Heat transfer effect on MHD flow of a micropolar fluid through porous medium with uniform heat source and radiation

Abstract The present study examines the effect of heat transfer on electrically conducting MHD micropolar fluid flow along a semi-infinite horizontal plate with radiation and heat source. The uniform magnetic field has applied along the principal flow direction. The obtained governing equations have been converted into a set of dimensionless differential equations and then numerically solved by using a well-known Runge-Kutta method with shooting technique. The velocity, microrotation, and temperature distribution are presented for various physical parameters. The numerical values of skin friction and Nusselt numbers at the plates are shown in tabular form, and the obtained results are compared with the results of a previous study. It has been found that the magnetic parameter increases the velocity profile whereas the boundary layer thickness reduces due to the inclusion of coupling parameter and inertia effect. The presence/absence of magnetic parameter and coupling parameter enable to enhance the angular velocity profile while it is worth to note that the backflow has generated in the vicinity of the plate.


Introduction
The knowledge of micropolar uids past a porous medium has signi cant practical applications across a wide range of areas namely polymer blends, porous rocks, alloys, foams and aerogels, microemulsions etc. One such application is in the eld of lubrication, since the gap or the clearance in a bearing may be comparable to the average grain or molecular size of a non-Newtonian lubricant. For example, in batch mixers, sodium, which displays non-Newtonian characteristics, may be used both as a heat transfer agent and as a lubricant in the journal bearing supporting the mixing screw. The theory of micropolar uid consisting of two di erent e ects such as microrotational and micro-inertia, rst proposed by Eringen [1]. The proposal led to the attention of the researchers in the magneto-hydrodynamic (MHD) micropolar uid past in a porous medium due to its vast applications in engineering problems namely geothermal, energy extractions, oil exploration and the boundary layer control in the eld of aerodynamics.
Many studies have examined the e ect of heat transfer and magnetic eld MHD micropolar uid through a porous medium with di erent ow geometry such as a vertical plate, semi-in nite plate, channels etc. [2-6]. Raptis and Kafousias [7] and Kim [8] investigated the in uence of magnetic eld on two-dimensional steady incompressible ow con ned in an in nite and moving vertical porous plate, respectively. Kim [8] reported that the velocity distributions of micropolar uids are comparable with the Newtonian uids. Additionally, it has shown that the surface skin friction decreased with increase in the moving vertical plate velocity. Beg et al. [9] studied the effect of heat generation/absorption on MHD free convection ow from a sphere to non-Darcian porous medium wherein the mathematical problem has been solved using network simulation method. There are some studies available in the open literature where the non-Newtonian uid in a porous medium have investigated with di erent ow geometry [10-16]. Takhar et al. [17] have studied the unsteady three-dimensional MHD-boundary-layer ow due to the impulsive motion of a stretching surface. Baag et al. [18] investigated the MHD boundary layer ow over an exponentially stretching sheet past a porous medium with the uniform heat source. Mixed convection from a vertical surface embedded in a porous medium saturated with a non-Newtonian nano uid has studied by Gorla et al. [19]. Chamkha et al. [20] has presented the non-similar solutions for mixed convection along a wedge embedded in a porous medium saturated by a non-Newtonian nano uid. Similarity solutions for MHD thermosolutal Marangoni convection over a at surface in the presence of heat generation or absorption e ects have observed by Mudhaf and Chamkha [21]. RamReddy et al. [22] has examined soret impact on mixed convection ow in a nano uid under convective boundary condition. Ramesh et al. [23] has presented the magnetohydrodynamic ow of a non-newtonian nano uid over an impermeable surface with heat generation/absorption. Chamkha and Rashad [24] have observed the MHD forced convection ow of a nano uid adjacent to a nonisothermal wedge. Recently, Raju et al. [25,26] studied the radiation e ect of Casson uid over a moving wedge lled with gyrotactic microorganism and Carreau nano uid over a cone packed with alloy nanoparticles, respectively. A molecular dynamics study on transient non-Newtonian MHD Casson uid ow dispersion over a vertical radiative cylinder with entropy heat generation has studied by Reddy et al. [27]. The studies mentioned above concluded that the uniform magnetic eld enables to control the heat generation of electrically conducting uid.
The viscous dissipation is one of the signi cant parameters in uid dynamics and usually very challenging to incorporate into the mathematical model. However, many researchers studied the e ect of viscous dissipation on hydromagnetic ow through a saturated porous medium and proposed a di erent mathematical model based on many ways [28][29][30][31][32][33][34]. Chamkha et al. [35] have presented non-Darcy natural convection ow for non-Newtonian nano uid over cone saturated in a porous medium with uniform heat and volume fraction uxes. Gorla and Chamkha [36] has observed natural convective boundary layer ow over a horizontal plate embedded in a porous medium saturated with a nano uid. Al-Hadhrami et al. [37] proposed a new viscous dissipation model for a porous medium, which is most suitable for most practical applications. It has found that the porous medium enables to enhance the thermal performance when the aspect ratio of the channel or the thermal conductivity of the channel matrix increased.
In the last few years, many researchers are devoted to investigating the unsteady micropolar uid through a porous medium in the presence or absence of a uniform magnetic eld [38,39]. Mohanty et al. [40] numerically studied the e ect of heat and mass transfer on MHD micropolar uid using Runge-Kutta fourth order method followed by shooting technique. It has found that the Lorentz force produced by the magnetic eld which enables the resistance to momentum eld in the presence/absence of porous matrix. Using the similar numerical techniques (Mohanty et al. [40]), the e ect of heat source parameter on electrically conducting micropolar uid has investigated by Mishra et al. [41]. Due to the presence/absence of heat source parameter, the two layers variation has observed in the thermal boundary layer. Later, Tripathy et al. [42] studied the e ect of chemical reaction on MHD micropolar uid considering moving the vertical porous plate. The thermal radiation signi cantly in uences the temperature distribution and heat transfer of electrically conducting micropolar uid. But here, it is noted that there are very few publications available in the literature which analysed the e ect of thermal radiation and heat transfer of micropolar uid and demands more systematic investigation in future. Rashedi et al. [43] proposed analytical solutions based on homotopy analysis method (HAM) for micropolar uid past through a porous medium.
In the analysis of ow through a porous medium, Darcy's law usually has been considered to be the fundamental equation (Muskat [44]; Collins [45]). The principle of Darcy's law states that the velocity components are directly proportional to the pressure gradient where the convective acceleration term of uid does not exist. Hence, this law is only valid for low-speed ows. The force of the uid which is also proportional to the velocity components may deviate from the Darcy drag force (Bear [46]). The generalisation proposed by Brinkman considered the convective force. To study the ow through a highly porous medium (such as bres) the generalised Darcy law is recommended. Hence, in the present study, generalised Darcy law has been used to account for the porosity of the medium. Consequently, the novelties of the present study placed down as follows: 1.
The momentum transport equation has been modi ed by inclusion of two terms i.e. σB φ ρ (U − u) magnetic parameter and Cφ(U −u ) (Non-Darcian) which account for the e ect of permeability of the medium on the ow phenomena.

2.
The heat energy equation has been generalised by considering the uniform heat source/sink, not taken care of by Rashidi et al. [43] which a ect the heat transport phenomena.
Motivated by these applications the present study explores the e ects of the magnetic parameter, heat source/sink parameter and inertia e ect of micropolar uid by modifying the momentum and energy equations, which primarily constitutive the ow model of any liquid. To achieve this aim, we considered the electrically conducting MHD micropolar ow past through a semiin nite horizontal plate. The porous medium with variable suction velocity and viscous dissipation are discussed in the analysis. The e ect of viscous dissipation is modelled by following the suggestion proposed by Al-Hadhrami et al. [37]. It is worth to mention that, the proposed mathematical model is highly non-linear, hence only approximate numerical solution is possible in contrast to the analytical solution. Finally, the obtained results are compared with the analytical results reported by Rashidi et al. [43] as a particular case.

Mathematical formulation
A steady two-dimensional MHD ow of a micropolar uid through a porous medium past a semi-in nite horizontal plate in the presence of uniform heat source was considered in the present study. The e ect of radiation was analysed by modifying the energy equation. The semi-in nite horizontal plate is placed along the x-direction, and y-axis is perpendicular to it. We assumed that a uniform transverse magnetic eld of strength B employed to the principle direction of ow. The transverse magnetic eld can be negligible due to the small magnetic Reynold number. A schematic of the ow geometry along with coordinate system is shown in Fig. 1. All the physical parameters in the proposed model have been independent along the x-direction as we considered an in nite plate along the vertical direction (xdirection). To investigate the e ects of heat absorption, a heat source parameter has been applied to the ow. Hence, the set of governing equations of the problem can be expressed as: By assuming Rosseland approximation, the radiative heat ux is taken as where σ * is the Stefan-Boltzmann constant and k* is the mean absorption coe cient. Assuming that the temperature di erences within the uid in the boundary layer are su ciently small, we can express the term T as a linear function of temperature. Hence, using expanding T in a Taylor series about Tw and neglecting higher-order terms, we obtain. Then, qr can be expressed as

∂T ∂y
The associated boundary conditions of the present mathematical problem can be de ned as follows:

Solution of the problem
The following non-dimensional variables have been used to achieve the dimensionless form of the governing equations: Using Eq. (6) into the Eqs.
(2) -(4), the nondimensional governing equations can be written as: Hence, the boundary conditions (Eq. 5) can be expressed as: In the present study, the local skin friction coe cient (C f ), the local wall couple stress (Mwx) and the local heat transfer coe cient (Nux) can be de ned by the following relation:

Numerical solution
The set of coupled non-linear di erential Eqs. (7) -(9) have been solved numerically using robust shooting method with Runge-Kutta scheme and considering the boundary conditions mentioned in Eq. (10). By following the way of superposition the Eq. (7) -(9) were expressed as 1 st order simultaneous equations of seven unknown variables. We considered that y = f , y = f , y + = f , y = g, y = g , y = θ, y = θ . Hence, the Eqs.
The simulation has been repeated for a considerable value of η∞ up to two consecutive values of f ( ), f ( ), g ( ) and θ ( ) only diverse when the digit indicates the limit of the boundary towards η. For a particular set of parameters, the nal value of η∞ is chosen based on the limit η → ∞. The simulation has repeated till the accuracy of − is attended.

Results and discussion
A numerical approach such as Runge-Kutta method with shooting technique has been used to solve the non-linear self-similar governing equations (2) to (4), where Eq. (5) denote the two-point boundary value problem (BVP). To convert the BVP into IVP, we assign some guessed value with unknown initial conditions and initiate the process of computation. The step size of the current simulation has xed at η = .
. Here it is noted that the present solution a ected by two additional parameters namely magnetic parameter (M) in the momentum equation and uniform heat source/sink parameter in the energy equation (not considered by Rashidi et al. [43]). and G = , respectively. It is observed that the velocity pro le enhanced with the increase of magnetic parameter in the absence/presence of coupling parameter (∆ = and ∆ = ). The higher value of the magnetic parameter (M = 2) increases the velocity distribution, hence, the velocity boundary layer thickness decreased. It is because the magnetic eld applied along to the normal of ow direction. This magnetic eld gives rise to a resistive force and slows down the movement of the uid. The velocity pro le was remaining invariant from η ≥ . The velocity pro le for the dimensionless parameters M and Kp is shown in Fig. 3 with the xed values of ∆ = . , F = R = . and G = . The velocity pro le observed to decelerates in the boundary layer in the presence/absence of magnetic parameters due to the porous matrix. It is also noted that the reverse e ect appears to be right in the case of the magnetic parameter. Fig. 4 illustrates the e ect of M and F on velocity pro le. It is clear that the increase in microinertia coe cient increases the velocity pro le at all points in the ow domain for both presence/absence of magnetic parameter. The present result is in good agreement with the outcome of Rashidi et al. [43] by withdrawing the magnetic parameter, i.e. M = from the velocity pro le. , F = R = . and G = . It is clear that in the presence/absence of M and ∆, angular velocity pro le increases and the back ow is generated in the vicinity of the plate after this it meets the boundary conditions. It has appeared that the magnetic eld retards the angular velocity pro le with coupling parameter. The e ect of M and Kp on angular velocity is exhibited in Fig. 6. It is interesting to note that magnetic eld enhances the pro le whereas, Kp retards it. A similar observation has well marked that the back ow occurs in the boundary layer. It is remarked that the angular velocity has linear behaviour within the ow domain η < . and after that, it rises to meet the boundary conditions. We observed that the increase in microinertia coe cient had enhanced the pro le at all points in the boundary layer. Due to the Lorentz force (often treated as a resistive force), a reduction in the velocity observed whereas the reverse e ect has encountered with the higher values of M. The critical aspect of the present study is the e ect of uniform heat source on the temperature pro le in the presence/absence of the M, ∆, Kp and R parameters with the xed values of other pertinent parameters.
From Fig. 8, it is remarked that increase in heat source parameter retards the temperature pro le at all points in the presence of M and ∆ (M = . and ∆ = ). In contrast for and, the maximum temperature has been observed at the boundary-layer which is parabolic in nature. It is because the resistive lorenze force which retards a certain amount of energy stored up in the thermal boundary layer, therefore, the temperature pro le increased.  Fig. 9 illustrates the e ect of the uniform heat source in the presence/absence of M and Kp. It is evident that growth in M and Kp the temperature pro le irrespectively decelerates for di erent values of the heat source. When M = , S = and Kp (without porous matrix) the prole attaining the maximum value and tends to meet the boundary conditions for η → ∞. The thermal radiation e ect has been observed in both the presence/absence of heat source and shown in Fig. 10. There two-layer variations in temperature pro le have been remarked for S = and S = at di erent values of radiation parameter (R). From Fig. 10, it can be observed that the temperature decreases in the thermal boundary layer with an increase in R and S. The present results make a good agreement with the outcome of Rashidi et al. [23] by withdrawing the parame- ter S(S = ) from the energy equation. Thermal radiation emitted by a hot body depends not only on its temperature but also on the material of the body, its shape and the nature of its surface.
The ow characteristics at the boundary surface are vital in determining the ow stability and hence calculation of skin friction, wall couple stress and Nusselt number are essential. The numerical values of aforesaid parameters have been shown in Table 1 for di erent pertinent parameters. It can appear that the skin friction and rate of heat transfer increases with the increase of magnetic parameter while the reverse e ect has been observed for couple stress. The inclusion of porous matrix opposes the ow and heat transfer aspect resulting in retardation in skin friction coe cient and Nusselt number, but couple stresses enhanced it. In the unsteady case, the presence of heat source parameter increases the values of f ( ), g ( ) and −θ ( ).

Conclusions
In the present study, the e ect of heat transfer on an electrically conducting MHD micropolar uid through a porous medium has been analysed considering radiation and uniform heat source. The present simulations results highlighted the following facts: 1.
The velocity pro le enhances at all points in the boundary layer with an increase in the magnetic parameter for the absence/presence of coupling parameter.

2.
The velocity distribution increases with increase in microinertia coe cient for both presence/absence of magnetic parameter.

3.
The temperature distributions in the thermal boundary layer follow a declining trend as the radiation and source parameter increases. 4.
The increase in magnetic parameter and porous parameter decelerates the temperature pro le irrespectively for the values of the heat source parameter.

5.
The presence/absence of magnetic parameter and coupling parameter enable to enhance the angular velocity pro le while it is worth to note that the backow has generated in the vicinity of the plate. 6.
In the absence of heat source, i.e. S = , present result is in good agreement with Rashidi et al. [43]. subscripts ω condition at wall ∞ condition at free stream