2D FLOW OF CASSON FLUID WITH NON-UNIFORM HEAT SOURCE/SINK AND JOULE HEATING

In this paper, two-dimensional magnetohydrodynamic (MHD) flow of Casson fluid over a fixed plate under non-uniform heat source/sink and Joule heating is analyzed by the homotopy analysis method (HAM). The governing boundary-layer equations have been reduced to the ordinary differential equations (ODEs) through the similarity variables. The current HAM-series solution is compared and successfully validated by the previous studies. Furthermore, the effects of thermo-physical parameters on the current solution are precisely examined. It is found that the skin friction coefficient and local Nusselt number are greatly affected by the Hartmann number. It can be concluded that employing the Casson fluid together with the suction effect can minimize the rate of heat and mass transfer.


INTRODUCTION
In general, fluid mechanics can be categorized into two main types: hydraulics and hydrodynamics which are developed through the experimental and theoretical analyses, respectively (Falkner and Skan, 1931). In recent decades, there have been many research studies concerning the hydrodynamics as well as heat and mass transfer theory. In this way, Khan and Azam (2017) investigated unsteady flow of Carreau fluid over a permeable stretching wall with the Lorentz force and suction/injection effect. They solved the governing boundary-layer equations through the bvp4c function in MATLAB and found that the skin friction coefficient increases with an increase in the Weissenberg number. They also showed that the nanoparticle concentration boundarylayer thickness is significantly affected by the Lewis number. Borrelli et al. (2017) presented a model dealt with the Oberbeck-Boussinesq approximation for three-dimensional (3D) stagnation-point flow of Newtonian fluids. They found that the skin friction coefficient increases with an increase in the Hartmann number. They also illustrated that the reversed flow without the effect of buoyancy force occurs at the minimum value of Hartmann number (i.e., 0.7583). They finally proved that their findings are fully consistent with those of Ramachandran et al. (1988) and Ishak et al. (2008). Rahman et al. (2014) investigated forcedconvection flow of fluids over an exponentially permeable shrinking/stretching wall based on the Buongiorno mathematical model in which the effect of thermophoresis and Brownian motion had been taken into account. They could develop those of Kuznetsov and Nield (2013) and found that the momentum boundary-layer thickness decreases with an increase in the second order slip parameter. Ranjit and Shit (2017) analytically examined the combined effects of Joule heating and zeta potential on the flow past a peristaltically induced microchannel which was supported by those of Tripathi (2013). They also employed the Debye-Hückel approximation technique and found that although the viscous dissipation increases with a decrease in the Brinkman number, the local Nusselt number is a decreasing function of this number. Besthapu et al. (2017) numerically studied the MHD mixed-convection flow of stratified nanofluids using the finite difference method (FDM). They showed that the nanoparticle concentration boundary-layer thickness increases with an increase in the resistive Lorentz force. Furthermore, they found that the effect of thermal stratification parameter can be neglected at the surface. Sheikholeslami and Ganji (2017) applied the Koo-Kleinstreuer-Li (KKL) model to investigate the MHD flow of CuO-H2O nanofluid over a permeable annulus. They formulated the averaged Nusselt number in terms of inclination angle, Hartmann and Rayleigh numbers, and showed a consistency with those of Khanafer et al. (2003). It is to be noted that mode details are set out in Hedayati and Domairry, 2015;Khoshrouye Ghiasi and Saleh, 2018a, 2018b As discussed above, analytical and numerical models play a leading role in solving boundary-layer differential equations. This paper focuses on how the HAM (Liao, 1992(Liao, , 2003 may be implemented to give a solution for MHD flow of Casson fluid combined with the non-uniform heat source/sink, inclined Lorentz force and Joule heating based on the Buongiorno mathematical model (Buongiorno, 2006). It is found that the current findings are in agreement with those of previous studies. In addition, some tables and graphs are provided to signify the effects of thermo-physical parameters on the current solution. To the best of author's knowledge, there have been no reports of this problem being solved to date.

GOVERNING EQUATIONS
In rheology, one of the well-known non-Newtonian models is the Casson fluid which is defined by the following constitutive equation for an isotropic incompressible flow (Casson, 1959):

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org where is the shear stress tensor, is the plastic dynamic viscosity of the fluid, is the yield stress, is the ( , )th component(s) of the strain rate, (= ) is the product of strain rate component(s) and is the critical value of .
For the 2D flow in the Cartesian coordinate system, the velocity, temperature and nanoparticle concentration fields can be expressed as follows: where and are the velocity components along and directions, respectively, is the temperature and is the nanoparticle concentration.
Utilizing the aforementioned assumptions, the governing continuity, momentum, energy and nanoparticle concentration equations yield: where is the kinematic viscosity, is the Casson fluid parameter, is the electrical conductivity, 0 is the magnetic field strength, is the density, is the inclination angle of magnetic field, is the thermal is the ratio of effective heat capacity of the nanoparticle to effective heat capacity of the base fluid, is the Brownian diffusion coefficient, is the thermophoresis diffusion coefficient, ∞ is the ambient temperature, is the specific heat at constant pressure and is the non-uniform heat source/sink. The associated boundary conditions are given by: where ( ) is the rate of mass transfer, is the wall temperature, is the nanoparticle concentration at the wall, ∞ is the free stream velocity and ∞ is the ambient nanoparticle concentration.
In order to derive the similarity solution of equation (3), the following variables can be outlined: where is the stream function which is governed by the continuity equation (i.e., = and = − ), is the similarity function, is the similarity parameter, is the non-dimensional temperature and is the non-dimensional nanoparticle concentration.
The non-uniform heat source/sink involved in equation (3) is given by (Abo-Eldahab and Aziz, 2004;Nandeppanavar et al., 2010;Subhas et al., 2007): where and are the coefficients of space and temperature-dependent heat source/sink, respectively. By substituting equation (5) into equation (3), the following system of equations can be stated as: along with the boundary conditions: { at = 0: = , = 0, = 1, = 1, is the Eckert number, = is the Lewis number and is the mass suction parameter. It should be emphasized that the underlined term on the left-hand side of equation (7) indicates the effect of magnetic entropy generation. The physical quantities of interest are the skin friction coefficient, local Nusselt number and local Sherwood number which can be defined as follows: where, Substituting equations (5) and (10) into equation (9) yields: where = ∞ is the local Reynolds number.

SOLUTION METHOD
This section provides an overview of the HAM for finding analytical solution of equation (7) together with the boundary conditions given in equation (8). To this end, the initial guesses and auxiliary linear operators can be chosen as follows: which have the following properties: where 1 , 2 , …, 7 are the arbitrary constants. Using Liao's theorem (1992), the following zeroth-order deformation equations can be constructed: in which, with the boundary conditions: where 0 ≤ ≤ 1 is an embedding parameter, ℎ , ℎ and ℎ are the auxiliary parameters, and , and are the nonlinear operators. For = 0 and = 1, equation (14) is assumed to be: Expanding , and into the Taylor series with respect to gives: in which, If the initial guesses, auxiliary linear operators and auxiliary parameters are properly chosen, one would get at = 1: Differentiating equation (14) times with respect to , setting = 0 and dividing them by ! gives the following th-order deformation equations: in which, with the boundary conditions: The general solutions for equation (21) in terms of particular solutions (i.e., ⋆ , ⋆ and ⋆ ) can be expressed as follows: { ( ) = ⋆ ( ) + 1 + 2 + 3 − , ( ) = ⋆ ( ) + 4 + 5 − , ( ) = ⋆ ( ) + 6 + 7 − , in which, The square residual errors can be defined as (Liao, 2010): where = 20 and = 0.5.

RESULTS AND DISCUSSION
This section deals with the previously outlined HAM-series solution for heat and mass transfer analysis in the MHD flow of Casson fluid subjected to inclined Lorentz force and Joule heating. In this way, the comparisons and parametric studies are made to investigate the validity and accuracy of the current solution. In this paper the pertinent parameters, unless stated otherwise, are listed as follows: Table 1 represents the variation of auxiliary parameters and its square residual errors at any order of approximation. From this table, it is seen that the allowable values of auxiliary parameters can be chosen by minimizing the square residual errors. Therefore, the current findings are provided using the optimized 20th-order of approximation (i.e., ℎ = -0.8169, ℎ = -0.8246 and ℎ = -1.0962).
Effect of the Hartmann number on the skin friction coefficient − 1 2 is depicted in Fig. 1 for different values of inclination angle of magnetic field . As this figure shows, − 1 2 increases with an increase in which is due to the presence of resistive Lorentz force. It should be noted that the Casson fluid is affected by the viscous force while the Lorentz force tends to decelerate flow of the fluid and retards its motion (Khoshrouye Ghiasi and Saleh, 2018c). Moreover, unlike = 0°, − 1 2 increases with an increase in . Since = 0° the effect of Hartmann number on the momentum boundary-layer thickness is negligible.
To investigate the effect of non-uniform heat source/sink on the temperature distribution, the variation of ( ) for different values of and is depicted in Fig. 4. As this figure shows, ( ) increases with an increase in or which is due to an increase in the thermal boundarylayer thickness. Furthermore, Fig. 4 emphasizes that the surface temperature (0) increases with an increase in . Table 3 Values of − ( ) =0 compared with those of Khan and Pop (2010) and Wang (1989) Khan and Pop (2010) Wang (   increases with an increase in , which is due to an interaction between the Brownian motion and thermal conductivity. Also, Table 4 shows that − ( ) =0 clearly decreases with an increase in . Therefore, it is essential to account for the effect Brownian diffusion and thermophoresis in the Buongiorno mathematical model.
A comparison between the current solution and those of Afify and Elgazery (2016) is reported in Table 5 to determine the effect of and on the variation of − ( ) =0 . It is noted that the pertinent parameters utilized by Afify and Elgazery (2016) (i.e., → ∞, = = 10 and = = = = = = 0) agree with those presented in this table. It is found that the relative error between the current solution and those of Afify and Elgazery (2016) equals to 0.331% in all cases. Fig. 5 illustrates the variation of nanoparticle concentration ( ) for different values of Lewis number . From this figure, it is seen that ( ) decreases with an increase in which is due to the effect of .

CONCLUSIONS
This paper aimed to study the effect of non-uniform heat sink/source, Joule heating and inclined Lorentz force on flow of Casson fluid based on the Buongiorno mathematical model. The governing boundary-layer equations correspond to the continuity, momentum, energy and nanoparticle concentration equations are derived, and solved through the HAM. It was found that the rate of heat transfer is affected by the Brownian diffusion and thermophoresis. Moreover, the HAM findings were compared and validated by those previous studies available in the literature. One can also conclude that this paper would be worthwhile to further explore this problem with different boundary conditions, solution methodologies and geometries.