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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 16, Issue 1

# Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel

Zeeshan Khan
/ Haroon Ur Rasheed
/ Tawfeeq Abdullah Alkanhal
• Department of Mechatronics and System Engineering, College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia
• Other articles by this author:
• Department of Mathematics, Islamia College University of Pesshawar, KP, 2500, Peshawar Pakistan
• Other articles by this author:
/ Ilyas Khan
/ Iskander Tlili
• Energy and Thermal Systems Laboratory, National Engineering Scholl of Monastir, Street Ibn El Jazzar, 5019 Monastir, Tunisia
• Other articles by this author:
Published Online: 2018-12-31 | DOI: https://doi.org/10.1515/phys-2018-0113

## Abstract

The effect of magnetic field on the flow of the UCMF (Upper-Convected-Maxwell Fluid) with the property of a heat source/sink immersed in a porous medium is explored. A shrinking phenomenon along with the permeability of the wall are considered. The governing equations for the motion and transfer of heat of the UC MF along with boundary conditions are converted into a set of coupled nonlinear mathematical equations. Appropriate similarity transformations are used to convert the set of nonlinear partial differential equations into nonlinear ordinary differential equations. The modeled ordinary differential equations have been solved by the Homotopy Analysis Method (HAM). The convergence of the series solution is established. For the sake of comparison, numerical (ND-Solve method) solutions are also obtained. Special attention is given to how the non-dimensional physical parameters of interest affect the flow of the UCMF. It is observed that with the increasing Deborah number the velocity decreases and the temperature inside the fluid increases. The results show that the velocity and temperature distribution increases with a porous medium. It is also observed that the magnetic parameter has a decelerating effect on velocity while the temperature profiles increases in the entire domain. Due to the increase in Prandtl number the temperature profile increases. It is also observed that the heat source enhance the thermal conductivity and increases the fluid temperature while the heat sink provides a decrease in the fluid temperature.

PACS: 45.10.-b; 47.10.-g; 47.55.pb; 47.56.+r

## Nomenclature

β Deborah number

η Similarity variable

ν Kinematic viscosity (m2s−1)

θ Dimensionless temperature

φ Stream function

λ1 Internal heat generation/absorption parameter

λR Relaxation time (t)

C Dimensional constant (t−1)

f Similarity variable

K Thermal conductivity

M Magnetic field (Te)

Nux Nusselt number

p Density of the fluid (kg m3)

Pr Prandtl number

Q Hear source/sink parameter

S Suctions/injection parameter(ms−1)

T Fluid’s temperature (K)

T Ambient temperature (K)

Tw Wall temperature (k)

U, V Velocity components (ms−1)

V1, V2 Wall velocity

x,y Cartesian coordinates (I)

## 1 Introduction

This technique has already been used for the solution of various problems [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. Ghadikolaei et al. [38] studied nonlinear thermal radiation effect on magneto Casson nanofluid flow with joule heating effect over an inclined porous stretching sheet. Hosseinzadeh et al. [39] investigated a numerical investigation on ethylene glycoltitanium dioxide nanofluid convective flow over a stretching sheet in presence of heat generation/absorption. Rahmati et al. [40] studied dimultaneous investigations the effects of non-Newtonian nanofluid flow in different volume fractions of solid nanoparticles with slip and no-slip boundary conditions. Ghadikolaei et al. [41] carried out analytical and numerical solution of non-Newtonian second-grade fluid flow on a stretching sheet. Sheikholeslami et al. [42] studied heat transfer improvement and pressure drop during condensation of refrigerant-based nanofluid; an experimental procedure. Guo et al. [43] studied numerical study of nanofluids’ thermal and hydraulic characteristics considering Brownian motion effect in a micro fin heat sink. Amini et al. [44] investigated thermal conductivity of highly porous metal foams using experimental and image based finite element analysis. Tian et al. [45] studied heat conduction investigation of functionally graded material plates with variable gradient parameters under exponential heat source load [46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67].

Furthermore, for the sake of clarity the proposed method is also compared with ND-Solve method [68] and ADM [70, 71]. The effect of modeled parameters such as internal heat generation/absorption, porosity, magnetic parameter, suctions/injection, and Deborah number on the solutions has been shown graphically and discussed. Furthermore, the present result was also compared with published work. A table has been constructed in order to represent the numerical values of a local Nusselt number of different involved physical quantities.

## 2 Statement of the problem

The system deals with two dimensional steady flow of an Upper-Convected-Maxwell fluid over pa shrinking surface subject to a constant transverse magnetic field B0. A porous channel is used to show the effects of suction/junction. The fluid saturates the porous medium fo y > 0 and the flow occupies the positive region y-axis. The porous medium saturates for > 0 . The shrinking velocity of the wall is V1 = −Hw1 where and the suction/blowing parameter is represented by V2. It is pointed out here that V2 < 0 represents suction phenomena while V2 > 0 corresponds to a blowing situation. It is also assumed that the temperature of the wall and free stream conditions are θw and θ, respectively. The heat generation or absorption q is also taken into account. In view of these assumptions, the governing basic flow equations are given in the following [1, 2, 3, 4, 5]:

$∂u∂x+∂v∂y=0,$(1)$u∂u∂x+v∂u∂y+λRu2∂2u∂x2+v2∂2u∂y2+2uv∂2u∂x∂y=v∂2u∂y2−σB02ρfu+λRv∂u∂y−vMpu+λRv∂u∂y,$(2)$u∂T∂x+v∂T∂y=α∂2T∂y2+qρcpT−T∞,$(3)

with the following boundary conditions

$u=V1,v=V2,T=Twaty=0,$(4)$u→0,T=T∞aty→∞.$(5)

u and v are the velocity components along the x and y axes, respectively. Here V1 = −cx is the shrinking velocity of the wall where c > 0 and V2 is the suction/blowing. Furthermore, (x, y) represents the coordinate system, λR is relaxation time, σ is electrical conductivity, B0 is magnetic field strength, ρf is fluid density, M p is permeability of porous media, T is fluid temperature, α is thermal diffusivity, and cp is the specific heat. Eqs. (2) and (3) can be transformed into a set of nonlinear ordinary differential equations by introducing the following similarity variables. $u=\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }y}$,

$u=∂ϕ∂y,v=∂ϕ∂x,ϕ=(cv)1/2xf(η),η=cv,De=λRc,M2=σB02cρf,S=−V2/(vc)1/2,Mp=vcMp,θ(η)=T−T∞Tw−T∞,α=kρfcp,Pr=vα,λH=qcρfcp.$(6)

The transformed ordinary differential equations are

$f‴−f′2+ff′+De2ff′f″−f2f‴+M2Deff″−M2f′−Mpf′+MpDeff″=0,$(7)$θ″+Prfθ′+PrλHθ=0.$(8)

The boundary conditions (4) and (5) become

$f0=S,f′0=−1,f′∞=0,$(9)$θ(0)=1,θ(∞)=0.$(10)

Here De = λRc represents the Deborah number in terms of relaxation time, ${M}^{2}=\frac{\sigma {{B}_{0}}^{2}}{c{\rho }_{f}}$is the magnetic parameter, S = −V2/(vc)1/2 is the suction/blowing parameter where S > 0 means wall mass suction and S < 0 means wall mass injection,${M}_{p}=\frac{v}{c{M}_{p}}$is the porosity parameter, $Pr=\frac{v}{\alpha }$represents the Prandtl number, ${\lambda }_{H}=\frac{q}{c{\rho }_{f}{c}_{\rho }}$is the internal heat generation/absorption parameter, and prime represents differentiation with respect to η.

The local Nusselt number Nux is the physical quantity of interest for the readers. It is defined as

$Nux=xqwkTw−T∞,$(11)

where qw (the wall heat flux) is defined as

$qw=−k∂T∂Ty=0.$(12)

In dimensionless form we can write the above expression as

$Nux/Rex1/2=−θ′(0).$(13)

## 3 HAM solution

In order to solve Equation (7) and (8) under the boundary conditions (8) and (10), we utilize the homotopy analysis method (HAM) with the following procedure. The solutions having the auxiliary parameters ħ regulate and control the convergence of the solutions. The initial guesses are selected as follows:

We select the initial approximation ssuch that the boundary conditionsaare satisfied as follows:

$f0(η)=s−1+e−ηandθ0(η)=e−η.$(14)

The linear operators are introduced as ${\mathrm{\Im }}_{f}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}{\mathrm{\Im }}_{\theta }:$

$ℑf(f)=f′andℑθ(θ)=θ″.$(15)

With the following properties:

$ℑf(c1+c2η+c3η2+c4e−η)=0 and ℑθ(c5+c6e−η)=0,$(16)

where ci(i = 1 − 6) are arbitrary constants in general solution.

The nonlinear operators, according to (7) and (8), are defined as:

$ℵff(η;p)=∂3f(η;p)∂η3−∂f(η;p)∂η2+f(η;p)∂f(η;p)∂η+De2f(η;p)∂f(η;p)∂η∂2f(η;p)∂η2−f(η;p)2∂3f(η;p)∂η3+M2Def(η;p)∂2f(η;p)∂η2−M2+Mp∂f(η;p)∂η+MpDef(η;p)∂2f(η;p)∂η2=0,ℵθf(η;p),θ(η;p)=∂2θ(η;p)∂η2+Prf∂θ(η;p)∂η+PrλHθ(η;p).$(17)

The auxiliary function becomes

$Hf(η)=Hθ(η)=e−η.$(18)

The symbolic software Mathematica is employed to solve ith order deformation equations:

$ℑffi(η)−χifi−1(η)=ℏfHff(η)Rf,i(η),ℑθθi(η)−χiθi−1(η)=ℏθHθ(η)Rθ,i,$(19)

where is auxiliary non-zero parameter and

$Rf,i(η)=fm−1′′′−∑k=0m−1fm−1−k′fk′+∑k=0m−1fm−1−kfk′+2De∑k=0m−1fm−1−k∑l=0kfk−l′fl′′−∑k=0m−1fm−1−k∑l=0kfk−l′fl′′′+DeM2+Mp∑k=0m−1fm−1−k′fk′′−M2+Mpf′,Rθ,i(η)=θm−1′′+Pr∑k=0m−1fm−1−kθk′+PrλHθm−1,$(20)$χi=0,if i≤11,if i>1,$

mation about the different steps of HAM see [16, 17, 18, 19, 20, 21, 22, 23, 24]).

To control and speed the convergence of the approximation series by the help of the so-called h-curve, it is significant to choose a proper value of auxiliary parameter. The calculation are carried out on a personal computer with 4GB RAM and 2.70GHz CPu. The h-curves of f (0) and θ(0) obtained by the 18th order of HAM solution are shown in Figure 1 which take approximately less than a minute in the exicution. To obtain the optimal values of auxiliary parameters, the averaged residual errors are defined as

Figure 1

The ħ-curve of f (0) and θ ̗(0) obtained by the 18th order of HAM solution when De = Mp = λH = 0.1, M = 0.2, R = 0.3 and Pr = 0.5

$Ref=d3f(η)δη3−df(η)δη2+f(η)df(η)δη+De2f(η)df(η)dηd2f(η)∂η2−f(η)2d3f(η)dη3+M2Def(η)d2f(η)dη2−M2+Mp∂f(η)dη+MpDef(η)d2f(η)dη2=0,$(21)$Reθ=d2θ(η)dη2+Prfdθ(η)dη+PrλHθ(η).$(22)

In order to survey the accuracy of the present method, the residual errors for the 20th order of HAM solutions of (21) and (22) are illustrated in Figures 2 and 3, and listed numerically in Table 1. An efficient numerical (ND-Solve) method is also applied to solve the transformed equation (7) and (8) correspond ding to the boundary conditions given in equation (9) and (10) and compared with a HAM solution graphically and numerically as shown in Figures 4-5 and Tables 1-2, respectively.

Figure 2

Residual error of Eq. (21) when De = Mp = λH = 0.1, M = 0.2, R = 0.3 and Pr = 0.5

Figure 3

Residual error of Eq. (22) when De = Mp = λH = 0.1, M = 0.2, R = 0.3 and Pr = 0.5

Figure 4

Comparison between HAM and Numerical solutions for velocity profile f (η) when s = 1.5, De = Mp = 0.1, λH = 0.2, M = 0.2 and R = 0.3

Figure 5

Comparison between HAM and Numerical solutions for temperature profile θ(η) when s = 1, De = Mp = 0.1, λH = 0.2, M = 0.2, R = 0.3 and Pr = 0.5

Table 1

Numerical comparison of HAM and numerical method for f (η) when s = 1.5

Table 2

Numerical comparison of HAM and numerical method for θ(η) when s = 1, De = Mp = 0.1, λH = 0.2, M = 0.2, R = 0.3 and Pr = 0.5

## 4 Results and discussion

The transformed equation (7) and (8) subject to the boundary conditions (9) and (10) are solved analytically

by the homotopy analysis method. An efficient numerical method called ND-Solve method is also used for the sake of comparison. In this section, numerical values are assigned to the physical parameters involved in the velocity, temperature

and local Nusselt number. The paper examined the effects of governing parameters on the transient velocity profile, temperature profile and local Nusselt number. For this purpose the SRM approach has been applied for various values of flow controlling parameters De = 0.1, Mp = 0.3, λH = 0.2, M = 0.2, R = 0.3 and Pr = 0.5, to obtain a clear insight into the physics of the problem. Therefore, all the graphs and tables correspond to the values above and the rest will be mentioned.

Figures 6 and 7 show the influence of Deborah number De on the velocities f and f ' respectively. It is obvious from these figures that the boundary layer thickness decreases for larger values of De, with an increase in η. Physically, viscous effects increase for the larger Deborah numbers. These retard the flow in the entire domain and consequently the momentum boundary layer will be thinner. Since the Deborah number defines the difference between the solids and liquid (or fluids), the material shows fluid like behavior for a small Deborah number and for larger values of Deborah number the material behaves like a viscoelastic solid such as rubber, jelly, polymers etc. From the present analysis it is quite obvious that the velocity field decelerating for larger numbers.

Figure 6

Effect of De on velocity profile f when S = 1,Mp = 0.3, M = 0.1

Figure 7

Effect of De on velocity profile f ' when S = 1, Mp = 0.3, M = 0.1

Figure 8 and 9 depict the effects of porosity parameter Mp on f and f ' respectively. It is observed that the velocity profile increases with increasing porosity parameter. Small variation is observed for f '. Furthermore, under the influence of constant magnetic field M = 0.3, the momentum boundary layer decreases due to the Lorentz force.

Figure 8

Effect of Mp on velocity profile f when S = 1, De = 0.5, M = 0.1

Figure 9

Effect of Mp on velocity profile f ' when S = 1, De = 0.5, M = 0.1

Figures 10 and 11 present the effects of suction/injection parameter s on f and f '. It is noticed that the velocity fields f and f ' satisfy the far field boundary conditions (9) asymptotically, thus verifying the analytical and numerical results obtained.

Figure 10

Effect of S on velocity profile f when Mp = 0.2, De = 0.5, M = 0.1

Figure 11

Effect of S on velocity profile f ' when Mp = 0.2, De = 0.5, M = 0.1

Figure 12 displays the influence of magnetic field on the velocity profiles f with s = 1, De = 0.1, Mp = 0.3, M = 0.2 and R = 0.3. Due to the inflection of the vertical magnetic field to the electrically conducting fluid, the

Figure 12

Effect of M on velocity profile f when Mp = 0.2, De = 0.5, S = 1

Lorentz force is produced. This force has the tendency to slow down the flow and as a result the velocity profile decreases. Figures 13-?? show the effect of different parameters on the temperature profiles. The effect of Prandtl number on the temperature distribution is shown in Figure 13. Based on the Prandtl number’s definition Pr = , this parameter is defined as the ratio between the momentum diffusion to thermal diffusion. Thus, with the increase of Prandtl number the thermal diffusion decreases and so the thermal boundary layer becomes thinner as seen in Figure 13. It physically means that the flow with a large Prandtl number dissipates the heat faster to the fluid as the temperature gradient gets steeper and hence increases the heat transfer coefficient between the surface and the fluid. Figure 14 depicts the variation of Deborah number on the

Figure 13

Effect of Pr on temperature profile θ when Mp = 0.2, De = 0.5, S = 2, M = 0.3, λH = 0.2

Figure 14

Effect of De on temperature profile θ when Mp = 0.2, S = 2, Pr = 0.5, M = 0.3, λH = 0.2

temperature distribution. Increasing the Deborah number De increases the temperature inside the fluid. Also, it is observed that for larger values of Deborah number the boundary layer thickness decreases with increasing η.

The influence of magnetic parameter Mon the temperature distribution is shown in Figure 15. It is observed that the temperature distribution increases with increasing values of magnetic parameter M. Physically, the Lorentz force due to the transverse magnetic field has the property of relaxing the fluid velocity and temperature distributions. Accordingly, the velocity and temperature boundary layer thickness decreases as the magnetic parameter increases.

Figure 15

Effect of M on temperature profile θ when Mp = 0.2, S = 2, Pr = 0.5, De = 0.2, λH = 0.2

The effect of the porous permeability parameter Mp on the fluid temperature distribution is shown in Figure 16.

Figure 16

Effect of Mp on temperature profile θ when M = 0.1, S = 2, Pr = 0.5, De = 0.2, λH = 0.2

The result shows that as the porous permeability parameter increases, there is a corresponding increase in the fluid temperature due to increased diffusion of heat within the flow channel.

The effect of heat source (λH > 0) and heat sink (λH < 0) parameters are shown in Figures 17 and 18 respectively. It is observed that a heat source enhances the thermal conductivity and increases the fluid temperature as shown in Figure 13. Figure 14 incorporates the effects of the heat sink parameter. As expected, a heat sink provides a decrease in the fluid temperature.

Figure 17

Effect of heat generation λH > 0 on temperature profile θ when M = 0.1, S = 2, Pr = 0.5, De = 0.2

Figure 18

Effect of heat absorption λH < 0 on temperature profile θ when M = 0.1, S = 2, Pr = 0.5, De = 0.2

Table 3 shows different values of skin friction parameter −f (0) for several values of permeability parameter M p. We see from this table that the magnitude of shear stress in the boundary is smaller for injection in comparison to the case of suction. These results are in accordance with the physical situation because the injection of the fluid amounts to an increase of fluid velocity, resulting in a decrease of the frictional force. The permeability parameter introduces additional shear stress on the boundary. Note that the increase of permeability parameter leads to the increase of skin friction parameter in all the cases of suction, blowing, and impermeability of the surface.

Table 3

Values of skin friction parameter −f (0) for several values of permeability parameter Mp when De = 0.1, M = 0.2

In order to discuss the results of local Nusselt number against different physical quantities including Deborah number, Prandtl number, and internal heat generation/absorption quantity, we have prepared Table 4. It is evident from this table that the local Nusselt number decreases due to an increase in Deborah number and quantity oif internal heat generation/absorption, whereas the local Nusselt number increases due to an increase in Prandtl number.

Table 4

Effect of various physical quantities on local Nusselt number −θ'(0) including Deborah number, Prandtl number Pr and internal heat generation/absorption parameter λH

## 5 Conclusion

In this research paper, the semi-analytical/numerical technique known as HAM has been implemented to solve the transformed nonlinear differential equations describing the MHD flow of an Upper-Convected-Maxwell fluid with the influence of the internal heat generation/absorption. The dynamics of the magneto-hydrodynamic fluid flow in porous medium over a porous wall are investigated. The present semi-numerical/analytical simulations agree closely with the previous studies for some special cases. HAM has been shown to be a very strong and efficient technique in finding analytical solutions for nonlinear differential equations. HAM is shown to illustrate excellent convergence and accuracy and is currently being employed to extend the present study to mixed convective heat transfer simulations. The convergence of the series solution is established. Furthermore, the present method is also compared with an efficient numerical technique so called ND-Solve method. The effects of different physical key parameters which effect fluid motion such as Deborah number, magnetic parameter, suction/injection parameter, heat generation/absorption parameter, and porosity parameter are plotted and discussed. It is observed that with the increasing Deborah number the velocity decreases and the temperature inside the fluid increases. The results show that the velocity and temperature distribution increases with a porous medium. It is also investigated that the magnetic parameter has a decelerating effect on velocity while the temperature profiles increases in the entire domain. Due to increase in Prandtle number the temperature profile increases. It is observed that the heat source enhance the thermal conductivity and increases the fluid temperature while the heat sink provides a decrease in the temperature fluids.

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Tian J.H., Jiang K., Heat conduction investigation of the functionally graded materials plates with variable gradient parameters under exponential heat source load, Int. J. Heat Mass Transf., 2018, 122, 22-30.

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Tlili I., Khan W., Ramadan K., Entropy Generation Due to MHD Stagnation Point Flow of a Nanofluid on a Stretching Surface in the Presence of Radiation, J. Nanofluids, 2018, 7(5), 879-890.

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Khan I., Abro K.A., Mirbhar M.N., Tlili I., Thermal analysis in Stokes’ second problem of nanofluid: Applications in thermal engineering, Case Studies Therm. Eng., 2018, 12, 271-275.

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Khan W., Gul T., Idrees M., Islam S., Khan I., Dennis L.C.C. Thin Film Williamson Nanofluid Flow with Varying Viscosity and Thermal Conductivity on a Time-Dependent Stretching Sheet, Appl. Sci., 2016. Google Scholar

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Accepted: 2018-06-05

Published Online: 2018-12-31

Citation Information: Open Physics, Volume 16, Issue 1, Pages 917–928, ISSN (Online) 2391-5471,

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