The Effect of Physical Parameters on Flow Variables of an Electrically Conducting Viscoelastic Fluid

In this paper the effect physical parameters on flow variables of unsteady, incompressible, electrically conducting viscoelastic fluid flowing between a pair of infinite vertical Couette porous channel walls embedded in a porous medium is analyzed. A uniform magnetic field is applied perpendicular to the channel walls. The temperature of the moving wall varies periodically. The temperature difference between the two walls is high enough due to thermal radiation. The solution of the governing equations is obtained using regular perturbation techniques. This technique is applied on partial differential equations that are difficult to solve. These partial differential equations are reduced to a set of ordinary differential equations in dimensionless form and thus they can be solved analytically. The effects of physical parameters on the flow variables are studied and the results have been discussed. The physical parameters considered include Hartmann number, viscoelastic parameter, Permeability of porous medium, chemical reaction parameter, radiative parameter, thermal Grashof number for heat transfer, modified Grashof number for mass transfer, frequency parameter, Prandtl number, mass diffusivity and Schmidt number. The flow variables considered include velocity, temperature and concentration. The theoretical results have been supported by simulation study. The observations include: (i) velocity decreases with increasing values of frequency, Hartmann number and viscoelastic parameter (ii) velocity increases with increasing values of temperature, thermal Grashof number, modified Grashof number and permeability of porous medium, (iii) the temperature decreases near the moving channel wall when the radiative parameter increases (iv) the temperature approaches to zero in the region near to the boundary layer of the stationary channel wall when the radiative parameter increases (v) concentration decreases with an increment in both chemical reaction and Schmidt number and (vi) The velocity of fluid increases as thermal Grashof number and modified Grashof number increases.


Introduction
The change of shear stress versus strain rate inside a fluid depending on viscosity can be classified as a linear, nonlinear or plastic response. If the shear stress is linearly proportional to the strain rate then it is known as Newtonian fluid, and the constant of proportionality is called as the coefficient of dynamic viscosity. In case of non-Newtonian fluid, the shear stress is not only nonlinear to strain rate but also can be time dependent and hence coefficient of dynamic viscosity can't be a constant. The properties of non-Newtonian fluid flow are treated by rheology, the science of studying deformation and flow of a substance. A fluid is said to have Thixotropic property if its viscosity decreases only with time but neither with the shear stress nor with strain rate. Also, when shear stress is independent of strain rate then the fluid material possesses plastic deformation.
Viscoelastic is the property of a fluid material that combines both viscous and elastic characteristic while the fluid is undergoing deformation. The behavior of viscoelastic fluid is different from that of both Newtonian and inelastic non-Newtonian fluids. This behavior includes the presence of normal stresses in shear flows those are sensitive to deformation. Some materials that possess viscoelastic properties include: shampoos, hand creams, tooth pastes, yoghurts, polymers and paints. The factors that can affect the properties of viscoelastic are temperature, time and shear rate. Unsteady, incompressible, and electrically conducting 'viscoelastic fluid' embedded in a porous medium is of great importance. Such fluid has several applications in industry including polymer production, ceramics, lubrication, electromagnetic propulsion, and magneto-chemistry.
In literature it can be observed that there have been a very few studies conducted on the present subject. To fill the gap in this study it has been considered the effects of thermal radiation, chemical reaction, Permeability of porous medium and electrically conducting viscoelastic fluid. However, second order effects in elasticity, plasticity and fluid dynamics have been analyzed [1]. The Magneto hydrodynamic free convection laminar flow of incompressible viscoelastic fluid was studied [2]. The MHD combined free and forced convection flow through two parallel porous walls were studied [3]. An oscillatory mixed convection flow of electrically conducting viscoelastic fluid in an infinite vertical channel filled with porous medium was analyzed [4]. MHD free convection flow of viscoelastic fluid past an infinite porous plate was investigated [5]. The unsteady flow and heat transfer through an elastic viscous liquid along an infinite hot vertical porous moving plate with variable free stream and suction were examined [6]. Flow and heat transfer of an electrically conducting viscoelastic fluid between two horizontal squeezing and stretching plates have been studied [7]. Heat and mass transfer in MHD viscoelastic fluid flow through a porous medium over a stretching sheet with chemical reaction was studied [8]. MHD mixed convective viscoelastic fluid flow in a permeable vertical channel with Dufour effect and chemical reaction parameter was studied [9]. Viscoelastic fluid flow past an infinite vertical porous plate in the presence of first order chemical reaction was investigated [10]. MHD flow through a porous medium past a stretched vertical permeable surface in the presence of heat source or sink and a chemical reaction has been studied [11]. The three dimensional free convection flows through porous medium in a vertical channel with heat source and chemical reaction was studied [12]. The unsteady two dimensional flows and heat transfer through an elastic viscous liquid along an infinite hot vertical porous surface bounded by porous medium was studied [13]. MHD flows in non-Newtonian fluids has been studied [16].
The effects of radiation on free convection MHD flow have attracted many scholars due to its applications in space science. Such radiations occur due to high temperatures and design of relevant equipments. Furthermore, heat and mass transfer with thermal radiation on free convection MHD flow are significant in the plate heat transfer. Recently, many scholars focused on this field of study due to its applications in gas cooled nuclear reactors, nuclear power plants, gas turbines, space vehicles, and supersonic flights. The unsteady convective flow in a moving plate with thermal radiation was studied [17][18]. The combined effects of radiation and buoyancy force past a vertical plate has been studied [19]. The influence of thermal radiation on convective flows over a porous vertical plate was analyzed [20]. The effects of thermal radiation on heat and mass transfer over a moving vertical plate were studied [21]. The effects of thermal radiation on mixed convection flow over a permeable vertical plate with magnetic field have been studied [22].
Studies of energy and concentration transfer due to chemical reaction have significant applications in many branches of science and technology. The effects of concentration transfer with the involvement of chemical reaction are important in chemical process which helps to obtain high value products from cheaper raw materials. The quality of production and the flow behavior of the fluid is affected by the concentration goes through some kind of chemical reaction with the surrounding fluid.
The effects of mass transfer over a vertical oscillating plate with chemical reaction were studied [23]. Chemical reaction and thermal radiation effects on MHD convective flow in a porous medium in the presence of suction studied [24]. The effects of magnetic field on MHD Couette flow of a thirdgrade fluid with chemical reaction studied [25]. Even though, the effects of thermal radiation, chemical reaction, permeability of porous medium and electrically conducting viscoelastic fluid in vertical Couette channel wall embedded in a porous medium are important they have not been considered in these studies [15]. Hence the present study has been motivated to verify such effects.
The fluid considered here has the properties viz., unsteady, incompressible, electrically conducting viscoelastic, chemical reacting and thermal radiating. Free stream velocity of the fluid is assumed to be fluctuating. Furthermore, it is assumed that temperature and concentration of the fluid do also fluctuate with time. The main objective of the present study is to analyze the effect of physical parameters on flow variables of unsteady, incompressible, and electrically conducting viscoelastic fluid on free convection oscillatory fluid flow bounded between two infinite vertical Couette channel walls embedded in a porous medium when the temperature and concentration oscillates with time in the presence of thermal radiation, chemical reaction and permeability of porous medium.

Mathematical Modeling and Analysis of the Problem
Note that here and in what follows fluid means unsteady, incompressible, electrically conducting viscoelastic fluid, chemical reacting and radiating fluid.
Consider two dimensional free convection Couette fluid flows bounded by two infinite vertical channel walls separated by a distance embedded in a porous medium. The geometrical representation of the model and coordinate system are shown in Figure 1. The ̅ axis is parallel to the gravitational acceleration vector but in opposite direction. The axis is transverse to the channel walls.
The vertical moving channel wall is located at = 0 along ̅ where the temperature is and the concentration is ̅ . The other stationary channel wall is located at = where the temperature is and the concentration is ̅ .
Initially at ̅ = 0, the stationary channel wall and the fluid are at the same temperature and concentration level of the fluid ̅ is the same at all points. At ̅ 0, the temperature of the moving wall and concentration of the fluid are raised to and ̅ respectively and they are maintained at constant values. This result is due to porous medium i.e., porous medium is used to insulate a heated body by maintaining its temperature. As time changes temperature changes leading to buoyancy force acting in fluid flow so the channel wall at 0 starts moving in its own plane along positive ̅ direction with velocity . And for all the later times ̅ 0 the channel wall continues to be in motion. But due to viscosity of the fluid and acceleration due to gravity the velocity will decrease and consequently it starts moving in opposite direction after some time ̅ the moving channel wall being heated by supplying external heat so as to maintain its temperature constant . However, the temperature of the wall at rises to an upper constant value . The governing equations of the model are derived based on the following assumptions: (i) The fluid properties such as temperature, viscosity, pressure, specific volume, specific gravity are constant (ii) the density varies with both temperature and concentration in the body force term or Boussinesq's approximation. It is a variable but not a constant (iii) the flow of the fluid is unsteady, laminar, incompressible, electrically conducting viscoelastic, thermal radiating and chemical reacting (iv) the magnetic field applied is transversal between the channel walls (v) the induced magnetic field of the fluid is negligible and hence the magnetic Reynolds number is a very small constant, the external electric field is supposed to be zero (vi) the electric field due to polarization of charges is negligible (vii) viscosity is considered with the constant permeability of porous medium (viii) there exists a homogeneous chemical reaction of first order with constant rate between diffusing concentration or species and the fluid in the moving wall. Note that a reaction is said to be of first order if the rate of reaction is directly proportional to concentration.
The expression for free stream velocity can be formed as Here in (1), is the mean constant free stream velocity, is the frequency and ̅ is the time. We now derive the governing equations of the present model. The momentum equation of the model takes the form as In (2), the vector cross product 2 3 is the Lorentz force. This term is a body force corresponding to magneto hydrodynamics flow. The minus sign in &4"5 6 ⁄ 8 indicates that the fluid flows along the direction from higher to lower potential. The plus sign in shear stress of viscoelastic fluid 94 6 : ; ⁄ 84<= >$ < ⁄ 8? is tensile stress or pulling force. Using ohm's law, the expression for the density of current can be constructed 2 @ A B̅ 3 . Upon substituting A 0, since electric field is assumed to be negligible, it reduces to 2 @ B̅ 3 . Also, the expression for Lorentz force reduces and takes the form as 2 3 &@3 C 5 . In view of these result, the equation (2) In (3) = >$ is the component of the shear stress of the viscoelastic fluid given in [26] as Here F is the coefficient of dynamic viscosity and G is the modulus of rigidity. If G approaches to infinity or if the component of shear stress is independent of time that is at steady state, the fluid behaves like a viscous fluid without elasticity. Solving for = >$ in terms of the velocity component 5 we obtain The higher order terms are neglected since their contributions are small enough as G approaches to infinity. In view of these the equation (3) after some algebraic manipulations reduces to The energy equation of the model can be expressed as The concentration equation of the model can be expressed as amplitude of free stream velocity; Frequency of oscillation; U dimension less temperature; g Thermal conductivity; " Kinematic viscosity; F Dynamic viscosity; J Thermal diffusivity; and @ Electric conductivity.
The set of equations (3) -(5) govern the present model represented in Figure 1.
In (4), the term 941 ; f ⁄ 8 <c _ < ⁄ ? is the thermal radiation effect. The minus sign of the radiative heat flux term indicates the emission of energy away from the objects that is heat is lost. Furthermore, in this equation, J > 0 shows heat generation and J < 0 indicates heat absorption i.e., thermal energy diffuses more rapidly through substances with high thermal diffusivity J and it diffuses slowly through those with low thermal diffusivity J.
In (5), the minus sign indicates that the mass diffusion represents 'Generative chemical reaction'. Here chemical reaction is an exothermic reaction and in that case heat is generated. However, the chemical reaction is said to be generative or exothermic if the chemical reaction parameter is negative i.e. 6 _ < 0.If 6 _ > 0 the chemical reaction is said to be destructive or endothermic chemical reaction. Also 6 _ ̅ − ̅ is the generative chemical reaction term.
It is assumed that the radiation heat flux is unidirectional along positive direction. Using the Roseland approximation for radiative heat transfer and the Roseland approximation for diffusion and also following the other scholarly works [27], the expression for radiative heat flux can be given as c _ = − 4@ 36 o ⁄ < p < ⁄ . Here, the parameters @ and 6 o represent the Stefan Boltzmann constant and the Roseland mean absorption coefficient respectively.
We now assume that the temperature differences within the fluid flow are sufficiently small so that p can be expressed as a linear function of using Taylor

Non Dimensionalization of the Model
In order to solve the governing equations (3) -(5) of the model it is convenient and easy to deal with its dimensionless form. Hence, we find the dimensionless form of the model by introducing the following non dimensional quantities.
The corresponding boundary conditions (6)

Analytical Solution of the Problem
To find the analytical solution of the non dimensional form of the present model we consider equations (11) -(15) and we assume the amplitudes of the free stream velocity, temperature and concentration variation are very small quantities. Hence, the amplitude of free stream velocity needs to be considered 0 < ≪ 1 Using perturbation techniques the solution of the model has the following form 5 , = 5 : Also, the free stream velocity takes the form as = 1 + On substituting equations (16) -(19) into (11) -(13), and equating harmonic and non-harmonic terms and neglecting higher orders of we obtain Various symbols are expressed in the Appendix.

Simulation Study of the Problem
Here simulation study of different flow parameters is conducted using Mat lab code. We consider fluid flow bounded between two infinite vertical Couette channel walls embedded in a porous medium when the temperature and concentration oscillates with time in the presence of thermal radiation, chemical reaction, permeability of porous medium and electrically conducting viscoelastic fluid. For simplicity we have not presented the graphs of skin friction, Nusselt number, Prandtl number, mass diffusivity and Sherwood number.
The effects of physical parameters such as viscoelastic parameter, permeability of porous medium, thermal Grashof number, modified Grashof number, Hartmann number, frequency parameter, radiation parameter, chemical reaction parameter and Schmidt number on flow variables such as velocity, temperature and concentration of the model have been studied using simulation. In Figure 2, the simulated results of the influence of frequency parameter on velocity have been presented. The graph in 5 plane represents respectively the distance between the channel walls and velocity. Other parameters are held constant. It can be observed that the velocity 5 starts with a minimum value near the moving channel wall and then it increases as it goes towards the stationary channel wall. Furthermore, the velocity decreases with increasing values of frequency. resistance of the magnetic field of the fluid flow will also increase more; and as a result the velocity also will decrease more. Furthermore, the decrement in velocity results in decreasing the temperature. It can be put in simple words that whenever the Hartmann number increases then the velocity as well as thermal boundary layer will decrease.
In Figure 4, the simulated results of the influence of viscoelastic parameter 6 : on velocity have been presented. The graph in 5 plane represents respectively the distance between the channel walls and velocity. Other parameters are held constant.
The result shows that for a fixed value of viscoelastic parameter 6 : , the value of the fluid velocity denoted by 5 starts from a minimum constant value at the moving plate, and as increases the velocity 5 increases and reaches a maximum constant value at the stationary channel wall. Generally, it can be concluded from the simulation study that as viscoelastic parameter 6 : increases the velocity profile decreases i.e. flow of the fluid retards with increasing of viscoelastic parameter.
In Figure 5, the simulated results of the influence of temperature U on velocity have been presented. The graph in 5 plane represents respectively the distance between the channel walls and velocity. Other parameters are held constant. The result shows that for a fixed value of temperature U, the value of the fluid velocity denoted by 5 starts from a minimum constant value at the moving plate, and as increases the velocity 5 increases and reaches a maximum constant value at the stationary plate. Generally, it can be concluded from the simulation study that as temperature U increases the velocity increases.
In Figure 6, the simulated results of the influence of thermal Grashof number ST on velocity have been presented. The graph in 5 plane represents respectively the distance between the channel walls and velocity. Other parameters are held constant. The result shows that for a fixed value of thermal Grashof number ST, the value of the fluid velocity denoted by 5 starts from a minimum constant value at the moving channel wall, and as increases the velocity 5 increases and reaches a maximum constant value at the stationary channel wall. Generally, it can be concluded from the simulation study that as thermal Grashof number increases the velocity profile increases. Physically, this means that the increment of the buoyancy force leads to increase the vertical component of the velocity.
In Figure 7, the simulated results of the influence of Modified Grashof number SV on velocity have been presented. The graph in 5 plane represents respectively the distance between the channel walls and velocity. Other parameters are held constant.
The result shows that for a fixed value of Modified Grashof number SV, the value of the fluid velocity denoted by 5 starts from a minimum constant value at the moving channel wall, and as increases the velocity 5 increases and reaches a maximum constant value at the stationary channel wall. Generally, it can be concluded from the simulation study that velocity of the fluid increases as modified Grashof number increases. In Figure 8, the simulated results of the influence of permeability of porous medium 6 on velocity have been presented. The graph in 5 plane is representing respectively the distance between the channel walls and velocity. Other parameters are held constant.
The result shows that for a fixed value of permeability of porous medium 6 , the value of the fluid velocity denoted by 5 starts from a minimum constant value at the moving channel wall, and as increases the velocity 5 increases and reaches a maximum constant value at the stationary channel wall. Generally, it can be concluded from the simulation study that as permeability of porous medium increases the velocity of the fluid increases. Physically, this means that the resistance dominated by the porous medium reduces as the permeability of the medium increases because of which the velocity increases. In Figure 9, the simulated result of the influence of Modified Grashof number SV on the concentration of the fluid has been studied by simulation. Naturally, the number SV and the concentration are inversely proportional to each other i.e. as SV increases the velocity of the fluid increases and hence the concentration decreases. The same fact has been even supported by the present simulation study.
In Figure 10, the simulated result of the influence of radiation parameter d on transfer of the temperature has been presented. The graph in U plane is representing respectively the distance between the channel walls and temperature. Other parameters are held constant.
The result shows that for a fixed value of thermal radiation parameter d, the temperature U starts from a constant value at the moving channel wall. Also as increases the temperature decreases till it reaches a minimum value. Nevertheless, the measure of temperature at the moving channel wall is always at a higher value than that at the stationary channel wall.  As the temperature of the moving channel wall is higher than that of the stationary channel wall, the temperature is expected to transfer from moving channel wall towards stationary channel wall. This fact has been supported in this simulation.
Also it can be observed from that simulation: (i) the value of temperature does not change with the change in d at the moving channel wall, (ii) the minimum value of temperature, occurs between the two channel walls, decreases with the increase of d i.e. minimum value of temperature is inversely proportional to that of d, and (iii) the value of temperature approaches zero with the increase of d at the stationary channel wall.
In Figure 11, the simulated result of the influence of Chemical reaction parameter ŸT on concentration has been presented. The graph in % plane is representing respectively the distance between the channel walls and concentration. Other parameters are held constant. The result shows that for a fixed value of chemical reaction parameter ŸT , the value of fluid concentration denoted by % starts from a maximum constant value at the moving, channel wall and as increases the concentration % decreases and reaches zero constant value at the stationary channel wall. Nevertheless, the measure of fluid concentration at the moving channel wall is always at a higher value than that at the stationary channel wall.
As the concentration of fluid at the moving channel wall is higher than that at the stationary channel wall, the concentration is expected to move from moving channel wall towards stationary channel wall. Fluids diffuse from the locations of higher concentrations to that of lower concentrations. This fact has been supported in this simulation.
Also it can be observed from that simulation: (  In Figure 12, the simulated result of the influence of Schmidt number `V on concentration has been presented. The graph in V plane is representing respectively the distance between the channel walls and concentration. Other parameters are held constant.
The result shows that for a fixed value of Schmidt number `V the value of fluid concentration denoted by V starts from a maximum constant value at the moving channel wall, and as increases the concentration V decreases and reaches zero constant value at the stationary channel wall. Nevertheless, the measure of fluid concentration at the moving channel wall is always at a higher value than that at the stationary channel wall.
As the concentration of fluid at the moving plate is higher than that at the stationary channel wall, the concentration is expected to move from moving channel wall towards stationary channel wall. Fluids diffuse from the locations of higher concentrations to that of lower concentrations. This fact has been supported in this simulation. Also it can be observed from that simulation: (i) the value of concentration of fluid does not change with the change in `V at the moving channel wall, (ii) the value of concentration of fluid, occurring between the two channel walls, decreases with the increase of `V i.e. the value of concentration of fluid is inversely proportional to that of `V, and (iii) the value of concentration of fluid goes to zero at the stationary channel wall irrespective of the value of `V. In general, the concentration decreases with an increase in `V. Physically, this is true since the increase in `V implies decrease of molecular diffusivity implies decrease in concentration boundary layer.

Conclusions
In this paper the effect of physical parameters on flow variables of fluid flowing between a pair of infinite vertical Couette porous channel walls embedded in a porous medium is analyzed. The effects of physical parameters Viz. Hartmann number, Viscoelastic parameter, Permeability of porous medium, Chemical reaction parameter, radiative parameter, thermal Grashof number for heat transfer, modified Grashof number for mass transfer, frequency parameter and Schmidt number on flow variables Viz. velocity, temperature and concentration has been discussed. The solution of the governing equations is obtained using regular perturbation techniques. This technique is used to transform partial differential equations that are difficult to solve in closed form or explicit form. Mat lab code is used to solve and simulate the graphs of higher order ordinary differential equations. Hence we obtain the following results: i. Velocity decreases with increasing values of frequency, Hartmann number, Viscoelastic parameter. ii. Velocity increases with temperature, thermal Grashof number, modified Grashof number and permeability of porous medium. iii. Thermal radiation parameter increases, the temperature decreases near the moving plate while it approaches to zero in the region near to the boundary layer of the stationary plate. iv. An increment in both chemical reaction and Schmidt number results in decreasing in concentration.