis of Heat Transfer in Fluids and Solids II

ye+1  ∂w  ∂u     ∂u   M  ∂u  1 0 (24) w1 ( u + mw ) − w1 ( Grθ + Gmφ ) dy −  w1    =  + w1   + ∫  ∂y  ∂y  ∂t  1 + m 2 ∂y   y     ye     e y e+1 ye+1     ∂w    ∂w2  ∂w  M  ∂w    0, w2 ( mu − w ) dy −  w2  (25)  + w2   = + ∫  ∂y  ∂y  ∂t  1 + m 2 ∂y   y     ye     e ye+1

ye+1

∫

ye

ye+1

∫

ye

 (1 + R )  ∂w3   ∂θ     Pr  ∂y   ∂y

(

)

(

)

  ∂θ  + w3   ∂t 

(1 + R )  w  ∂θ   3  dy − Pr   ∂y 

ye+1

 0,  =   ye

(26)

ye+1

 1  ∂w4   ∂φ   1   ∂φ    ∂φ  0.      + w4   − ( γφ ) w4 dy −  w4    = Sc   ∂y   y  ∂t   Sc  ∂y   ∂y   e

(27)

Finite Element Formulation The finite element model may be obtained from Eqs. (24) − (27) by substituting appropriate finite element approximations of the form: 2

u = ∑ u ejψ ej , j =1

2

2

j =1

j =1

w = ∑ wejψ ej , θ = ∑ θ ejψ ej ,

2

φ = ∑ φ ejψ ej ,

(28)

j =1

w= w= w= ψ ej (= j 1, 2 ) where u ej , wej , θ ej and φ ej are, respectively, primary fluid with w= 1 2 3 4 velocity, secondary fluid velocity, fluid temperature and fluid concentration at j th node of the eth

element ( ye , ye +1 ) and ψ j ' s are the shape functions for a typical element ( ye , ye +1 ) and are taken as:

ψ 1e =

ye +1 − y y − ye and ψ 2e = , ye +1 − ye ye +1 − ye

( ye ≤ y ≤ ye+1 ) .

(29)

For the computational purpose the boundary condition y → ∞ is fixed at ymax = 5, where ymax represents infinity, i.e., external to the momentum, energy and concentration boundary layers. The whole domain is divided into a set of 100 line elements of equal width 0.05, each element being two nodded. Using Eqs. (28) and (29) into Eqs. (24) to (27), after assembly of all the element equations by inter-element connectivity conditions, we obtain a matrix of order 404 × 404. These obtained system of equations are non-linear therefore an iterative scheme has been used to solve this system of equations. After imposing the boundary conditions (19) only a system of 396 equations remains and these equations has been solved by using Gauss elimination method by maintaining the desired accuracy 0.0005. Results Validation To validate the present results, a comparison has been made by considering primary and secondary skin frictions τ x and τ z in the absence of radiation and chemical reaction with the results of Rajput and Kanaujia [25] obtained by analytical solution (Laplace transform technique) shown in table 1 and 2. An excellent agreement is noticed in this comparison which validates the present numerical scheme. This justifies the correctness of the results presented in the manuscript.

Defect and Diffusion Forum Vol.401

53

Table 1: Comparison of primary skin friction between the present results and results by Rajput and Kanaujia [25] in the absence of radiation and chemical reaction, i.e., R = 0 and γ = 0. Present Results by results Rajput and Gm Pr Sc Gr m ωt t M by FEM Kanaujia [25]

10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

10.0 10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0

0.71 0.71 0.71 7.00 0.71 0.71 0.71 0.71 0.71

2.01 2.01 2.01 2.01 5.00 2.01 2.01 2.01 2.01

2.0 2.0 2.0 2.0 2.0 3.0 2.0 2.0 2.0

0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5

30 30 30 30 30 30 30 45 30

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.25

τx

τx

1.1780 1.5309 3.1769 1.0070 0.6887 0.9621 1.3388 1.7315 1.5749

1.1778 1.5307 3.1768 1.0063 0.6887 0.9621 1.3387 1.7314 1.5744

Table 2: Comparison of secondary skin friction between the present results and results by Rajput and Kanaujia [25] in the absence of radiation and chemical reaction, i.e., R = 0 and γ = 0. Present Results by results Rajput and Gr Gm Pr Sc m t ωt M by FEM Kanaujia [25]

10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

10.0 10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0

0.71 0.71 0.71 7.00 0.71 0.71 0.71 0.71 0.71

2.01 2.01 2.01 2.01 5.00 2.01 2.01 2.01 2.01

2.0 2.0 2.0 2.0 2.0 3.0 2.0 2.0 2.0

0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5

30 30 30 30 30 30 30 45 30

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.25

τz

τz

0.2149 0.2210 0.2560 0.2108 0.1975 0.3065 0.2790 0.1988 0.2490

0.2150 0.2208 0.2560 0.2107 0.1972 0.3062 0.2795 0.1983 0.2489

Results and Discussion In order to determine the effect of physical parameters on the flow fields such as magnetic parameter M , Hall parameter m, thermal Grashof number Gr , mass Grashof number Gm , Prandtl number Pr , Schmidt number Sc , radiation parameter R, chemical reaction parameter γ , phase angle ωt and time t , numerical computations have been carried out for primary velocity u , secondary velocity w, temperature θ , concentration φ , primary skin friction τ x , secondary skin friction τ z , Nusselt number Nu and Sherwood number Sh. These obtained numerical results have been presented through the graphs and tables and then discussed. Figure 2 illustrates the effects of thermal Grashof number Gr on the primary and secondary velocities. The thermal Grashof number Gr signifies the ratio of thermal buoyancy force to viscous hydrodynamics force. As increase in Gr tends to decrease drag forces and hence, fluid velocity increases. This implies that increase in the thermal Grashof number tends to accelerate fluid velocity in both primary and secondary flow directions. Here, the positive values of thermal Grashof number

54

Computational Analysis of Heat Transfer in Fluids and Solids II

correspond to cooling of the plate. Figure 3 shows the effects of mass Grashof number Gm on the primary and secondary velocities. The mass Grashof number Gm signifies the ratio of species buoyancy force to the viscous hydrodynamic force. As mass Grashof number increases, the viscous hydrodynamic force decreases as a result momentum of the fluid increases. This implies that fluid velocity in both primary and secondary flow directions tends to increase with increasing Gm . Also, it is noticed that velocity components suddenly raise near to the plate and after this velocity components asymptotically decreases to zero as y → ∞. Figure 4 represents the effect of magnetic parameter M on the primary and secondary velocities. The application of a magnetic field perpendicular to the flow direction of an electrically conducting fluid, it experiences an electric field and produces current perpendicular to both magnetic field and flow direction. The product of electric field and magnetic field creates a force which is known as Lorentz force. The direction of the Lorentz force is always opposite to the direction of the flow in the absence of applied electric field which opposes the fluid velocity. This implies that increase in magnetic parameter tends to retard the fluid velocity in the primary flow direction whereas it tends to enhance the fluid velocity in the secondary flow direction. This tendency of the magnetic field is clearly supported by the physical reality. The effects of the Hall parameter m on the primary and secondary velocities are depicted in Fig. 5. It is observed that an increase in the Hall parameter tends to accelerate the fluid velocity in both primary and secondary flow directions. This situation clearly supports the fact that Hall current induces a cross-flow in the boundary layer. Figure 6 displays the effect of Prandtl number Pr on the primary and secondary velocities. As increase in Pr correspond to stronger momentum diffusivity and weaker thermal diffusivity. This implies that increase in the Prandtl number tends to decelerate fluid velocity in both primary and secondary flow direction. The influence of radiation parameter R on the primary and secondary velocities is shown in Fig. 7. It is seen that both primary and secondary velocities enhance with increasing radiation parameter. This is consistent with the definition of R. Figure 8 shows the effects of Schmidt number Sc on the primary and secondary velocities. As Sc increases fluid velocity in both primary and secondary flow directions is expected to reduce since increasingly momentum is diffused at a lower rate than species. The effects of chemical reaction parameter γ on the primary and secondary velocities are presented in Fig. 9. It is observed that an increase in γ tends to decrease the fluid velocity in both primary and secondary flow direction. Figure 10 demonstrate the effects of phase angle ωt on the primary and secondary velocities. It can be seen that increase in the phase angle tends to decelerate the fluid velocity in both primary and secondary flow directions. Physically, as ωt increase buoyancy force tends to decrease and hence, decrease the fluid momentum. The influence of time parameter t on the primary and secondary velocities is presented in Fig. 11. As time progresses, fluid velocity in both primary and secondary flow directions are getting accelerated due to increasing buoyancy effects. Figure 12 depicts the effect of Prandtl number Pr on the fluid temperature. Prandtl number signifies the ratio of momentum to thermal diffusivities. As Pr increases thermal boundary layer thickness decreases. This implies that increase in Prandtl number tends to decrease the fluid temperature across the boundary layer. The effects of radiation parameter R on the fluid temperature are shown in Fig.13. It is observed that increase in the radiation parameter tends to enhance the fluid temperature in the boundary layer. As time progress, the fluid temperature θ increases as seen in Fig. 14. The effects of Schmidt number Sc on the fluid concentration φ are presented in Fig. 15. To be more realistic, the values of Schmidt number are chosen to represent the diffusing chemical species of most common interest like hydrogen ( Sc = 0.23), water-vapor ( Sc = 0.64), sulfur dioxide

( Sc = 1.20) and naphthalene ( Sc = 2.23). It is clear that an increase in Schmidt number tends to decrease the fluid concentration. Physically, increase in Sc decrease molecular diffusivity which results in a decrease of concentration boundary layer. Figure 16 displays the effect of chemical

Defect and Diffusion Forum Vol.401

55

reaction parameter γ on the fluid concentration. It is seen that increase in γ tends to decrease concentration of species in the boundary layer since large values of γ reduce the solutal boundary layer thickness and increase the mass transfer. The variation of fluid concentration φ with progression of time t is depicted in Fig.17. It is noticed that fluid concentration increases with increasing time t.

Fig. 2: Primary and secondary velocities for varying Gr when = Gm 10.0, = M 1.0, = m 0.5, = Pr 0.71, = R 1.0, = Sc 0.22, = γ 0.5, = ωt 300 at t = 0.2.

Fig. 3: Primary and secondary velocities for varying Gm when = Gr 10.0, = M 1.0, = m 0.5, = Pr 0.71, = R 1.0, = Sc 0.22, = γ 0.5, = ωt 300 at t = 0.2.

Fig. 4: Primary and secondary velocities for varying M when = Gr 10.0, = Gm 10.0, = m 0.5, = Pr 0.71, = R 1.0, = Sc 0.22, = γ 0.5, = ωt 300 at t = 0.2.

56

Computational Analysis of Heat Transfer in Fluids and Solids II

Fig. 5: Primary and secondary velocities for varying m when = Gr 10.0, = Gm 10.0, = M 1.0, = Pr 0.71, = R 1.0, = Sc 0.22, = γ 0.5, = ωt 300 at t = 0.2.

Fig. 6: Primary and secondary velocities for varying Pr when Gr 10.0, Gm 10.0, M 1.0, m 0.5, R 1.0, Sc 0.22, γ 0.5, ωt 300 at t = 0.2. = = = = = = = =

Fig. 7: Primary and secondary velocities for varying R when = Gr 10.0, = Gm 10.0, = M 1.0, = m 0.5, = Pr 0.71, = Sc 0.22, = γ 0.5, = ωt 300 at t = 0.2.

Defect and Diffusion Forum Vol.401

Fig. 8: Primary and secondary velocities for varying Sc when = Gr 10.0,= Gm 10.0, = M 1.0, = m 0.5, = Pr 0.71, = R 1.0, = γ 0.5, = ωt 300 at t = 0.2.

Fig. 9: Primary and secondary velocities for varying γ when = Gr 10.0, = Gm 10.0, = M 1.0, = m 0.5, = Pr 0.71, = R 1.0, = Sc 0.22, = ωt 300 at t = 0.2.

Fig. 10: Primary and secondary velocities for varying ωt when = Gr 10.0, = Gm 10.0, = M 1.0, = m 0.5, = Pr 0.71, = R 1.0, = Sc 0.22, = γ 0.5 at t = 0.2.

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Computational Analysis of Heat Transfer in Fluids and Solids II

Fig. 11: Primary and secondary velocities for varying t when = Gr 10.0, = Gm 10.0, = M 1.0, = m 0.5, = Pr 0.71, = R 1.0, = Sc 0.22, = γ 0.5 and ωt = 300.

Fig. 12: Temperature profiles with Pr when R = 1.0 at t = 0.2.

Fig. 14: Temperature for varying time t when Pr = 0.71 and R = 1.0.

Fig. 13: Temperature profiles with R when Pr = 0.71 at t = 0.2.

Fig. 15: Concentration profiles with Sc when γ = 0.5 at t = 0.2.

Defect and Diffusion Forum Vol.401

Fig. 16: Concentration profiles with γ when Sc = 0.22 at t = 0.2.

59

Fig. 17: Concentration for varying time t when Sc = 0.22 and γ = 0.5.

The numerical values of primary skin friction (τ x ) , secondary skin friction (τ z ) , Nusselt

number ( Nu ) and Sherwood number ( Sh ) for variations in Gr , Gm , Pr , Sc , M , m, R, γ , ωt and t are

presented in tables 3 to 6, respectively. From table 3 and 4, it is noticed that an increase in Gr , Gm , m, R and t tends to increase both primary and secondary skin frictions whereas an increase in Pr , Sc and γ tends to decrease both primary and secondary skin frictions. Magnetic parameter tends to decelerate primary skin friction and accelerate secondary skin friction whereas phase angle has opposite effect. As seen from table 5, that an increase in Pr tends increase Nusselt number whereas it decreases with increasing R and t. Table 6 reveals that increase in Sc and γ tends to increase Sherwood number whereas it decreases with increasing t.

Gr

Gm

10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

10.0 10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

Table 3: Numerical values of primary skin friction. ωt γ Sc Pr m t M R (degrees) 0.71 0.71 0.71 7.00 0.71 0.71 0.71 0.71 0.71 0.71 0.71

0.22 0.22 0.22 0.22 0.62 0.22 0.22 0.22 0.22 0.22 0.22

2.0 2.0 2.0 2.0 2.0 3.0 2.0 2.0 2.0 2.0 2.0

0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5

1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5

30 30 30 30 30 30 30 30 30 45 30

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3

τx 1.187770 1.448654 2.611304 1.034078 0.847760 0.866006 1.251342 1.213240 1.158930 1.278982 1.322450

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Computational Analysis of Heat Transfer in Fluids and Solids II

Gr

Gm

10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

10.0 10.0 20.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0

Table 4: Numerical values of secondary skin friction. ωt γ Sc Pr t m M R (degrees) 0.71 0.71 0.71 7.00 0.71 0.71 0.71 0.71 0.71 0.71 0.71

0.22 0.22 0.22 0.22 0.62 0.22 0.22 0.22 0.22 0.22 0.22

2.0 2.0 2.0 2.0 2.0 3.0 2.0 2.0 2.0 2.0 2.0

0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5 0.5 0.5

1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0 1.0 1.0 1.0

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 0.5 0.5

30 30 30 30 30 30 30 30 30 45 30

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.3

τz 0.182128 0.199436 0.282328 0.169832 0.148496 0.438436 0.237312 0.184796 0.180124 0.170260 0.207208

Table 5: Numerical values of Nusselt number.

Pr

R

t

Nu

0.71 7.00 0.71 0.71

1.0 1.0 2.0 1.0

0.2 0.2 0.2 0.3

0.147746 0.445450 0.120620 0.088648

Table 6: Numerical values of Sherwood number. γ Sc t Sh 0.22 0.62 0.22 0.22

0.5 0.5 1.0 0.5

0.2 0.2 0.2 0.3

0.262394 0.433564 0.290562 0.229272

Conclusions In this study, the unsteady MHD radiating and reacting mixed convection flow past an impulsively started oscillating vertical plate with Hall current has been presented. The leading governing equations of the problem have been solved numerically by using FEM. The significant findings of the study are summarized as follows: • Primary fluid velocity increases with increasing Gr , Gm , m, R and t whereas it decreases with increasing M , Pr , Sc , γ and ωt. •

Secondary fluid velocity increases with increasing Gr , Gm , M , m, R and t whereas it decreases with increasing Pr , Sc , γ and ωt.

•

An increase in Gr , Gm , m, R and t tends to increases both primary and secondary skin frictions whereas both primary and secondary skin frictions decreases with increasing Pr , Sc and γ .

Defect and Diffusion Forum Vol.401

• • •

61

An increase in M tends to decrease primary skin friction whereas it has opposite effect on secondary skin friction and ωt has reverse effect to that of M on primary and secondary skin frictions. Nusselt number increases with increasing Pr whereas it decreases with increasing R and t. An increase in Sc and γ tends to increases Sherwood number whereas it decrease with increasing t.

References [1]

V. M. Soundalgekhar, H. S. Takhar, Radiation effects on free convection flow past a semiinfinite vertical plate, Modeling Measurement and Control, B (51) (1993) 31 - 40.

[2]

A. Raptis, C. Perdikis, Radiation and free convection flow past a moving plate, Appl. Mech and Eng, 4(4) (1999) 817 - 821.

[3]

R. Muthucumaraswamy, G. S. Kumar, Heat and mass transfer effects on moving vertical plate in the presence of thermal radiation, Theo. Appl. Mech, 31(1) (2004) 35 - 46.

[4]

O. D. Makinde, MHD mixed-convection interaction with thermal radiation and nth order chemical reaction past a vertical porous plate embedded in a porous medium, Chemical Engineering Communications, 198 (4) (2011) 590-608.

[5]

H. S. Takhar, R. S. Gorla, V. M. Soundalgekhar, Radiation effects on MHD free convection flow of a radiating gas past a semi-infinite vertical plate, Int. J. Num. Heat and Fluid Flow, 6(2) (1996) 77 - 83.

[6]

R. Muthucumaraswamy, B. Janakiraman, MHD and radiation effects on moving isothermal vertical plate with variable mass diffusion, Theo. Appl. Mech, 33(1) (2006) 17 - 29.

[7]

V. Rajesh, S. V. K. Varma, Radiation effects on MHD flow through a porous medium with variable temperature and mass diffusion, Int. J. Appl. Math and Mech, 6(1) (2010) 39 - 57.

[8]

S. Das, B. Tarafdar, R.N. Jana, O.D. Makinde, Influence of rotational buoyancy on magnetoradiation–convection near a rotating vertical plate, European Journal of Mechanics-B/Fluids 75(2019) 209-218.

[9]

U. S. Rajput, S. Kumar, Radiation effect on MHD flow through porous media past an impulsively started vertical plate with variable heat and mass transfer, Int. J. Math. Arch, 4(10) (2013) 106 - 114.

[10] U. S. Rajput, G. Kumar, Radiation effect on MHD flow past an inclined plate with variable temperature and mass diffusion, Int. J. Appl. Sci and Eng, 14(3) (2017) 171 - 183. [11] R. Nandkeolyar, G. S. Seth, O. D. Makinde, P. Sibanda, M. S. Ansari, Unsteady hydro-magnetic natural convection flow of a dusty fluid past an impulsively moving vertical plate with ramped temperature in the presence of thermal radiation, ASME, Journal of Appl. Mech, 80, 061003 (2013) (9pages). [12] S. Das, R. N. Jana, O. D. Makinde, Magneto-convective boundary layer slip flow of nanofluid past a convectively heated vertical plate, Journal of Nanofluids, 4(4) (2015) 494 - 504. [13] A. Muhammad, O. D. Makinde, Thermo-dynamic analysis of unsteady MHD mixed convection with slip and thermal radiation over a permeable surface, Defect and Diffusion Forum, 374 (2017) 29 - 46. [14] P. O. Olanrewaju, O. D. Makinde, Effects of thermal diffusion and diffusion thermo on chemically reacting MHD boundary layer flow of heat and mass transfer past a moving vertical plate with suction /injection, Arabian Journal of Science and Engineering, 36 (2011) 16071619.

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[15] O. D. Makinde, P. Sibanda, Effects of chemical reaction on boundary layer flow past a vertical stretching surface in the presence of internal heat generation, International Journal of Numerical Methods for Heat & Fluid Flow, 21(6) (2011) 779-792. [16] R. Muthucumaraswamy, P. Ganesan, First order chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux, Acta Mechanica, 147 (2001) 45 - 57. [17] O. D. Makinde, Chemically reacting hydro-magnetic unsteady flow of radiating fluid past a vertical plate with constant heat flux, Zeitschrift f’ur Naturforschung, 67a (2012) 239 - 247. [18] R. Jayakar, B. Kumar, O. D. Makinde, Thermo-diffusion effects on MHD chemically reacting fluid flow past an inclined porous plate in a slip flow regime, Defect and Diffusion forum, 387 (2018) 587 - 599. [19] T. C. Cowling, Magnetohydrodynamics, Wiley Inter Science, New York, (1957). [20] J. Fife, Hybrid-PIC Modeling and electrostatic probe survey of Hall thrusters-PhD Thesis, Department of Aeronautics and Astronautics, MIT, USA, (1998). [21] P. A. Davidson, Magneto-hydrodynamics in material processing – Annual review Fluid Mech, 31 (1999) 273 - 300. [22]S. Gandluru, D.R.V. Prasad Rao, O.D. Makinde, Hydromagnetic-oscillatory flow of a nanofluid with Hall effect and thermal radiation past vertical plate in a rotating porous medium, Multidiscipline Modeling in Materials and Structures, 14(2) (2018) 360–386. [23] A. N. Maguna, N. M. Mutua, Hall current effects on free convection flow and mass transfer past a semi infinite vertical flat plate, Int. J. Mathematics and Statistics studies, 1(4) (2013)1-22. [24] M. Thamizhsudar, J. Pandurangan, R. Muthucumaraswamy, Hall effects and rotation effects on MHD flow past an exponentially accelerated vertical plate with combined heat and mass transfer effects, Int. J. Appl. Mech and Eng, 20( 3) (2015) 605 - 616. [25] U. S. Rajput, N. Kanaujia, MHD flow past a vertical plate with variable temperature and mass diffusion in the presence of Hall current, Int. J. Appl. Sci. Eng, 14 (2) (2016) 115 – 123. [26] B. Prabhakar Reddy, Hall effect on MHD transient flow past an impulsively started infinite horizontal porous plate in a rotating system, Int. J. Appl. Mech. Eng, 23(2) (2018) 471 – 483. [27] R.P. Sharma, K. Avinash, N. Sandeep, O.D. Makinde, Thermal radiation effect on nonNewtonian fluid flow over a stretched sheet of non-uniform thickness. Defect and Diffusion Forum, 377 (2017) 242-259. [28] O.C. Zienkiewiez, The finite element method in engineering sciences - 2nd edition., McGrawHill, New York, (1971). [29] J. N. Reddy, An introduction to the finite element method, McGraw-Hill, New York, (1985). [30] K. J. Bathe, Finite element procedures, Prentice-Hall, New Jersey, (1996).

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 63-78 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-03-11 Accepted: 2020-04-15 Online: 2020-05-28

Analytical and Numerical Study on Cross Diffusion Effects on Magneto-Convection of a Chemically Reacting Fluid with Suction/Injection and Convective Boundary Condition S. Eswaramoorthi1, M. Bhuvaneswari2, S. Sivasankaran3,c*, O.D. Makinde4,d Department of Mathematics, Dr.N.G.P. Arts & Science College, Coimbatore 641048, Tamil Nadu, India

1

Department of Mathematics, Kongunadu Polytechnic College, D.Gudalur, Dindigul, Tamilnadu, India 3Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

2

Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

4

[email protected], [email protected]

c

Keywords: MHD; Soret/Dufour effects, heat generation, suction/injection, chemical reaction, convective boundary condition

Abstract: The purpose of this paper is to investigate the Soret and Dufour effects on unsteady mixed convective boundary layer flow of a viscous fluid over a stretching surface in a porous medium in the presence of magnetic field with heat generation/absorption, chemical reaction, suction/injection and convective boundary condition. The governing time-dependent partial differential equations are transformed into non-linear ordinary differential equations using similarity transformations. These equations subject to the appropriate boundary conditions are solved analytically by homotopy analysis method (HAM) and numerically by Runge-Kutta fourth order method and shooting technique. The numerical solution is compared with analytical solution. The influence of the different parameters on velocity, temperature and concentration profiles are discussed in graphical as well as in tabular form. It is observed that the fluid velocity and temperature increase on increasing the buoyancy ratio parameter and heat generation/absorption parameter. Also found that the surface heat and mass transfer rates increase on increasing the suction/injection and heat generation/absorption parameters. Nomenclature a,b,c positive constants A unsteady parameter B0 magnetic strength Bi Biot number C concentration of the fluid Cf local skin friction coefficient Ci, (i = 1 to 7) arbitrary constants cp specific heat Cr chemical reaction parameter cs concentration susceptibility De mass diffusivity fw suction (> 0) or injection (< 0) parameter Df Dufour number g acceleration due to gravity Gr local Grashof number Ha Hartmann number

hf,hθ,hϕ non-zero auxiliary parameters jw surface mass flux k1 permeability of the porous medium k2 coefficient of chemical reaction km mass transfer coefficient kT thermal diffusion ratio Lf,Lθ,Lϕ linear operators N buoyancy ratio parameter Nf,Nθ,Nϕ non-linear operators Nu local Nusselt number Pr Prandtl number p embedding parameter Q heat generation (> 0) or absorption (< 0) qr radiative heat flux qw surface heat flux Re local Reynolds number

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Computational Analysis of Heat Transfer in Fluids and Solids II

hc convective heat transfer coefficient Ri Richardson number H(η) non-zero auxiliary function Sc Schmidt number Hg heat generation (> 0) or absorption (< 0) Sh local Sherwood number parameter Sr Soret number Greek Symbols T temperature of the fluid αm thermal diffusivity Tf temperature of the hot fluid βC concentration expansion coefficient Tm mean fluid temperature βT thermal expansion coefficient u,v velocity components χm characteristic function uw stretching surface velocity η similarity variable Vw suction (> 0) or injection (< 0) µ dynamic viscosity x,y space coordinates ν kinematic viscosity ρ fluid density Subscripts σ fluid electrical conductivity w surface conditions τw surface shear stress ∞ free stream conditions Introduction The boundary layer flow and heat transfer of a viscous fluid over a stretching sheet are occurring in several industrial applications. Some of the practical applications are flow through packed beds, underground disposable of radioactive waste materials, metal spinning, exothermic and endothermic reactions, etc. Kumari et al. [1] studied the unsteady free convection of a viscous fluid over a vertical surface. The study of unsteady boundary layer flow of a viscous fluid over a stretching surface was investigated by many authors, like, [2-6] in various situations. The Magneto-hydro-dynamics (MHD) was applied in many areas of science, engineering and technology, such as MHD power generators, MHD flow meters, MHD pumps, polymer industry, spinning of filaments, etc. Since the above important applications, many researchers have investigated the behaviour of a MHD boundary layer flow with different fluids. Liao [7] found the analytical solution of MHD boundary layer flow of a non-Newtonian fluid over a stretching sheet. He found that the skin friction decreases on increasing the magnetic field parameter. Many researchers like [8-19] were studied the MHD boundary layer flow over a stretching surface with different conditions. In combined heat and mass transfer processes, the thermal energy flux due to the concentration gradient is called the Dufour or diffusion-thermal effect and the mass flux under the temperature gradient is referred as the Soret or thermo-diffusion effect. These effects are significantly used in geosciences and chemical engineering. Soret and Dufour effects of MHD natural convective boundary layer flow over a stretching surface were studied by Postelnicu [20]. Many studies were reported the effects of Soret and Dufour number in various physical situations, see, [21-26]. Convective boundary condition over a stretching surface have considered by [27-28]. Recently, Eswaramoorthi et al. [29] studied the heat transfer analysis of a viscoelastic fluid over a stretching surface. They found that the Biot number increases the surface heat transfer rate and decreases the surface mass transfer rate. This paper is an extension work of Chamkha and Ben-Nakhi [21] to include magnetic field, internal heat generation/absorption, Soret number, Dufour number and convective boundary condition. Mathematical Formulation Consider an unsteady two-dimensional boundary layer flow and heat transfer of an incompressible viscous fluid over a stretching sheet in a porous medium. The x−axis is taken along the stretching surface in the direction of the motion and y−axis is perpendicular to it. A uniform magnetic field of strength B0 is applied on the y direction and the induced magnetic field is neglected, which is justified for MHD flow at small magnetic Reynolds number (see Figure 1).

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65

Figure 1: Schematic diagram of physical configuration. ax , the surface temperature is 1 − ct bx bx taken as Tw = T∞ + and surface concentration is C w = C ∞ + , where a, b and c are the 1 − ct 1 − ct positive constants with ct < 1 which is higher than the free stream temperature T∞ and free stream concentration C∞. The pressure gradient is neglected. The fluid phase is assumed to be heat generating or absorbing. The Soret and Dufour effects are included to study the heat and mass transfer. The first order homogeneous chemical reaction is taking place in the flow. The bottom surface of the stretching sheet is heated by convection from a hot fluid at temperature Tf, which provides a heat transfer coefficient hc see, Eswaramoorthi et al. [30]. Under the above assumptions and usual boundary layer approximation, the MHD mixed convective viscous flow, heat and mass transfer are governed by the following equations, see Chamkha and Ben-Nakhi [21].

It is assumed that the sheet is moving with velocity U w =

∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u ∂u ∂ 2u  υ σB02  u + gβT (T − T∞ ) + gβ C (C − C∞ ) , υ = − + +u +v ρ  ∂t ∂x ∂y ∂y 2  k1

(2)

∂T ∂T ∂T ∂ 2T D k ∂ 2 C Q +u +v = αm 2 + e T (T − T∞ ) , + 2 ∂t ∂x ∂y C s C p ∂y ρc p ∂y

(3)

∂C ∂C ∂C ∂ 2 C D k ∂ 2T +u +v = De 2 + e T − k 2 (C − C ∞ ) . Tm ∂y 2 ∂t ∂x ∂y ∂y

(4)

The boundary conditions for this problem can be expressed as, ∂T u = U w ( x), v = Vw , − k = h f (T f − T ), C = C w ( x), at y = 0, ∂y u → 0,

∂u → 0, T → T∞ , C → C ∞ , as y → ∞. ∂y

(5)

Now, we introduce following dimensionless similarity variables,

η=y

ax aυ a f ′(η ), v = − f (η ), , u= υ (1 - ct) 1 − ct 1 − ct

T − T∞ C − C∞ θ (η ) = , φ (η ) = . T f − T∞ C f − C∞

(6)

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Computational Analysis of Heat Transfer in Fluids and Solids II

Substituting equation (6) into equations (2)-(4), we have

η   f ′′′ + ff ′′ − f ′2 − A f ′ + f ′′  − (K + Ha ) f ′ + Ri(θ + Nφ ) = 0, 2  

(7)

η   θ ′′ + Pr ( fθ ′ − f ′θ ) − Pr Aθ + θ ′  + Pr Hgθ + Pr Dfφ ′′ = 0,

(8)

 η  φ ′′ + Sc( fφ ′ − f φ′ ) − ScA φ + φ ′  − ScCrφ + ScSrθ ′′ = 0.

(9)

2



2







The boundary conditions (5) becomes f (0) = f w , f ′(0) = 1, θ ′(0) = − Bi[1 − θ (0)], φ ′(0) = 1, f ′(∞) = 0, f ′′(∞) = 0, θ (∞) = 0, φ (∞) = 0,

where

A=

is unsteady parameter,

c a

(10)

K = υx k 1 U w is the permeability parameter,

M = σB x ρU w is the magnetic field parameter, Ri = Gr Re 2 is the Richardson number with 2 0

Gr = gβT (T f − T∞ )x 3 υ 2 is the local Grashof number and Re = U w x υ is the local Reynolds

number, N = β c (C w − C ∞ )x 3 β T (T f − T∞ ) is the buoyancy ratio parameter, Pr = υ α m is the Prandtl number,

the internal heat generation/absorption parameter, Hg = Qx U w is Df = De k T (C w − C ∞ ) c s c pυ (T f − T∞ ) is the Dufour number, Sc = υ De is the Schmidt number, Cr = k2 x U w is dimensionless chemical reaction parameter, Sr = De k T (T f − T∞ ) Tmυ (C w − C ∞ ) is

the Soret number, fw is the suction/injection parameter and Bi = h f

υ

a

k is the Biot number. The

reduced local skin friction coefficient, local Nusselt number and local Sherwood number are important physical parameters given by 1 2

1

C f Re 2 = f ′′(0),

1

1

Nu / Re 2 = −θ ′(0), Sh / Re 2 = −φ ′(0).

(11)

xq w xj w , Sh = , k (Tw − T∞ ) De (C w − C ∞ )

(12)

where Cf =

τw

ρU / 2 2 w

,

Nu =

 ∂u   ∂T   is the surface heat flux, and τ w = µ   is the wall shear stress, qw = −k  y ∂ y ∂   y =0   y =0

 ∂C   is the surface mass flux. jw = − De   ∂y  y = 0 Homotopy Analysis Method The governing ordinary differential equations (7) - (9) with the boundary conditions (10) are analytically solved using homotopy analysis method. The initial approximations of HAM are f0(η) = 1− e−η, θ0(η) = Bie−η /(1+Bi), ϕ0(η) = e−η and the linear operators are Lf = d3f/dη3 – df/dη, Lθ = d2θ/dη2 − θ, Lϕ = d2ϕ/ dη2 − ϕ. The mth-order deformation problem containing the auxiliary parameters hf, hθ and hϕ. These parameters have a key role to adjust and control the convergence of the solutions. The hf, hθ and hϕ curves are displayed in the Figure 2 for 15th order of approximations. It is observed that the range for admissible values of hf, hθ and hϕ are −0.7 ≤ hf, hθ ≤−0.2 and −0.7 ≤ hϕ ≤−0.3. It is found from our computation that the series solution convergence in the whole region of η when hf = hθ = hϕ = −0.5.

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Figure 2: h curves of f′′(0), θ′(0) and ϕ′(0) with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, fw = 0.5, Hg = −0.5, Df = 0.1, Bi = 0.5, Cr = 1.0 and Sr = 0.2. Numerical Solution The higher order non-linear ordinary differential equations (7) – (9) with the boundary conditions (10) are converted into a first order equations and they are numerically solved by employing a Runge-Kutta method and shooting technique with initial guessing f′′(0), θ′(0) and ϕ′(0). This process is continued until we get the desired accuracy. The analytical and numerical results are compared with the results available in the literature and they are depicted in Table 1 - Table 4. This provides the accuracy of our analytical and numerical results. Results and Discussion In this section we present the velocity profile (f′(η)), temperature profile (θ(η)) and concentration profile (ϕ(η)), local skin friction coefficient (1/2CfRe 1/2), local Nusselt number (Nu/Re1/2) and local Sherwood number (Sh/Re1/2) for various parameters. The value of Prandtl number (Pr =1.0) and Schmidt number (Sc = 0.62) are fixed throughout our study. Table 1 presents the comparison of f′′(0) between our results and available results in literature. It is found that these results are in good agreement. The Table 2 presents the local skin friction coefficient for different values of fw, Df, Bi, Cr and Sr for steady and unsteady flows. It is observed that the local skin friction coefficient increases on increasing the values of Dufour number, Biot number and homogeneous chemical reaction parameter and it decreases on increasing the values of suction/injection parameter and chemical reaction parameter for both flows. Table 1: Comparison of −f′′(0) with those reported by Chamkha and Ben-Nakhi [21] & Sharida et al. [31].

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Computational Analysis of Heat Transfer in Fluids and Solids II

Table 2: Variation of the local skin friction coefficient for steady and unsteady flows.

The local Nusselt number for different values of fw, Df, Bi, Cr and Sr for steady and unsteady flows are presented in the Table 3. It is seen that the surface heat transfer rate increases on increasing the suction/injection parameter, Dufour number, Biot number and chemical reaction parameter and it decreases on increasing the Soret number for both flows. From the Table 4, it is found that the surface mass transfer rate increases on increasing the suction/injection parameter, Dufour number and chemical reaction parameter. However, it is a decreasing function of Bi and Sr for steady and unsteady flows.

Defect and Diffusion Forum Vol.401

Table 3: Variation of the local Nusselt number for steady and unsteady flows.

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Computational Analysis of Heat Transfer in Fluids and Solids II

Table 4: Variation of the local Sherwood number for steady and unsteady flows.

Figures 3(a-c) show the effect of suction/injection parameter on velocity, temperature and concentration profiles for steady and unsteady flows. The suction causes to draw the amount of the fluid into the surface. This causes to thinner the momentum boundary layer thickness for both flows. In addition, the thermal and solutal boundary layers get depressed by increasing the suction parameter. However, the injection produces the opposite effect, namely increasing the fluid velocity, temperature and concentration for steady and unsteady flows. The velocity and temperature profiles for different values of the Richardson number are plotted in the Figures 4(a-b) for steady and unsteady flows. It is observed that the momentum and thermal boundary layer thicknesses increase on increasing the Richardson number for both flows. Figures 5(a-b) illustrate the various values of the buoyancy ratio parameter on the velocity and temperature profiles for steady and unsteady flows. It is observed that the momentum and thermal boundary layer thicknesses become thicker with the buoyancy ratio parameter.

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Figure 3: The steady and unsteady flows of velocity profile (a), temperature profile (b) and concentration profile (c) for different values of fw with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, Hg = −0.5, Df = 0.1, Bi = 0.5, Cr = 1.0 and Sr = 0.2.

Figure 4: The steady and unsteady flows of velocity profile (a) and temperature profile (b) for different values of Ri with A = 0.5, K = 0.1, Ha = 0.5, N = 0.5, fw = 0.5, Hg = −0.5, Df = 0.1, Bi = 0.5, Cr = 1.0 and Sr = 0.2.

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Figure 5: The steady and unsteady flows of velocity profile (a) and temperature profile (b) for different values of N with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, fw = 0.5, Hg = −0.5, Df = 0.1, Bi = 0.5, Cr = 1.0 and Sr = 0.2. The velocity and temperature profiles for different values of the heat generation or absorption parameter are shown in Figures 6(a-b) for steady and unsteady flows. It is observed that the momentum boundary layer thickens by increasing the presence of heat generation/absorption parameter. The presence of heat generation parameter increases the fluid temperature which causes thicken the thermal boundary layer thickness. On the contrary, presence of heat absorption parameter decreases the fluid temperature which causes thinner the thermal boundary layer thickness. Figure 7 displays the effect of the Dufour number on the temperature profile. Increasing the Dufour number tends to enhances the thermal boundary layer due to the decrease of the viscous force. Figure 8 displays the effect of the Biot number on the temperature profile for steady and unsteady flows. It is absorbed that the fluid temperature increases on increasing the Biot number. The concentration profile for different values of the chemical reaction parameter are plotted in the Figure 9. Increasing the chemical reaction parameter produces a decrease in the species concentration. This causes to thinner the solutal boundary layer thickness. Figure 10 illustrates the different values of Soret number on concentration profile. It is observed that the solutal boundary layer thickness increases on increasing the Soret number.

Figure 6: The steady and unsteady flows of velocity profile (a) and temperature profile (b) for different values of Hg with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, fw = 0.5, Df = 0.1, Bi = 0.5, Cr = 1.0 and Sr = 0.2.

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Figure 7: The steady and unsteady flows of temperature profile for different values of Df with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, fw = 0.5, Hg = −0.5, Bi = 0.5, Cr = 1.0 and Sr = 0.2.

Figure 8: Temperature profile for different values of Bi with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, fw = 0.5, Hg = −0.5, Df = 0.1, Cr = 1.0 and Sr = 0.2.

Figure 9: Concentration profile for different values of Cr with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, fw = 0.5, Hg = −0.5, Df = 0.1, Bi = 0.5, and Sr = 0.2.

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Figure 10: Concentration profile for different values of Sr with A = 0.5, K = 0.1, Ha = 0.5, Ri = 2.0, N = 0.5, fw = 0.5, Hg = −0.5, Df = 0.1, Bi = 0.5, and Cr = 1.0. The variations of skin friction coefficient for different values of parameters are presented in Figures 11(a-c). It is found that the surface shear stress reduces due to the effect of suction/injection, whereas it enriches due to the Dufour number, heat generation/absorption parameter and Biot number. It is observed from the Figures 12(a-d) that the surface heat transfer rate reduces on increasing the Soret number. However, it is a increasing function of suction/injection, Dufour number, heat generation/absorption parameter and Biot number under the combination of low Dufour number and Biot number. Figures 13(a-d) show the variation of Sherwood number for different values of parameters. It is observed that the surface mass transfer rate increases on increasing the suction/injection, Dufour number and heat generation/absorption parameter, while it decreases on increasing the Biot number and Soret number.

Figure 11: Variation of the skin friction coefficient for different values of fw, Df, Hg and Bi.

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Figure 12: Variation of the Nusselt number for different values of fw, Df, Hg, Bi and Sr.

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Figure 13: Variation of the Nusselt number for different values of fw, Df, Hg, Bi and Sr. Conclusions The present study focused on the Soret and Dufour effects on unsteady mixed convective boundary layer flow of a viscous fluid over a stretching surface in a porous medium in the presence of magnetic field with heat generation/absorption, chemical reaction, suction/injection and convective boundary condition. The governing partial differential equations are converted into a system of nonlinear ordinary differential equations by similarity transformation. Convergent series solution is found through the homotopy analysis method. Our results can be summarized as follows: • The fluid velocity and its momentum boundary layer thickness increases on increasing the buoyancy ratio parameter, heat generation/absorption parameter. Opposite trend is found on increasing the suction/injection parameter. • The thermal boundary layer thickness enhances with increasing the buoyancy ratio parameter, heat generation/absorption parameter, Dufour number, Biot number. The suction/injection parameter suppress the heat transfer inside the thermal boundary layer. • The Soret number boost up the solutal boundary layer thickness. however, it suppresses on increasing the suction/injection parameter and chemical reaction parameter. • The surface shear stress increases with increasing the heat generation or absorption parameter, Biot number, Dufour number and it decreases by increasing the suction/injection parameter, respectively. • The surface heat transfer rate increases with increasing the suction/injection parameter, heat generation/absorption parameter Dufour number, Biot number and it decreases by increasing the Soret number, respectively. • The surface mass transfer rate increases with increasing the suction/injection parameter, heat generation/absorption parameter Dufour number and it decreases on increasing the Biot number, respectively.

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References [1]

M. Kumari, A. Slaouti, H.S. Takhar, S. Nakamura, G. Nath, Unsteady free convection flow over a continuous moving vertical surface, Acta Mech. 116 (1996), 75-82.

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E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over an unsteady stretching surface, Heat Mass Transf. 41(1) (2004), 1-4.

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M.M. Rashidi, S.A. Mohimanian Pour, Analytic approximate solutions for unsteady boundarylayer flow and heat transfer due to a stretching sheet by homotopy analysis method, Nonlinear Anal. Model. Control 15(1) (2010), 83-95.

[4]

P.M. Patil, E. Momoniat, S. Roy, Influence of convective boundary condition on double diffusive mixed convection from a permeable vertical surface, Int. J. Heat Mass Transf. 70 (2014), 313-321.

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M.R. Safaei, G. Ahmadi, M.S. Goodarzi, A. Kamyar, S.N. Kazi, Boundary layer flow and heat transfer of FMWCNT/water nanofluids over a flat plate, Fluids 1(31) (2016), 1-13.

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A. Ahmed, J.I. Siddique, Dual solutions in a boundary layer flow of a power law fluid over a moving permeable flat plate with thermal radiation, viscous dissipation and heat generation/absorption, Fluids 3(6) (2018), 1-16.

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S.J. Liao, On the analytic solution of magneto-hydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech. 488 (2003), 189-212.

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A.A. Afify, MHD free convective flow and mass transfer over a stretching sheet with chemical reaction, Heat Mass Transf. 40 (2004), 495-500.

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P. Sibanda, O.D. Makinde, On steady MHD flow and heat transfer past a rotating disk in a porous medium with ohmic heating and viscous dissipation, Int. J. Num. Meth. Heat Fluid Flow 20 (3), 269-285, 2010.

[10] T. Javed, Z. Abbas, M. Sajid, N. Ali, Heat transfer analysis for a hydromagnetic viscous fluid over a non-linear shrinking sheet, Int. J. Heat Mass Transf. 54 (2011), 2034-2042. [11] S.E. Ahmed, A.K. Hussein, H.A. Mohammed, S. Sivasankaran, Boundary layer flow and heat transfer due to permeable stretching tube in the presence of heat source/sink utilizing nanofluids, Appli. Math. Compu. 238 (2014), 149-162. [12] A. Zaib, S. Shafie, Thermal-diffusion and diffusion-thermo effects on unsteady MHD free convection flow over a stretching surface considering Joule heating and viscous dissipation with thermal stratification chemical reaction and Hall current, J. Franklin Inst. 351 (2014), 12681287. [13] M. Turkyilmazoglu, Three dimensional MHD flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid with various physical effects, Int. J. Heat Mass Transf. 78 (2014), 150-155. [14] S. Eswaramoorthi, M. Bhuvaneswari, S. Sivasankaran, S. Rajan, Effect of radiation on MHD convective flow and heat transfer of a viscoelastic fluid over a stretching surface, Procedia Eng. 127 (2015), 916-923. [15] A. J. Benazir, R. Sivaraj, O.D. Makinde, Unsteady Magnetohydrodynamic Casson fluid flow over a vertical cone and flat plate with non-uniform heat source/sink, Int. J. Engin. Research Africa 21 (2016), 69-83. [16] S. Karthikeyan, M. Bhuvaneswari, S. Sivasankaran, S. Rajan, Soret and Dufour effects on MHD mixed convection heat and mass transfer of a stagnation point flow towards a vertical plate in a porous medium with chemical reaction radiation and heat generation, J. Appli. Mech. 9(3) (2016), 1447-1455.

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[17] K. Jagan, S. Sivasankaran, M. Bhuvaneswari, S. Rajan, O.D. Makinde, Soret and Dufour effect on MHD Jeffrey nanofluid flow towards a stretching cylinder with triple stratification, radiation and slip, Defect Diffusion Forum. 387 (2018), 523-533, [18] K. Jagan, S. Sivasankaran, M. Bhuvaneswari, S. Rajan, Effect of second order slip and nonlinear thermal radiation on mixed convection flow of MHD Jeffrey nanofluid with double stratification under convective boundary condition, IOP Conf. Ser. Mater. Sci. Eng. 390 (2018), 1-6. [19] K. Loganathan, S. Sivasankaran, M. Bhuvaneswari, S. Rajan, Second-order slip, cross-diffusion and chemical reaction effects on magneto-convection of Oldroyd-B liquid using CattaneoChristov heat flux with convective heating, Journal of Thermal Analysis and Calorimetry, 136 (2019), 401-409. [20] A. Postelnicu, Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transf. 47 (2004), 1467-1472. [21] A.J. Chamkha, A. Ben-Nakhi, MHD mixed convection-radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour’s Effects, Heat Mass Transf. 44 (2008), 845-856. [22] A.A. Afify, Similarity solution in MHD: Effects of thermal-diffusion and diffusion thermo on free convective heat and mass transfer over a stretching surface considering suction or injection, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009), 2202-2214 [23] A. Mahdy, Soret and Dufour effect on double diffusion mixed convection from a vertical surface in a porous medium saturated with a non-Newtonian fluid, J. NonNewtonian Fluid Mech. 165 (2010), 565-575. [24] R.V. Rathish Kumar, S.V.S.S.N.V.G. Krishna Murthy, Soret and Dufour effects on doublediffusive free convection from a corrugated vertical surface in a non-Darcy porous medium, Trans. Porous Media 85 (2010), 117-130. [25] S. Shatey, S.S. Motsa, P. Sibanda, The effects of thermal radiation hall currents Soret and Dufour on MHD flow by mixed convection over a vertical surface in porous media, Math. Probl. Eng. 2010 (2010), 1-20. [26] M.S. Alam, M.M. Rahman, M.A. Samad, Dufour and Soret effects on unsteady MHD free convection and mass transfer flow past a vertical porous plate in a porous medium, Nonlinear Anal. Model. Control 11(3) (2011), 217-226. [27] O.D. Makinde, A. Aziz, MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition, Int. J. Thermal Sci. 49 (2010), 1813-1820. [28] S.V. Subhashini, N. Samuel, I. Pop, Double-diffusive convection from a permeable vertical surface under convective boundary condition, Int. J. Heat Mass Transf. 38 (2011), 1183-1188. [29] S. Eswaramoorthi, M. Bhuvaneswari, S. Sivasankaran, S. Rajan, Soret and Dufour effects on viscoelastic boundary layer flow heat and mass transfer in a stretching surface with convective boundary condition in the presence of radiation and chemical reaction, Scientia Iranica, Transaction B: Mech. Engin. 23(6) (2016), 2575-2586. [30] S. Eswaramoorthi, M. Bhuvaneswari, S. Sivasankaran, H. Niranjan, S. Rajan, Effect of partial slip and chemical reaction on convection of a viscoelastic fluid over a stretching surface with Cattaneo-Christov heat flux model, IOP Conf. Ser. Mater. Sci. Eng. 263 (2017), 1-8. [31] S. Sharidan, T. Mahmood, I. Pop, Similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet, Int. J. Appl. Mech. Eng. 11(3) (2006), 647-654.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 79-91 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-05-13 Revised: 2019-07-07 Accepted: 2019-09-18 Online: 2020-05-28

Physical Aspects on MHD Micropolar Fluid Flow Past an Exponentially Stretching Curved Surface K. Anantha Kumar1,a, V. Sugunamma2,b*, N. Sandeep3,c, S. Sivaiah4,d Department of Mathematics, Sri Venkateswara University, Tirupati-517 502, A.P., India.

1,2

Department of Mathematics, Central University of Karnataka, Kalaburagi-585 367, India.

3

Department of Humanities and Sciences, School of Engineering and Technology, Gurunanak Institutions Technical campus, Telangana-501506, India.

4

[email protected], [email protected], [email protected], d [email protected]

a

Keywords: MHD, Microplar fluid, Joule’s heat, non-linear radiation, irregular heat source/sink.

Abstract. The present analysis is composed of heat transfer characteristics on MHD free convective stagnated flow of micropolar liquid due to stretching of an exponential curved sheet. The flow is supposed to be time-independent and not turbulent. The impact of non-linear radiation, unequal heat source/sink, Joule heating and variable thermal conductivity are supposed. Appropriate alterations are mused to change the original PDEs as ordinary ones and then solved by shooting and fourth order Runge-Kutta-Fehlberg integration schemes. Graphs are outlined to inspect the impacts of sundry nondimensional variables on the distributions of velocity, micro rotation and temperature. We discern that there is an augmentation in the fields of heat with Eckert number, nonlinear radiation and irregular hear parameters. Also it is motivating to comment that material parameter is a decreasing function of velocity. We establish the consequences in this analysis evidence to be extremely agreeable with the obtainable consequences. Introduction Recently, sundry studies have fluid flow past stretching surfaces, but recent ten years, many works studying micropolar fluid flow over solid surfaces appeared, because of the appearance of the significant this analysis in our life and more significance for manufacturing and industrial applications. There are many articles and many studies about micropolar liquids and thermal transport. Lately, many works stated the significance of the consequence of micro sized fluids in many fields; in the industry, like glass, in the army, in solar energy, in nuclear reactors, in the flow of urine in the kidneys and the bladder, etc. The works which can define the assets of non-Newtonian liquids, the fluid dynamics past an elongated stretched sheet, are significant in sundry practical solicitations such as extrusion of rubber sheets, glass blowing, metal spinning, production of paper and drawing plastic films, these are some illustrations. Okechi et al. [1] discussed the characteristics of viscous liquid oven an elongated curved surface and found that the curvature parameter has a propensity to diminish the momentum boundary layer thickness. An analytical solution is presented for the problem of boundary layer flow of micropolar liquid across a solid sheet with chemical response was examined by Sheikholaslami et al. [2]. The transformed equations are tackled by Homotopy analysis method. Rehman et al. [3] investigated the features of stagnation point flow of non-Newtonian liquid across an exponentially stretching sheet. The unsteady flow of shear thickening fluid due to an elongated curved surface in the appearance of porous medium was scrutinized by Saleh et al. [4]. Naveed et al. [5] bestowed simultaneous solutions for hydro magnetic viscous liquid subject to a solid surface. Recently, authors [6-7] conducted a numerical scrutiny to examine the timedependent Williamson liquid heat flow past a curved surface with irregular heat fall/raise. The study of the magnetohydrodynamics has important claims in astrophysics, geology, drug industries and mechanical engineering. MHD generators, pumps, bearings and boundary layer control are some notable applications of Lorentz force. The flow and heat transfer features of MHD Newtonian fluid flow over a stretching surface was examined by Abbas et al. [8]. Alshomrani and

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Gul [9] presented simultaneous solutions for Cu and Al2O3 based water nanoliquids across a stretched cylinder in the attendance of Lorentz force. The heat transfer features of an electrically accomplishing and time-dependent motion of shear thickening fluid across a stretching surface were investigated by Sandeep and Malvandi [10]. It was concluded that Lorentz force strongly controls the momentum layer thickness of non-Newtonian liquid. The influence of drag force on time-dependent bio convective motion of Carreau liquid across a coagulated surface was reported by Anantha Kumar et al. [11-12]. The influence of non-linear radiation on free convective flows show a massive character in space technology and in many industrial processes involving higher temperatures, such as cooling of metallic pieces, petroleum pumps, the manufacture of paper plates and making of electronic chips. Naveed et al. [13] and Sandeep et al. [14] examined the heat transfer features of micropolar fluid due to stretching of a surface in the attendance of linear radiative heat flux and found that the temperature is an increasing function of radiation parameter. The influence of radiative heat flux on magneto hydrodynamic flow of non-Newtonian liquid across a curved sheet was examined by Abbas et al. [15] and Hayat et al. [16]. Zeeshan et al. [17] reported the influence of radiation and drag force on the motion of ferrofluid across a stretched sheet. R.K. Fehlberg amalgamation scheme is utilized to solve the coupled equations. Numerical exploration on MHD oblique motion of micropolar liquid across a stretching sheet was deliberated by Anantha Kumar et al. [18] in the attendance of nonlinear radiation. The mechanism of irregular heat raise/fall has extensive range of applications in medicine, bioscience and countless industrial happenings like radiated diffusers, crude oil recovery, and the target of insertion bearing. Mehmood et al. [19] addressed about the heat and mass transport behaviour of non-Newtonian liquids using variable heat generation/absorption. The influence of first order chemical reaction on MHD flow of micropolar fluid over a curved surface was examined by Hayat et al. [20]. The features of mass transfer on non-Newtonian fluid over a stretching sheet under the impact of irregular heat parameters was reported by the authors [21-22] and concluded that the heat transfer performance can be controlled by the variable heat source/sink parameters. Tetbirt et al. [23] studied the influence of Joule heating on non-Newtonian liquid flow subject to a vertical channel in the attendance of Lorentz force. The influence of slip, frictional heat on MHD flow of micropolar fluid due to stretching of a convective surface in the attendance of linear radiative heat flux was reported by Ramzan et al. [24]. It was concluded that the Eckert number and the Biot number has a tendency to enhance the distribution of temperature. All the above said articles are carried out into account of constant physical properties of the ambient fluid but various practical situations demand for physical properties with variable characteristics. Temperature dependent Thermal conductivity is one of such properties, which is assumed to very linear with temperature. Initially, Chiam [25] examined the influence of temperature dependent thermal conductivity on the boundary layer flow due to stretching of a sheet. The work of Chiam [25] was extended by Abel and Mahesh [26] with radiation and irregular heat raise/fall. The magneto hydrodynamic viscoelastic liquid over a stretching surface in the attendance of thermal conductivity is investigated. Adegbie et al. [27] and Malik et al. [28] investigated the variable properties of MHD non-Newtonian fluid flow over a stretching surface. It was concluded that the fluid temperature enhances with variable thermal conductivity parameter. The impact of variable viscosity and temperature on MHD non-Newtonian fluid over a melting surface in the attendance of thermal stratification were examined by Anantha Kumar et al. [29]. Srinivasacharya and Jagadeeswar [30] studied the influence of Biot number on MHD flow of viscous liquid over an exponentially stretching surface with variable fluid properties and heat source/sink. It was concluded that the rate of heat transport decreases with thermal conductivity parameter. Makinde and Animasaun [31-32] described a theoretic study on MHD bio convective motion of Newtonian fluid past a solid surface. A new heat flux model is exploited to analyze the influence of heat transfer on MHD flow past a stretching sheet was studied by Makinde et al. [33]. Hence, inspired by the above studies, here we made an effort to examine the free convective boundary layer flow considering variable thermal conductivity, nonlinear Rossland approximation,

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irregular heat fall/raise and Joule heating over an exponentially stretching curved surface. The Mathematical model is developed for the current analysis and solved numerically after applying the suitable similarity transformations. The impact of various governing non-dimensional parameters on the distribution of velocity, microrotation, and temperature are studied with the aid of graphical illustrations. Mathematical Formulation Consider the 2D boundary layer flow of an incompressible and electrically conducting micropolar liquid over an exponentially stretching curved surface with nonlinear thermal radiation. The fluid motion is laminar and time independent. Assume that the co-ordinate system (𝑟𝑟, 𝑠𝑠), where 𝑠𝑠 axis is considered in the flow direction and 𝑟𝑟 axis is orthogonal to it. Let 𝑑𝑑 be the radius of the circle. The 𝑠𝑠 surface is stretching with the velocity 𝑢𝑢𝑤𝑤 (𝑠𝑠) = 𝐴𝐴𝑒𝑒 𝐿𝐿 along 𝑠𝑠 direction, ( A, L > 0) where A the initial stretching is rate and L is the reference length. The constant magnetic field of strength 𝐵𝐵0 is deployed in the radial direction as shown in the Fig. 1. The following are some assumptions on the flow model. (i) Micropolar liquid model. (ii) Ohmic heating and magnetic Reynolds number are very small. (iii) Joule heating, space dependent heat source/sink and variable thermal conductivity are deemed. (iv) Convective boundary condition is applied to the boundary.

Fig.1: Flow geometry. Owing to the above declared assumptions, the flow equations will be ∂ ∂u 0, ( ( r + d )v ) + d = ∂r ∂s ∂p ρ 2 = u , ∂r r + d  ∂u ∂u ∂N d uv  d ∂p ρ v + + − −κ − σ B02u + u = ∂r (r + d ) ∂s  ∂r (r + d ) ∂s (r + d )   ∂ 2u  u 1 ∂u − (µ + κ )  2 + , 2  (r + d ) ∂r (r + d )   ∂r

(1) (2)

(3)

 ∂2 N  ∂N  ∂N  ∂u d 1 ∂N  u  + = Γ u ρ jv (4)  2 +  − κ  2N + +  , ∂r (r + d )  (r + d ) ∂r   ∂r (r + d ) ∂s    ∂r  ∂T d u ∂T  1 ∂ ∂T  ∂  2 2 ρC p  v + =   (r + d )k (T )  − ( (r + d )q f )  − σ B0 u + q′′′, (5)  ∂r  ∂r   ∂r (r + d ) ∂s  (r + d )  ∂r 

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Here (𝑢𝑢, 𝑣𝑣) respectively, the velocity components along (𝑠𝑠, 𝑟𝑟) directions, 𝜌𝜌 is density, 𝑝𝑝 is the pressure (dimensional), 𝜎𝜎 is electrical conductivity, 𝜇𝜇 is the viscosity (dynamic), 𝛤𝛤 is the spin gradient viscosity, 𝑁𝑁 is the micro-rotation (dimensional), κ is the vortex viscosity, 𝐶𝐶𝑝𝑝 is the heat capacitance, T is the fluid temperature k (T ) is the temperature dependent thermal conductivity and 𝑞𝑞𝑓𝑓 is the radiative heat flux. Let us define 𝛤𝛤 and k (T ) as 𝜅𝜅

𝛼𝛼

𝛤𝛤 = �𝜇𝜇 + 2� 𝑗𝑗 = 𝜇𝜇 𝑗𝑗 �1 + 2 �

𝑘𝑘(𝑇𝑇) = 𝑘𝑘∞ �1 + 𝜀𝜀 �

Here j =

2υ L s

𝑇𝑇−𝑇𝑇∞

𝑇𝑇𝑤𝑤 −𝑇𝑇∞

�� = 𝑘𝑘∞ (1 + 𝜀𝜀𝜀𝜀(𝜂𝜂))

(6)

�

𝜅𝜅

is the micro-inertia parameter, 𝛼𝛼 = 𝜇𝜇 is the micropolar parameter, 𝑘𝑘∞ is ambient

Ae L fluid thermal conductivity and ε is the variable thermal conductivity parameter. In Eqn. (5), 𝑞𝑞𝑓𝑓 is taken to examine the heat transport performance and it can be defined as

4σ * ∂T 4 16σ * 3 ∂T , qf = T − ∗ = − 3k ∂r 3k ∗ ∂r

(7)

Here (σ * , k * ) respectively, Stefan-Boltzmann constant and mean absorption coefficient. In Eqn. (5), q′′′ is added to discuss the concept of non-uniform heat source or sink and it is given by = q′′′

k (T )uw ( s ) ∗ A (Tw − T∞ ) f ′ + B∗ (T − T∞ ) ) , ( 2Lυ

(8)

The negative and positive values of 𝐴𝐴∗ , 𝐵𝐵 ∗ respectively corresponds to heat sink/source. Using Eqs. (6)-(8), Eqn. (4) can be altered as

  ∂ ∂T − σ B02u 2 +   + ( k∞ (1 + εθ ) ) ∂r   ∂r  (9) * k∞ (1 + εθ )uw ( s ) ∗ 16σ  ∂  3 ∂T  1  ∗ 3 ∂T  T ( A (Tw − T∞ ) f ′ + B (T − T∞ ) ) ,   T + + ∂r  3k *  ∂r  ∂r  (r + d ) 2Lυ   ∂T d u ∂T +  ∂r (r + d ) ∂s

ρC p  v

 ∂ 2T  1 ∂T = + (1 ) k εθ  2 +  ∞ (r + d ) ∂r   ∂r

For the present study, boundary conditions suggested are 𝜕𝜕𝜕𝜕

𝑢𝑢 = 𝑢𝑢𝑤𝑤 (𝑠𝑠), 𝑣𝑣 = 0, 𝑁𝑁 = −𝑀𝑀𝑟𝑟 𝜕𝜕𝜕𝜕 , 𝑇𝑇 = 𝑇𝑇𝑤𝑤 𝑢𝑢 → 0,

𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

→ 0, 𝑁𝑁 → 0, 𝑇𝑇 → 𝑇𝑇∞

at

as

𝑟𝑟 = 0,

𝑟𝑟 → ∞,

�

(10)

Here 𝑀𝑀𝑟𝑟 micro-rotation parameter. (0 ≤ 𝑀𝑀𝑟𝑟 ≤ 1). We mention that 𝑀𝑀𝑟𝑟 = 0 suggests that 𝑁𝑁 = 0. It means the microelements in the fluid are not able to rotate near the wall surface, because microelements concentration is strong. Let us define the similarity variable ( η ) s

η=

Aυ e L r, 2L

Let us define the expressions (𝑢𝑢, 𝑣𝑣, 𝑝𝑝, 𝑁𝑁 and 𝑇𝑇) in terms of similarity variables

(11)

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d Aυ e − u= Ae f ′(η ), v = r+d 2L s

L

L

83



2s

ρ A2 e L P(η ),  ( f (η ) + η f ′(η ) ) , p = 

N= A

(12)

   

s

Aυ e L s L e g (η ), T = T∞ (1 + (θ w − 1)θ (η ) ) , θ w = Tw / T∞ , 2L

Here ( F ′(η ), g (η ), P(η ), θ (η ) ) respectively, dimensionless velocity, micro-rotation, pressure and temperature and θ w is the temperature ratio parameter. With the help of similarity renovations, Eqn. (1) is satisfied routinely and the Eqs. (2), (3), (4) and (9) becomes 2

dP 1  df  =   , dη (a + η )  dη 

(13) 2

 d3 f 4a 1 d2 f 1 a η dP df  2a + η  df  P= η  + + − (1 + α )  3 + − 2 2 (a + η ) dη a + η (a + η ) dη  (a + η ) 2  dη   dη (a + η ) dη a d2 f a df dg df f f + −α −M + λθ , 2 2 (a + η ) dη (a + η ) dη dη dη 2 1 dg  a  dg df  α  d g + − 1 +   2 + f  2   dη (a + η ) dη  (a + η )  dη dη

  d2 f 1 df  g  − α  2g + 2 + 0, (15) = dη (a + η ) dη   

2 2  d 2θ  dθ   df   a dθ 1 dθ ∗ df ∗  +A + B θ +ε  − Pr MEc  (1 + εθ )  2 + Pr f  +   dη dη (a + η ) dη  dη  a + η  dη    dη   2 2   dθ    1 dθ   3 d θ 2  + N r  (1 + (θ w − 1)θ (η ) )  2 + 0  + 3  (1 + (θ w − 1)θ (η ) ) (θ w − 1)     , =  (a + η ) dη    dη     dη  

s

Ae L Here a = d 2υ L

λ=

g βT (T − T∞ )2 L (uw ) 2υ

Ec =

(uw ) 2 2 L C p (Tw - T∞ )e

s

L

is the curvature parameter, M = is

the

convection

(14)

parameter,

(16)

σ B02 is magnetic field parameter, Aρ

Pr =

µC p

is

k

the

Prandtl

number,

16σ ∗T∞3 is the Eckert number and N r = is nonlinear radiation parameter. 3k∞ k *

The altered boundary conditions are df d 2 f dθ = = − Bi (1 − θ ), f = 0, 1, g = -M r , dη dη 2 dη 2

df d f → 0, 2 → 0, g → 0, θ → 0, dη dη

at as



η= 0,    η → ∞,  

Now we eliminate the pressure form the Eqs. (13) and (14), we get

(17)

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Computational Analysis of Heat Transfer in Fluids and Solids II

 d4 f d2 f df  α  dg d 2 g  2 d3 f 1 1 a + − + − + + η ( )    4 dη 2   (a + η ) dη 3 (a + η ) 2 dη 2 (a + η )3 dη  4a  dη  dη  a  d3 f d2 f df  df  1 df d 2 f   1 1  (18) f + + − + 2  +  = 0,  f −3  3 2 2 dη  (a + η ) dη dη   (a + η )   dη (a + η ) dη (a + η ) dη    a  dθ d2 f   M − λ ,    dη 2  (a + η )  dη

(1 + α ) 

The significant uses of current problem in manufacturing, engineering and industrial processes are friction factor, local Nusselt number and couple stress coefficient. These are defined by

= CF

τw Mw s jw , , , = = C Nu S 2 1 µ j uw k∞ (Tw − T∞ ) 2 ρ (u w )

(19)

  κ   ∂N  u   ∂u   ∂T  Here τ s = −k (T )  µ +  j  , qs =  + q f are  ( µ + κ )  ∂r − r + d  + κ N  , M w = 2   ∂r r 0=     ∂r r 0  r =0 = the surface shear stress, couple stress and surface heat flux. From Eqn. (12), Eqn. (19) becomes     d2 f  d2 f  1 df  1 Re s 2 CF = 2  (1 + α )  − −α Mr   ,  2 2   a dη η 0=  dη  dη η 0   =   α   dg  −1   3  dθ  2 CS = , Re s Nu = ,  − (1 + N rθ w )  1 +     2 d d η η    η 0=  η 0  =

where Re s =

uw ( s ) L

υ

(20)

is the local Reynolds number.

Results and Discussion The system of nonlinear and mutual ODEs (13)-(16) with the boundary conditions (17) is elucidated mathematically with the support of Shooting and Runge-Kutta-Fehlberg integration scheme. The influence of several dimensionless parameters on the flow fields (velocity, microrotation and temperature) is viewing via graphs. Results are got by assigning the values of dimensionless parameter as 𝛼𝛼 = 1.5, 𝑀𝑀 = 0.5, 𝐴𝐴∗ = 𝐵𝐵 ∗ = 0.02, 𝑎𝑎 = 0.2, 𝑃𝑃𝑃𝑃 = 7, 𝜆𝜆 = 0.5, ∈= 0.5, 𝜂𝜂 = 1, 𝐸𝐸𝐸𝐸 = 1, 𝑁𝑁𝑟𝑟 = 0.1, 𝜃𝜃𝑤𝑤 = 1.5 and 𝑀𝑀𝑟𝑟 = 0.2. In all the figures 𝑓𝑓 ′ , 𝑔𝑔 and θ (functions of η) indicate the distributions of velocity, micro-rotation and temperature. Table 1 conveys the validation of the current outcomes for friction factor with diverse values a with the available outcomes of Okechi et al. [1]. We establish a good agreement with their consequences. It is witnessed from that the progressive values of a lessening the friction factor. 1

Table 1: Validation of the results of Re s 2 CF for different values of a when α = 0 , * * ε = λ N= θ= M B= 0 and Pr = = Ec = Bi = A= = 0. r w a Okechi et al. [1] Present Results 5 1.4196 1.4189 10 1.3467 1.3466 20 1.3135 1.3136 30 1.3028 1.3028 40 1.2975 1.2975 50 1.2944 1.2944 100 1.2881 1.2881

Defect and Diffusion Forum Vol.401

85

Figs. 2-4 are erected to discern the effect of 𝑀𝑀 on 𝑓𝑓 ′ (𝜂𝜂), 𝑔𝑔(𝜂𝜂) and θ(𝜂𝜂). The velocity and microrotation distributions reduce with cumulative values of magnetic parameter. Actually, high magnetic field origins a resistive type of force in the flow, named as Lorentz force. This force works against the flow direction. Due to this a decrease in the distributions of velocity and micro-rotation is noticed. It is observed from the Fig. 4 that, the fluid temperature enhances with larger magnetic field parameter. This is owing to the Lorentz force. It generates some additional heat energy in the flow, which causes an enhancement in the fluid temperature. The influence of 𝑃𝑃𝑃𝑃 on the distribution of heat (θ(η)) is portrayed in Fig. 5. For rising values of 𝑃𝑃𝑃𝑃 yields a reduction in the curves of temperature. Owing to high conductivity, the heat function and its layer thickness enhances. Generally, Prandtl number is the ratio between momentum and thermal diffusivities. So, increasing 𝑃𝑃𝑃𝑃 causes a decrement in the curves of heat and the corresponding layer thickness. Figure 6 is sketched to discuss the impact of Biot number ( Bi ) against the heat function ( θ (η ) ). We see that rising values of Bi improves the fluid temperature. Essentially, 𝐵𝐵𝐵𝐵 is the relation between the internal conductive resistances to the surface convective resistance. So, profile θ (η ) and the corresponding layer thickness becomes stronger for increase in Bi . 1

0.6

0.9 0.5

0.8 0.7

0.4

) g(

0.5

M = 1, 2, 3, 4

I

f ( )

0.6 0.3

M = 1, 2, 3, 4

0.4 0.2

0.3 0.2

0.1

0.1 0

0 0

0.5

1

1.5

2

2.5

3

3.5

0

4

Fig. 2: Velocity profiles with M.

2

1

3

4

5

6

7

Fig. 3: Micro-rotation profiles with M.

0.9

0.5

0.8

0.45

0.7

0.4 0.35

0.6

0.3

(

( )

)

0.5

0.25

0.4

Pr = 1, 2, 3, 4

0.2

0.3

0.15

0.2

0.1

0.1

0.05

M = 1, 2, 3, 4 0

0

0

1

2

3

4

5

6

7

Fig. 4: Temperature profiles with M.

0

1

2

3

4

5

6

7

8

Fig. 5: Temperature profiles with Pr.

The impact of convective parameter on the flow fields is depicted in Figs. 7-9. It is observed from the figures 7 and 8 that, both the velocity and micro-rotation enhances with an escalation in the values of 𝜆𝜆. But an opposite trend is noticed for the curves of temperature (see Fig. 9). Generally, 𝜆𝜆 is the ratio of buoyancy to viscous forces. Moreover, higher values of 𝜆𝜆 causes a hike in the viscous forces, which causes an enhancement in the profiles of 𝑓𝑓 ′ (𝜂𝜂) and 𝑔𝑔(𝜂𝜂). Figure 10 is plotted to analyse the influence of Eckert number on the distribution of heat function (θ(η)). It is evident from the figure that, the temperature is an increasing function of 𝐸𝐸𝐸𝐸. Basically, 𝐸𝐸𝐸𝐸 is the ratio of advective transport

86

Computational Analysis of Heat Transfer in Fluids and Solids II

to heat dissipation potential. So, increasing 𝐸𝐸𝐸𝐸 causes an enhancement in the curves of heat and the corresponding layer thickness. The influence of 𝐴𝐴∗ , 𝐵𝐵 ∗ on the heat function θ(𝜂𝜂) is depicted via Figs.11 and 12. It was detected that amassing values of A* , B* consequences a hike in the distribution of temperature. Primarily, an inflammation in the values of irregular heat parameters assists as an agent to produce temperature in the flow. Owing to this, we detected that a growth in the fluid temperature for growing values of A* , B* . 0.8

1 0.9

0.7

0.8 0.6

0.7

f ( )

0.6 0.5

I

0.4

0.4

0.3

0.3 0.2

0.2 0.1

0.1

= 1, 2, 3, 4

Bi = 0.5, 1.0, 1.5, 2.0

0 0

1

3

2

4

5

6

0

7

0.4

0.2

0

Fig. 6: Temperature profiles with Bi.

0.8

0.6

1

1.2

1.4

1.8

1.6

2

Fig. 7: Velocity profiles with λ.

2

0.6

1.8 0.5

1.6 1.4

0.4

1

( )

g(

)

1.2

= 1, 2, 3, 4

0.3

0.8 0.2

0.6 0.4

0.1

0.2

= 1, 2, 3, 4

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

5

0.5

0

Fig. 8: Micro-rotation profiles with λ.

1

1.5

2

2.5

3

3.5

4

Fig. 9: Temperature profiles with λ.

0.8

0.7

0.7

0.6

0.6

0.5

0.5

)

0.4

(

)

0.4 (

( )

0.5

0.3 0.3

0.2

0.2

0.1

0.1

*

A = 0.1, 0.3, 0.5, 0.7

Ec = 0.1, 0.3, 0.5, 0.7

0 0

1

0 2

3

4

5

Fig. 10: Temperature profiles with Ec.

6

0

1

2

3

4

5

6

Fig. 11: Temperature profiles with A*.

Defect and Diffusion Forum Vol.401

0.7

87

1 0.9

0.6

0.8

0.5

0.7 0.6

M = 0, 0.3, 0.6, 0.9 r

0.5

I

f ( )

( )

0.4

0.3

0.4 0.3

0.2

0.2

0.1 0.1

*

B = 0.1, 0.2. 0.3. 0.4 0 0

1

2

3

4

5

0

6

0

Fig. 12: Temperature profiles with B*.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 13: Velocity profiles with Mr.

7

1 0.9

6

0.8

5

0.7 0.6 0.5

I

g(

f ( )

)

4

3

= 0.0, 0.5, 1.0, 1.5

0.4 0.3

2

0.2

1 0.1

M = 0, 0.5, 1, 1.5 r 0 0

1

2

3

4

5

6

7

8

Fig. 14: Micro-rotation profiles with Mr.

0 0

0.5

1

1.5

2

2.5

3

Fig. 15: Velocity profiles with α.

Figures 13 and 14 are portrayed to know the impact of micro-rotation parameter ( M r ) on the distributions of velocity and micro-rotation. As likely, for boosting values of M r the distribution of micro-rotation and the layer thickness enhance but a contradictory development is noticed for fluid velocity. The impact of micropolar parameter ( α ) on the fields of f ′(η ) and g(η ) is depicted via Figs. 15 and 16. It is fascinating to observe that the fluid velocity is reducing factor of α . But the micro-rotation profile is proportional to micropolar parameter. The influence of curvature parameter ( a ) on the velocity and temperature fields is portrayed via Figs. 17 and 18. It is easy to detect that the distribution of temperature is a decreasing function of larger a but an opposite result is noticed for fluid velocity. Generally, uplifting values of a , enriches the radius of the exponential surface as a result the fluid velocity enhances and the temperature decreases.

88

Computational Analysis of Heat Transfer in Fluids and Solids II

1

6

0.9 5

0.8 0.7

4

f ( )

g(

3

0.5

I

)

0.6

0.4 2

0.3 0.2

1

0.1

= 0.0, 0.5, 1.0, 1.5

0 0

0.5

1

1.5

a = 0.1, 0.2, 0.3, 0.4

0 2

2.5

3

4

3.5

4.5

0

5

0.2

Fig. 16: Micro-rotation profiles with α.

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Fig. 17: Velocity profiles with a. 0.5

0.7

0.45 0.6

0.4 0.5

0.35 0.3

(

(

)

)

0.4

a = 0.1, 0.2, 0.3, 0.4

0.3

0.25 0.2 0.15

0.2

0.1 0.1

0.05 0

0 0

1

2

3

4

5

Fig. 18: Temperature profiles with a .

6

N = 0.0, 0.5, 1.0, 1.5 r 0

1

2

3

4

5

6

7

Fig. 19: Temperature profiles with Nr.

Figs. 19 and 20 are sketched to know the essence of non-linear radiation ad temperature ratio parameter on the distribution of heat function ( θ (η ) ). It is interesting to note that, fluid temperature is an escalating factor of non-linear radiation parameter. Generally, radiation is a heat transfer mechanism which procures the energy via liquid grains. So it improves thermal energy in the flow. Owing to this, we perceived an improvement in the heat function ( θ (η ) ) for larger N r . It is noteworthy that distribution of temperature boosts with mounting values of θ w . The essence of temperature dependent thermal conductivity ( ε ) on the distribution of heat function can be seen in the Fig. 21. As likely, larger ε provides an enhancement in the thermal field.

Defect and Diffusion Forum Vol.401

0.5

89

0.45

0.45

0.4

0.4

0.35

0.35 0.3

0.25 )

0.25

(

(

)

0.3

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05 w

= 0, 1, 2, 3

= 0.0, 0.5, 1.0, 1.5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Fig. 20: Temperature profiles with θw.

5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 21: Temperature profiles with ∈.

Conclusions The free convective flow and heat transfer features are scrutinized in the stagnation point flow of micropolar fluid subject to an exponentially stretching curved surface. The impacts of drag force, irregular heat raise/fall, non-linear radiation, Joule heating and temperature dependent thermal conductivity are deemed. Graphical results are presented to understand to interpret the influence of physical parameters on flow fields. Some notable conclusions are listed below. • An increase in microrotation parameter decreases the distribution of velocity and increases the micro-rotation field. • Heat function ( θ (η ) ) and Eckert number are straight proportional to each other. • Enhancing values of irregular heat raise/fall parameters worked as a proxy of heat source. • The distribution of velocity is enhances with higher values of curvature parameter but a reverse trend is perceived for higher values of magnetic field and micropolar parameters. • An increase in the temperature dependent thermal conductivity parameter causes an enhancement in the distribution of temperature. • The distribution of temperature is enhances with higher values of non-linear radiation Biot number, and temperature ratio parameters but a reverse trend is perceived for higher values of Prandtl number and convection parameter. References [1]

N.F. Okechi, M. Jalil, M., S. Asghar, Flow of viscous fluid along an exponentially stretching curved surface, Res. Phys., 7 (2017) 2851-2854.

[2]

M. Sheikholeslami, M. Hatami, D.D. Ganji, Micropolar fluid flow and heat transfer in a permeable channel using analytical method, J. Mol. Liq., 194 (2014) 30-36.

[3]

A. Rehman, S.Nadeem, M.Y. Malik, Boundary layer stagnation-point flow of a third-grade fluid over an exponentially stretching sheet, Braz. J. Chem. Eng. 30 (2013) 611-618.

[4]

S.H.M. Saleh, N.M. Arifin, R. Nazar, I. Pop, Unsteady Micropolar fluid over a permeable curved stretching shrinking surface, Math. Prob. Eng., 2017 (2017)Article ID 3085249.

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M. Naveed, Z. Abbas, N. Sajid, J. Hasnain, Dual solutions in hydromagnetic viscous fluid flow past a shrinking curved surface, Arab. J. Sci. Eng., 43 (2018) 1189-1194.

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Computational Analysis of Heat Transfer in Fluids and Solids II

[6]

K. Anantha Kumar, J.V.R. Reddy, V. Sugunamma, N. Sandeep, Simultaneous solutions for MHD flow of Williamson fluid over a curved sheet with non-uniform heat source/sink, Heat Transf. Res. 50 (2019) 581-603.

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K. Anantha Kumar, J.V.R. Reddy, V. Sugunamma, N. Sandeep, MHD flow of chemically reacting Williamson fluid over a curved/flat surface with variable heat source/sink, Int. J. Fluid Mech. Res., In Press, (2019).

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Z. Abbas, M. Naveed, M. Sajid, Heat transfer for stretching flow over a curved surface with magnetic field, J. Eng. Thermophys. 22(4) (2013) 337-345.

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[10] N. Sandeep, A. Malvandi, Enhanced heat transfer in liquid thin film flow of non-Newtonian nanofluids embedded with graphene nanoparticles, Adv. Powder Tech. 27 (2016) 2448-2456, 2016. [11] K. Anantha Kumar, B. Ramadevi, V. Sugunamma, Impact of Lorentz force on unsteady bio convective flow of Carreau fluid across a variable thickness sheet with non-Fourier heat flux model, Def. Diff. Forum, 387 (2018) 474-497. [12] K. Anantha Kumar, V. Sugunamma, N. Sandeep, J.V.R. Reddy, Impact of Brownian motion and thermophoresis on bio convective flow of nanoliquids past a variable thickness surface with slip effects, Multi. Model. Mater. Struc. 15 (2019) 103-132. [13] M. Naveed, Z. Abbas, M. Sajid, MHD flow of micropolar fluid due to a curved stretching surface with thermal radiation, J. Appl. Fluid Mech. 9(1) (2016) 131-138. [14] N. Sandeep, C. Sulochana, B. R. Kumar, Unsteady MHD radiative flow and heat transfer of a dusty nanofluid over an exponentially stretching surface, Eng. Sci. Tech., Int. J., 19 (2016) 227240, 2016. [15] Z. Abbas, M. Naveed, M. Sajid, Hydrodynamic slip flow of nanofluid over a curved stretching surface with heat generation and thermal radiation, J. Mol. Liq. 215 (2016) 756-762. [16] T. Hayat, M.M. Rashidi, M. Imtiaz, A. Alsaedi, MHD convective flow due to a curved surface with thermal radiation and chemical reaction, J. Mol. Liq., 225 (2017) 482-489. [17] A. Zeeshan, A. Majeed, R. Ellahi, Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation, J. Mol. Liq. 215 (2016) 549-554. [18] K Anantha Kumar, V. Sugunamma and N. Sandeep, Impact of non-linear radiation on MHD non-aligned stagnation point flow of micropolar fluid over a convective surface, J. Non-Equil. Thermodyn (2018) https://doi.org/10.1515/jnet-2018-0022. [19] T. Hayat, S. Farooq, B. Ahmad, A. Alsaedi, Homogeneous-Heterogeneous reactions and heat source/sink effects in MHD flow of micropolar fluid with Newtonian heating in a curved channel, J. Mol. Liq. 223 (2016)469-488. [20] K. Mehmood, S. Hussain, M. Sagheet, Mixed convection flow with non-uniform heat source/sink in a doubly stratified magnetonanofluid, AIP Adv. 6 (2016) https://doi.org/10.1063/1.4955157. [21] K. Gangadhar, K.R. Venkata, O.D. Makinde, B.R. Kumar, MHD flow of a Carreau fluid past a stretching cylinder with Cattaneo-Christov heat flux using spectral relaxation method, Defect and Diffusion Forum, 387 (2018) 91–105.

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[22] T.S. Kumar, B.R. Kumar, O.D. Makinde, A.G. Vijaya Kumar, Magneto-convective heat transfer in micropolar nanofluid over a stretching sheet with non-uniform heat source/sink, Defect and Diffusion Forum, 387 (2018) 78–90. [23] A. Tetbirt, M. N. Bouaziz, M. T. Abbes, Numerical study of magnetic field effect on the velocity distribution field in a micro/macro-scale of a micropolar and viscous fluid in verticle channel, J. Mol. Liq. 216 (2016) 103-110. [24] M. Ramzan, M. Farooq, T. Hayat, J. D. Chung, Radiative and Joule heating effects in the MHD flow of a micropolar fluid with partial slip and convective boundary condition, J. Mol. Liq., 221 (2016) 394-400. [25] T.C. Chiam, Heat transfer in a fluid with variable thermal conductivity over a linearly stretching sheet, Acta Mech., 129(1/2) (1982) 63-72. [26] M.S. Abel, N. Mahesha, Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Appl. Math. Model. 32 (2008) 1965-1983. [27] S. K. Adegbie, O. K. Kọrikọ, I. L. Animasaun, Melting heat transfer effects on stagnation point flow of micropolar fluid with variable dynamic viscosity and thermal conductivity at constant vortex viscosity. Journal of the Nigerian Mathematical Society, 36 (1) (2016) 34-47. [28] M.Y. Malik, M. Bibi, F. Khan, T. Salahuddin, Numerical solution of Williamson fluid flow past a stretching cylinder and heat transfer with variable thermal conductivity and heat generation/absorption, AIP Adv., 6 (2016) Article ID: 035101. [29] K. Anantha Kumar, J.V.R. Reddy, V. Sugunamma, N. Sandeep, Impact of cross diffusion on MHD viscoelastic fluid flow past a melting surface with exponential heat source, Multi. Mod. Mat. Str., (2018) DOI 10.1108/MMMS-12-2017-0151. [30] D. Srinivasacharya, P. Jagadeeshwar, Effect of variable properties on the flow over an exponentially stretching sheet with convective thermal conditions, Model. Measure. Control B, 87, (2018) 7-14. [31] O.D. Makinde, I. L. Animasaun, Bio convection in MHD nanofluid flow with nonlinear thermal radiation and quartic autocatalysis chemical reaction past an upper surface of a paraboloid of revolution., Int. J. Therm. Sci. 109 (2016) 159-171. [32] O.D. Makinde, I. L. Animasaun, Thermophoresis and Brownian motion effects on MHD bioconvection of nanofluid with nonlinear thermal radiation and quartic chemical reaction past an upper horizontal surface of a paraboloid of revolution, J. Mol. Liq. 221 (2016) 733-743. [33] O. D. Makinde, N. Sandeep, I.L.Animasaun, M. S. Tshehla, Numerical exploration of CattaneoChristov heat flux and mass transfer in magnetohydrodynamic flow over various geometries, Defect Diff. Forum, 374 (2017) 67-82.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 92-106 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-04-03 Revised: 2019-10-02 Accepted: 2019-10-02 Online: 2020-05-28

MHD Flow of Non-Newtonian Molybdenum Disulfide Nanofluid in a Converging/Diverging Channel with Rosseland Radiation J. Raza1,a, F. Mebarek-Oudina2,b*, P. Ram3,c, S. Sharma3,d, Department of Mathematics and Statistics, ISP Multan, Pakistan

1

Department of Physics, Faculty of Sciences, University of 20 août 1955 - Skikda, B.P. 26 Route El-Hadaiek, Skikda 21000, Algeria

2

Department of Mathematics, National Institute of Technology, Kurukshetra, India

3

[email protected], [email protected],

b

Keywords: MHD, Molybdenum Disulfide nanofluid, Casson fluid, thermal radiation

Abstract: The steady two-dimensional flow of an incompressible non-Newtonian Molybdenum Disulfide nanofluid in the presence of source or sink between two stretchable or shrinkable walls under the influence of thermal radiation is investigated numerically. A generalized transformation is applied to convert the constructed set of partial differential equations (PDEs) into the system of nonlinear coupled ordinary differential equations (ODEs). The obtained system of ODEs is solved by using Runge-Kutta 4th and 5th order. The influence of physical parameters, shrinking/ stretching parameter, Casson parameter, Hartmann number, Reynolds number, solid volume fraction, opening angle of the channel and radiation parameter on the velocity and temperature distribution are observed for converging and diverging channels. It is noticed that thermal boundary layer thickness is diminished for increased thermal radiation resulting in gradual temperature fall. The results also reveal that velocity and temperature profile both are elevated on raising the stretching parameter and Hartmann number. A comparative analysis is made out to validate the present results. Introduction The fluid flow between two inclined plates is an essential and integral part of the physical model due to its importance in the various fields such as mechanical, industrial, biological, physical and several other applications. Blood flow through arteries and capillaries, flows through canals, cavity flow and river flow are the examples of converging /diverging channel. The viscous incompressible fluid between converging /diverging channel separated by an angle and driven by source or sink at the apex is called as Jeffery-Hamel flow after pioneering research by Jeffery [1] and Hamel [2], respectively. The solution of Jeffery-Hamel flow in terms of Jacobean elliptic function is given by Rosenhead [3]. Roy and Nayak [4] discussed steady incompressible laminar visco-elastic flow in converging/diverging channel with suction and injection. Such type of flow is affected due to the presence of magnetohydrodynamics (MHD) [5-7]. Dib et al. [8] used Duan–Rach approach (DRA) to obtain a purely analytical solution of MHD Jeffery-Hamel flow. Their findings establish the Duan– Rach approach (DRA) as an accurate and efficient alternative to the standard Adomian Decomposition Method (ADM). Rashidi et al. [9] investigated heat transfer of steady, incompressible water-based nanofluid flow over a stretching sheet in the presence of a transverse magnetic field with thermal radiation. They observed that the rising of the buoyancy parameter increases the velocity and decreases the temperature of the nanofluid. The entropy for an unsteady MHD nanofluid flow past a stretching permeable surface studied by Abolbashari et al. [10]. They reported that the increase in the nanoparticle volume fraction parameter, unsteadiness parameter, magnetic parameter, suction parameter, Reynolds number, Brinkman number, and Hartmann number causes an increase of the entropy generation number. The steady two-dimensional, viscous incompressible water based MHD nanofluid flow from a source or sink between two stretchable or shrinkable walls with thermal radiation effect is investigated analytically using Duan–Rach Approach (DRA) by Dogonchi et al. [11]. They reported that the fluid velocity and temperature distribution increase with the increasing of stretching parameter.

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During the past few decades, the study of boundary layer flow and heat transfer over a stretched surface has also attracted the researchers due to its significant applications in different fields like hot rolling, glass production, production of plastic sheets, artificial fibres and continuous stretching of plastic films, etc. A large amount of literature is available on the boundary layer flow of non- Newtonian fluids over the stretching/shrinking surfaces. The flow behaviour on the boundary for quiescent fluid past a stretching sheet was the first time analyzed by Crane [12]. From that point forward, many authors from everywhere throughout the world have utilized the said problem to present and formulate the different properties of flows over the shrinking surfaces. Mustafa et al. [13] considered the Bodewadt flow and heat transfer of nanofluid over a stretched plate. Numerical results revealed that stretching of the disk reduces the boundary layer thickness. Moreover, the skin friction coefficient and heat transfer rate enhance with the increase in volume fraction. The rheology of Copper-Kerosene nanofluid in a channel over stretching sheet under the influence of a magnetic field is examined by Reza et al. [14]. They used a 3-stage Lobatto IIIA formula for solving the non-linear ordinary differential equations [14-15]. The results show that the solid volume fraction reduces the velocity of nanofluid particles near the lower wall of the channel and increases the thermal boundary layer thickness of the channel. The study of unsteady nanofluids flow and heat intensification over rotating or stationary stretching surfaces under the combined effects of a magnetic field, variable viscosity is exhaustively examined and captured by Ram et al. [16-19]. They analyzed the variable viscosity effects on time dependent magnetic nanofluid flow past a stretchable rotating sheet. Alkasassbeh et al. [20] investigated heat transfer of convective fin with temperature-dependent internal heat generation using a hybrid block method. Nanofluid is the colloidal suspension of magnetic particles in a carrier liquid having surfactant coatings. The most important property of the nanofluids is the high thermal conductivity relative to the pure fluid. Due to this characteristic of nanofluids, it has a wide spread importance in many areas of engineering and industries such as thermal power generator system, nuclear reactors, storage device, gas turbine rotors [21], medical equipment, air cleaning machines, crystal growth process etc. Mebarek-Oudina [23] investigated the fluid flow and heat transfer of Titania nanofluid filled in a cylindrical annulus with different base fluids. He solved the Maxwell model for convective heat transfer by using finite volume method coupled with the SIMPLER algorithm. Sheikholeslami et al. [24] studied MHD nanofluid flow and heat transfer between two horizontal parallel plates in a rotating system. They calculated effective thermal conductivity and viscosity of the nanofluid by KKL (Koo– Kleinstreuer–Li) correlation. They reported that the magnitude of the skin friction coefficient increases as the magnetic parameter, rotation parameter and Reynolds number increase, but decreases as the nanoparticle volume fraction increases. Recently, Ram et al. [25] have discussed and enumerated several nanofluid problems in detail to analyze the flow behaviour and heat transfer characteristics. The natural convection heat transfer stability of fluid flow with applied magnetic field for deceleration of the fluid is investigated by Mebarek-Oudina et al. [26-28]. Hamid et al. [29] examined the influence of molybdenum disulfide (MoS2) nanoparticles shapes on a rotating flow of nanofluid along an elastic stretched sheet. They observed that with an increase in the thermal radiation parameter and Prandtl number, local Nusselt number decreases while it is increased with an increase in variable thermal conductivity parameter. Alam et al. [30] numerically studied the effects of magnetic field on entropy generation rate of conducting fluid flow in convergent-divergent channel. Many other numerical investigations with respect to fluid flow and heat transfer in a convergentdivergent channel can be reported in [31-34]. The aim of the present paper is to study the two-dimensional flow of an incompressible nonNewtonian viscous fluid in the presence of source or sink between two stretchable or shrinkable walls. The effect of thermal radiation is to be taken into account. The resulting nonlinear coupled equations of momentum and energy are simplified using similarity transformation. The influence of the stretching/shrinking parameter, Hartmann number, Reynolds number, volume fraction of nanofluid, radiation parameter, etc. are investigated on flow and heat transfer characteristics for converging and diverging channels.

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Computational Analysis of Heat Transfer in Fluids and Solids II

Model Problem Consider the steady two-dimensional fully developed flow of an incompressible conductive viscous fluid from a source or sink between two stretchable or shrinkable walls that meet at an angle 2α as shown in Fig. 1.

Fig. 1: Physical model of the proposed problem. The constitutive equation for the Casson fluid can be written as [35, 40, 41] 𝜏𝜏𝑦𝑦 ⎧ 2 �𝜇𝜇𝐵𝐵 + � 𝑒𝑒𝑖𝑖𝑖𝑖 , 𝜋𝜋 > 𝜋𝜋𝑐𝑐 , ⎪ √2𝜋𝜋 (1) 𝜏𝜏𝑖𝑖𝑖𝑖 = ⎨2 �𝜇𝜇𝐵𝐵 + 𝜏𝜏𝑦𝑦 � 𝑒𝑒𝑖𝑖𝑖𝑖 , 𝜋𝜋 < 𝜋𝜋𝑐𝑐 . ⎪ �2𝜋𝜋𝑐𝑐 ⎩ where 𝜇𝜇𝐵𝐵 is the plastic dynamic viscosity of the non-Newtonian fluid, 𝜏𝜏𝑦𝑦 is the yield stress of the fluid, 𝜋𝜋 is the product of the component of deformation rate with itself, and 𝜋𝜋𝑐𝑐 is critical value of 𝜋𝜋 based on non- Newtonian model. The walls are supposed to radially stretch or shrink in accordance with 𝑢𝑢 = 𝑢𝑢𝑤𝑤 = 𝑠𝑠⁄𝑟𝑟 , (2) where s is the stretching/ shrinking rate. The walls are considered to be divergent if 𝛼𝛼 > 0 and convergent if 𝛼𝛼 < 0. We assume that the velocity is purely radial and depends on r and θ and further there is no magnetic field in the z-direction. The reduced forms of continuity, Navier–Stokes and energy equations in polar coordinates are: 1 𝜕𝜕(𝑟𝑟𝑟𝑟) (𝑢𝑢𝑢𝑢) = 0, 𝜌𝜌𝑛𝑛𝑛𝑛 𝑟𝑟 𝜕𝜕𝜕𝜕 𝜎𝜎𝑛𝑛𝑛𝑛 𝐵𝐵°2 1 𝜕𝜕𝜕𝜕 𝜇𝜇𝑛𝑛𝑛𝑛 𝜕𝜕𝜕𝜕 1 𝜕𝜕 2 𝑢𝑢 𝑢𝑢 1 𝜕𝜕𝜕𝜕 1 𝜕𝜕 2 𝑢𝑢 �1 + � � 2 − + 𝑢𝑢 =− + + �− 𝑢𝑢 𝜌𝜌𝑛𝑛𝑛𝑛 𝜕𝜕𝜕𝜕 𝜌𝜌𝑛𝑛𝑛𝑛 𝑟𝑟 𝑟𝑟 𝜕𝜕𝜕𝜕 𝑟𝑟 2 𝜕𝜕𝜃𝜃 2 𝜕𝜕𝜕𝜕 𝛽𝛽 𝜕𝜕𝑟𝑟 𝜌𝜌𝑛𝑛𝑛𝑛 𝑟𝑟 2 1 𝜕𝜕𝜕𝜕 2 𝜇𝜇𝑛𝑛𝑛𝑛 1 𝜕𝜕𝜕𝜕 �1 + � − 2 , 𝑟𝑟 𝜌𝜌𝑛𝑛𝑛𝑛 𝜕𝜕𝜕𝜕 𝑟𝑟 𝜌𝜌𝑛𝑛𝑛𝑛 𝛽𝛽 𝜕𝜕𝜕𝜕

(3)

(4)

(5)

𝑘𝑘𝑛𝑛𝑛𝑛 𝜕𝜕𝜕𝜕 𝜕𝜕 2 𝑇𝑇 1 𝜕𝜕𝜕𝜕 1 𝜕𝜕 2 𝑇𝑇 1 𝜕𝜕 1 𝜕𝜕 �𝑢𝑢 � = � 2+ + 2 2� − �𝑟𝑟𝑞𝑞𝑟𝑟,𝑟𝑟𝑟𝑟𝑟𝑟 � − �𝑟𝑟𝑞𝑞𝜃𝜃,𝑟𝑟𝑟𝑟𝑟𝑟 � 𝑟𝑟 𝜕𝜕𝜕𝜕 𝑟𝑟 𝜕𝜕𝜃𝜃 𝜕𝜕𝜕𝜕 �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛 𝜕𝜕𝑟𝑟 �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛 𝑟𝑟 𝜕𝜕𝜕𝜕 �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛 𝑟𝑟 2 𝜕𝜕𝜕𝜕 +

𝜇𝜇𝑛𝑛𝑛𝑛

�𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛

1 𝜕𝜕𝜕𝜕 2 𝑢𝑢 2 1 𝜕𝜕𝜕𝜕 2 �1 + � �2 �� � + � � � + � � �, 𝛽𝛽 𝜕𝜕𝜕𝜕 𝑟𝑟 𝑟𝑟 𝜕𝜕𝜕𝜕

(6)

where u=u(r, θ) is the velocity, p is the fluid pressure, σ is the conductivity of the fluid, B0 is the electromagnetic induction, T is the fluid temperature and qrad is the radiative heat flux. Using Roseland approximation for radiation (see Rashidi et al. [36]) we have:

Defect and Diffusion Forum Vol.401

𝑞𝑞𝑟𝑟,𝑟𝑟𝑟𝑟𝑟𝑟 = − �

95

4𝜎𝜎 ∗ 𝜕𝜕𝑇𝑇 4 , ∗ � 3𝑘𝑘𝑛𝑛𝑛𝑛 𝜕𝜕𝜕𝜕

(7)

4𝜎𝜎 ∗ 𝜕𝜕𝑇𝑇 4 𝑞𝑞𝜃𝜃,𝑟𝑟𝑟𝑟𝑟𝑟 = − � ∗ � . (8) 3𝑘𝑘𝑛𝑛𝑛𝑛 𝜕𝜕𝜕𝜕 Here σ* is the Stefan–Boltzmann constant and knf* is the mean absorption coefficient of the nanofluid. Further we assume that the temperature difference within the flow is such that T4 may be expanded in a Taylor series. Hence, expanding T4 about T∞ and neglecting higher order terms we get: 𝑇𝑇 4 ≅ 4𝑇𝑇∞3 𝑇𝑇 − 3𝑇𝑇∞4 .

Using Eqs. (7) – (9) in (6), we get: 𝜕𝜕𝜕𝜕

�𝑢𝑢 𝜕𝜕𝜕𝜕 � =

𝜇𝜇𝑛𝑛𝑛𝑛

�𝜌𝜌𝐶𝐶𝑝𝑝 �

𝑛𝑛𝑛𝑛

where ρ nf

𝑘𝑘𝑛𝑛𝑛𝑛

�𝜌𝜌𝐶𝐶𝑝𝑝 �

1

𝑛𝑛𝑛𝑛

𝜕𝜕2 𝑇𝑇

1 𝜕𝜕𝜕𝜕

1 𝜕𝜕2 𝑇𝑇

3 16𝜎𝜎∗ 𝑇𝑇∞

�𝜕𝜕𝑟𝑟 2 + 𝑟𝑟 𝜕𝜕𝜕𝜕 + 𝑟𝑟 2 𝜕𝜕𝜃𝜃2 � + 3𝑘𝑘 ∗ 𝜕𝜕𝜕𝜕 2

𝑢𝑢 2

1 𝜕𝜕𝜕𝜕 2

�1 + 𝛽𝛽� �2 ��𝜕𝜕𝜕𝜕 � + �𝑟𝑟 � � + �𝑟𝑟 𝜕𝜕𝜕𝜕� �,

𝑛𝑛𝑛𝑛 �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛

𝜕𝜕2 𝑇𝑇

1 𝜕𝜕𝜕𝜕

3 16𝜎𝜎∗ 𝑇𝑇∞

�𝜕𝜕𝑟𝑟 2 + 𝑟𝑟 𝜕𝜕𝜕𝜕 � + 3𝑘𝑘 ∗

𝑛𝑛𝑛𝑛 �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛

𝜕𝜕2 𝑇𝑇

(9)

�𝜕𝜕𝜃𝜃2 � +

(10)

is the effective density of the nanofluid, µnf is the effective dynamic viscosity of the

(

nanofluid, ρ C p

)nf is the heat capacitance and knf

is the thermal conductivity of the nanofluid, are

given as in Rashidi et al. [36] 𝜌𝜌𝑛𝑛𝑛𝑛 = 𝜌𝜌𝑓𝑓 (1 − 𝜑𝜑) + 𝜌𝜌𝑠𝑠 , 𝜇𝜇𝑓𝑓 𝜇𝜇𝑛𝑛𝑛𝑛 = , (1 − 𝜑𝜑)2.5

�𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛 = �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑓𝑓 (1 − 𝜑𝜑) + �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑠𝑠 𝜑𝜑, 𝑘𝑘𝑛𝑛𝑛𝑛 𝑘𝑘𝑠𝑠 + 2𝑘𝑘𝑓𝑓 − 2𝜑𝜑�𝑘𝑘𝑓𝑓 − 𝑘𝑘𝑠𝑠 � = , 𝑘𝑘𝑓𝑓 𝑘𝑘𝑠𝑠 + 2𝑘𝑘𝑓𝑓 + 2𝜑𝜑�𝑘𝑘𝑓𝑓 − 𝑘𝑘𝑠𝑠 �

σ  3  s − 1 ϕ σ f  σ nf   , = 1+ σf  σs   σs  + 2 −  − 1 ϕ   σ f  σ f   

(11)

(12)

(13)

(14𝑎𝑎) (14𝑏𝑏)

Here 𝜑𝜑 is the solid volume fraction, 𝜑𝜑𝑠𝑠 is for nano-solid particles, 𝜑𝜑𝑓𝑓 is for base fluid, 𝜌𝜌𝑛𝑛𝑛𝑛 is effective density, 𝜇𝜇𝑛𝑛𝑛𝑛 is the effective dynamic viscosity, �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛 is heat capacitance and 𝑘𝑘𝑛𝑛𝑛𝑛 thermal conductivity of the nanofluid. The governing equations are accompanied by the boundary conditions, due to the symmetry assumption at the channel centreline (θ=0) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑢𝑢𝑐𝑐 =0= , 𝑢𝑢 = , (15) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑟𝑟 and due to the stretching/shrinking convergent/divergent wall condition at the plates (θ=±α)

𝑠𝑠 𝑇𝑇𝑤𝑤 , 𝑇𝑇 = 2 , (16) 𝑟𝑟 𝑟𝑟 where 𝑢𝑢𝑐𝑐 is the centreline rate of movement and Tw is the constant wall temperature. We remark that under general conditions where asymmetry is also allowed, infinitely many solutions exist [37]. To preserve the symmetry with respect to the centreline, we impose the same stretching/shrinking rates on both walls. Considering only radial flow, Eq. (2) implies that 𝑢𝑢 = 𝑢𝑢𝑤𝑤 =

𝑓𝑓(𝜃𝜃) = 𝑟𝑟𝑟𝑟(𝑟𝑟, 𝜃𝜃).

(17)

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Computational Analysis of Heat Transfer in Fluids and Solids II

Using the dimensionless variables: 𝑓𝑓(𝜂𝜂) =

𝑓𝑓(𝜃𝜃) , 𝑓𝑓𝑚𝑚𝑚𝑚𝑚𝑚

𝜃𝜃(𝜂𝜂) = 𝑟𝑟 2

𝑇𝑇 𝜃𝜃 where 𝜂𝜂 = . 𝑇𝑇𝑤𝑤 𝛼𝛼

(18)

Substituting these into the governing equations and eliminating the pressure term yield the nonlinear third-order ordinary differential equation for the flow from the radial momentum equation: 𝑓𝑓 ′′′ (𝜂𝜂) + 2𝛼𝛼𝛼𝛼𝐴𝐴1 (1 − 𝜑𝜑)2.5 𝑓𝑓(𝜂𝜂)𝑓𝑓 ′ (𝜂𝜂) + (4 − (1 − 𝜑𝜑)2.5 𝐻𝐻𝐻𝐻𝛽𝛽0 )𝛼𝛼 2 𝑓𝑓 ′ (𝜂𝜂) = 0, 𝐴𝐴2 (1 + 𝑁𝑁)𝜃𝜃 ′′ (𝜂𝜂) + 2𝛼𝛼 2 � 𝑃𝑃𝑃𝑃𝑃𝑃(𝜂𝜂) + 2 + 2𝑁𝑁� 𝜃𝜃(𝜂𝜂) 𝐴𝐴3 1 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 2 �4𝛼𝛼 2 𝑓𝑓 2 (𝜂𝜂) + 𝑓𝑓 ′ (𝜂𝜂)� = 0 , + 2.5 𝐴𝐴3 (1 − 𝜑𝜑) 𝑅𝑅 �1 + 1� � 𝛽𝛽 where 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 are defined as: 𝜌𝜌𝑛𝑛𝑛𝑛 𝜌𝜌𝑠𝑠 𝐴𝐴1 = = (1 − 𝜑𝜑) + 𝜑𝜑 , 𝜌𝜌𝑓𝑓 𝜌𝜌𝑓𝑓 �𝜌𝜌𝐶𝐶𝑝𝑝 �𝑛𝑛𝑛𝑛

𝐴𝐴2 =

�𝜌𝜌𝐶𝐶𝑝𝑝 �𝑓𝑓

= (1 − 𝜑𝜑) +

�𝜌𝜌𝐶𝐶𝑝𝑝 �𝑠𝑠

�𝜌𝜌𝐶𝐶𝑝𝑝 �𝑓𝑓

(19) (20) (21)

𝜑𝜑 ,

(22)

𝜅𝜅𝑛𝑛𝑛𝑛 𝜅𝜅𝑠𝑠 + 2𝜅𝜅𝑓𝑓 − 2𝜑𝜑�𝜅𝜅𝑓𝑓 − 𝜅𝜅𝑠𝑠 � = , 𝜅𝜅𝑓𝑓 𝜅𝜅𝑠𝑠 + 2𝜅𝜅𝑓𝑓 + 2𝜑𝜑�𝜅𝜅𝑓𝑓 − 𝜅𝜅𝑠𝑠 �

𝐴𝐴3 =

(23)

subject to the appropriate boundary conditions: 𝑓𝑓(0) = 0,

where C =

𝑓𝑓 ′ (0) = 0,

𝜃𝜃 ′ (0) = 0,

𝑓𝑓(1) = 𝐶𝐶,

𝜃𝜃(1) = 1

(24)

s 𝛼𝛼𝑢𝑢𝑐𝑐 is the stretching ( C > 0 ) or shrinking ( C < 0 ) parameter, 𝑅𝑅 = �𝜈𝜈𝑓𝑓 is the Reynolds uc

number, Pr =

uc ρ C p kf

is the Prandtl number, Ec =

α uc2

C pf Tw

is the Eckert number and N =

16σ *T∞3 * knf 3knf

is the radiation parameter. It is noted that setting C to zero at this stage leads to stationary wall condition for the traditional Jeffery–Hamel flow. The elliptic function solutions as given in the literature may be fulfilled here for nonzero C, but it is thought that this is not very convenient. Numerical Results and Discussion The resultant self-similar nonlinear ordinary differential equations (ODEs) (19) and (20) are solved numerically using Runge-Kutta Method of 4th and 5th order subject to the boundary conditions (24). A computer-aided mathematical software Matlab is used in order to compute the numerical results. For this, we first transformed our ODEs (19)-(20) into 1st order initial value problem (IVPs) using reduction of order and then apply the shooting method to compute the missing initial conditions [35-41]. Once the missing initial conditions are found then apply Runge-Kutta Method of 4th and 5th order to compute the rest of the solutions. Table 1: Thermophysical properties of water and nanoparticle [42]. 𝝆𝝆 𝒌𝒌� Properties 𝜷𝜷 × 𝟏𝟏𝟏𝟏𝟓𝟓� �𝒌𝒌𝒌𝒌. 𝒎𝒎−𝟑𝟑 𝑪𝑪𝒑𝒑� −𝟏𝟏 𝑾𝑾. 𝒎𝒎 . 𝑲𝑲 𝑱𝑱. 𝒌𝒌𝒌𝒌−𝟏𝟏 . 𝑲𝑲 𝑲𝑲−𝟏𝟏 Water 991.1 4179 0.613 21 3 MoS2 397.21 904.4 2.8424 5.06 × 10

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Table 2: Comparison of the numerical values those from Ref. [37] and [11] 𝜶𝜶

𝑹𝑹

𝜷𝜷

𝑯𝑯𝑯𝑯

-5

50

∞

0

𝑪𝑪

-2 -1 0 1 2

Ref. [11] −5.130921689 −4.652183982 −2.833915413 0 3.654305033

𝒇𝒇′ (𝜼𝜼) Ref. [37]

−5.1309222926 −4.6521591354 −2.8339514330 0 3.669711185

Present

-5.13093174 -4.65215094 -2.83394249 0 3.66969312

Table 2 illustrates the comparison of skin friction results for a special case of this present study to the one already reported in the literature for 𝛽𝛽 => ∞, 𝛼𝛼 = −5, 𝐻𝐻𝐻𝐻 = 0 and 𝑅𝑅 = 50. It is seen that our numerical results are in perfect agreement with that of [11, 37](see table 2). This validates the numerical accuracy of our results. Table 3 shows the effects of stretching/shrinking parameter variation on the skin friction coefficients. It is observed that skin friction coefficient decreases monotonically as stretching/shrinking parameter varies from −1 to 1. Effects of Casson parameter on skin friction coefficient for diverging/converging channel are presented in tables 4-5. The divergent channel skin friction decreases gradually with a boost in the strength of Casson parameter. Meanwhile, the trend is opposite for the case of convergent channel. A rise in Reynolds number lessens the skin friction for divergent channel, while the skin friction amplifies for convergent channel as the Reynolds number increases (see tables 6-7). The effects of stretching/shrinking parameter on the heat transfer rate for the different values of thermal radiation parameter are shown in table 8. It is observed from the numerical values that heat transfer rate increases with an increase in the stretching/shrinking parameter for 𝑁𝑁 = 0.1 and 𝑁𝑁 = 0.6. Table 3: Effects of stretching/shrinking parameter on skin friction coefficient 𝑓𝑓 ′ (1) for 𝑅𝑅 = 50, 𝛽𝛽 = 0.3, 𝛼𝛼 = 2.5, 𝐻𝐻𝐻𝐻 = 100. 𝐶𝐶

-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

𝜑𝜑

0.05

0.10

𝑓𝑓 ′ (1)

3.955548235 2.923218199 1.919567677 0.945016763 0 3.962228864 2.934703416 1.931619671 0.953280215 0

Table 4: Effects of Casson parameter on skin friction coefficient 𝑓𝑓 ′ (1)for 𝑅𝑅 = 50, 𝜑𝜑 = 0.05, 𝛼𝛼 = 2.5, 𝐻𝐻𝐻𝐻 = 100 (For Divergent Channel). 𝛽𝛽

0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5

𝐶𝐶 -0.5

0.5

𝑓𝑓 ′ (1)

2.923217499 2.87566349 2.843325628 2.819918989 2.802191964 0.945016763 0.910188238 0.886152941 0.868566617 0.855141161

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Table 5: Effects of Casson parameter on skin friction coefficient 𝑓𝑓 ′ (1) for 𝑅𝑅 = 50, 𝜑𝜑 = 0.05, 𝛼𝛼 = −2.5, 𝐻𝐻𝐻𝐻 = 100 (For Convergent Channel). 𝛽𝛽

𝐶𝐶 -0.5

0.3 0.6 0.9 1.2 1.5 0.3 0.6 0.9 1.2 1.5

0.5

𝑓𝑓 ′ (1)

3.095921404 3.1561325 3.197431546 3.227512987 3.250396046 1.060104001 1.097089388 1.122144301 1.140237723 1.15391039

Table 6: Effects of Reynolds number on skin friction coefficient 𝑓𝑓 ′ (1) for 𝛽𝛽 = 0.3, 𝜑𝜑 = 0.05, 𝛼𝛼 = 2.5, 𝐻𝐻𝐻𝐻 = 100 (For Divergent Channel). 𝑅𝑅 50 100 150 200 50 100 150 200

𝐶𝐶 -0.5 0.5

𝑓𝑓 ′ (1)

2.923219499 2.838028458 2.753744867 2.67047588 0.945017463 0.885922947 0.825777418 0.764578312

Table 7: Effects of Reynolds number on skin friction coefficient 𝑓𝑓 ′ (1) for 𝛽𝛽 = 0.3, 𝜑𝜑 = 0.05, 𝛼𝛼 = −2.5, 𝐻𝐻𝐻𝐻 = 100 (For Convergent Channel). 𝑅𝑅 50 100 150 200 50 100 150 200

𝐶𝐶 -0.5 0.5

𝑓𝑓 ′ (1)

3.095920304 3.183227965 3.271039334 3.359258276 1.060103101 1.11612499 1.171155292 1.225216141

Table 8: Effects of stretching/shrinking parameter on heat transfer rate −𝜃𝜃 ′ (1) for 𝛽𝛽 = 0.3, 𝜑𝜑 = 0.05, 𝛼𝛼 = 2.5, 𝐻𝐻𝐻𝐻 = 100. 𝐶𝐶

-1 -0.5 0 0.5 1

𝑁𝑁 = 0.1

0.01488701 0.017230463 0.01977974 0.022533638 0.025490855

𝑁𝑁 = 0.6

0.012611426 0.014220092 0.015969692 0.017859329 0.019888112

Figures 2 and 3 elucidate the effects of stretching parameter C on the velocity 𝑓𝑓(𝜂𝜂) and the temperature distribution 𝜃𝜃(𝜂𝜂) of converging and diverging channels for the fixed values of the various physical parameters. It is observed that both velocity and temperature profiles are enhanced with an increase the in stretching parameter. This may be attributed to a rise in thermal boundary layer

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thickness as the internal heating take place due to an increase in stretching parameter. Effects of Casson parameter 𝛽𝛽 on velocity profile 𝑓𝑓(𝜂𝜂) for the diverging and converging channel can be seen from Figures 4 and 5 respectively. From Fig. 4, it is observed that velocity boundary layer thickness decreases which lead the profile bending downwards as Casson parameter 𝛽𝛽 increase. Opposite trend is observed for the case of the converging channel (See Fig.5). Figures 6-9 elucidate the effect of the Hartmann number Ha on velocity and temperature profile 𝜃𝜃(𝜂𝜂) for the case of divergent and convergent channel respectively. It is observed that velocity and temperature profile increase by increasing the strength of the Hartmann number Ha. From Fig.6, it is evident that transport rate reduced by increasing the numerical values of the Hartmann number Ha. Physically, we can conclude that it is due to the transverse magnetic field which is applied normally to the channel and opposes the phenomenon of transport. It is because an increase in the Hartmann number produces more Lorentz force which in turn causes resistance in the fluid, thereby restricting the movement of the fluid particles within the said topology. The effects of Reynolds number R on velocity profile 𝑓𝑓(𝜂𝜂) for both converging and diverging channel for the case of shrinking channel is depicted from Figures 10-11. It is depicted from these profiles that an increase in the Reynolds number R fluid velocity decreases for diverging channel and reversal flow exists. However, totally opposite behaviour is observed for the case of converging channel. Figures 12 and 13 show the effect of 𝛼𝛼 on velocity profile 𝑓𝑓(𝜂𝜂) for stretching and shrinking channel walls respectively. It is clearly seen from these profiles that fluid velocity increases as 𝛼𝛼 → −∞ and decreases as 𝛼𝛼 → ∞. In the same way, thermal boundary layer thickness increases gradually by increasing the variations of 𝛼𝛼 for shrinking and stretching channel walls (See Figs. 14-15). Figures 16-19 explain the effects of solid volume fraction 𝜑𝜑 on velocity and temperature profile for converging/diverging channel and stretching/shrinking channel walls respectively. It is observed from these profile that increase in the strength of the solid volume fraction 𝜑𝜑 in stretching/shrinking divergent channel lead to decrease the velocity of the fluid particles and temperature distribution. From the physical point of view, we can conclude that as the nanoparticles injected in the base fluid, the density of the fluid increases gradually, and fluid become thicker so that it can have more difficulty to move within the channel freely. Figures 20-21 show the effect of thermal radiation N on temperature profile 𝜃𝜃(𝜂𝜂) for stretching/shrinking divergent/convergent channel respectively. It is observed from these figures that thermal boundary layer thickness reduced as thermal radiation N increased, so that temperature distribution falls gradually.

Fig. 2: Effect of Shrinking parameter on velocity profile.

Fig. 3: Effect of Shrinking parameter on Temperature profile.

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Fig. 4: Effect of Casson Parameter on velocity profiles for divergent channel.

Fig. 5: Effect of Casson parameter on velocity profiles for convergent channel.

Fig. 6: Effect of Hartmann number on velocity profile for divergent channel.

Fig. 7: Effect of Hartmann number on temperature profile for divergent channel.

Fig. 8: Effect of Hartmann number on velocity profile for convergent channel.

Fig. 9: Effect of Hartmann number on temperature profile for convergent channel.

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Fig. 10: Effect of Reynolds number on velocity profile for shrinking case.

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Fig. 11: Effect of Reynolds number on temperature profile for shrinking case.

Fig. 12: Effect of 𝛼𝛼 on velocity profile C=0.5. Fig. 13: Effect of 𝛼𝛼 on velocity profile C=-0.5.

Fig. 14: Effect of 𝛼𝛼 on temperature profile for divergent channel.

Fig. 15: Effect of 𝛼𝛼 on temperature profile convergent channel.

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Fig. 16: Effect of solid volume fraction on velocity for C=0.5.

Fig. 18: Effect of solid volume fraction on velocity for divergent channel.

Fig. 20: Effect of Radiation parameter on temperature profile for convergent channel

Fig. 17: Effect of solid volume fraction on velocity for C= -0.5.

Fig. 19: Effect of solid volume fraction on temperature for divergent channel.

Fig. 21: Effect of Radiation parameter on temperature profile for divergent channel.

Nomenclature Latin Symbols Radial velocity, (m/s) 𝑢𝑢 𝑢𝑢𝑐𝑐 𝑇𝑇

𝑇𝑇𝑤𝑤

Centerline rate of movement Fluid temperature (K or °C) Constant wall temperature

Greek symbols 𝛼𝛼 Channel angle, divergent (𝛼𝛼 > 0) or convergent (𝛼𝛼 < 0), deg Electrical conductivity, (Ω⋅m) σ Plastic dynamic viscosity of the non𝜇𝜇𝐵𝐵 Newtonian fluid Yield stress of the fluid 𝜏𝜏𝑦𝑦

Defect and Diffusion Forum Vol.401

T∞ 𝑝𝑝 𝑠𝑠

Temperature of free fluid stream (K or °C) Fluid pressure (Pa) Stretching/shrinking rate

𝑟𝑟

Characteristic length (m)

𝐶𝐶𝑝𝑝

Specific heat at constant pressure (J/(kg K)

B0 𝐶𝐶

Electromagnetic induction (T) Stretching (C>0) or Shrinking (C 1). The thick lines correspond to skin tissue blood flow velocity while the das lines relate to the surrounding air flow velocity. In figure 2, it is observed that a positive increase in thermal buoyancy parameter Gr1 due to a rise in the temperature gradient between the arterial blood and the surrounding environment boosts the tissue blood flow rate and enhance the velocity of blood towards the skin tissue. Consequently, perspiration may occur, leading to more air flow from the surroundings toward the skin surface for cooling process. Similar trend is observed in figure 3 with increases thermal Biot number Bi due to enhance heat transfer rate from the arteries to the tissue, leading to a rise in the skin tissue blood flow and air flow from the surrounding environment. Moreover, an increase in tissue permeability (with Da increases) improves the blood flow rate into the skin tissue as well as air flow towards the skin surface as shown in figure 4. This scenario may occur whenever tissue capillaries dilate due to environmental influence. Interestingly, a fall in the surrounding environment temperature as shows in figure 5, lessens the skin tissue blood flow rate and the surrounding air flow towards the skin surface also diminishes. Figure 6 and 7 reveal that both the suction Reynolds number (Re) and tissue blood flow Eckert number (Ec) enhance the blood flow rate and air flow towards the skin surface. Moreover, an increase in Re boosts the blood suction rate from the arteries into the tissue while an increase in Ec enhances the internal heat generating due to amplification of velocity gradient, consequently, the blood flow rate into the skin tissue increases.

Fig.2: Velocity Profiles with increasing Gr1. Fig.3: Velocity Profiles with increasing Bi.

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Fig.4: Velocity Profiles with increasing Da. Fig.5: Velocity Profiles with increasing δ.

Fig.6: Velocity Profiles with increasing Re. Fig.7: Velocity Profiles with increasing Ec. Effects of Parameters Variation on Temperature Profiles Figures 8-13 illustrate the effects of various biophysical parameters on temperature profiles both within the skin tissue in the region 0 ≤η≤ 1 and the surrounding air in the region η > 1. The thick lines correspond to skin tissue temperature profiles while the das lines indicate the temperature distribution within the surrounding air in the environment. Figures 8-10 revealed that the tissue temperature rises as the parameter values of Gr1, Bi, and Da increase. This is expected, since as the buoyancy parameter Gr1, thermal Biot number Bi and Darcy number Da increase, the temperature gradient between the arterial blood and the surrounding environment escalations, consequently, the tissue capillaries dilate and heat flows from the arteries to the tissues. Meanwhile, a fall in surrounding air temperature lessens the skin tissue temperature as shown in figure 11. The low ambient air temperature serves as a heat sink for the skin tissue, leading to a rise in body heat loss. In figures 12 and 13, it is observed that the skin tissue temperature ascends with increasing suction Reynolds number Re and Eckert Ec. This may be attributed to a boost in internal heat generation due to an elevation in blood velocity gradient within the skin tissue coupled with an increase in surrounding air temperature.

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Fig.8: Temperature Profiles with increasing Gr1. Fig.9: Temperature Profiles with increasing Bi.

Fig.10: Temperature Profiles with increasing Da. Fig.11: Temperature Profiles with increasing δ.

Fig.12: Temperature Profiles with increasing Re. Fig.13: Temperature Profiles with increasing Ec.

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Human Skin Friction Coefficient and Nusselt Number Figures 14-17 demonstrate the influence of biophysical parameters on the coefficient of friction and the heat transfer rate at the surface of skin tissue (η=1) due to interaction between the skin tissue surface and the surrounding environmental conditions. In figures 14 and 15, it is observed that the friction at the skin tissue surface escalations with increasing values of suction Reynolds number Re, thermal buoyancy parameter Gr1, Eckert number Ec and Darcy number Da. As these parameters increase, both the tissue blood flow rate and surrounding air flow velocity gradient at the skin surface amplified, consequently, the shear stress rate at the skin surface rises. This extreme scenario may damage unprotected skin surface leading to a pathological condition. Moreover, it is interesting to note that the friction at the skin surface lessened with a rise in the heat sink parameter δ and thermal Biot number Bi. This may be attributed to a decline in the surrounding air flow velocity gradient at the skin tissue surface. The effects of various parameters on heat transfer rate at the skin surface are depicted in figures 16 and 17. A boost in Nusselt number is observed with a rise in parameter values of Da, Re, δ, Gr1 and Ec. As these parameters increases, the temperature gradient at the skin tissue surface rises, consequently, the rate of heat loss to the ambient surrounding from the skin surface increases. The trend is opposite with increasing values of thermal Biot number, leading to a decrease in Nusselt number at the skin tissue surface.

Fig. 14: Skin friction with increasing Re, Da, δ. Fig. 15: Skin friction with increasing Gr1, Ec, Bi.

Fig. 16: Nusselt number with increasing Re, Da, δ. Fig. 17: Nusselt number with increasing Gr1, Ec, Bi.

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Conclusions A two phase flow model that described the impact of thermal buoyancy caused by environmental temperature variation on human skin tissue’s blood flow rate and thermoregulation is theoretically investigated. Both analytical and numerical methods are employed to tackle the model boundary value problem. Our results can be summarised as follows: • Increase in Bi, Da, Re, Gr1, Ec boost the tissue blood flow velocity coupled with surrounding air flow rate while increase in δ lessens it. • Increase in Bi, Da, Re, Gr1, Ec boost the tissue temperature coupled with surrounding air flow temperature while increase in δ lessens it. • Increase in Da, Re, Gr1, Ec boost skin tissue surface friction while increase in Bi, δ lessens it. • Increase in Da, Re, δ, Gr1, Ec enhances Nusselt number while increase in Bi lessens it Finally, since the biophysical and pathological conditions of human body may be clinically established through skin temperature, adequate knowledge of skin tissue blood flow rate and thermoregulation characteristics coupled with its dependent on environmental influence are essential in medical treatment of many illnesses. Our results will no doubt be of biomedical interest. References [1]

P.O. Fanger, Thermal comfort, Danish Technical Press, Copenhagen, (1970).

[2]

A. B. Howmik, R. Singh, R. Repaka, S. C. Mishra, Conventional and newly developed bioheat transport models in vascularised tissues: a review, J. Therm. Biol. 38 (2013) 107–125.

[3]

S. B. Wilson, V. A. Spence, A tissue heat transfer model for relating dynamic skin temperature changes to physiological parameters, Phys. Med. Biol. 33 (1988) 895–912.

[4]

M. P. Çetingül, C. Herman, A heat transfer model of skin tissue for the detection of lesions: sensitivity analysis, Phys. Med. Biol. 55 (2010) 5933–5951.

[5]

Z. S. Deng, J. Liu, Mathematical modelling of temperature mapping over skin surface and its implementation in the male disease diagnostics, Comput. Biol. Med. 34 (2004) 495–521.

[6]

O. Prakash, O.D. Makinde, S. P. Singh, N. Jain, D. Kumar, Effects of stenosis on nonNewtonian flow of blood in blood vessels. International Journal of Biomathematics, 8(1) (2015) #1550010 (pp1-13).

[7]

O. D. Makinde, Effect of variable viscosity on arterial blood flow. Far East Jour. Appl. Maths. 4(1) (2000) 43-58.

[8]

J. Prakash, O. D. Makinde, Radiative heat transfer to blood flow through a stenotic artery in the presence of erythrocytes and magnetic field. Latin American Applied Research, 41 (2011) 273277.

[9]

D.A. Nield, A. Bejan, Convection in porous media, Springer- Verlag, New-York (1992).

[10] T. Chinyoka, O. D. Makinde, Computational dynamics of arterial blood flow in the presence of magnetic field and thermal radiation therapy. Advances in Mathematical Physics, 2014 (2014) Article ID 915640 (pp1-9). [11] S. Das, T.K. Pal, R.N. Jana, O. D. Makinde, Temperature response in living skin tissue subject to convective heat flux. Defect and Diffusion Forum, 387 (2018) 1–9. [12] K. P. Ivanov, The development of the concepts of homeothermy and thermoregulation, J. Therm. Biol. 31 (2006) 24-29. [13] O. D. Makinde, E. Osalusi, Second law analysis of laminar flow in a channel filled with saturated porous media, Entropy, 7(2) (2005)148-160.

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[14] O. D. Makinde, Non-perturbative solutions of a nonlinear heat conduction model of the human head, Scientific Research and Essays, 5 (6) (2010) 529-532. [15] H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm,J. Appl. Physiol.1 (1948) 93–122. [16] E. H. Wissler, A mathematical model of the human thermal system, Bulletin of Mathematical Biophysics 62 (1964) 66-78. [17] M. M. Chen, K. R. Holmes, Microvascular contributions in tissue heat transfer Ann. New York Acad. Sci. 335 (1980) 137–150. [18] S. Weinbaum, L.M. Jiji, A new simplified bioheat equation for the effect of blood flow on local average tissue temperature. ASME Journal of Biomechanical Engineering, 107 (1985) 131– 139. [19] S. Weinbaum, L.M. Jiji, D.E. Lemons, Theory and experiment for the effect of vascular microstructure on surface tissue heat transfer. Part I. Anatomical foundation and model conceptualization, ASME Journal of Biomechanical Engineering, 106 (1984) 321–330. [20] N. L. Nilsson, Blood flow, temperature, and heat loss of skin exposed to local radiative and convective cooling, J. Invest. Dermatol. 88 (1987) 586–593. [21] M. Stańczyk, J.J. Telega, Modelling of heat transfer in biomechanics – a review. Part I. Soft tissues. Acta of Bioengineering and Biomechanics, 4(1) (2002) 31–61. [22] J.W. Valvano, Bioheat transfer. Encyclopedia of Medical Devices and Instrumentation, Second Edition, Wiley (2005) 1-10. [23] M. Fu, W. Weng, W. Chen, N. Luo, Review on modelling heat transfer and thermoregulatory responses in human body. J. Therm. Biology B 62 (2016) 189–200.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 117-130 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-05-24 Revised: 2019-10-02 Accepted: 2020-04-15 Online: 2020-05-28

Turbulent Heat Transfer Characteristics of a W-Baffled Channel Flow Heat Transfer Aspect Y. Menni1,a*, A.J. Chamkha2,b, O.D. Makinde3,c Unit of Research on Materials and Renewable Energies, Department of Physics, Faculty of Sciences, Abou Bekr Belkaid University, P.O. Box 119-13000-Tlemcen Algeria

1

Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia

2

Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

3

a*

[email protected], [email protected], [email protected]

Keywords: Forced convection, Nusselt number, thermal enhancement, w-baffle, heat transfer aspect.

Abstract. In this work, the thermal behavior of a turbulent forced-convection flow of air in a rectangular cross section channel with attached W-shaped obstacles is investigated. The continuity, momentum and energy equations employed to control the heat and velocity in the computational domain. The turbulence model of k-ε is employed to simulate the turbulence effects. The finite volume method with SIMPLE algorithm is employed as the solution method. The results are reported temperature, local and average Nusselt numbers, and mean velocity contours. The subject is relevant and important for industrial applications. Introduction The arrangement of obstacles, such as baffles, fins and ribs, within channels are among the effective methods used by many researchers and investigators in their numerical and experimental studies. Muszyński and Kozieł [1] carried out two-dimensional numerical investigations of the fluid flow and heat transfer for the laminar flow of the louvered fin-plate heat exchanger, designed to work as an air-source heat pump evaporator. The simulations were performed for different geometries with varying louver pitch, louver angle and different louver blade number. The maximum heat transfer improvement interpreted in terms of the maximum efficiency was obtained for the louver angle of 16° and the louver pitch of 1.35 mm. Park et al. [2] systematically presents the results of heat transfer and friction factor data measured in five short rectangular channels with turbulence promoters. They investigated the combined effects of the channel aspect ratio, rib angleof-attack, and flow Reynolds number on heat transfer and pressure drop in rectangular channels with two opposite ribbed walls. Their experimental results were also compared with literature values. Experimental and numerical studies were conducted by Wong et al. [3] to investigate the dynamic and thermal behavior of a turbulent airflow in a horizontal air-cooled rectangular duct, with inclined square-sectioned cross-turbulators mounted on its bottom surface. Effects of varying the angle formed by the cross-turbulators between 30° and 120° on the convective heat transfer and pressure drop were studied. An optimum range of angles formed by the cross-ribs corresponding to a maximum enhancement of forced convection was observed. According to the experimental and numerical results obtained, its value would be between 60° and 70°. Nasiruddin and Kamran Siddiqui [4] indicated that the convective heat transfer in a heat exchanger tube may be enhanced by placing a baffle inside the tube. The investigators considered a comparative study between three different baffle orientations. The first case examined a vertical baffle. The second case investigated a baffle inclined towards the upstream end, and the third one considered a baffle inclined towards the downstream end. The results suggested that a baffle inclined towards the downstream side with a smaller inclination angle (15° in their study) is a better choice as it enhances the heat transfer by a similar magnitude with a minimal pressure loss. Hwang and Liou [5] investigated heat transfer and loss friction in a rectangular channel with symmetrically mounted solid and fully perforated ribs on

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parallel broad walls (rib open-area ratio: β = 50%; rib pitch-to-height ratio: Pi/H = 5-20; rib heightto-channel hydraulic diameter ratio: H/De = 0.081 and 0.162; rib-to-channel height ratio: H/2B = 0.13 and 0.26; rib height-to-channel hydraulic diameter ratio: H/De = 0.081 and 0.162; Reynolds number: Re = 10,000-50,000). The results indicated that the perforated ribs had the advantages of eliminating the hotspots and providing a superior heat transfer performance. Karwa and Maheshwari [6] investigated fully (open area ratio of 46.8%) and half (open area ratio of 26%) perforated baffles covering Re values ranging from 2,700 to 11,150. The study showed an enhancement of 79-169% in Nusselt number over the smooth duct for the fully perforated baffles and 133-274% for the half perforated baffles while the friction factor for the fully perforated baffles are 2.98-8.02 times of that for the smooth duct and is 4.42-17.5 times for the half perforated baffles. In general, the half perforated baffles are thermo-hydraulically better to the fully perforated baffles at the same pitch. Of all the configurations studied, the half perforated baffles at a relative roughness pitch of 7.2 give the greatest performance advantage of 51.6-75% over a smooth duct at equal pumping power. Sahel et al. [7] presented a new baffle design to eliminate the formation of lower heat transfer areas (LHTAs), particularly in the downstream regions of solidtype baffles. This design concerns a perforated baffle having a row of four holes placed at three different positions. These positions are characterized by a ratio called the pore axis ratio (PAR) which is taken equal to 0.190, 0.425 or 0.660. In their study, the baffle perforated with PAR = 0.190 was found to be as the best design, which reduces significantly the LHTAs, since it ensures an increase in the thermal transfer rate from 2% to 65 %, compared with simple baffle. However, the pressure loss may decrease until 12 times compared with the simple baffle. Khan et al. [8] described experimental investigation of heat transfer with turbulent flow in a rectangular channel with inclined solid and perforated baffles combined with rib turbulators. Combining ribs with perforated and inclined baffles yielded an increase in average Nusselt number, albeit with a pressure drop penalty. In situations where rate of heat transfer is critical to the performance of a device, combining ribs with baffle is a viable solution. Se Kyung et al. [9] have done a numerical study on the heat transfer and frictional characteristics of airflow inside a rectangular channel fitted with different types of inclined baffles (type I: solid baffle; type II: 3 hole baffle; type III: 6 hole baffle; and type IV: 9 hole baffle). The numerical results of the flow field showed that the flow patterns around the different baffle configurations are entirely different and these significantly affect the local heat transfer characteristics. The heat transfer and friction factor characteristics are significantly affected by the perforation density of the baffle plate. It was found that the heat transfer enhancement of baffle type II has the best values. An experimental investigation was carried out by Ko and Anand [10] to measure the heat transfer coefficients and pressure loss in a uniformly heated rectangular channel with wall mounted staggered porous baffles. The experiments were conducted in Reynolds number range of 20,000-50,000. The use of porous baffles resulted in heat transfer enhancement as high as 300% compared to heat transfer in straight channel with no baffles. The experimental result analysis showed that the heat transfer enhancement per unit increase in pumping power was less than one for the range of parameters studied. Guerroudj and Kahalerras [11] simulated the influence of porous block shape on the laminar mixed convective heat transfer and airflow characteristics inside a two-dimensional parallel plate channel when the buoyant and forced flow effects are simultaneously present. The influence of the buoyancy force intensity, the porous blocks shape going from the rectangular shape to the triangular shape, their height, the porous medium permeability, the Reynolds number and the thermal conductivity ratio was analyzed. The results revealed essentially, that the shape of the blocks can alter substantially the flow and heat transfer characteristics. A computer code was developed by Kamali and Binesh [12] to study the turbulent heat transfer and friction in a square duct with various-shaped ribs mounted on one wall. The simulations were performed for four rib shapes, i.e., square, triangular, trapezoidal with decreasing height in the flow direction, and trapezoidal with increasing height in the flow direction. The results showed that features of the inter-rib distribution of the heat transfer coefficient are strongly affected by the rib shape and trapezoidal ribs with decreasing height in the flow direction provide higher heat transfer enhancement and pressure drop than other shapes.

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Sripattanapipat and Promvonge [13] simulated the laminar periodic flow and heat transfer in a two dimensional horizontal channel with isothermal walls and with staggered diamond-shaped baffles. They reported that the diamond shape of the baffle with different tip angles (5 to 35°) may enhance the heat transfer from 200 to 680% for Reynolds number ranging from 100 to 600. However, this intensification is associated with enlarged friction loss ranging from 20 to 220 times above the smooth channel. Saini and Saini [14] conducted an experimental prediction on the turbulent flow and convective heat transfer characteristics in a rectangular air channel with arc-shaped elements attached to the underside of a heated plate. The effects of various dimensionless parameters such as arc angle and height on the Nusselt number and friction factor were studied for Reynolds numbers ranging from 2,000 to 17,000. Their results suggested that a significant heat transfer coefficient enhancement in a solar air channel can be achieved by introducing arc-shaped ribs into the flow. The maximum enhancement in Nusselt number was obtained as 3.80 times corresponding to the relative arc angle of 0.3333 at relative roughness height of 0.0422, with the minimum pressure loss. Stehlik et al. [15] compared heat transfer and friction loss correction factors of an optimized segmental baffle heat exchanger to those of a helical baffle heat exchanger. In their studies, the correction factors for helical baffles were examined as a function of baffle inclination angle to gain an understanding of the underlying transport phenomena as well as to characterize the baffle for design purpose. An improved structure of ladder-type fold baffle was proposed by Wen et al. [16] in order to block the triangular leakage zones in original heat exchangers with helical baffles. They numerically showed that the distribution of shell-side velocity and temperature in improved heat exchanger are more uniform and axial short circuit flow is eliminated. The fluid flow and heat transfer characteristics of the improved heat exchanger and the original heat exchanger were also experimentally studied. They showed that the shell-side heat transfer coefficient and overall heat transfer coefficient are improved by 22.3-32.6% and 18.1-22.5%, respectively. A numerical investigation for fully developed turbulent flow in a square duct fitted with 45° in-line V-baffle pairs mounted on both upper and lower walls was conducted by Fawaz et al. [17] in order to examine the changes in flow structure and thermal performance, using air as the working fluid at Re ranging from 5,000 to 25,000. Effect of various baffle blockage ratios (BR = 0.2, 0.4 and 0.6) and baffles pitch ratios (PR = 0.5,1 and 1.5) on flow behavior and heat transfer were investigated. They found that the TEF of the V-baffle pointing upstream at BR = 0.2 is higher than that at larger BR and the TEF of this same baffle at PR = 0.5 is higher than that at higher PR, at the lowest Re value. Sriromreun et al. [18] reported experimental and numerical investigations of the heat transfer and flow friction characteristics for a solar air heater channel with in-phase and out-phase Z-shaped baffles in the turbulent regime from Re = 4,400 to 20,400. The Z-baffles inclined to 45° relative to the main flow direction are characterized at three baffle-to channel-height ratios (e/H = 0.1, 0.2 and 0.3) and baffle pitch ratios (P/H = 1.5, 2 and 3). The effects of e/H and P/H ratios were more significant for the in-phase Z-baffle than for the out-phase Z-baffle. In addition, numerical [19] and experimental [20-25] works on air baffled channels were conducted to analyze the heat transfer coefficient and pressure loss. In those studies, different structural parameters of the model and various operating parameters were used. In this paper, a computational thermal analysis of the turbulent forced-convection fluid stream behavior in the presence of two staggered, transverse, solid-type, double V-form baffles (or Wshaped obstacles) is conducted in order to improve the heat transfer phenomenon within thermal devices. The thermal aspect is shown for flow Reynolds numbers based on the hydraulic diameter of the channel ranging from 12,000 to 32,000. To perform a detailed analysis of the aerodynamic and thermal fields within this model of W-baffled channel, the finite volume approach, by means of Commercial CFD software FLUENT, for a steady-state, incompressible, and two-dimensional flow is used, and the SIMPLE-algorithm is implemented for all calculations. The analysis results are presented in terms of mean velocity, fields and profiles of temperature, local and average numbers of normalized Nusselt, and thermal enhancement factor for various axial and transverse channel

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stations. Small attack angle value (α = 45°) and high Reynolds number (Re = 32,000) values lead to the best functioning regime in a channel containing W-shaped obstacles. Problem Definition The particularity of this computational thermal analysis is the forced-convection heat transfer aspect. The main aim of this paper is to examine the turbulent flow field and convective heat transfer characteristics of a constant property Newtonian fluid inside a two-dimensional horizontal channel of rectangular form, containing two transverse, staggered, solid-type, double V-form (or Wshaped) baffles, where a constant surface temperature is applied on all solid boundaries of the duct, Fig. 1.

Figure 1. Geometry under examination (dimensions in m). This channel section due to of their specific geometry, accelerate the flow disturbance and improve the heat exchange between the working fluid and the hot walls. The flow Reynolds number is varied between 12,000 and 32,000. The physical models for fluid flow in the rectangular cross section channel provided by double V-baffles were developed under the following assumptions: 1- Steady two-dimensional heat transfer and fluid flow; 2- Flow is turbulent and incompressible; 3- Physical properties of air fluid (Cp, μ, λf, ρ) and solid (λs) are constants; 4- Fluid is viscous Newtonian; 5- Temperatures applied to the lower and upper surfaces of the channel (Tw) are considered constants; 6- Velocity profile at the inlet of the channel is uniform; 7- Radiation heat transfer is not considered; 8- Thicknesses of the bottom and top wall surfaces of the channel are neglected; and 9- Standard k-ε turbulence model proposed by Launder and Spalding [26], by means of Commercial CFD software FLUENT is applied in this thermal analysis to simulate the fluid flow behavior. The governing flow equations, i.e., continuity, momentum and energy equations, used to simulate the incompressible steady fluid flow and heat transfer in the whole domain treated are given as [4, 29, 30] (1) ∇V = 0 , 2 (2) ρ V .∇V = −∇P + µ f ∇ V ,

(

) ρc (V .∇T ) = k ∇ T . p

f

2

(3)

The standard k-ε model is defined by two transport equations, one for the turbulent kinetic energy, k and the other for its dissipation rate ε, as given below [26]    ∂ (4) (ρkui ) = ∂  µ + µt  ∂k  + Gk − ρε σ k  ∂x j  ∂xi ∂x j  and

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 µt  µ + σε  The turbulent viscosity, μt is modeled as follows: ∂ (ρεui ) = ∂ ∂xi ∂x j

 ∂ε  ε ε2   + C1ε Gk − C2ε ρ k k  ∂x j 

121

(5)

k2 (6) ε In these equations, Gk represents the generation of turbulence kinetic energy due to the mean velocity gradients. The related constant parameters are [26] (7) C1ε = 1.44, C2ε = 1.92, Cµ = 0.09, σ k = 1.0, σ ε =1.3

µt = ρCµ

The Reynolds number is defined as ρ U Dh µ The local Nusselt number, Nux which can be written as h D Nu x = x h Re =

λf

(8)

(9)

The average Nusselt number, Nu can be obtained by 1 (10) Nu = ∫ Nu x ∂x L where hx is the local convective heat transfer coefficient. In the entrance region of the test section, 1- A uniform velocity profile is applied, u = Uin and v = 0; 2- At the bottom (y = -H/2) and top (y = H/2) channel walls, the non-slip and impermeability boundary conditions are implemented over the channel wall as well as the W-baffle surfaces, that is u = v = 0; 3- In the channel outlet (x = L) it is prescribed the atmospheric pressure, P = Patm; 4- The inlet temperature of air Tin is considered to be uniform at 300 K; 5- A constant temperature of Tw = 375 K is applied on the entire walls of the channel as the thermal boundary condition. CFD Solution The two-dimensional, incompressible Navier–Stokes equations and the turbulence model equations are discretized using the finite volume method, details of which can be found in Patankar [27]. For the momentum equations, the pressure and velocities are linked together based on the SIMPLE-algorithm [27]. Considering the characteristics of the flow, the Quick-scheme [28] was applied to the interpolations, while a Second-order upwind scheme [27] was used for the pressure terms. In order to verify the accuracy of numerical results obtained in this numerical study with the computer code Fluent, a validation of our analysis was made by comparing with the results of Demartini et al. [25] that are available in the literature. These authors studied a similar problem for the circulation of air through a rectangular channel but with simple baffles. Under the same conditions, we conducted a comparison in terms of the axial velocity profiles for a Reynolds number equal to Re = 8.73×104 at axial location x = 0.159 m from the entrance, shown in Fig. 2. The comparison of results by our numerical method and results of Demartini et al. [25] for steadystate flow conditions shows very good agreement as illustrated in the figure.

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Figure 2. Numerical validation of dimensionless axial velocity profiles at station x = 0.159 m for Re = 8.73×104. Results and Discussion Fig. 3 shows the contour plots of mean velocity fields for different Re numbers, i.e., Re = 12,000, 17,000, 22,000, 27,000, and 32,000. It is clearly noted that the fluid velocity values are almost negligible near the two W-obstacles, particularly in the downstream areas this is caused by the presence of the recirculation zones. Far from these regions, the current lines become parallel, which leads to the progressive development of the flow. It is also worth noticing that the velocity increases in the region extending from the end of each W-obstacle to the wall of the channel. This rise in velocity is caused by the presence of the W-obstacles and also by the presence of recycling; hence, an abrupt change in the direction of the flow comes out. One can also notice that the largest velocity values are found near the top of the channel. The flow starts accelerating just after the second W-obstacle. The velocity of the flow is also influenced by the Re number value. If the Re number is increased, then the flow accelerates in the vicinity of the W-obstacle faces, and this causes the convective heat transfer rate to rise. The analysis of the isotherms presented in Fig. 4 for various values of Reynolds number, shows that the fluid temperature significantly increases with the presence of the double V-baffles compared to that of the situation with no baffles. In the downstream region of the two double V-baffles, recirculation cells with relatively high temperatures are observed. In the space between the tip of each double V-baffle and the walls of the channel, the temperature is decreased. The total temperature profiles are shown in Fig. 5 (a) to (d) for different transverse sections of the channel. The numerical results show that the temperature of the air in the recirculation zone is substantially high compared to that obtained in the same region without baffles. This observation is confirmed by Nasiruddin and Kamran Siddiqui [4]. We also note that the hottest areas are, mostly, located near the solid boundaries of the test channel (lower and upper walls of the channel, and double V-baffle surfaces). It is also found that the temperature value at the heated wall level decreases with increasing the flow velocity. On the other hand, the results analysis allowed associating to the fluid temperature elevations, the effect of double V-baffles and fins. At the output of each free segment between the tip of each double V-baffle and the channel walls, the air in flow does not encounter any metal obstacles; its velocity decreases due to the sudden enlargement, and the lack of W-baffles constitutes a supplementary factor of local attenuation of the turbulence in these areas. However, the air temperature increases as soon as the fluid once again finds in contact with the double V-baffles, and this is repeated in an analogous manner from one cell to another. What was also noticed, the fluid temperature is inversely related with the increase in the Reynolds number. The graphical representations of the temperature variation as a function of Reynolds number (Re = 12,000; 22,000; and 32,000) in the transverse section situated between both the W-baffles at x = 0.255 m, is shown in Fig. 6. These results certify that the heat exchange between the heat transfer fluid and the heated walls in the channel with staggered 45° double V-

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baffles is more important with low flow Reynolds numbers. It is clear that for high Reynolds numbers, the fluid temperature significantly decreases i.e., there exists an inverse proportionality between increasing flow Reynolds number and the total temperature in each channel cross section. Additionally, according to analysis of our numerical results on the velocity profiles [see Fig. 6] and the total temperature profiles for different sections of the channel [see Figs. 5 and 6), it is found that the fluid temperature is related to the flow velocity.

(a)

(b)

(c)

(d)

(e)

Figure 3. Mean velocity fields for various values of flow rate: (a) Re = 12,000, (b) Re = 17,000, (c) Re = 22,000, (d) Re = 27,000, and (e) Re = 32,000.

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(a)

(b)

(c)

(d)

(e)

Figure 4. Temperature fields for various values of flow rate: (a) Re = 12,000, (b) Re = 17,000, (c) Re = 22,000, (d) Re = 27,000, and (e) Re = 32,000.

(a)

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(b)

(c)

(d)

Figure 5. Profiles of fluid temperature (a) upstream of the first W-baffle at x = 0.159 m and x = 0.179 m, (b) between the first and the second W-baffles at x = 0.255 m and x = 0.285 m, (c) before the second W-baffle at x = 0.315 m and x = 0.335 m, and (d) near the channel outlet at x = 0.525 m, Re = 12,000.

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Figure 6. Variation of fluid temperature profiles with Reynolds number for x = 0.255 m.

Figure 7. Distribution of normalized local Nusselt number along the bottom and top walls of the channel for Re = 12,000. The heat transfer rate, characterized by the normalized local Nusselt number, is then determined and shown along the lower and upper walls of the channel with Re = 12,000 in Fig. 7. These profiles present in all cases (bottom or top walls) a minimum and a maximum of the Nusselt number. It is found that the heat transfer rate minimums are observed at the level of base of these double V-baffles and that the Nusselt number increases along the baffle and reaches its maximum on its upper face. The effect of the Reynolds number on the normalized local Nusselt number evolution is seen in Fig. 8 (a) and (b) for both bottom and top walls of the channel, respectively. The results show that the heat transfer rate is increased with the increase of flow Reynolds number, because when the Re increases, the turbulence increases and the recirculation region become stronger and consequently the heat dissipation increases. If we think in terms of no-dimensional average Nusselt number, as shown in Fig. 9, there is increasing almost linearly Nusselt number as a function of Reynolds number. There is a linear increment between average heat transfer ratio and flow Reynolds number value. It may be generated by the augmentation of the flow acceleration causing by increasing the flow velocity. It is then found that the double V-baffles play an effective factor to dissipate the heat from the solid walls and the temperature of the flow increases in the regions occupied by the double V-baffles and between the W-deflectors. This enhances the heat transfer.

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(a)

(b)

Figure 8. Distribution of normalized local Nusselt number along the (a) bottom and (b) top walls of the channel for different flow Reynolds numbers.

Figure 9. Variation of normalized average Nusselt number with Reynolds number. Fig. 10 shows the variations of the thermal enhancement factor (TEF) as a function of the Reynolds number at the lower and upper channel walls. In the figure, the TEF value tends to increase with augmenting the Re number for both channel surfaces. The upper wall containing the first 45° W-shaped baffle provides the highest TEF value while the lower wall fitted with the

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second 45° W-shaped baffle yields the lowest one. It is found that the TEF values vary between 0.853-1.037; and 1,277-1,644 for lower and upper walls of the channel, respectively, depending on the Re values. In the case of bottom surface, the TEF was found to be much larger than unity; its maximum value was around 1,644 for Re = 32,000. This value of TEF is decreased by 36,952 % in the case of the bottom channel wall at same Re number.

Figure 10. Evaluation of thermal-aerodynamic performance for various Reynolds numbers. Conclusion The most important conclusions that can be drawn from this study are as follows: • The velocity of the flow is influenced by the Re number value. If the Re number is increased, then the flow accelerates in the vicinity of the W-obstacle faces, and this causes the convective heat transfer rate to rise; • The temperature of the air in the recirculation zone is substantially high compared to that obtained in the same region without baffles; • In the space between the tip of each double V-baffle and the walls of the channel, the temperature gradient is increased; • At the output of each free segment between the tip of each double V-baffle and the channel walls, the air in flow does not encounter any metal obstacles; its velocity decreases due to the sudden enlargement, and the lack of W-baffles constitutes a supplementary factor of local attenuation of the turbulence in these areas. • The fluid temperature is inversely related with the increase in the Reynolds number. • The heat transfer rate minimums are observed at the level of base of these double V-baffles and that the Nusselt number increases along the baffle and reaches its maximum on its upper face • The double V-baffles play an effective factor to dissipate the heat from the solid walls and the temperature of the flow increases in the regions occupied by the double V-baffles and between the W-deflectors. This enhances the heat transfer. • The TEF values vary between 0.853-1.037; and 1,277-1,644 for lower and upper walls of the channel, respectively, depending on the Re values.

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References [1]

T. Muszyński, S. M. Kozieł, Parametric study of fluid flow and heat transfer over louvered fins of air heat pump evaporator, Archives of Thermodynamics, 37(3) (2016) 45-62.

[2]

J. S. Park, J. C. Han, Y. Huang, S. Ou, R. J. Boyle, Heat transfer performance comparisons of five different rectangular channels with parallel angled ribs, International Journal of Heat and Mass Transfer, 35(11) (1992) 2891-2903.

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T. T. Wong, C. W. Leung, Z. Y. Li, W. Q. Tao, Turbulent convection of air-cooled rectangular duct with surface-mounted cross-ribs, International Journal of Heat and Mass Transfer, 46 (2003) 4629-4638.

[4]

Nasiruddin, M. H. Kamran Siddiqui, Heat transfer augmentation in a heat exchanger tube using a baffle, International Journal of Heat and Fluid Flow, 28 (2007) 318-328.

[5]

J. J. Hwang, T. M. Liou, Heat transfer in a rectangular channel with perforated turbulence promoters using holographic interferometry measurement, International Journal of Heat and Mass Transfer, 38(17) (1995) 3197-3207.

[6]

R. Karwa, B. K. Maheshwari, Heat transfer and friction in an asymmetrically heated rectangular duct with half and fully perforated baffles at different pitches, International Communications in Heat and Mass Transfer, 36 (2009) 264-268.

[7]

D. Sahel, H. Ameur, R. Benzeguir, Y. Kamla, Enhancement of heat transfer in a rectangular channel with perforated baffles, Applied Thermal Engineering, 101 (2016)156-164.

[8]

J. A. Khan, J. Hinton, S. C. Baxter, Enhancement of heat transfer with inclined baffles and ribs combined, Enhanced Heat Transfer, 9(3/4) (2002) 137-151.

[9]

O. Se Kyung, B. K. P. Ary, S. W. Ahn, Heat transfer and frictional characteristics in rectangular channel with inclined perforated baffles, World Academy of Science, Engineering and Technology, 3(1) (2009) 13-18.

[10] K. H. Ko, N. K. Anand, Use of porous baffles to enhance heat transfer in a rectangular channel, International Journal of Heat and Mass Transfer, 46 (2003) 4191-4199. [11] N. Guerroudj, H. Kahalerras, Mixed convection in a channel provided with heated porous blocks of various shapes, Energy Conversion and Management, 51 (2010) 505-517. [12] Kamali, A. R. Binesh, The importance of rib shape effects on the local heat transfer and flow friction characteristics of square ducts with ribbed internal surfaces, International Communications in Heat and Mass Transfer, 35 (2008) 1032-1040. [13] S. Sripattanapipat, P. Promvonge, Numerical analysis of laminar heat transfer in a channel with diamond-shaped baffles, International Communications in Heat and Mass Transfer, 36 (2009) 32-38. [14] S. K. Saini, R. P. Saini, Development of correlations for Nusselt number and friction factor for solar air heater with roughened duct having arc-shaped wire as artificial roughness, Solar Energy, 82 (2008) 1118-1130. [15] P. Stehlik, J. Nemcansky, D. Kral, L. W. Swanson, Comparison of correction factors for shelland-tube heat exchangers with segmental or helical baffles, Heat Transfer Engineering, 15 (1994) 55-65. [16] J. Wen, H. Yang, S. Wang, Y. Xue, X. Tong, Experimental investigation on performance comparison for shell-and-tube heat exchangers with different baffles, International Journal of Heat and Mass Transfer, 84 (2015) 990-997.

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[17] H. E. Fawaz, M. T. S. Badawy, M. F. Abd Rabbo, A. Elfeky, Numerical investigation of fully developed periodic turbulent flow in a square channel fitted with 45° in-line V-baffle turbulators pointing upstream, Alexandria Engineering Journal 57(2) (2018) 633-642. [18] P. Sriromreun, C. Thianpong, P. Promvonge, Experimental and numerical study on heat transfer enhancement in a channel with Z-shaped baffles, International Communications in Heat and Mass Transfer, 39 (2012) 945-952. [19] K. Boukhadia, H. Ameur, D. Sahel, M. Bozit, Effect of the perforation design on the fluid flow and heat transfer characteristics of a plate fin heat exchanger, International Journal of Thermal Sciences, 126 (2018) 172-180. [20] R. Ben Slama, The air solar collectors: comparative study, introduction of baffles to favor the heat transfer, Solar Energy, 81 (2007) 139-149. [21] R. K. Karwa, Experimental studies of augmented heat transfer and friction in symmetrically heated rectangular ducts with ribs on heated wall in transverse, inclined, v-continuous and vdiscrete pattern, International Communications in Heat and Mass Transfer, 30 (2003) 241250. [22] S. Tamna, S. Skullong, C. Thianpong, P. Promvonge, Heat transfer behaviors in a solar air heater channel with multiple V-baffle vortex generators, Solar Energy, 110 (2014) 720-735. [23] R. Kumar, A. Kumar, R. Chauhan, M. Sethi, Heat transfer enhancement in solar air channel with broken multiple V-type baffles, Case Studies in Thermal Engineering, 8 (2016) 187-197. [24] F. Wang, X. Chen, J. Chen, Y. You, Experimental study on a debris-flow drainage channel with different types of energy dissipation baffles, Engineering Geology, 220(30) (2017) 4351. [25] L. C. Demartini, H. A. Vielmo, Q. V. Möller, Numeric and experimental analysis of turbulent flow through a channel with baffle plates, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 26 (2004) 153-159. [26] B. E. Launder, D. B. Spalding, The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering, 3(2) (1974) 269-289. [27] S. V. Patankar, Numerical heat transfer and fluid flow, McGraw-Hill, New York, (1980). [28] B. P. Leonard, S. Mokhtari, ULTRA-SHARP nonoscillatory convection schemes for highspeed steady multidimensional flow, NASA TM 1-2568, NASA Lewis Research Center, (1990). [29] M. Sankar, S. Kiran, G.K. Ramesh, O.D. Makinde, Natural convection in a non-uniformly heated vertical annular cavity, Defect and Diffusion Forum, 377 (20187) 189-199. [30] O. D. Makinde, Z.H. Khan, R. Ahmad, W. A. Khan, Numerical study of unsteady hydromagnetic radiating fluid flow past a slippery stretching sheet embedded in a porous medium, Physics of Fluids, 30 (2018) 083601 (7pages).

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 131-139 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-03-12 Revised: 2019-10-21 Accepted: 2019-10-21 Online: 2020-05-28

MHD Boundary Layer Flow over a Cone Embedded in Porous Media with Joule Heating and Viscous Dissipation S. Devi1,a, M.K. Sharma1,b* Department of Mathematics, Guru Jambheshwar University of Science and Technology Hisar-125001, India

1

[email protected], [email protected]

a

Keywords: Mixed convection, cone, boundary layer, Joule heating, convective boundary condition.

Abstract: Aim of the paper is to study the Magnetohydrodynamic boundary layer flow over a cone under the effect of joule heating and viscous dissipation. The surface of the cone is cooled and heated by the flowing fluid having constant temperature Tf along with variable heat transfer coefficient hf(x). The surface of the cone is subjected under the convective heat flux. The governing equation for MHD boundary layer flow are non-linear partial differential equations, are transformed into ordinary differential equations using similarity techniques. The reduced ordinary coupled equations are solved with Runge-Kutta’s fourth order method followed by shooting techniques. The effects on flow and heat convection of various physical parameters pertinent to the modeled problem are computed and analyzed and shown through graphs. Introduction Mixed convection MHD boundary layers have very important applications in various engineering processes. The combined effect of free an forced convection flow have many important applications in transport processes. For examples solar collectors , heat exchangers, electronics apparatus, spinning of fibers etc.. The combined effect of forced and free convection flow becomes important in the presence of large buoyancy forces that caused by temperature difference in between the solid surfaces and free stream become vast which create the effects on temperature and flow field. The effects of laminar mixed convection flows about heated surfaces were investigated for variable surface heat flux and variable wall temperature by Ramachandran et al.[1]. The similarity solutions of mixed convection boundary layer for prescribed heat flux were investigated by Merkin and Mahmood [2]. Unsteady mixed convection boundary layer flow adjacent to vertical surfaces was examined by Devi et al.[3]. The problem of free convection around vertical flat plate in porous medium was investigated by Cheng and Minkowycz [4]. The mixed convection boundary layer flow over an isothermal cone in porous medium with the effects of radiations was investigated by Yih [5]. Mixed convection boundary layer flow past vertical surfaces in porous medium was investigated by Aly et al.[6]. A through description on the flow through porous media is given by Nield and Bejan [7]. The dual solution for the mixed convection boundary layer flow past a vertical porous plate was obtained by Ishak et al.[8]. The solution for thermal boundary layer past a plate with convective surface boundary condition was obtained by Aziz [9]. The study of mixed convection boundary layer flow on a vertical flat plate in porous medium was done by Harris et al.[10]. The study of an anisotropy effect on steady mixed convection flow on a vertical flat plate in porous medium was done by Bachok et al.[11]. Natural convection MHD boundary layer flow of low Prandtl number with the effect of variable thermal conductivity, viscous dissipation with Ohmic heating was analyzed by Sharma and Singh [12]. Makinde and Aziz [13] obtained the numerical solution for the mixed convection thermal boundary layer with convective boundary conditions over a vertical plate. Unsteady flow and heat transfer over shrinking sheet with suction in nanofluid was analyzed by Rohini et al.[14]. The study of boundary layer flow of a nanofluid over an exponentially stretching sheet with convective boundary conditions was done by Mustafa et al. [15]. Effects of variable thermal conductivity and viscous dissipation on MHD boundary layer over stretching sheet embedded in porous medium were studied by Dessie and Kishan [16]. MHD boundary layer flow of nanofluid

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with effect of Joule heating and thermal radiation over an exponentially stretching sheet was studied by Rao et al.[17]. The Mixed convective flow over a cone embedded in porous medium was analyzed by Rosali et al.[18].Other relevant published articles can be found in [19-25]. The aim of the present study is to investigate the mixed convection MHD boundary layer flow with the effect of Joule heating and viscous dissipation having variable permeability of the porous medium and variable heat transfer coefficient. Model Problem Considering a cone having semi vertical angle φ , is embedded in a porous medium of variable permeability defined by K = K 0 exp{− y (U ∞ / xα )1 / 2} with the effect of joule heating having variable permeability embedded in porous medium having local radius r and semi vertical angle φ . The x-axis is taken along generating line of cone and y-axis is considered normal to it. The uniform static transverse magnetic field is applied normal to the cone. It is considered that surface of the cone is cooled and heated by the flowing fluid having constant temperature T f along with variable heat transfer coefficient h f ( x ) . The surface of the cone is subjected under the convective heat flux. The ambient fluid temperature is T ∞ . Here, T f > T ∞ represents heated cone whereas, T f < T ∞ represents cooled cone. The uniform free stream velocity of the fluid U ∞ is considered in upward direction. The governing boundary layer equations are written as: x

r

g

B

y

Fig.1: Physical model and coordinate system The equation of continuity ∂ (ru ) ∂ (rv ) + = 0, ∂x ∂r

(1)

The equation of motion

 σ B 2 e − y (U ∞ / xα ) K e − y (U ∞ / xα )  ∂p u=− 0 u,  − ρgβ (T − T ∞ ) cos φ  − µ µ   ∂x 1/ 2

The energy equation

1/ 2

(2)

Defect and Diffusion Forum Vol.401

u

133

2 ∂T ∂T µ  ∂u  ∂ 2T σ B 2 u 2   , =α + + +v ∂y ∂x ρ c p  ∂y  ρcp ∂y 2

(3)

At free stream condition the pressure gradient is related with

∂p µU∞ = − − σ B2 U ∞ . K ∂x

(4)

In view of equation (2) reduces u =U∞+

g K 0 exp{− y (U ∞ / xα )1 / 2} β (T − T ∞ ) cos φ σ B 2 K 0 exp{− y (U ∞ / xα )1 / 2} − (u − U ∞) ,

ν

(5)

µ

where, r ( x) = x sin φ , u and v are component of velocity, respectively in x and y direction, T is fluid temperature, υ is the kinematic viscosity, σ is the electrical conductivity of the fluid, ρ is the density of the fluid, α is the thermal diffusivity of the fluid, c p is the specific heat at constant temperature. Under the prescribed conditions, boundary conditions for flow velocity and temperature are given by u = v= 0

at y=0

u → U∞, −κ

T → T∞,

(

∂T = h f ( x) T f − T ∞ ∂y

)

as y → ∞,

(6)

at y=0

where, κ is thermal conductivity. Method of Solution Introducing the following transformations u=

1 ∂ψ 1 ∂ψ and v = − r ∂x r ∂y

(7)

where, ψ is the stream function. Considering following similarity transformation, Roshali et al.[18]

ψ = αr (Pe x ) 2 f (η ), θ (η ) = (T − T ∞ ) / (T f − T ∞ ) 1

,

(8)

η = (Pe x ) 2 ( y / x) , 1

where, Pe x =

U∞ x

denotes the local Peclet number. Substituting Eq. (8) into Eq. (3) and (5), we get α the following coupled ordinary non-linear differential equations

f ' (η ) = {(1 + e −η M / G ) + λ *θ (η )} /(1 + e −η M / G ) ,

θ '' +

3 f θ ' + Pr .Ec f ' '2 + M .Ec f '2 = 0 , 2

(9) (10)

where, prime denotes derivatives w.r.t. η, and the corresponding boundary conditions reduces to

f (0) = 0 , f ' (∞) = 1 , θ (∞) = 0 , θ ' (0) = − Bi[1 − θ (0)] ,

(11)

where λ * denotes constant mixed convection parameter and Bi denotes Biot number, defined as

134

Computational Analysis of Heat Transfer in Fluids and Solids II

λ* =

Ra x

1/ 2 , Bi = h f ( x) x , Pe x k (U ∞ / α )1 / 2

Ra x = gβ K 0 (T f − T ∞) x cos φ αν is the local Rayleigh number, Pr =

(12)

µC p κ

is the Prandtl number,

2 σ 2 x  νx U∞ Ec = is the Eckert number, G = is the porosity parameter and M =  B  C p (T f − T ∞ ) U∞K0  ρU ∞  is the magnetic parameter. For energy Eq.(10) to have a similarity solution, convective heat transfer parameter must be a constant Roshali et al.[18]. This condition can be obtained if heat transfer coefficient hf is proportional to x −1 / 2 and we have

h f ( x) = a x

(13)

−1 / 2

where a is constant and the thermal Biot number in Eq. (12) then become Bi =

a 1/ 2

k (U ∞ / α )

.

(14)

The coupled non-linear ordinary differential boundary layer equations (9) and (10) along with boundary conditions (11) are solved using Runge-Kutta fourth order method following shooting techniques. The effects of various physical parameters on velocity, temperature profiles are shown through graphs. The numerical values of skin friction coefficient, heat transfer coefficients are computed for various physical parameters pertinent to the problem. Skin-Friction Coefficient: The dimensionless shear stress at the surface of the cone is defined as Skin- friction coefficient which is given by Cf =

µ  ∂u  



2  ∂y  ρU ∞   y =0

.

(15)

Using Eq.(8) into (15), we obtain

(Pe x )12 C f / Pr =

f ' ' (0),

(16)

Nusselt Number: The dimensionless coefficient of heat transfer is defined by Nusselt Number which is given by: Nu

x

=−

x T f −T ∞

(

 ∂T  ) ∂y 

(17) y =0

Using Eq.(8) into (17) , we obtain

(Pe x )−12 Nu x = −θ ' (0) .

(18)

Results and Discussion Figure 2 depicts the trend of velocity profile for the variation of magnetic parameter M. It exhibits that the velocity of the flow decreases with increase of the magnetic parameter. So the shear stress decreases with increase of the magnetic parameter. Figure 3 depicts the trend of temperature profile for the variation of magnetic parameter M. It exhibits that the temperature of the flow increases with increase of the magnetic parameter. Figure 4 depicts the trend of velocity profile for the variation of Biot number Bi. It exhibits that the velocity of the flow increases with increase of the Biot number Bi.

Defect and Diffusion Forum Vol.401

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Figure 5 depicts the trend of temperature profile for the variation of Biot number Bi. It exhibits that the temperature of the flow increases with increase of the Biot number Bi. Figure 6 depicts the trend of velocity profile for the variation of Eckert number Ec. It exhibits that the velocity of the flow increases with increase of the Eckert number Ec. Figure 7 depicts the trend of temperature profile for the variation of Eckert number Ec. It exhibits that the temperature of the flow increases with increase of the Eckert number Ec. Figure 8 depicts the trend of velocity profile for the variation of mixed convection parameter λ * . It exhibits that the velocity of the flow increases with increase of the mixed convection parameter λ * . Figure 9 depicts the trend of temperature profile for the variation of mixed

convection parameter λ * . It exhibits that the velocity of the flow decreases with increase of the mixed convection parameter. Figure 10 depicts the trend of velocity profile for the variation of porosity parameter G. It exhibits that the velocity of the flow increases with increase of the porosity parameter G. Figure 11 depicts the trend of temperature profile for the variation of porosity parameter G. It exhibits that the temperature of the flow decreases with increase of the porosity parameter G. Figure 12 depicts the trend of velocity profile for the variation of Prandtl number Pr. It exhibits that the velocity of the flow decreases with increase of the Prandtl number Pr. Figure 13 depicts the trend of velocity profile for the temperature of Prandtl number Pr. It exhibits that the temperature of the flow decreases with increase of the Prandtl number Pr. Table 1 depicts the effects of various embedded parameters on the plate surface temperature and Nusselt number. Interestingly, the both the heat transfer rate and the plate surface temperature amplify that rising values of M, Bi, Ec and Pr but diminish with a rising values of λ* and G. This implies that convective heat transfer and magnetic field boost the temperature gradient at the plate surface while both of the thermal buoyancy and medium permeability lessen it.

136

Computational Analysis of Heat Transfer in Fluids and Solids II

Table 1: Computational results for surface temperature and Nusselt number Bi Ec G Pr θ (0) (Pe x )− 1 2 Nu x λ*

M

0.5 1.5 5

0.5 0.1 0.5 1

0.001

0.001 0.1 0.5

0.1

1.2

-0.5 1.2

0.5

0.5 1 5

0.72

0.72 5 7

0.831 0.912 1 0.112 0.831 2.779 0.831 1.013 2.073 1.292 0.831 0.831 0.794 0.734 1.013 1.394 1.735

0.915 0.956 1 0.112 0.915 3.729 0.915 1.0065 1.5365 1.146 0.915 0.915 0.897 0.877 1.0065 1.197 1.3675

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138

Computational Analysis of Heat Transfer in Fluids and Solids II

Conclusions In this problem Runge-Kutta’s fourth order method is used to analyze the problem of mixed convection Magnetohydrodynamic boundary layer flow under the effect of joule heating and viscous dissipation having variable permeability embedded in porous medium. We attain the following observation of the above problem: An increase in the Magnetic parameter decreases the velocity while increases the temperature of the flow. As the value of Biot number increases, the velocity and temperature of flow also increases. When, the value of Eckert numbers increases, the velocity and temperature both increases. As the value of mixed convection parameter increases, the temperature of the flow decreases. When, the value of porosity parameter increases, the velocity of the flow increases while the temperature of the flow decreases. Then temperature and the velocity of the flow both decrease as the value of the Prandtl numbers increases. Acknowledgement We are grateful to the University Grant Commission, New Delhi for financial support as JRF. References [1]

N. Ramachandran, T.S. Chen, B.F. Armaly, Mixed convection in stagnation flows adjacent to vertical surfaces, ASME J. HeatTransf.110 (1988)373–377.

[2]

J.H. Merkin, T. Mahmood, Mixed convection boundary layer similarity solution for prescribed heat flux, J. Appl. Math. Phys.40 (1989) 51–68.

[3]

C. D. S. Devi, H. S. Takhar, G. Nath, Unsteady mixed convection flow in stagnation region adjacent to a vertical surface, Heat Mass Transf.26 (1991) 71–79.

[4]

P. Cheng, W.J. Minkowycz,. Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. J. Geophy. Res.82 (1997) 2040-2044.

[5]

K.A. Yih, Radiation effect on mixed convection over an isothermal cone in porous media, Heat Mass Transf. 37 (2001) 53–57.

[6]

E. H. Aly, L. Elliott, D. B. Ingham, Mixed convection boundary-layer flow over a vertical surface embedded in a porous medium. Eur. J. Mech. B Fluids22 (2003) 529–543.

[7]

D. A. Nield, A. Bejan, Convection in porous media, 3rd ed., Springer, New York, 2006.

[8]

A. Ishak, R. Nazar, N. M.Arifin, I. Pop, Dual solutions in mixed convection flow near a stagnation point on a vertical porous plate, Int. J. Therm. Sci .47 (2008) 417–422.

[9]

A. Aziz, A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition, Commun. Nonlinear Sci. Numer. Simul.14 (2009) 1064–1068.

[10] S. D. Harris, D.B. Ingham, I. Pop, Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium Brinkman model with slip. Transp. Porous Media, 77 (2009) 267-285. [11] N. Bachok, A. Ishak, I. Pop, Mixed convection boundary layer flow near the stagnation point on a vertical surface embedded in a porous medium with an isotropy effect, Transp. Porous Media 82 (2010) 363–373. [12] P.R. Sharma, G. Singh, Effects of variable thermal conductivity, viscous dissipation on steady MHD natural convection flow of low Prandtl fluid on an inclined porous plate with Ohmic Heating. Meccanica, 45 (2010) 237-247. [13] O.D. Makinde, A. Aziz, Mixed convection from a convectively heated vertical plate to a fluid with internal heat generation, ASME, Journal of Heat Transfer, 133 (2011) 122501(6pages).

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[14] A.M. Rohni, S. Ahmad, I. Pop, Flow and heat transfer over an unsteady shrinking sheet with suction in nanofluids, Int. J. Heat Mass Transf. 55 (2012) 1888–1895. [15] M. Mustafaa, T. Hayat, S. Obaidat, Boundary layer flow of a nanofluid over an exponentially stretching sheet with convective boundary conditions, International Journal of Numerical Methods for Heat and Fluid Flow, 23 (2013) 945-959. [16] H. Dessie, N. Kishan, MHD effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity, viscous dissipation and heat source/sink Ain Shams Engineering Journal, 5(3) (2014) 967-977. [17] J.A. Rao, G. Vasumathi, J. Mounica, Joule heating and thermal radiation effects on MHD boundary layer flow of a nanofluid over an exponentially stretching sheet in a porous medium. World Journal of Mechanics, 5(2015) 151-164. [18] H. Rosali, A. Ishak, R. Nazar, I. Pop, Mixed convection boundary layer flow past a vertical cone embedded in a porous medium subjected to a convective boundary condition. Propulsion and Power Research, 5(2) (2016) 118-122. [19] O.D. Makinde, W.A. Khan, Z.H. Khan, Stagnation point flow of MHD chemically reacting nanofluid over a stretching convective surface with slip and radiative heat, Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 231(4) (2017) 695–703. [20] O. D. Makinde, S. R. Mishra, Chemically reacting MHD mixed convection variable viscosity Blasius flow embedded in a porous medium, Defect and Diffusion Forum, 374 (2017) 83-91. [21] A. S. Eegunjobi, O. D. Makinde, S. Jangili, Unsteady MHD chemically reacting and radiating mixed convection slip flow past a stretching surface in a porous medium, Defect and Diffusion Forum, 377 (2017) 200-210. [22] O. D. Makinde, MHD mixed-convection interaction with thermal radiation and nth order chemical reaction past a vertical porous plate embedded in a porous medium, Chemical Engineering Communications, 198(4) (2011) 590-608. [23] O. D. Makinde, On MHD mixed convection with Soret and Dufour effects past a vertical plate embedded in a porous medium. Latin American Applied Research, 41 (2011) 63-68. [24] O. D. Makinde, Heat and mass transfer by MHD mixed convection stagnation point flow toward a vertical plate embedded in a highly porous medium with radiation and internal heat generation. Meccanica, 47 (2012) 1173-1184. [25] O. D. Makinde, W.A. Khan, Z.H. Khan, Analysis of MHD nanofluid flow over a convectively heated permeable vertical plate embedded in a porous medium, Journal of Nanofluids 5(4) (2016) 574-580.

Defect and Diffusion Forum ISSN: 1662-9507, Vol. 401, pp 140-147 © 2020 Trans Tech Publications Ltd, Switzerland

Submitted: 2019-05-26 Revised: 2019-10-02 Accepted: 2020-04-15 Online: 2020-05-28

Squeeze Film Lubrication on a Rigid Sphere and a Flat Porous Plate with Piezo-Viscosity and Couple Stress Fluid B.N. Hanumagowda1,a*, C.K. Sreekala2,b, Noorjahan1,c, O.D. Makinde3,d School of Applied Sciences, REVA University, Bengaluru-560064, India

1

Department of Mathematics, KNS Institute of Technology, Bangalore, India

2

Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

3

[email protected], [email protected], c [email protected], [email protected]

a

Keywords: Porous plate, rigid sphere, piezo-viscosity, couple stress fluid, film lubrication

Abstract. The paper aims to conduct a theoretical analysis about squeeze film behaviour with piezoviscosity on a fluid lubricated with polar additives on a rigid sphere and a flat porous plate. By taking into account of micro-continuum theory of stokes, Barus formula and Darcy’s equations, Reynolds equation in modified form is derived by bearing in mind the variation of viscosity along film thickness. An approximate analytical solution for fluid film pressure, dimensionless load and squeezing time is derived. The results are presented numerically and graphically. To make obvious the accuracy, the results are compared with accessible literature and a remarkable similarity is reported Introduction Traditionally, a large number of investigations about squeeze film technology on couple-stress fluid is restricted to iso-viscous lubricants [1-3]. The Stokes theory [4] is the simplest theory of fluids which allows polar effects such as the occurrence of couple stresses, body couples, and nonsymmetric tensors. The squeeze film characteristics between a sphere and a flat surface were studied by Conway and Lee [5]. They found that the effect of increase in lubricant viscosity with pressure causes large increase of pressure near the central area as compared to the pressure obtained for isoviscous lubricant. Lin [6] studied the squeeze film characteristics with couple stress fluid model between a sphere and a flat plate and observed that it improves the squeeze film characteristics of the system. But viscosity is not constant in many practical situations, and may affect the lubrication process of bearing significantly. The influence of pressure viscosity oils on temperature, pressure and thickness of film for elasto-hydrodynamic rolling contacts was discussed by Bartz and Ehlert [7]. Hanumagowda [8] examined the characteristics of thin film for circular step bearing and considered the viscosity variation of couple stress fluids. Many investigators have studied the combined impact of PDV and couple stress fluids on various bearings. [9-16]. Permeable bearings are useful because of their self-lubricating characteristics and low cost. The lubricant penetrates the pores and remains effectual throughout the bearing life. Hydrodynamic lubrication of porous metal bearing was first studied by Morgan [17]. The effect of squeeze film on porous rectangular plates has been studied by Wu [18]. Prakash and Vij [19] revealed that porosity may lead to a decrease in load carrying capacity of inclined slider bearing. The porosity inversely influences the bearing characteristics, but it can be overcome by numerous design and maintenance advantages. Cusano [20] has shown that the seepage through the edges of porous bearings may be reduced using porous housings of different permeability to enhance the bearing performance. To the best of authors knowledge, no literatures has been done about combined impact of PDV and couple stress fluids on porous sphere and a flat surface. This encourage the present study, where the key intention is to examine combined impact of the effects of PDV and couple stresses on squeeze film lubrication of porous sphere and a flat surface.

Defect and Diffusion Forum Vol.401

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Mathematical Formulation The squeeze film mechanism between a rigid sphere of radius R and an infinite porous plate is represented in Figure 1 under a constant load W. The viscosity 𝜇𝜇 varies with respect to pressure only.

Figure1. Physical model of the problem Based on the usual thin film lubrication assumptions, the equations governing motion are 1 ∂ ∂w 0, = ( ru ) + r ∂r ∂z

(1)

∂  ∂u  ∂ 4u ∂p , −  −µ  +η 4 = ∂z  ∂z  ∂z ∂r

(2)

∂p = 0. ∂z

(3)

The thickness of the oil-film is given by r2 . = h hm + 2R

(4)

The pressure dependent viscosity given by Barus is

µ = µ0 eα p .

(5)

The boundary conditions for the components of velocity are At z = 0,

u = 0,

At z = h, u = 0 ,

∂ 2u ∂z

∂ 2u ∂z

2

2

=0 ,

w = − w* ,

∂h =0 , w= . ∂t

(6) (7)

The Darcy’s law for porous material is −k ∂p∗ , µ (1 − β ) ∂r

(8)

−k ∂p∗ . w = µ (1 − β ) ∂z

(9)

u∗ = ∗

Making use of equations (5) in equation (2) and solve using boundary conditions (6) and (7) we get

u =

−2α p

e 2 µ0

  2z − h   cosh  −0.5α p    2 ∂p  z − hz 2l   e 2 2  α p + 2l − 2l , h ∂r  e   cosh  −0.5α p   2l   e

(10)

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Computational Analysis of Heat Transfer in Fluids and Solids II

where l = η / µ0

The modified Reynolds equation obtained by applying boundary conditions and integrating continuity equation (1) over film thickness is

1 ∂  ∂p  dh , ϑ (h, l , α , p)r  = 12µ0 r ∂r  ∂r  dt

(11)

where e −α p h3 − 12l 2 he −2α p + 24l 3 e −2.5α p tanh ( e0.5α p h / 2l ) + ϑ ( h, l , α , p ) =

12δ ke−α p . (1 − β )

(12)

Using the following non dimensional parameters in (11) we get

µ α R ( ∂h / ∂t ) * hm ph02 r * r = , P = , ζ = , hm = , − − 0 R h0 µ0 R ( ∂h / ∂t ) h02 *

* = h* h= / h0 , δ h= l / h0 0 / R, l Hence the dimensionless Reynolds equation becomes

* ∂  * * * 12 *  * * ∂p  = − ( h , l , , p , ) r r ϑ ζ ψ   δ ∂r *  ∂r * 

(13)

where *

*

*

*

*3 −ζ p*

h e ϑ (h , l , ζ , p ,ψ ) =

*2 * −2ζ p*

− 12l h e

*3 −2.5ζ p*

+ 24l e

tanh ( e

12ψ eζ p h / 2l ) + (1 − β )

0.5ζ p* *

*

*

.

(14)

Equation (14) is a non-linear equation. Therefore by perturbation, the film pressure is expanded for 0 ≤ ζ 1 a/c = 1.4, 1.6, 1.7

a/c = 0.3, 0.2, 0.1, 0 a/c = 1.4, 1.6, 1.8

1.2

1.6

a/c < 1,

a/c > 1

1.0

0.8

1.2 f '(y)

h'(y) 0.6

0.8

0.4

a/c = 0.1, 0.2, 0.3

0.4

0.2

0.0 0

1

2

3

5 y 6

4

7

8

9

0.0

10

0

1

Fig.3(a):Axial velocity profile for different values of a/c with Pr = 1, Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1,  = 0.1, r = -5,  = 2,  = 0.5.

2

4 y

3

5

6

7

8

Fig.3(b):Transverse velocity profile for different values of a/c with Pr = 1, Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1,  = 0.1, r = -5,  = 2,  = 0.5.

1.0

0.5 a/c < 1,

a/c < 1,

a/c > 1

0.4

0.8

0.3

0.6

 (y)

 (y)

0.4

0.2

a/c = 1.8, 1.6, 1.4, 0.3, 0.2, 0.1, 0

a/c = 1.8, 1.6, 1.4, 0.3, 0.2, 0.1, 0 0.1

0.2

0.0

0.0

0

1

2

3

y

4

5

0

6

Fig.3(c):Temperature profile for different values of a/c with Pr = 1, Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1,  = 0.1, r = -5,  = 2,  = 0.5.

1

2

1.4 r = -2,

y

3

4

5

6

Fig.3(d):Concentration profile for different values of a/c with Pr = 1, Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1,  = 0.1, r = -5,  = 2,  = 0.5.

1.0 r = -5,

a/c > 1

r = -1.5

r = -5,

1.2

0.8

r = -2,

r = -1.5

 = 10, 5, 2

1.0 0.8

0.6

h'(y) 0.6

f '(y)  = 10, 5, 2

0.4

0.4 0.2

0.2

0.0 -0.2

0.0 0

1

2

y

3

4

Fig.4(a):Axial velocity profile for different values of r and  with Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1, Pr = 1, a/c = 0,  = 0.1,  = 0.5.

5

0

1

2

3

4

y

5

6

7

Fig.4(b):Transverse velocity profile for different values of r and  with Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1, Pr = 1, a/c = 0.0,  = 0.1,  = 0.5.

8

192

Computational Analysis of Heat Transfer in Fluids and Solids II

0.5

r = -5,

r = -2,

1.0

r = -1.5

0.4

r = -2,

r = -1.5

0.8

0.3

0.6  = 2, 5, 10

 (y)

 (y)

 = 2, 5, 10

0.2

 = 2, 5, 10  = 2, 5, 10

0.4

0.1

0.0

r = -5,

0.2

0

1

2

3

y

4

5

Fig.4(c):Temperature profile for different values of r and  with Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1, Pr = 1, a/c = 0,  = 0.1,  = 0.5.

6

0.0 0

1

2

3

y

4

5

6

Fig.4(d):Concentration profile for different values of r and  with Sc = 1, Nt = 0.1, Nb = 0.1,  = 0.1, Pr = 1, a/c = 0,  = 0.1,  = 0.5.

The impact of Brownian motion parameter Nb and the Thermophoresis parameter on the temperature and concentration profiles are depicted in Fig 5(a-b). With the increase of values of Nb , more and more heat is transferred and hence there is an enhancement in the thermal boundary layer thickness and in turn temperature profile upsurges. On the other hand, the reverse phenomenon is seen for the concentration profile. Thermophoresis force gets enhanced with Nt , which results in the augmentation of concentration boundary layer thickness and concentration profile. In this case, the nanoparticles move away from the hot sheet towards the ambient fluid. Hence, the same trend is observed in both the profiles, namely temperature as well as concentration. (see Fig 5(a-b)). From Fig.6, it is noticed that for higher values of Biot number (  = 0.5,1,10,100 ) , the thickness of the thermal boundary layer gets enhanced and consequently, a remarkable rise in the temperature profile is observed. As is well known, the Biot number  is the ratio of convective to conductive heat transfer coefficient. So,   1 reveals that the heat transfer is mainly due to conduction, while   1 indicates that the convection is the key source of heat transfer. The impact of Pr and 1 on the temperature profile  ( ) is presented in Fig. 7. As Pr increases, the temperature decreases and hence the thermal boundary layer thickness decreases, while the reverse tendency is observed in the case of 1 . The reason behind this phenomenon is the assumption of temperature-dependent thermal conductivity (i.e., k = k (T )) in the energy equation, which decreases the magnitude of transverse velocity by a quantity k (T ) / y . The effect of increasing values of Sc and  2 is presented in Fig. 8. As

Sc ( =   / D ) increases, there is a decrease in the molecular diffusivity D , which results in a

reduction in the concentration boundary layer thickness and therefore, the concentration profile decreases and exactly the opposite trend occurs with an increase in  2 . Finally, streamlines graph has been plotted for  = −5, 0,5 and depicted in Fig. 9(a-c). The reduced form of skin friction coefficients ( f (0) and h (0) ) , Nusselt number  (0) , and

Sherwood number  (0) are charted in Table 3. Skin friction f ''(0) decreases for the increasing values of both  r and  , whereas it increases for increasing values of a c . Increasing values of a c

and  r have a twofold consequence on skin friction coefficient h (0) , initially near the sheet it grows then falls as moved away from the sheet. Nusselt number  '(0) gains for growing values of  r , Nb ,

Nt ,  and 1 , whereas an opposite tendency is witnessed for Pr and a c . With increasing values of  r ,  , Nt and  2 . The reduced Sherwood number  (0) rises while it declines in the case of a c , Nb and Sc , respectively.

Defect and Diffusion Forum Vol.401

193

1.0

Nt = 0.5,

0.6

Nt = 1,

Nt = 0.5,

Nt = 1.5

Nt = 1,

Nt = 1.5

0.8

0.5

0.4

0.6

 (y)

 (y)

0.3

Nb = 0.5, 1

Nb = 1, 0.5

0.4

0.2

0.2 0.1

0.0

0.0

0

1

2

y

3

4

5

0

6

1

2

Fig.5(a):Temperature profile for different values of Nt and Nb with Sc = 1,  = 0.1,  = 2, r = -5, Pr = 1, a/c = 0.01,  = 0.1,  = 0.5.

1.0 Pr = 1,

y

3

4

5

6

Fig.5(b):Concentration profile for different values of Nt and Nb with Sc = 1,  = 0.1,  = 2, r = -5, Pr = 1, a/c = 0.01,  = 0.1,  = 0.5.

Pr = 2

 = 0.1,

0.5

0.9

 = 0.4

0.8

0.4 0.7 0.6

0.3

 (y)

0.5

(y)

Pr = 5, 2, 1, 0.72

 = 0.5, 1, 10, 100

0.4

0.2

0.3 0.2

0.1

0.1 0.0 0

1 2 3 4 5 y Fig.6:Temperature profile for different values of Pr and  with Sc = 1.0 Nt = 0.1, Nb = 0.1, a/c = 0, r = -5, 1 = 0.1, 2 = 0.1,  = 2.

0.0

6

0

y 1 2 3 4 5 Fig.7:Temperature profile for different values of Pr and 1 with Sc = 1, a/c = 0.1, Nt = 0.1, Nb = 0.1, 2 = 0.1, r = -5,  = 2,  = 0.5.

1.0  = 0.1,

6

 = 0.4

(a)

3.0

0.8

 = -0.5

2.5

0.6

2.0

y

(y)

1.5

0.4

Sc = 2, 1, 0.6, 0.4, 0.22 1.0

0.2

0.5

0.0 0

1 2 3 y 4 5 6 7 Fig.8:Concentration profile for different values of Sc and 2 with Pr = 1, a/c = 0.1, Nt = 0.1, Nb = 0.1, 1 = 0.1, r = -5,  = 2,  = 0.5.

0.0 3

2

1

0

x

1

2

3

194

Computational Analysis of Heat Transfer in Fluids and Solids II

(b)  = 0

3.0

(c)  = 0.5

3.0

2.5

2.5

2.0

y

y

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0 3

2

1

0

1

2

3

3

2

1

x

0

1

2

3

x

Fig. 9(a,b,c): Streamlines with Pr = Sc = 1, Nb = Nt = 1 =  2 = a / c = 0.1,  = 2 and  r = −5. Table 3: Numerical values of the physical quantities f ''(0), h(0),  (0) and  (0) , for various values of pertinent parameters.

h ( 0 )

  ( 0)

  (0)

-0.869926

-0.124339

-0.263151

-0.464874

-0.9664

-0.0231614

-0.260945

-0.456914

-1.00771 -1.07149 -1.07154 -1.07158

0.0199124 0.373965 0.373988 0.374040

-0.260138 -0.261248 -0.261181 -0.261225

-0.453866 -0.486877 -0.461677 -0.441482

-1.07160

0.367658

-0.261381

-0.424788

-0.869926

-0.124339

-0.263151

-0.464874

-0.886308

-0.124877

-0.352663

-0.430137

2.0

-0.913978

-0.128347

-0.501181

-0.377583

2.0

-0.918367

-0.125867

-0.522698

-0.371084

0.1

2.0

-0.918768

-0.1256

-0.5247

-0.370508

1000

0.1

2.0

-0.918818

-0.125568

-0.524951

-0.370436

0.1 0.5 1.0

5000 0.5 0.5

0.1 0.1 0.1

2.0 2.0 2.0

-0.918858 -0.871291 -0.875205

-0.12553 -0.105845 -0.106312

-0.525152 -0.229806 -0.197257

-0.370378 -0.509068 -0.559801

0.5

1.5

0.5

0.1

2.0

-0.880311

-0.105394

-0.153608

-0.587624

-5.0

0.5

1.0

0.5

0.1

2.0

-0.875205

-0.106312

-0.197257

-0.559801

-5.0

1.0

1.0

0.5

0.1

2.0

-0.876537

-0.109383

-0.185414

-0.560494

0.1

-5.0

1.5

1.0

0.5

0.1

2.0

-0.877877

-0.112566

-0.173355

-0.573595

0.1

0.1

-5.0

0.1

0.1

0.5

0.1

2.0

-0.835959

-0.207349

-0.264956

-0.472786

1.0

0.1

0.1

-2.0

0.1

0.1

0.5

0.1

2.0

-0.898192

0.121602

-0.263899

-0.468805

1.0

1.0

0.1

0.1

-1.5

0.1

0.1

0.5

0.1

2.0

-0.679792

-0.0753735

-0.272926

-0.511864

1.0 1.0 1.0

1.0 1.0 1.0

0.1 0.1 0.1

0.0 0.1 0.2

-5.0 -5.0 -5.0

0.1 0.1 0.1

0.1 0.1 0.1

0.5 0.5 0.5

0.1 0.1 0.1

0.1 0.1 0.1

-1.07083 -1.07154 -1.07222

0.373899 0.373988 0.374052

-0.264879 -0.261181 -0.257592

-0.456769 -0.461677 -0.466439

Pr

Sc

a/c

1

r

Nt

Nb



2



f  ( 0 )

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

0.5

0.1

2.0

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

0.5

0.1

5.0

1.0 1.0 1.0 1.0

1.0 1.0 1.0 1.0

0.0 0.1 0.1 0.1

0.1 0.1 0.1 0.1

-5.0 -5.0 -5.0 -5.0

0.1 0.1 0.1 0.1

0.1 0.1 0.1 0.1

0.5 0.5 0.5 0.5

0.1 0.0 0.1 0.2

10.0 0.1 0.1 0.1

1.0

1.0

0.1

0.1

-5.0

0.1

0.1

0.5

0.3

0.1

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

0.5

0.1

2.0

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

1.0

0.1

2.0

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

10

0.1

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

100

0.1

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

500

1.0

1.0

0.0

0.1

-5.0

0.1

0.1

1.0 1.0 1.0

1.0 1.0 1.0

0.0 0.01 0.01

0.1 0.1 0.1

-5.0 -5.0 -5.0

0.1 0.5 0.5

1.0

1.0

0.01

0.1

-5.0

1.0

1.0

0.01

0.1

1.0

1.0

0.01

0.1

1.0

1.0

0.01

1.0

1.0

1.0

Conclusions Analysis of oblique stagnation point flow of Casson nanofluid over a stretching sheet in the presence of variable fluid properties has been carried out. Obtained nonlinear partial differential equations (PDEs) are converted to dimensionless nonlinear ordinary differential equations (ODEs) via similarity transformations and are solved using Optimal Homotopy Analysis Method (OHAM). Following are some interesting conclusions: ➢ Axial velocity decreases with an increase in the variable viscosity parameter θr and the Casson parameter β whereas, it increases for increasing values of a/c . On the other hand, the dual effect of variable viscosity parameter  r and a/c is observed on transverse velocity.

Defect and Diffusion Forum Vol.401

195

➢ Thermal boundary layer thickness enhances for increasing values of variable viscosity parameter θr, Casson parameter β, thermal conductivity parameter  1 , thermophoresis parameter Nt , Brownian motion parameter Nb and Biot number  . On the contrary, the constant a/c and Prandtl number Pr decrease the thermal boundary layer thickness. ➢ Influence of variable viscosity parameter θr, Casson parameter β , thermophoresis parameter Nt and variable species diffusivity parameter  2 is to enhance the concentration boundary layer thickness and the reverse trend is observed for the constant a/c, the Brownian motion parameter Nb and the Schmidt number Sc . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15]

[16]

K. Hiemenz, Die Grenzschicht an einem in den gleichformigen Flussigkeitsstrom eingetauchten geraden Kreiszylinder, Dinglers Polytech. J. 326 (1911) 321–324. Y. Matunobu, Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. I, J. Phys. Soc. Japan. 42 (1977) 2041–2049. Y. Matunobu, Structure of pulsatile Hiemenz flow and temporal variation of wall shear stress near the stagnation point. II, J. Phys. Soc. Japan. 43 (1977) 326–329. K. Tamada, Two-dimensional stagnation-point flow impinging obliquely on a plane wall, J. Phys. Soc. Japan. 46 (1979) 310–311. H. Niimi, M. Minamiyama, S. Hanai, Steady axisymmetrical stagnation-point flow impinging obliquely on a wall, J. Phys. Soc. Japan. 50 (1981) 17–18. T. Chiam, Stagnation-point flow towards a stretching plate, J. Phys. Soc. Japan. 63 (1994) 24432444. T.R. Mahapatra, A.S. Gupta, Heat transfer in stagnation-point flow towards a stretching sheet, Heat Mass Transf. 38 (2002) 517–521. R. Nazar, N. Amin, D. Filip, I. Pop, Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet, Int. J. Eng. Sci. 42 (2004) 1241–1253. M. Reza, A.S. Gupta, Steady two-dimensional oblique stagnation-point flow towards a stretching surface, Fluid Dyn. Res. 37 (2005) 334. Q. Wu, S. Weinbaum, Y. Andreopoulos, Stagnation-point flows in a porous medium, Chem. Eng. Sci. 60 (2005) 123–134. F. Labropulu, D. Li, I. Pop, Non-orthogonal stagnation-point flow towards a stretching surface in a non-Newtonian fluid with heat transfer, Int. J. Therm. Sci. 49 (2010) 1042–1050. O. D. Makinde, Heat and mass transfer by MHD mixed convection stagnation point flow toward a vertical plate embedded in a highly porous medium with radiation and internal heat generation. Meccanica, 47 (2012) 1173-1184. O. D. Makinde, W. A. Khan, Z. H. Khan, Buoyancy effects on MHD stagnation point flow and heat transfer of a nanofluid past a convectively heated stretching/shrinking sheet. International Journal of Heat and Mass Transfer 62 (2013), 526-533. W. A. Khan, O. D. Makinde, Z. H. Khan, Non-aligned MHD stagnation point flow of variable viscosity nanofluids past a stretching sheet with radiative heat. International Journal of Heat and Mass Transfer, 96 (2016) 525-534. W. Ibrahim, O.D.Makinde, Magnetohydrodynamic stagnation point flow of a power-law nanofluid towards a convectively heated stretching sheet with slip. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 230(5), (2016), 345-354. O.D. Makinde, W.A. Khan, Z.H. Khan, Stagnation point flow of MHD chemically reacting nanofluid over a stretching convective surface with slip and radiative heat. Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering, 231(4) (2017) 695–703.

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[17] S. Nadeem, R. Mehmood, N.S. Akbar, Optimized analytical solution for oblique flow of a Casson-nano fluid with convective boundary conditions, Int. J. Therm. Sci. 78 (2014) 90–100. [18] O. D. Makinde, A. Aziz, MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition, International Journal of Thermal Sciences, 49(2010) 1813-1820. [19] O. D. Makinde, P. O. Olanrewaju, Buoyancy effects on thermal boundary layer over a vertical plate with a convective surface boundary condition, Transaction of ASME-Journal of Fluid Engineering, 132 (2010) 044502(4pages). [20] O. D. Makinde, A. Aziz, Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition, International Journal of Thermal Sciences, 50 (2011) 13261332. [21] O. D. Makinde, Similarity solution for natural convection from a moving vertical plate with internal heat generation and a convective boundary condition, Thermal Science, 15(1) (2011) S137-S143. [22] A. Ghaffari, T. Javed, F. Labropulu, Oblique stagnation point flow of a non-Newtonian nanofluid over stretching surface with radiation: A numerical study, Therm. Sci. 21 (2017) 2139–2153. [23] K. V. Prasad, K. Vajravelu, H. Vaidya, M.M. Rashidi, Z.B. Neelufer, Flow and heat transfer of a Casson liquid over a vertical stretching surface: Optimal solution, Am. J. Heat Mass Transf. 5 (2018) 1–22. [24] K. Vajravelu, K. V. Prasad, H. Vaidya, Influence of Hall current on MHD flow and heat transfer over a slender stretching sheet in the presence of variable fluid properties, Commun. Numer. Anal. 2016 (2016) 17–36. [25] K.V. Prasad, H. Vaidya, K. Vajravelu, P.S. Datti, V. Umesh, Axisymmetric mixed convective MHD flow over a slender cylinder in the presence of chemically reaction, Int. J. Appl. Mech. Eng. 21 (2016) 121–141. [26] K. V Prasad, H. Vaidya, K. Vajravelu, M.M. Rashidi, Effects of variable fluid properties on MHD flow and heat transfer over a stretching sheet with variable thickness, J. Mech. 33 (2017) 501–512. [27] K. V. Prasad, K. Vajravelu, H. Vaidya, B.T. Raju, Heat transfer in a non-Newtonian nanofluid film over a stretching surface, Journal of Nanofluids. 4 (2015) 536-547. [28] K. Vajravelu, K. V. Prasad, H. Vaidya, Influence of Hall Current on MHD flow and Heat transfer Over a Slender Stretching sheet in the Presence of Variable Fluid Properties, J. Communications in Numerical Analysis. 1 (2016) 17-36. [29] R. Mehmood, M.K. Nayak, N.S. Akbar, O.D. Makinde, Effects of thermal-diffusion and diffusion-thermo on oblique stagnation point flow of couple stress Casson fluid over a stretched horizontal Riga plate with higher order chemical reaction, Journal of Nanofluids 8(1) (2019) 94-102. [30] I. Ullah, T. Abdullah Alkanhal, S. Shafie, K.S. Nisar, I. Khan, O.D. Makinde, MHD slip flow of Casson fluid along a nonlinear permeable stretching cylinder saturated in a porous medium with chemical reaction, viscous dissipation, and heat generation / absorption, Symmetry 11(4) (2019) 531(27pages). [31] S. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 2003–2016. [32] R.A. Van Gorder, Optimal homotopy analysis and control of error for implicitly defined fully nonlinear differential equations, Numer. Algorithms. (2018) 1–16. doi:10.1007/s11075-0180540-0.

Keyword Index A Analytical Solution

1, 14

B Boundary Layer Brownian Motion

92 63 47 131 63, 131 183 14 25 140

1 63, 164

E Effective Length Environmental Influence Extended Surface

14 107 1

F Film Lubrication Flow Transfer Forced Convection Functionally Graded Material

140 25 117 14

Injection Irregular Heat Source/Sink

63 79

Joule Heating Joule’s Heat

131 79

K Kummel Function

14

M Magnetic Parameter Maxwell Fluid MHD Microplar Fluid Mixed Convection Mixed Convection Flow Molybdenum Disulfide Nanofluid Moving Fin

47 148 47, 63, 79, 92 79 131 164 92 14

N Nanofluid Non-Linear Radiation Non-Newtonian Fluid Non-Similar Analysis Numerical Solutions Nusselt Number

183 79 25 25 25 107, 117

O

G Groundwater Aquifer

107

J

D DTM Dufour Effect

148

I 131 183

C Casson Fluid Chemical Reaction Chemical Reaction Parameter Cone Convective Boundary Condition Convective Heating Convective-Radiative Fin Correlation Expression Couple Stress Fluid

Homogeneous-Heterogeneous Reaction Human Tissue

36

Oldroyd-B Nanofluid Optimal Homotopy Analysis Method (OHAM)

164 164

H Hall Current Heat Absorption Heat Generation Heat Transfer Heat Transfer Aspect

47 164 63, 164 1, 25 117

P Piezo-Viscosity Porous Plate Powell-Eyring Viscosity Model

140 140 25

198

Computational Analysis of Heat Transfer in Fluids and Solids II

R Radiation Parameter Rigid Sphere

47 140

S Similar Analysis Skin Friction Soil Erosion Rate Soil Heat Transfer Rate Soil Water Percolation Soret Soret Effect Stagnation Point Flow Suction Surface Runoff

25 107 36 36 36 164 63 183 63 36

T Thermal Buoyancy Thermal Enhancement Thermal Radiation Thermophoresis Effect Thermoregulation

107 117 92 183 107

V Variable Fluid Properties Variable Thermal Conductivity Variable Thickness

183 148 148

W W-Baffle

117

Author Index A Abd-Alhameed, R.A. Adeola, L.H. Ahmad, A. Ahmed, B. Ahmed, Y. Animasaun, I.L.

14 107 25 25 14 25

N Narayana, P.V.S. Ndlovu, L.P. Noorjahan,

148 1 140

O Oguntala, G.A.

14

B Basha, H. Basha, N.Z. Bhuvaneswari, M.

164 183 63

117

D Deng, J.M.M. Devi, S.

36 131

E Eswaramoorthi, S.

63

H Hanumagowda, B.N.

140

J Jyothi, P.K.

Prasad, K.V.

164, 183

R

C Chamkha, A.J.

P

148

Rahimi-Gorji, M. Ram, P. Raza, J. Reddy, B.P.

164 92 92 47

S Sandeep, N. Sarojamma, G. Sharma, M.K. Sharma, S. Sivaiah, S. Sivasankaran, S. Sobamowo, G. Sreekala, C.K. Sreelakshmi, K. Sugunamma, V. Sunzu, J.M.

79 148 131 92 79 63 14 140 148 79 47

K Khan, R. Kumar, K.A.

25 79

M Makinde, O.D. Manjunatha, G. Matao, P.M. Mebarek-Oudina, F. Menni, Y. Moitsheki, R.J. Monaledi, R.L.

36, 63, 107, 117, 140 164, 183 47 92 117 1 25

V Vaidya, H. Vajravelu, K. Vishwanatha, U.B.

164, 183 164, 183 183

W Wakif, A.

25, 183

Z Zaydan, M.

25