Heat Absorption and Joule Heating Effects on Transient Free Convective Reactive Micropolar Fluid Flow Past a Vertical Porous Plate

: Mathematical model for an unsteady, incompressible, electrically conducting micropolar fluid past a vertical plate through porous medium with constant plate velocity has been investigated in the present study. Heat absorption, Joulian dissipation, and first-order chemical reaction is also considered. Under the assumption of low Reynolds number, the governing transport equations are rendered into non-dimensional form and the transformed first order differential equations are solved by employing an efficient finite element method. Influence of various flow parameters on linear velocity, microrotation velocity, temperature, and concentration are presented graphically. The effects of heat absorption and chemical reaction rate decelerate the flow is particularly near the wall. Skin friction and wall couple stress increases as heat absorption increases but the reverse phenomenon is observed in the case of chemical reaction rate. Wall mass transfer rate increases for chemical reaction and Sherwood number increases for heat absorption. Finite element study is very versatile in simulating unsteady micropolar rheo-materials processing transport phenomena. However, a relatively simple reaction effects restricted to first order.

; Singh and Kumar (2016); Rahman and Sultana (2016); Siva Reddy and Shamshuddin (2016); Siva Reddy and Thirupathi (2016a, 2016b, 2016c; Thirupathi, Anwar Beg and Kadir (2017a); Thirupathi, Chamkha and Siva Reddy (2017b); Thirupathi and Mishra (2018); Shamshuddin and Thirupathi (2017)]. Boundary value problems of various non-Newtonian fluid models have been solved numerically and analytically, which has been proposed to characterize the real fluid behavior. One such fluid is micropolar fluid, which exhibits microstructural characteristics like rotatory motions, gyration of fluid microelements. These fluids can support couple stresses, shear stresses, body couples and, also exhibit micro rotational effects and inertia due fluid bear dilute suspension of rigid macromolecules with individual motions. In the current trend among non-Newtonian fluids, Micropolar fluid has acquired special attention in numerous applications. Best-proven theories of fluids with microstructure was first brought in and formulated by ] and Cowin [Cowin (1968)] which has been applied in various fields in recent years. Later by generalizing micropolar fluids to heat conduction and other thermal effects, Eringen ] extended his theory and developed a new unified theory of mechanical materials. The hydrodynamic flow of micropolar fluids and their aspects are documented in Stokes [Stokes (1984)] and Lukaszewicz [Lukaszewicz (1999)]. More comprehensive detail of this theory and its applications can be found in Ariman et al. Sylvester (1973, 1974)].
To the best of our knowledge, above mentioned studies combined effect of heat absorption, chemical reaction and Joule heating on micropolar flows have not been much explored, this article presents these flows to obtain an approximate solution via widely accepted and robust Galerkin finite element technique to the perturbation approximate form solution presented by Roja et al. [Roja, Reddy and Reddy (2013)]. The mathematical formulation, method of solution and validation are presented in Sections 2, 3 and 4 respectively. Results and discussions are explored scientifically and experimentally in Section 5. Important results are summarized in the conclusion section. Finally, figures and tables are shown after the references section.

Mathematical formulation
The free convective flow of an incompressible electrically conducting micropolar fluid past a vertical porous plate is considered. The plate considered is permeable and is moving with constant velocity Up .Physical configuration is illustrated in Fig. 1. Initially, the temperature w T and concentration w C thereafter maintained as constant temperature T * ∞ and constant concentration C * ∞ . Darcy's law and low Reynolds number flow are assumed. Uniform magnetic field of strength 0 B is applied which is transverse to the flow direction. Though the Magnetic Reynolds number is very small, in comparison to the applied magnetic field the induced magnetic field is negligible [Cowling (1957)]. In comparison to the direction of y * , the term radiative heat flux considered in x * direction is negligible. Since the plate is of infinite extent and electrically non-conducting all physical quantities, except pressure, depend on y * and * t only.

Figure 1: Flow configuration and coordinate system
By taking the aforesaid assumptions into consideration the governing boundary layer equations (see Roja et al. [Roja, Reddy and Reddy (2013)]) for unsteady convective micropolar fluid flow under Boussinesq's approximation are as follows: The following dimensional initial and boundary conditions are: The following form is taken from Roja et al. [Roja, Reddy and Reddy (2013) . The radiative heat flux term is described as follows It is difficult to obtain the solution since Eq. (8) is highly nonlinear. Earlier several authors have obtained this problem assuming small temperature differences within the fluid flow [Adunson and Gebhert (1972); Raptis and Perdikis (1998)]. Using Taylor's series expansion about * T neglecting higher order terms takes the form Differentiating Eq. (8) w.r.t * y and substituting in (10), we get Now we introduced the following dimensionless variables and parameters: Using Eqs. (6)-(11) in (2)-(5) and dropping stars the governing equations and the boundary conditions in the non-dimensional form are: Dimensionless initial and boundary conditions are The physical quantities of interest used in materials processing simulations and design are Skin friction coefficient at the porous plate is written as The couple stress coefficient at the porous plate is written as Heat transfer coefficient and mass transfer coefficient at the porous plate in terms of Nusselt number and Sherwood number, as follows Converting (17)-(19) in the non-dimensional form are obtained as follows Skin-friction is obtained as, Wall couple stress is defined as, The non-dimensional form of Nusselt number is computed as The non-dimensional form of Sherwood number is evaluated as , local Reynolds number.

Numerical solution
The set of unsteady, reduced, non-dimensional, coupled partial differential Eqs. (12)-(15) under initial and boundary conditions (16) are non-linear and coupled, and therefore cannot be solved analytically. The finite element method (FEM) is the most popular and adoptable method for solving ordinary as well as partial differential equations. It is equally versatile at solving Newtonian and non-Newtonian problems. The details of the variational fi nite element method are documented succinctly in Reddy et al. [Reddy (1985); Rao (1989); Beg, Rashidi and Bhargava (2011)]. The fundamental steps in finite-element analysis are 1) Finite element discretization 2) Derivation of element equations 3) Assembly of element equations 4) Imposition of boundary conditions 5) Solution of assembled equations, which are briefly described in Shamshuddin et al. [Shamshuddin, Beg, Ram et al. (2017)] and the resulting final matrix equation obtained can be solved by any efficient iterative scheme.

Variational formulation
The variational formulation associated with Eqs. (12)-(15) over a typical two-node linear is given by: where β w are arbitrary test functions and may be viewed as the variations in , u ω ,θ and φ respectively. After dropping the order of integration and non-linearity, we arrive at the following system of equations
These matrices are defined as follows:      In general, to verify that the converged solutions are indeed correct, i.e., to guarantee grid (mesh) independency, a grid refinement test is carried out by dividing the whole domain into successively sized grids 81×81, 101×101 and 121×121 in the z-axis direction. Furthermore, the finite element code is run for different grid sizes and for a grid size of 101×101 the solutions are observed to achieve mesh independence. Therefore, for all subsequent computations, a grid size of 101 intervals is elected, with a step size 0.01 and hence a set of 404non-linear equations are solved with an iterative scheme. Finally, the solution is assumed to be convergent when the iterative process is terminated when the following condition is fulfilled: and n denotes the iterative step. This criterion maintains high accuracy for coupled multi-physical boundary layer equations. Once the key variables are computed, many wall gradient functions may be automatically evaluated.

Validation of numerical results
To validate and accuracy of the numerical results obtained by adopting Galerkin finite element method, weighted residual approach in the present analysis, we have compared with the analytical result of earlier published [Roja, Reddy and Reddy (2013)] in the absence of heat absorption and chemical reaction parameter which is depicted in Tabs. 1 and 2 and these favorable comparisons lend high confidence in the present finite element code accuracy. Tabs. 3 and 4 depicts for the general model with all parameters invoked. From Tab. 3 clearly, it is noticed that as heat absorption parameter Q increases as f C , w C decreases and Nu increases.       Q > Fig. 2 depicts the effect of temperature on heat absorption parameter Q . Existence of the heat absorption effect has the tendency to repress the fluid temperature. These behaviors are obvious from Fig. 2 that temperature distributions decrease as Q increases. Fig. 3 exhibits the effects of the Eckert number Ec on the temperature profiles. Eckert number characterizes the heat dissipation which is advective transport with respect to heat dissipation potential. It is observed that an increasing Ec the temperature of the fluid increases in the thermal boundary layer. However, higher values of Eckert number, temperature decreases significantly (Fig. 3). It is since internal energy is increased in the free stream indicates that an adequately large infinity boundary condition has been imposed in the finite element model. The profiles of the temperature in the boundary layer for various values of Prandtl number r P are shown in Fig. 4. Prandtl number represents the relative rate of momentum diffusion to energy diffusion. For Pr 1 < energy, diffusion rate exceeds momentum diffusion. Also, fluids with higher Prandtl number possess greater viscosities and as r P increases from 0.71 through 1 to 5. Temperature is significantly suppressed with greater Prandtl number, as plotted in Fig. 4, Greater Prandtl number corresponds to lower thermal conductivity. This leads to a reduction in thermal energy convected through the fluid from the plate. A similar trend can be observed in many articles related to micropolar including Roja et al. [Roja, Reddy and Reddy (2013)]. , which appeared in the thermal diffusion term in Eq. (14) i.e., ( ) ( )

Results and discussion
.R<1 represents the thermal radiation is dominated by thermal conduction. R=1 represents the contribution of both the thermal properties are equal and R>1 shows thermal radiation dominates over thermal conduction. In present simulations, we confine our discussion considering all these three cases. It is noticed from the present study that as R increases temperature profiles are markedly decreases.

Figure 5: Effect of Radiation Parameter R on velocity u
The pattern of concentration profiles for chemical reaction parameter Kr and Schmidt number Scare presented in Figs. 6-7. The present study, the destructive type of homogenous chemical reaction is considered. It is interesting to note that a sharp fall in concentration profile is marked with an increase in a chemical reaction. It is due to the fact that, when the chemical reaction takes place, a destructive chemical reaction destroys the original species. The term / Sc v D = defines as the ratio of the momentum to the mass diffusivity. From the profiles, it is observed that the depression in concentration magnitudes is due to the reduction in molecular diffusivity which manifests in a stifled migration of species. Hence concentration profiles decrease as Sc increases. Figs. 8-9 illustrate the influence of β , the micro-rotation parameter on velocity and micro-rotation profiles respectively. It is evident that velocity distribution is greater for a Newtonian fluid ( β =0) with the given parameters, as compared with non-Newtonian fluid (micropolar fluid). As β increases the peak values are attained close to the plate and these migrate away from the plate, hence all profiles decay from the peak and disappear in the free stream velocity. In addition, the micro-rotation (Fig. 9) i.e., angular velocity takes negative values throughout the regime. Hence micro-rotation velocity profiles increase as β increases.
The response of Grashof number Gr and modified Grashof number Gm on velocity and microrotation profiles are depicted in Figs. 10-13. It is to note that, as Gr or Gm increases velocity profiles also increase. However, the reverse effect is encountered in case of micro-rotation profiles i.e., the profile decreases with an increasing value of both the buoyancy parameters. The said observation is due to the fact that Gr ,the relative measure of the magnitude of the thermal buoyancy and to that of the opposing frictional forces acting on the micropolar fluid and Gm , the relative measure of species buoyancy force to viscous hydrodynamic force. Figs. 14-15 shows the variations in the velocity and angular velocity for various values of dimensionless magnetic field parameter .
M Physically, in magneto-hydrodynamic material processing, the applied magnetic field caused by Lorentz force, is a resistive force which retards the velocity profile significantly. As a result, the boundary layer thickness also decreases. In the case of Fig. 15, the angular velocity profile enhances significantly with an increase in the magnetic parameter. The pattern of velocity and microrotation profiles for permeability K are shown in Figs. 16-17. It is originated from drag force (Darcian) term in the composite linear momentum Eqn.
(12), viz ( )u K / 1 − in a combined term N as mentioned in Eq. (12). As permeability quickly increases velocity near the wall of the porous plate increases. A similar trend is also found in case of microrotation i.e., as permeability increases microrotation velocity also increases. Figs. 18-19 presents the graphical representation of dimensionless velocity and angular velocity profiles for some representative values of plate velocity. It is to be noted that the peak value of velocity across the boundary layer increases near the porous plate, also we observe the magnitude of microrotation on porous plate increases as p U increases. Therefore, higher the plate velocity accelerates the linear flow whereas the micro-rotation decelerates.

Concluding remarks
In this work, a mathematical model for an unsteady, incompressible, electrically conducting micropolar fluid past a vertical plate in a porous media has been presented. Heat absorption in energy equation and chemical reaction effect in the solutal transfer equation have been included in the formulation. The conservation for momentum, angular momentum, energy, and concentration have been non-dimensionalized with appropriate variables. Based on the obtained solutions with the numerical scheme and using some graphical illustrations generated with the MATLAB software, for validation of present solutions, results are compared with the earlier published analytical solution of Roja et al. [Roja, Reddy and Reddy (2013)] and excellent correlation achieved. The main findings were the linear flow is accelerated and, in a consequence, the thickness of momentum boundary layer decreases with increasing values of microrotation parameter, plate velocity, thermal Grashof number, solutal Grashof number, and permeability parameter. A reverse phenomenon is observed in the case of the magnetic field parameter. Angular velocity (micro-rotation) is suppressed and micro-rotation boundary layer thickness increased with increasing of microrotation parameter, thermal Grashof number, solutal Grashof number, and permeability parameter. Conversely, angular velocity is elevated as plate velocity and magnetic field increases. Increasing Prandtl number, radiation-conduction parameter, heat absorption parameter, and Eckert number decrease temperature profiles. Increasing Schmidt number, first-order chemical reaction parameter decreases concentration values and reduces concentration boundary layer thickness. Sherwood number (wall mass transfer rate) is enhanced with increasing homogeneous chemical reaction, no variations in case of heat absorption parameter. Nusselt number (wall heat transfer rate) is enhanced with increasing heat absorption parameter, no variations in case of chemical reaction parameter. The current simulations have shown the strong potential of finite element methods (FEM) in simulating realistic transport phenomena in heat exchange rheo-materials processing. Further studies will investigate alternate non-Newtonian models Eg. nanofluid particles in one-dimensional two-phase model with n-order chemical reaction will be communicated imminently.