1. Introduction

The present study deals with the problem of heat transfer for the Couette flow of a third grade fluid between two parallel porous plates. Analytical solutions for the velocity and temperature profiles have been obtained in explicit forms. The related studies in the recent years dealing with third grade fluids are [1-5].

Many models described by nonlinear differential equations of real life problems are difficult to solve either analytically or numerically. Much effort have been made in the literature for different techniques to make the nonlinear model solvable. Among the methods that have been employed to solve nonlinear models are related to Adomian decomposition method (ADM) [6], Variation iterative method (VIM) [7], spectral collocation method [8], homotopy analysis method (HAM) [9], homotopy perturbation method (HPM) [10], optimal homotopy asymptotic method (OHAM) [11], etc.

In the present investigation, we construct analytical approximate solutions of the governing problem of heat transfer for the Couette flow of a third grade between two parallel porous plates using optimal homotopy asymptotic method (OHAM). The results reveal that the OHAM is valid not only for weakly nonlinear equations, but also for strongly nonlinear ones. Effects of the viscoelastic parameter and the suction/injection on velocity and temperature profiles are shown graphically.

2. Mathematical Model

For Couette flow problem, the lower plate is stationary and the upper plate is moving with a constant velocity and the effect constant pressure gradient is neglected. The movement of fluid is solely due to motion of the upper plate. The fluid is brought into motion through the action of viscous stress at the plate. The temperature of the lower plate is maintained at and that of the upper plate at . Mathematical model of the present flow problem are formulated as below:

(1)

(2)

subject to the boundary condition

(3)

(4)

Where are the material moduli, is the sunction/injection parameter, fluid velocity, is the fluid temperature, is the fluid viscosity, is the fluid density, is the specific heat capicity and is the thermal conductivity.

Introducing the dimensionless parameters

(5)

the non-dimensional problem, after dropping the asterisks, Eq. (1), (2) and boundary conditions (3) and (4) becomes

(6)

(7)

(8)

(9)

where , Pr is the Prandtl number, Ec is the Eckert number and λ is the Brinkman number.

3. Basic Idea of the Optimal Homotopy Asymptotic Method (OHAM)

We classifying the equations to solved for velocity u and temperature Θ:

(10)

where , are linear operator, are non-linear operator, are known function and B is a boundary operator.

By means of OHAM we first construct an optimal homotopy which satisfies the following equations:

(11)

(12)

where and is an embedding parameter, is a nonzero auxiliary function for and, and are unknown functions. Clearly, when and it holds that

(13)

(14)

Choose the auxiliary function in the form

(15)

where are constants to be determined. Construct a Taylor's series solution of equations in the form

(16)

where represent and represent . Substituting Eq. (16) into Eqs. (11) - (14) and collecting the same powers of q, and equating each coefficient of q to zero, we obtain

(17)

(18)

(19)

where is the coefficient of , obtained by expanding in series with respect to the embedding parameter .

(20)

The solution of equations (31) and (32) can be approximately obtained in the form

(21)

(22)

The residual follows as

(23)

If , then will be the exact solution. Generally such a case will not happen for nonlinear problems, but we can minimize the functional by the method of lease squares

(24)

where and belong to the domain of the problem. Finally, the unknown constants can be optimally identified from the conditions

(25)

with these known values of , the approximate solutions are well determined.

4. Solutions for Velocity and Temperature

Choosing the linear and nonlinear operators defined by

(26)

(27)

(28)

The correspond boundary conditions are

(29)

(30)

The zeroth order deformation:

(31)

(32)

The first order deformation:

(33)

(34)

The second order deformation:

(35)

(36)

By using the widely applied symbolic computational software MATHEMATICA. It is found that the solutions of Eqs. (31) - (36) are:

(37)

(38)

(39)

(40)

(41)

(42)

In view of Eq. (41), the second order approximation solution for equations (15) and (16):

(43)

(44)

Using Eq. (3), these result into the following residual

(45)

(46)

and the functional is defined as

(47)

(48)

The unknown constant and can be calculated from the conditions

(49)

(50)

The unknown constant and can be calculated from the conditions

(51)

(52)

(a) For fluid velocity, we take α = 0.5. Using the condition (25), we obtain the values

(53)

and

(54)

For no transpiration rate, , therefore from Eq. (43) ; the approximate solution of second order is found to be

(55)

The above mentioned expression is of the same type that was obtained by Siddiqui et al.[5] :

For transpiration rate . Therefore as a particular case, taking suction () and , we obtain , and the approximate analytical solution as

(56)

For injection case () and we obtain

(57)

The general form is given by

(58)

(b) For fluid temperature

Taking the values , , we obtained , , and the approximate solution of second order, in the form

(59)

and the general form

(60)

5. Analysis of Results

Fig. 1 shows the comparison between the present results, obtained by using OHAM and those Siddiqui et al. [5] obtained by employing HPM in non-porous media. We note that, the precision from the solution acquired through the current method, for velocity and temperature pro.les are excellent.

Fig. 2(a) shows the effect of the material constant β on temperature profile, in case of of suction, when and . It is observed from this figure that the fluid temperature increases with increase in the value of β. From Fig. 2 (b), for injection case, it is observed that the behavior of the temperature profile is slow to that in Fig. 2 (a).

Fig. 3 (a) shows the effect of the material constant β for velocity profile in case of suction, when is fixed. It is observed from this figure that the fluid velocity decreases with increase in the value of β. However, in Fig. 3 (b) for injection case, the behavior of the velocity profile is reversed to that in Fig. 3 (a).

Figure 1. Comparison between the results obtained from OHAMand Siddiqui et al. results [5] obtained from HPM in case (a) velocity and case (b) temperature when and

Figure 2. Effect of material constant β on the temperature profiles for the case of suction (a) and injection (b) when and

Figure 3. Effect of material parameter β on the velocity pro.les for the case of suction (a) and injection (b) when

6. Conclusions

Analytical explicit expressions for the velocity and temperature profiles have been obtained for the Couette flow of a third grade fluid between two parallel porous plates, which match well with the previous analytical solutions the HPM solutions [5]. It is found that the Couette flow features dominate as viscoelastic parameter is increased. It is also observed that the effects of the fluid/temperature injection or suction exert a great influence on the general flow pattern, by enhancing or suppressing the influence of the wall into the flow domain.