FINITE ELEMENT ANALYSYS OF RADIATIVE UNSTEADY MHD VISCOUS DISSIPATIVE MIXED CONVECTION FLUID FLOW PAST AN IMPULSIVELY STARTED OSCILLATING PLATE IN THE PRESENCE OF HEAT SOURCE

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INTRODUCTION
Current research has been mingled with a mixed convective flow of porous media.The mathematical developments were used to characterize the flow within porous media prior to 1969 reviewed.Flow in porous media is important in many areas of science and technology including biology, soil science, reaction engineering, waste treatment, and separation science.The mixed physical process of MHD convection flow has been gaining increasing research attention owing to its increased utilization in diverse physical chemical and engineering applications.
In the presence of strong magnetic fields, the Hall Effect becomes an important mechanism for electrical conduction in ionized gases and plasmas.Unlike metals, the number density of charge carriers in ionized gases is low, which results in anisotropic behavior of the electrical properties.Hence, a current is induced in the direction normal to both the electric and magnetic fields.The Hall Effect has important engineering applications, such as the Hall generators, Hall probes, and Hall Effect thrusters used for space missions.
The study of heat and mass transfer with chemical reactions is of great practical importance to engineers and scientists because of its almost universal occurrence in many branches of science and engineering.Combined heat and mass transfer in fluid-saturated porous media finds applications in a variety of engineering processes such as heat exchanger devices, petroleum reservoirs, chemical catalytic reactors and processes, porous plate in slip-flow regime with heat source.Chemical reaction and radiation effects on MHD free convection flow past an exponentially accelerated vertical porous plate is stated Sitamahalakshmi et al. (2019).Effects of thermal radiation on MHD chemically reactive flow past an oscillating vertical porous plate with variable surface conditions and viscous dissipation is expressed by Prabhakar Reddy et al. (2019).FDM and FEM correlative approach on unsteady heat and mass transfer flow through a porous medium is conveyed by Shankar Goud et al. (2020).Saddam Atteyia Mohammad (2020) is analyzed effects of variable viscosity on heat and mass transfer by MHD mixed convection flow along a vertical cylinder embedded in a non-darcy porous medium.Anil Kumar et al. (2020) described thermal radiation effect on MHD heat transfer natural convective nanofluid flow over an impulsively started vertical plate.Anil Kumar et al. (2021) 2020) by including the presence of above-mentioned flow parameters.This is not a simple extension of the previous work.It varies several aspects from that such as the presence of mass transfer in the momentum equation, radiation absorption inclusion in the energy equation and the addition of species diffusion equation.Apart from the modification of set of governing equations, we also changed the method of solution due to the existence of nonlinear-coupled partial differential equations, which are solved, by Galerkin finite element method, with its computational cost efficiency, has been employed for obtaining the solutions.The consequent changes and comportment of diverse aspects such as the concentration, velocity, temperature, and engineering parameters have been comprehensively focused on observing the possible changes in the behavior of the fluid. of strength  0 is applied in the direction perpendicular to the fluid flow.In the Cartesian co-ordinate system, the  ′ -axis is taken along the plate in the vertically upward direction, the  ′ -axis perpendicular to the direction of the plate and the  ′ -axis is normal to the  ′  ′ -plane.The physical model of the problem is shown in Fig. 1.Initially, at time  ′ ≤ 0 the temperature of the fluid and the plate is  ∞ ′ and the concentration of the fluid is  ∞ ′ .Subsequently, at time  ′ > 0, the plate starts oscillating in its own plane with frequency  ′ , the temperature of the plate and the concentration of the fluid, respectively are raised to   ′ and  ′ .It is assumed that the radiation heat flux in the  ′ -direction is negligible as compared to that in  ′ -direction.As the plate is of infinite extent and electrically nonconducting, all the physical quantities, except the pressure, are functions of  ′ and  ′ .The generalized Ohm's law on taking Hall current into account Cowling (1957) is given by

MATHEMATICAL ANALYSIS
, ,   and   are respectively,velocity vector, magnetic field vector, electric field vector, current density vector, electric conductively, cyclotron frequency and electron collision time.The equation of continuity . → = 0 gives  ′ = 0 everywhere in the flow since there is no variation of the flow in  ′  direction, where  → = ( ′ ,  ′ ,  ′ ) and  ′ ,  ′ ,  ′ are respectively, velocity components along the coordinate axes.
The magnetic Reynolds number is so small that the induced magnetic field produced by the fluid motion is neglected.The solenoid relation This constant is zero since   ′ ′ = 0at the plate which is electrically nonconducting.Hence,  ′ ′ = 0 everywhere in the flow.In view of the above assumption, Equation (1) yields  ′ = 0.This implies that   ′ =constant and  ′ = constant everywhere in the flow and choose this constant equal to zero, i.e.,   ′ =   ′ =0.Solving for   ′and  ′ from Equations (2)and(3), on using Taking into consideration the assumptions made above, under the Boussinesq's approximation, and using Equations (4)and( 5), the basic governing equations of the flow are derived as:

GOVERING EQUATIONS:
The description of the physical problem closely follows that of Rajput et al. (2016).This introduces unsteadiness in the flow field.The physical model and the coordinate system are shown Fig. 1.
The initial and boundary conditions for the problem are: The radiation heat flux r q under the Roseland approximationMagyari and Pantokratoras (2011) expressed by T into the Taylor series about '  T which, after neglecting higher-order terms, takes the form: Using Equation ( 14) and introducing the non-dimensional quantities: ) 9 ( and ) 13 ( the following are obtained in nondimensional form as follows The initial and boundary conditions Equation ( 10), in non-dimensional form become:

SOLUTION OF THE PROBLEM
The set of partial differential equations given ( 16)-( 19) are highly nonlinear therefore cannot be solved analytically.Thus, for the solution of this problem, Galerkin finite element method by Bathe (1996) and Reddy (1985) has been implemented.The finite element method is a powerful technique for solving differential or partial differential equations as well as for integral equations.This method is so general that it can be applied even for integral equations including heat transfer fluid mechanics, chemical processing, solid mechanics, electrical systems and other fields also.The steps involved in the finite element analysis are as in follows For equation ( 16), taking the linear element over two nodded ) (e , ) ( In equation ) 21 ( integrating the first term using by parts method and neglecting that term.After that, replace Galerkin finite element approximation over the two hugged linear variable )' ( ' e of the form , are the basis functions along the th j and th k nodes , velocity components The following difference strategy obtained when putting the row corresponding to the node i to zero.
The following system of equations is got after applying Crank-Nicholson method on ) 16 ( Here Index i designates to space and j for time.Theequations at every internal nodal point on a particular n-level constitute a tri-diagonal system of equations.They are solved by making use of the Thomas algorithm.A grid independent test is employed to get the solution with the least error.It is carried out by testing with various grid sizes.The equations at each internal nodal point on a particular n-level represent a tri-diagonal system of equations.So, in the equations ) 27 ( to ) 30 ( , taking  = 1(1) and using the boundary conditions ) 20 ( , the following tridiagonal system of equations are obtained.The tri-diagonal system is solved by making use of Thomas algorithm for which a numerical code is executed using MATLAB Program.To prove the convergence of the numerical scheme, the computation is carried out for small changed values of h and  and the iterations performed until a tolerance 10−8 is achieved.No notable change is observed in the values of , w ,   .Thus, the Galerkin finite element method is convergent and stable.The dimensionless primary and secondary skin frictions are given by From table 1 it is clear that skin friction decreases due to an increase in Hartmann number.it is noticed that the skin friction increases due to an increase in porous medium.

RESULT AND DISCUSSIONS
The parameters like    It is seen from figure-3 that the primary velocity falls when M increases.That is the primary fluid motion is retarded due to application of transverse magnetic field.This phenomenon clearly agrees with the fact that Lorentz force that appears due to interaction of the magnetic field and fluid velocity resists the fluid motion.It can be observed from the figure 5 that r G signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer.As expected, it is observed that there was a rise in the velocity due to the enhancement of thermal buoyancy force.Also, as r G increases, the peak values of the velocity increase rapidly near the porous plate and then decays smoothly to the free stream velocity.r G signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer.As expected, it is observed that there was a rise in the velocity due to the enhancement of thermal buoyancy force.Also, as r G increases, the peak values of the velocity increase rapidly near the porous plate and then decays smoothly to the free stream velocity.m G defines the ratio of the species buoyancy force to the viscous hydrodynamic force.As expected, the fluid velocity increases and the peak value is more distinctive due to increase in the species buoyancy force.The velocity distribution attains a distinctive maximum value in the vicinity of the plate and then decreases properly to approach the free stream value.It is noticed that the velocity increases with increasing values of m G .
From Figure 9 that primary velocity is increased due to increase in K .Physically, increase in K tends to decrease the resistance of the porous medium as a result increase the fluid velocity.Secondary velocity is increased due to increase in K .Physically, increase in K tends to decrease the resistance of the porous medium as a result increase the fluid velocity.Figure 13 shows that the primary fluid velocity is increased with the progression of time.Physically, buoyancy force gradually increases with time as a result fluid velocity enhances.It observed that there is a decrease in the temperature and temperature boundary layer as Pr increased.This is because the fluid is highly conductive for a small value of Pr.Physically, if Pr increases, the thermal diffusivity decreases, and this phenomenon leads to the decreasing manner of the energy transfer ability that reduces the thermal boundary layer.This is due to the fact that an increase in r K causes the concentration at the boundary layer to become thinner, which decreases the concentration of the diffusing species.This decrease in the concentration of the diffusing species diminishes the mass diffusion

CONCLUSIONS
In this paper, the unsteady MHD mixed convective radiating and chemically reacting fluid flow past an impulsively started oscillating vertical plate with Hall current, viscous dissipation, heat source is provided, which is embedded in porous medium.The Galerkin finite element method has been applied to solve the dimensionless governing equations of the flow.It has been found that thermal and mass buoyancy force, Hall parameter, radiation parameter and time tends to accelerate both u and w whereas an increase in Prandtl number, Schmidt number, and chemical reaction rate tends to decelerate both u and w.Increase in the magnetic parameter depreciate u and reverse trend is noticed on w.These parameters have similar effect on both primary and secondary skin frictions.The fluid temperature enhanced with increment in radiation parameter and time whereas reverse trend is noticed when Prandtl number is increased and opposite effect is noticed on the Nusselt number.The fluid concentration decline with increment in Schmidt number and chemical reaction rate whereas opposite trend is observed with progression of time and opposite effect is noticed on the Sherwood number.The value of the local skin-friction coefficient increases with increase in porous parameter.It is expected that the current study of the physics of flow over a vertical surface will serve as the foundation for many scientific and engineering applications involving the flow of electrically conducting fluids.The findings could be valuable in determining the flow of oil, gas, and water through an oil or gas field reservoir, as well as subsurface water migration and filtering and purification procedures.The results of this problem can be helpful in various devices subject to significant variations in gravitational force, its application on heat exchanger designs, wire and glass fiber drawing, and its application in nuclear engineering in connection with reactor cooling.
investigated effects of Soret, Dufour, Hall current and rotation on MHD natural convective heat and mass transfer flow past an accelerated vertical plate through a porous medium.Mateo et al. (2020) studied unsteady MHD radiating and reacting mixed convection past an impulsively started oscillating plate.Keeping in mind the work done by previous researchers, we attempted to analyze heat source and viscous dissipation effects on unsteady magneto hydrodynamic mixed convective heat and mass transfer flow of a fluid past an oscillating plate embedded in a porous medium in the presence of constant wall temperature and concentration.The novelty of this work is the consideration of heat source/sink and viscous dissipation in conservation of energy.We have extended the work of Matao et al. (

Figure 2 :
Figure 2: Primary velocity distribution with respect to

Figure 2
Figure 2 depict the comparison of the present work's velocity profile with the previous study done by Mateo et al. (2020).For fluid velocity, this figure shows excellent agreement (under some limiting conditions) between the current work and previously published work Mateo et al. (2020).

Figure 3 :
Figure 3: Primary velocity for varying M .

Figure 4 :
Figure 4: Secondary velocity for varying M .The secondary velocity increases with increasing M , because of less

Figure 5 :
Figure 5: Primary velocity for varying

Figure 6 :
Figure 6: Secondary velocity distribution with respect to

Figure 7 :
Figure 7: Primary velocity for varying m G .

Figure 8 :
Figure 8: Secondary velocity distribution with respect to

Figure 9 :
Figure 9: Primary velocity for varying K .

Figure 11 :
Figure 11: Primary velocity for varying

Figure 12 :
Figure 12: Secondary velocity for varying

Figure 13 :
Figure 13: Primary velocity for varying t.

Figure 14 :
Figure 14: Secondary velocity for varying t.Figure14shows that the secondary fluid velocity is increased with the progression of time.Physically, buoyancy force gradually increases with time as a result fluid velocity enhances in both directions.

Figure 14
Figure 14: Secondary velocity for varying t.Figure14shows that the secondary fluid velocity is increased with the progression of time.Physically, buoyancy force gradually increases with time as a result fluid velocity enhances in both directions.

Figure 15 :
Figure 15: Temperature for varying Pr.It observed that there is a decrease in the temperature and temperature boundary layer as Pr increased.This is because the fluid is highly conductive for a small value of Pr.Physically, if Pr increases, the thermal diffusivity decreases, and this phenomenon leads to the decreasing manner of the energy transfer ability that reduces the thermal boundary layer.

Figure 17 :
Figure 17: Temperature for varying c E .

Figure
Figure elucidates that the fluid temperature  decreases with increasing values of S .When S exists, thermal boundary layer is always starting to be thickened as result fluid temperature depreciate in the boundary layer.The temperature profile follows a trend that is quite similar to the one described in Sharma et al. (2022).

Figure 19
Figure 19 shows that the fluid temperature  increase with the progression of time t.It is seen from figure 20that the increasing values of Sc leads to fall in the concentration distribution.Physically, increase of Sc means decrease of molecular diffusivity D, this results in a decrease of concentration boundary layer.Hence, the concentration of the species is higher for small values of Sc and lowers for large values of Sc.

Figure 21 :
Figure 21: Concentration distribution for varying

Figure 21
Figure 21 shows a destructive type of chemical reaction because the concentration decreases for increasing r K which indicates that the diffusion rates can be tremendously changed by r K .

Figure 22 :
Figure 22: Concentration distribution for varying t.Concentration increases with progression of time t.Initially, species concentration takes the value 1 and afterward for large values of y it tends to zero with increase of t.
From the table 2 it is clear that h S increases with increasing values of

Table 1 :
Numerical values of Primary and Secondary Skin Frictions z 

Table 2 .
Numerical values of Nusselt number