^{1}

^{*}

^{1}

^{1}

^{1}

In this work, we analyze Couette flow problem for an unsteady magnetohydrodynamic (MHD) fourth-grade fluid in presence of pressure gradient and Hall currents. The existing literature on the topic shows that the effect of Hall current on Couette flow of an unsteady MHD fourth-grade fluid with pressure gradient has not been investigated so far. The arising non-linear problem is solved by the homotopy analysis method (HAM) and the convergence of the obtained complex series solution is carefully analyzed. The influence of pressure number, Hartmann number, Hall parameter and fourth-grade material parameters on the unsteady velocity is discussed through plots and on local skin-friction coefficient discussed through numerical values presented in tabular form.

In fluid mechanics, everyone is familiar with Couette flow problem, the flow between two parallel plates in which bottom plate is fixed and upper plate is initially at rest and is suddenly set into motion in its own plane with a constant velocity, is termed as Couette flow [

investigated by Hayat and Kara [

In order to understand the interaction of electric, magnetic, and hydrodynamic forces in the unsteady fourth-grade fluid, we considered a simple flow problem, known as the Couette flow. The effects of pressure gradient and Hall current on the flow are also taken into account. The complex analytic solution for non-linear problem is found by using the homotopy analysis method (HAM) [

Consider the unsteady flow of an electrically conducting incompressible fourth-grade fluid between two parallel flat plates, subjected to a uniform transverse magnetic field. We assume that the bottom plate is fixed and the top plate is stationary when

where

The equations governing the magnetohydrodynamic flow with Hall effect are:

The boundary and initial conditions are

where

where

where prime denotes differentiation with respect to

modified Hartmann number [

Now using Equations (1)-(3) and (7) the Equation (10) can be written in dimensionless variables as

where

The boundary conditions (9) lead us to take base functions for the velocity

The velocity

To start with the homotopy analysis method, due to the boundary conditions (9) it is reasonable to choose the initial guess approximation

and the auxiliary linear operator

with the property:

where

At

similarly

The total complex analytic solution in compact form is

where from initial guess in Equation (14) we obtain

all other unknown constants can be determined by utilizing first nine given in Equation (21) by using the recurrence relations, which we calculated but it is not possible to write here due to their length. We know that the auxiliary parameter

The discussion of emerging parameters on the dimensionless velocity

Figures 2 to 10 are plotted in absence of Hall currents and in

bottom plate to towards the moving top plate the velocity increases for all values of the time, even the fluid close to the upper plate moves with the same velocity as of the upper plate and the fluid close to the bottom plate has nearly zero velocity.

distribution is presented for the various values of third-grade parameters

It is observed from

in the absolute value of the skin friction coefficient. Increase in pressure number

The Couette flow between two parallel plates filled with MHD unsteady fourth-grade fluid is studied analytically. The effects of the pressure and Hall current are also incorporated. A non-linear fourth-grade model for the fluid is used. The model is invoked into the governing equations and the resulting one dimensional equation for unsteady MHD flow is derived. This equation is solved by HAM in general to study the sensitivity of the flow to

the parameters that are used in the fourth-grade model. The various dimensionless parameters seem to affect the velocity a lot. The velocity profile and local skin friction coefficient are greatly influenced by the Hall parameter, fourth-grade fluid parameters, pressure and Hartmann numbers.