Chemical Reaction and Heat Source Eﬀects on MHD Free Convective Flow over A Linearly Accelerated Moving Vertical Porous Plate

Objective : To make discussion on the chemical reaction and heat source eﬀects on MHD ﬂow on micro polar ﬂuid along vertical porous plate. Method : The governing dimensionless equations are involved analytically by using ﬁnite element scheme. Findings : The eﬀects of governing parameters on the ﬂow variables are discussed quantitatively with the help of graphs for the ﬂow ﬁeld, temperature ﬁeld, concentration ﬁeld, skin friction and Nusselt number. Novelty : The accuracy of the problem has been veriﬁed by comparing with the previous published work and the agreement between the results is excellent, which established conﬁdence in the numerical results reported in this study.


Introduction
Theoretical study of magneto-hydrodynamics (MHD) flow problem plays with a chemical reaction is of huge number of applications to scientists, engineers. In a lot of chemical engineering process chemical reaction take place involving a distant mass and working fluid in that the plate was touching. This process takes place in several manufacturing applications like developed of food processing, glassware, ceramic objects and polymer production. Foreign mass possibly here moreover naturally or mixed through the water or air. The existence of a distant mass in water and air effects few types of the chemical reactions. Chemical technology improved with the study of chemical effect. Such as food processing, generating the electric power and polymer fabrication. Ali et al. (1) examined the effects of Heat and Mass movement through free convection flow in a vertical plate. Reddy et al. (2) analyzed the chemical reaction effects on MHD natural convection. Sehra et al. (3) examined the Convection heat mass transfer over a vertical plate with chemical reaction. Nayak et al. (4) consideredthe chemical reaction results on stretching sheet. Kumar et al. (5) interpreted the thermal diffusion effects on MHD. Babu et al. (6) analyzed the impacts of chemical reaction, diffusion thermo and radiation on unsteady natural convective flow past an inclined vertical plate under aligned magnetic field. Babu et al. (7) considered the radiation and chemical reaction results on MHD free convention mass transfer fluid flow in a porous plate. Kumar et al. (8) studied the suction special effects on accelerated vertical on permeable plate. Hoq et al. (9) studied theerror analysis of heat conduction partial differential equations. Prasad et al. (10) considered radiation, viscous dissipation effects on the heated vertical plate in permeable medium. Magala et al. (11) has investigated dissipation effects on accelerated vertical plate through suction. Kandasamy et al. (12) interpreted the effects of heat and mass transfer along a wedge with heat source and concentration in the presence of suction. Lakshmi et al. (13) considered viscous dissipation effects on pours. Bhagya Lakshmi et al. (14) has studied the MHD free convective flow of dissipative fluid past an exponentially accelerated vertical plate. Shehzad et al. (15) studied the MHD flow medium. Pramod Kumar et al. (16) reviewed induced magnetic field in the free convective Radiating Stream above Permeable Laminate. Ekakitie et al. (17) interpreted the impact of chemical reaction and heat source on MHD free convection flow over an inclined porous Surface. Anupam et al. (18) studied the radiation effects on stretching sheet. Chamkha et al. (19) explained in a permeable medium. Sankar et al. (20) explained the radiation impacts in the semi-infinite permeable plate. Sweta et al. (21) analyzed fem of heat absorption effects on MHD Casson fluid flows above exponentially accelerate temperature through ramped surface concentration.
The objective of the present work is to study the chemical reaction and heat source effects on MHD flow on micro polar fluid along vertical porous plate. It has been noticed that the chemical reaction and heat source parameter had an effect on velocity profile, temperature profile, and concentration profile. The governing partial differential equations are solved by Galerkin Finite Element Method. We have extended the problem of Bhagya Lakshmi (14) in the presence of chemical reaction, heat source parameter and the accuracy of present problem have been verified by comparing with theoretical solution of Bhagya Lakshmi (14) through figures and the agreement between the results is excellent. This has established confidence in the numerical results reported in this paper.

Mathematical Formulations
The transitory magneto-hydrodynamic free convective flow of fluid over a growing accelerated plate with unstable temperature presented. Here x, y -axis are taken as along the plate in vertically straight way and normal to the plate. The plate is considering bound less in x-direction, all the flows quantity turn into self-similar missing from the leading edge. The total physical quantities turning into functions of t,y. By time t<=0. Fluids are at equal to t and C less important than the constant wall temperature, concentration correspondingly. By t>0, the plate was growing accelerated through a velocity= u exp (at) in its own plane and temperature of the plate, concentration are rising linearly with time-t. An unvarying magnetic playing field amount H is applying in the y-direction. Therefore the magnetic field and the velocity are taken by H=(0,H ), q=(u,v). The fluids electrically conducted the magnetic Reynolds number is to a great extent less than 1 and therefore induced magnetic field can be uncared for in comparison with applying magnetic field in the non appearance of input electric field. The heat because of viscous dissipation considered into an account below the above assumptions with Boussines mass, energy, momentum and species governing the free convection boundary layer flows over vertical plate can be studied as: Continuity equation: Momentum equation: The Energy equation: The Concentration equation: Boundary conditions can be written in dimensional form as Then the resultant non-dimensional equations are: Boundary conditions can be written in dimensionless form as

Method of Solution
The finite element method (FEM) is employed to solve the transformed, coupled boundary value problem defined by Equations. (7)-(9) under Eq.(10). The fundamental steps involved in finite-element analysis of a problem are as follows: Step 1: Discretization of the fluid domain into finite elements Step 2: Generation of element equations Step 3: Assembly of element equations Step 4: Imposition of boundary conditions Step 5: Solution of assembled equations Numerical study of a dissipative fluid flow The variation formulation associated with Equations. (7) -(9) are joined with boundary conditions (10), we represent the velocities u, temperature θ and concentration C. By applying the Galerkin finite element method for equation (7) over a typical two-nodded linear element(e) ( y j ≤ y ≤ y k ) is https://www.indjst.org/ We write the element equation for the elements y i−1 ≤ y ≤ y i andy i ≤ y ≤ y i+1 . Assembling these element equations, where r = k h 2 where h and k are the mesh sizes along y direction and time direction respectively. Index i, j refers space and time respectively. The mesh system consists of h=0.4 for velocity profiles and concentration profiles and k=0.5 has been considered for computations. In the above equations taking i=1(1) n and using initial and boundary conditions (10), the following system of equation are obtained.
Where A i 's are matrices of order n and X i and B ′ i s are column matrices having n-components. The solution of above system of equations are obtained using Thomas algorithm for velocity, angular velocity, temperature and concentration. In order to prove the convergence and stability of Galerkin finite element method, we are using C programming with slightly change of values h and k, then no significant difference was observed in the values of velocity, angular velocity, temperature , concentration. Hence, the Galerkin finite element method is steady and convergent.  Nusselt number:

Results and Discussion
In . We noticed that effect of Pr velocity profiles are decreases. Since fluids having high viscosity then we observed that it move slowly. Figure 2 (b) and Figure 1(b) showed that reaction of Eckert number (E) on the velocity fields, Temperature fields. Observed that, velocity profiles, Temperature profiles increases as E increases. The Eckert number effects on the flow of field are to improve energy, yielding a larger temperature and greater buoyancy force. The increases in the buoyancy force are because of increases in dissipation parameter to improve the velocity. Figure 1(c) shown that the effects of acceleration parameter (a) taking place the velocity fields. We noticed that rises in acceleration parameter leads to rises in velocity profile. Figure 1(d), Figure 1(e) shown that the effects of the mass Grashof (Gc), Grashof number (Gr) number on the velocity profiles. It is observed that greater cooling of the surface in Grashof number, mass Grashof number results raises in the velocity of air. It is due to the fact increase in the values of thermal Grashof number; mass Grashof number has the tendency to increase the thermal and mass buoyancy effect. This gives rise to an increase in the induced flow. Figure 1(f) shows the effect of time t on the velocity in cooling of the plate. It is obvious from the figure that the velocity increases with the increase of time t. Figure 1(g) shown that magnetic parameter (M) increases while decreasing in velocity profile. The presence of transverse magnetic field produces a resistive force on the fluid flow. This force is called the Lorentz force, which leads to slow down the motion of electrically conducting fluid. Figure 1(h) and Figure 3 (a) observed that velocity fields and concentration fields are decreases while increasing in Chemical reaction parameter (K 1 ). Figure 2(2a) observed that temperature profiles are decreases while increasing in Prandtl number (Pr). Physically, it is possible because fluids with have high viscosity and hence move slowly. Figure 1(i) and Figure 2(2.c) illustrate the effect of heat absorption S on temperature fields and velocity fields. We observed that velocity fields and temperature fields are increases with increasing in the source parameter. It is seen that heat absorption/heat generation tends to accelerate the motion of the fluid. The positive sign indicates the heat generation (heat source) whereas negative means heat absorption (heat sink). Heat source physically implies the generation of heat from the, which increases the temperature in the flow field. Therefore, as the heat source parameter increased, the temperature increases steeply. The influence of heat source parameter S>0 on velocity and temperature profiles is very much significantly related to the heat sink parameter S <0. These results are clearly supported from the physical point of view. Figure 3(3b) illustrates that the non-dimensional concentration fields for Schmidt number (Sc). A decrease in concentration with increasing Sc. Also it is noticed that the concentration boundary layer becomes thin as the Schmidt number increases.

Conclusion
The study on chemical reaction and heat source effects on the problem of Magneto hydrodynamics free convection of dissipative flow fluid past an exponentially accelerated plate. We studied velocity, concentration and temperature fields by different parameters like heat source parameter, Prandtl value, Schmidt number, modified Grashof and Grashof number, acceleration parameter, chemical reaction parameter. We observed that velocity fields decreases with respect to t=0.2,0.6 increases in magnetic parameter (M), Chemical reaction parameter K1, Modified Grashof number Gc, Prandtl number Pr, acceleration parameter a, Grashof number Gr. The following conclusions can be drawn The velocity, temperature fields increases with increasing the heat source parameter.