MHD Accelerated Flow of Maxwell Fluid in a Porous Medium and Rotating Frame

1 UTM Centre for Industrial and Applied Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia 2 Department of Mathematics, Faculty of Science, University of Kordofan, 51111 El Obeid, Sudan 3Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia 4Department of Fundamental and Applied Sciences, Universiti Teknologi Petronas, 31750 Tronoh, Perak, Malaysia


Introduction
Several fluids including butter, cosmetics and toiletries, paints, lubricants, certain oils, blood, mud, jams, jellies, shampoo, soaps, soups, and marmalades have rheological characteristics and are referred to as the non-Newtonian fluids.The rheological properties of all these fluids cannot be explained by using a single constitutive relationship between stress and shear rate which is quite different than the viscous fluids [1,2].Such understanding of the non-Newtonian fluids forced researchers to propose more models of non-Newtonian fluids.
In general, the classification of the non-Newtonian fluid models is given under three categories which are called the differential, the rate, and the integral types [3].Out of these, the differential and rate types have been studied in more detail.In the present analysis we discuss the Maxwell fluid which is the subclass of rate-type fluids which take the relaxation phenomenon into consideration.It was employed to study various problems due to its relatively simple structure.Moreover, one can reasonably hope to obtain exact solutions from Maxwell fluid.This motivates us to choose the Maxwell model in this study.The exact solutions are important as these provide standard reference for checking the accuracy of many approximate solutions which can be numerical or empirical in nature.They can also be used as tests for verifying numerical schemes that are being developed for studying more complex flow problems [4][5][6][7][8][9].
To the best of our knowledge, no investigation has been reported so far which discusses the accelerated flows of non-Newtonian fluids in a rotating frame.This is the objective of the present study.Here, we examine the rotating and MHD flow induced by an accelerated plate.Two explicit examples of acceleration subject to a rigid plate are taken into account.Constitutive equations of a Maxwell fluid are used and modified Darcy's law has been utilized.The exact solution to the resulting problem is developed by Fourier sine transform.With respect to physical applications, the graphs are plotted in order to illustrate the variations of embedded flow parameters.The mathematical results of many existing situations are shown as the special cases of the present study.

Formulation of the Problem
We choose a Cartesian coordinate system by considering an infinite plate at  = 0.An incompressible fluid occupying the porous space is electrically conducting by exerting an applied magnetic field  ∘ parallel to the -axis.The electric field is not taken into consideration, the magnetic Reynolds number is small, and the induced magnetic field is not accounted for.The Lorentz force J ×  ∘ under these conditions is equal to − 2 ∘ V. Here, J is the current density, V is the velocity field, and  is electrical conductivity of fluid.Both plate and fluid possess solid body rotation with a uniform angular velocity Ω about the -axis.
The governing equations are in which  is the fluid density,  is a radial vector with  2 =  2 +  2 ,  is the pressure, and R is Darcy's resistance.The extra stress tensor S for a Maxwell fluid satisfies where T is the Cauchy stress tensor, I is the identity tensor, L is the velocity gradient, A = L + L  is the first Rivlin-Eriksen tensor,  is the relaxation time,  is dynamic viscosity of fluid and / indicates the material derivative.
According to Tan and Masuoka [4], Darcy's resistance in an Oldroyd-B fluid satisfies the following expression: where   is the retardation time,  is the porosity, and  is the permeability of the porous medium.For Maxwell fluid   = 0, and hence, We seek a velocity field of the form: which together with (2) yields  = 0.Then, using ( 1) and ( 3) into ( 5) we arrive at where and   are and -components of Darcy's resistance ; and -component of (1) indicates that p ̸ = p(), and modified pressure p is p =  − (/2)Ω 2  2 .
Invoking ( 4) and ( 8) in ( 7) and then neglecting the pressure gradient, we arrive at Mathematically when the pressure gradient is ignored, the equation of motion becomes simplified and manageably solvable, and physically the fluid is still in motion as required by the boundary conditions.We have T = −I + S, and when the pressure gradient is negligible, the stress tensor T is equivalent to the extra stress tensor S, portraying the fluid motion.
The initial and boundary conditions for a constant accelerated plate are where  has a dimension of / 2 .

Solution for Constant Accelerated Flow
Setting  =  +  and introducing the following dimensionless quantities: The above problem statement reduces to After using Fourier sine transform, (14) becomes in which   (, ) indicates the Fourier sine transform of (, ).
The solution of (15) satisfying condition ( 16) can be expressed as or with Inversion of Fourier sine transform in (17) gives The above expression for hydrodynamic fluid  = 0 in a nonporous space 1/ = 0 is 2 . ( Letting  = 0 in (21) and ( 22), we arrive at The velocity field (, ) given by ( 21) is equivalent to that obtained by Fetecau et al. [3,Equation 25] and [5,Equation 31].

ISRN Mathematical Physics
Result (20) for a magnetohydrodynamic viscous fluid  = 0 in a porous space is In the above expression, if we put  = 0 and 1/ = 0, we obtain the equivalent solution which was obtained by Fetecau et al. [5,Equation 22], To see the variations of embedded flow parameters in the solution expressions, Figures 1 to 5

Solution for Variable Accelerated Flow
In this section, the problem statement consists of ( 9) to ( 10), (12) and where R has a dimension of / 3 .The flow problem after defining in terms of the dimensionless quantities, where 2 . (30) The previous expression for hydrodynamic fluid in a nonrotating  = 0 and nonporous space 1/ = 0 is  where For a magnetohydrodynamic viscous fluid  = 0 →  = 0 in a porous space, (31) takes the form In order to see the variations of embedded flow parameters in the solution expressions, Figures 6, 7

Results and Discussion
This section concerns the variations of embedded flow parameters in the solution expressions.Hence, Figures 1-10 velocity components.It is observed that the role of  on the magnitude of velocity component is qualitatively similar to that of .This is in accordance with the fact that the Lorentz force acts as a resistance force to the flow under consideration.On the other hand, the boundary layer thickness decreases when both  and  increase.The effects of  and the velocity components are sketched in Figure 3. Clearly the magnitude of velocity decreases when  is increased; that is, when the medium becomes less porous (porosity  decreases).Figure 4 plots the effects of the solid body rotation in terms of .sketched in Figure 5.It is found that the magnitude of velocity components increases with dimensionless time.Further, it is observed that the variations of parameters in constant accelerated flow (i.e., Figures 1-5) are qualitatively similar (i.e., in Figure 6, the magnitudes of an imaginary component of velocity are decreasing function of ) to the effects plotted for variable accelerated flow (i.e., .However, the velocity profiles in constant accelerated flow and variable accelerated flow are not similar quantitatively.Comparison shows that the velocity profiles in variable accelerated flow are larger when compared to those of constant accelerated flow.

Conclusions
In this research, the constant accelerated flow and variable accelerated flow of non-Newtonian fluid in a rotating frame are examined.The exact solutions are first established via constant and variable accelerated cases are similar in a qualitative sense.
have been displayed in order to illustrate such variations for the constant accelerated flow.Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.

Figure 1 :Figure 2 :
Figure 1: Velocity profiles for different values of .

Figure 3 :Figure 4 :
Figure 3: Velocity profiles for different values of .
, 8, 9, and 10 have been displayed in order to illustrate such variations for the variable accelerated flow.Further, in each Figure, panels (a) and (b) depict the behaviours of real and imaginary parts of dimensionless velocity.