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This work is concerned with the influence of uniform suction or injection on unsteady incompressible Couette flow for the Eyring-Powell model. The resulting unsteady problem for horizontal velocity field is solved by means of homotopy analysis method (HAM). The characteristics of the horizontal velocity field and wall shear stress are analyzed and discussed. Pade approximants and Taylor polynomials are also found for velocity profile and are used to make the maximum error as small as possible. The graphs of the error for the Pade approximation and Taylor approximation are drawn and discussed. Convergence of the series solution is also discussed with the help of *h*-curve and interval of convergence is also found.

The study of non-Newtonian fluids has generated much interest in recent years in view of their numerous industrial applications, especially in polymer and chemical industries. The examples of such fluids includes various suspensions such as coal-water or coal-oil slurries, molten plastics, polymer solutions, food products, glues, paints, printing inks, soaps, shampoos, toothpastes, clay coating, grease, cosmetic products, custard, blood, etc. Some interesting studies of non-Newtonian fluids are given by Hayat et al. [1-5], Asghar et al. [

Fang [

studied Couette flow of a third-grade fluid with variable magnetic field. Seth et al. [

The Eyring-Powell model [

Keeping this all in view, in the present paper, the authors envisage studying the time-dependent Couette flow of incompressible non-Newtonian Eyring-Powell model with porous walls. The resulting unsteady problem is solved by means of homotopy analysis method (HAM) [41-58], which is very powerful and efficient in finding the analytic solutions for a wide class of nonlinear differential equations. The method gives more realistic series solution that converges very rapidly in physical problems. The convergence region for the series solution is found with the help of. For a given amount of computational effort, one can usually construct a rational approximation that has smaller overall error in given domain than a polynomial approximation [

Consider an unsteady, incomprssible, non-Newtonian, Couette flow problem for the Eyring-Powell model, in which the bottom wall is fixed and subjected to a mass injection velocity and there is mass suction velocity at the top wall, correspond to injection and correspond to suction. The top plate is stationary when, there is only mass transfer in the transverse direction, say direction. At, the top wall is started impulsively to a constant velocity. The Eyring-Powell model is derived from the theory of rate processes, which describes the shear of a non-Newtonian flow. The Eyring-Powell model can be used in some cases to describe the viscous behavior of polymer solutions and viscoelastic suspensions over a wide range of shear rates. The stress tensor in the Eyring-Powell model for non-Newtonian fluids is given by [

where is the dynamic viscosity, and are the characteristics of the Eyring-Powell model. Taking the second order approximation of the function as

The governing equation for this problem can be obtained as

where is the kinematic viscosity, is the density of the fluid, bottom wall is located at, top wall is located at and is the velocity at the upper wall. Equations (3) and (4) can be non-dimensionalized by defining

Then Equations (3) and (4) become

where is the Reynolds number, is the fluid parameter and is the local non-Newtonian parameter based on velocity of plate. Using stream function relations with velocity [

where, is the reduced stream function and prime denotes ordinary derivative w. r. t. When, Equation (8) becomes

where is some arbitrary unknown function of.

To start with the homotopy analysis method it is very much important to choose an initial guess approximation and a linear operator. Therefore, due to the boundary conditions (9) it is reasonable to choose the initial guess approximation

and the linear operator

which satisfies the following property:

where and are arbitrary constants. If is an embedding parameter and is auxiliary non zero parameter then the so-called zero-order deformation equation is

subject to boundary conditions

where

and when and, then

As the embedding parameter increases from 0 to 1, varies (or deforms) from the initial approximation to the solution. Using Taylor’s theorem and Equation (16), one obtains

in which

Clearly, the convergence of the series (17) depends upon. Assume that is selected such that the series (17) is convergent at, then due to equation (16) we have

For the order deformation problem, we differentiate Equations (13) and (14) w.r.t and then setting and finally dividing it by the deformation equation for is given by

where

Following the HAM and trying higher iterations with the unique and proper assignment of the results converge to the exact solution:

using the symbolic computation software such as MATHEMATICA, MATLAB or MAPLE to solve the system of linear equations, (20), with the boundary conditions (21), and successively obtain

The auxiliary parameter gives the convergence region and rate of approximation for the homotopy analysis method for above problem. For this purpose, the is plotted for above problem. It is obvious from

Pade approximants make up the best approximation of a function in the form of a rational function of a given order. Pade approximation helps us in improving the ac curacy of approximate solution available in the form of a polynomial. Pade approximants are better approximation of a function than its Taylor series, they work even in those cases where Taylor series does not converge. Pade

approximations are also used to enlarge the interval of convergence of approximate series solution [

over for the Pade approximant is shown in

of degree and at, , , , obtained as

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difference between and is also invisible on this scale.

In this part we discuss the graphs for the variation of the horizontal velocity profiles and shear stress at the wall with distance from the wall for different values of Reynolds number, local non-Newtonian parameter, fluid parameter, homotopy parameter and time.

Figures 8 and 9 describe the variation of the horizontal velocity profiles with for several values of by keeping, , and fixed.

the bottom wall, with increase in fluid parameter, horizontal velocity profiles shows decreasing trend.

shows that for mass injection at the bottom wall, with increase in, horizontal velocity profiles shows increasing trend in magnitude but have negative values. From

suction at the top wall, with increase in, increases at all points and the reverse behavior is observed. Figures 14 and 15 describe the variation of the horizontal velocity profiles with for several values of time, for fixed values of, , and. Figures 14 and 15 are plotted for negative value of. From

fluid material parameters and enhance the magnitude of the velocity profile. In Figures 8 to 15 it is observed that the behavior of suction is the reverse of the injection in all the cases, which is a confirmation for the validity of our results. Graphs from 8 to 15 are plotted for large values of the parameter, and, because for small values it is observed that the curves of different profiles overlaps and behavior is not clear, whether it is increasing or decreasing.