Unsteady Incompressible Couette Flow Problem for the Eyring-Powell Model with Porous Walls

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 and interval of convergence is also found. -curve 


Introduction
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][2][3][4][5], Asghar et al. [6], Khan et al. [7,8], Cortell [9,10], Ayub et al. [11-13], Ariel et al. [14], Rajagopal [15][16][17], Erdogan [18], Siddiqui and Kaloni [19] and Fetecau [20].Couette flow is an important type of flow in the history of fluid mechanics.Researchers have deep interest in this flow and they study it in many ways.Some important studies about this flow are as follows: Fang [21] studied Couette flow problem for unsteady incompressible viscous fluid bounded by porous walls.Khaled and Vafai [22] considered Stokes and Couette flows due to an oscillating wall.Asghar et al. [23] discussed unsteady Couette flow in a second grade fluid with variable material properties.Hayat et al. [24] examined the axial Couette flow problem of an electrically conducting fluid in an annulus.Hayat and Kara [25] studied Couette flow of a third-grade fluid with variable magnetic field.Seth et al. [26] presented Couette flow problem for a porous channel.Bhaskara and Bathaiah [27] have analyzed Couette flow problem for flow through a porous straight channel with MHD and Hall effects.Das et al. [28] considered unsteady Couette flow problem in a rotating system.Ganapathy [29] presented a note on the oscillatory Couette flow in a rotating system.Guria [30,31] discussed Couette flow problem for rotating and oscillatory flow.Sigh [32] found a periodic solution for oscillatory Couette flow.
The Eyring-Powell model [33] although more mathematically complex, has certain advantages over the Second grade, Maxwell, Power-law and Micropolar fluid models.Eyring-Powell model is derived from the kinetic theory of liquids rather than the empirical relations.It correctly reduces to Newtonian behavior for low and high shear stress.Recently, Eldabe et al. [34] and Zueco and Beg [35] discussed the non-Newtonian fluid flow under the effect of couple stresses between two parallel plates using Eyring-Powell model.Prasad et al. [36] studied momentum and heat transfer of a non-Newtonian Eyring-Powell fluid over a non-isothermal stretching sheet.Patel and Timol [37] presented a numerical treatment of MHD Eyring-Powell fluid flow.Sirohi et al. [38] studied Eyring-Powell fluid flow past a 90˚ wedge.Javed et al. [39] discussed flow of an Eyring-Powell non-Newtonian fluid over a stretching sheet.Noreen and Qasim [40] analyzed peristaltic flow of MHD Eyring-Powell fluid in a channel.
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][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][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 [59].Our goal is to make the maximum error as small as possible.For this purpose, Pade approximants and Taylor polynomials are found.The graphs of the error for Pade approximants and Taylor polynomials are plotted and it is observed that maximum absolute error occurs at the end point The graphs for the horizontal velocity profile and shear stress at the wall for injection/suction are drawn and discussed in detail.The tables for the initial slope and wall shear stress are also constructed and discussed.More significantly, the series solution clearly demonstrates how various physical parameters play their part in determining properties of the flow.

Mathematical Description of the Problem
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 w 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 0 .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 [33] where  is the dynamic viscosity,  and c are the characteristics of the Eyring-Powell model.Taking the second order approximation of the function 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 0 y  , top wall is located at y h  and 0 U is the velocity at the upper wall.Equations ( 3) and ( 4) can be non-dimensionalized by defining 2 0 , , and .

 
, f Y T is the reduced stream function and prime denotes ordinary derivative w. r. t Y .When 0 T  , Equation (8) becomes where is some arbitrary unknown function of .

 
T  T

Analytic Solution
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 1 and 2 are arbitrary constants.If C C   0,1 p  is an embedding parameter and 1 is auxiliary non zero parameter then the so-called zero-order deformation equation is subject to boundary conditions , where , ; , ;  , ;   , ; , ; 1 , ; , ; , and when and , then As the embedding parameter increases from 0 to 1, varies (or deforms) from the initial approximation p  , ; Using Tay-lor's theorem and Equation ( 16), one obtains Clearly, the convergence of the series ( 17) depends upon 1 .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 , , 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 MA-THEMATICA, MATLAB or MAPLE to solve the system of linear equations, (20), with the boundary conditions (21), and successively obtain

Convergence of the Analytic Solution 5. Pade Approximation
The auxiliary parameter 1 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 Figure 1 that the range for the admissible values for 1 is 1 .The solution series converges in the whole region of and T for or 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 [61].A standard MATHEMATICA routine can be used to find Pade approximant for the function  is so small as to be invisible on this scale.The graph of the error .       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 , m  and T , 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.shows that for mass suction at the top wall, with increase in

Tables
Here Tables 1-2 are prepared for the variation of the initial slopes and dimensionless shear stress at the wall .These results are obtained for different values of 1 laying in the interval of convergence, for different order of approximations.

 
The diagonal Pade approximants can be used to investigate the mathematical behavior of the solution   f Y to determine the initial slope .It can be seen from  It can be seen from Table 2 that for mass injection at the bottom wall, for a fixed value of and

R
Y is so small as to be invisible on this scale   0,1 .• We observe that the maximum absolute error for Pade approximant and Taylor approximations occur at the end point 1 Y  .• It is observed that increase in the degree of Taylor polynomial increases the maximum absolute error.• For positive and negative values of local non-Newtonian parameter  , the variation of the horizontal velocity profiles is same.
• The fluid material parameters m and  enhance the magnitude of the velocity profile.• It is noted that the mass transfer has a dominant effect on the velocity profile and in all cases behavior of suction is the reverse of the injection.• The curves of the velocity profile for small values of m ,  and T overlaps and behavior is not ex-

R
with increase in  .
Figure 1.-curve for the stream function 1 

Figure 2 2 . From Figure 2
Figure 2 depicts the graph of   f Y and its Pade ap-

Figure 2 .
Figure 2. The graph of and its Pade approximation

Figure 4 4 P
Figure 4 illustrates that the difference between   f Y and   4 P Y is invisible on this scale.Figure 5 indicates the graph of the error

1 .
the Taylor approximation .It is observed that the largest absolute error occur at the end point, Figure6describes that the

Figure 10 1 
shows that when there is mass injection at the bottom wall, with increase in but have negative values, an inverted behavior is observed, which is consistent with what we expected.Figures 12 and 13 illustrate the variation of the horizontal velocity profiles   ,T Tf  Y with Y for several values of time , for fixed values of T  , , e and .Figures 12 and 13 are plotted for positive value of m R  .

Figure 12 showsFigure 10 .Figure 11 .Figure 12 . 1 Figure 13 .Figure 14 .
Figure 10.The graph of the horizontal velocity profiles    Tf Y T , with for several values of Y  and

Figure 16 and 1 .
17 elucidate the variation of the shear stress at the wall with the parameter Figure 16 is for mass injection e at the bottom wall and Figure 17 is for mass suction e

Figure 16
shows that with increase in , shear stress at the wall increases at all points for all values of m 0,T  Tf   and have positive values.

Figure 17 showsFigure 15 .Figure 16 . 21 TFigure 17 .Figure 18 .Figure 19 .
Figure 15.The graph of the horizontal velocity profiles    Tf Y T , with Y for several values of , for −ve value of

Figure 20 .
Figure 20.The graph of the shear stress at the wall   T and have positive and negative values both for all values of time .In Figures 16 to 21 it is observed that shear stress for suction has reverse behavior of injection.

5 ,RFigure 21 .
Figure 21.The graph of the shear stress at the wall    Tf T 0, with for several values of T  and e R 5   .

1 ,
with the increase in the shear stress at the wall increases for all values of 0 .It is observed that with the increase of mass injection shear stress at the wall also increases.It is also noted that for fixed value of , 1 , e and m , with the increase inT  R shear stress at the wall increases.For all the parameters there is an increase is observed.
In this study, a series of solutions for the horizontal velocity field of unsteady incompressible Couette flow with Eyring-Powell model are constructed.The results are discussed under the effects of parameters , m  , 1 and e through graphs and tables.We have following observations about the effects of pertinent parameters in the flow field on the horizontal velocity, shear stress at the wall and on initial slope of  R   f Y .• The solution series converges in the whole region of Y and T We have considered the general Pade and Taylor approximations of   f Y .The polynomials of the rational approximations are given in analytic form.• We note that the difference between the HAM solution   f Y and Pade approximate solution     2,2 plainable.•For mass suction and injection at the bottom and top wall shear, stress at the wall increases in all cases but has opposite sign.Initial slope of  f Y for Pade approximants ,

Table 1
that for a fixed value of e ,