The Influence of Magnetohydrodynamic Flow and Slip Condition on Generalized Burgers’ Fluid with Fractional Derivative

Abstract:  This paper investigates the effect of magnetohydrodynamic (MHD) of an incompressible generalized burgers’ fluid including a gradient constant pressure and an exponentially accelerate plate where no slip hypothesis between the burgers’ fluid and an exponential plate is no longer valid. The constitutive relationship can establish of the fluid model process by fractional calculus, by using Laplace and Finite Fourier sine transforms. We obtain a solution for shear stress and velocity distribution. Furthermore, 3D figures are drawn to exhibit the effect of magneto hydrodynamic and different parameters for the velocity distribution.

Y. Liu et al (11), has studied the effect of radiation with heat transfer and magnetohydrodynamic flow by burgers' fluids because of the exponential accelerating plate. Ebaid (12) has studied the Effect of MHD on peristaltic transport of a Newtonian fluid in an asymmetric channel with slip condition .Yaqing Liu .et al (13) investigated the magnetohydrodynamic which generalized Maxwell fluid that induced of moving plate and effects of second-order slip. Ghada et al, (14) investigated the magneto hydrodynamic flow of burgers' fluid by flowing and accelerating plate under the influence of the gradient pressure .Shihao et al (15), investigated effect of slip on 3D flow on fluid Burger between two side of wall generated by accelerated plate and a constant gradient pressure , Zheng et al (16) investigated studied slip effect on magnetohydrodynamic which be an Oldroyd_B fluid beginning by an accelerated plate.
In this research, we discuss the Influence of magneto hydrodynamic (MHD) on 3D Flow for Burgers' Fluid with Slip Condition between the two side of the walls that generated by a constant pressure gradient and exponential accelerated plate. The best solution of the velocity distribution and shear stress are acquired by using the Fourier Sine and Laplace transformations.

The description of the problem and its solution
Suppose that the generalized burgers' fluid with fractional derivative between the two sides of wall which occupy the complete space over the plate that 151 is perpendicular to the side of wall. The fluid starts to move because of the exponentially accelerated plate with a movement of this velocity Exp(-t) and a pressure constant B and by the presence of the slip condition .The related boundary condition and initial condition are as follows: u(y, z, 0) = ∂ u(y, z, 0) = ∂ 2 u(y, z, 0) ∂t 2 = 0 , represents the distance between the two side walls.
The equalization of an incompressible and generalized burger's fluid are presented by (6): = − + , Where -pI represents the Indeterminate Spherical Stress , T represents the Cauchy Stress Tensor, S represents the extra stress tensor , 2 a new material constant of burgers' fluid , 1 represents the relaxation time and 3 represents the retardation times , represents viscosity coefficient, = + ( ) is the first Rivlin Ericksen Tensor , = grad is the velocity gradient, , are the fractional calculus parameters, Where is the gradient operator and V represent the velocity vector , denoted the fractional operator is defined by (7): Here Gamm(.) denotes the Gamma function. We assume the stress and velocity of the form = ( , , ), = u( , , )̂ … (9) where ̂ is the unit vector along the x-coordinate direction and using the initial condition ( , , 0) = 0, we find And, suppose that a burger fluid is penetrated by a magnetic field B 0 that is applied parallel to the y-axis while the magnetohydrodynamic ignored by taking a fewest magnetic Reynolds number. Hence, the MHD body force caused by the external magnetic field takes the form Where represents the pressure gradient along x-axis and = 1, = 2 are the tangential stresses different from zero.
Then from Equation (10), Equation (11) and Equation (12), we obtain the governing equation for the generalized fractional Burgers' fluid We get the solution of velocity distribution by using the Finite Fourier Sine and Laplace transformations with series fractional derivative (6). Now, by multiplying two sides of Equation (13) through sin( ) and integrate w.r.t z from 0 to d, we obtain the equation: Using the Laplace transform of Equation (14), we obtain it We obtain the Equation (17) by utilize the ordinary differential equations to Equation (15) .

Solution of Shear Stress
Utilized the Laplace transformation to Equation (10) and Equation (11), we get the equations Applying the invers Finite Fourier sine transform to Equation (17)and source (17)  We can obtain the form of Equation (26) by using the Fox H-function (18).  As to the shear stress 2, it can be obtained from Equation (23) and Equation (17)by implementing the same count steps with those of 1

Analysis and Results
That work, we discussed the analytic solution for the magnatohydrodynamic flow of Burgers' fluid with fractional derivative because of a constant pressure gradient and exponential acceleration plate between two sides wall. That be by means of the Finite Fourier sine and Laplace transformations, so the solutions are acquired in terms of the Fox H-function. Several Figures are drawn to display the influence of various parameters of the Burger's fluid , Fig.1 shows the effects of increasing magnetic field N results in the increasing of the velocity surfaces . Figure 2, 3 the variation of the fractional derivative of parameters α,β that show ,the effects of increasing α is retard the velocity increasing surface, so increasing β has the adverse effect of α. Figure 4, 5 show the material parameters 1 , 3 , the effect of decreasing 1 3 increase is the ( ) increasing velocity surface that generalized burger's fluid. Figure 6 show the difference of velocity surface of different value of time.
It is apparent that the velocity flow is increase with increase of time t. Figure 7 displays influence of slip coefficient , the fluid flows increase with decreasing slip coefficient .