Comparison of OHAM and HPM methods for MHD flow of an upper-convected Maxwell fluid in a porous channel

This paper presents comparison between optimal homotopy asymptotic method and homotopy perturbation method for the solution of two-dimensional steady flow of an incompressible Maxwell fluid over a transpiration channel within a porous medium due to a impulsively moving wall. The results obtained using HPM for injection case fails to be in accordance with the available analytical data.


Introduction
Maxwell fluid is a subclass of non-Newtonian fluid of rate type with property to describe the relaxation time effects.Due to its ability in effectively taking various non-Newtonian effects, it has been the topic of several research Qi and Xu [2007], Hayat and Sajid [2007], Hayat et al. [2008], Jamil et al. [2011], Salah et al. [2011] and Hayat et al. [2011].The governing equations are very complicated and highly nonlinear as compared to those for Newtonian fluids.There are few analytical solutions of the equations involving Newtonian fluids and such solutions become even rarer when equations for non-Newtonian fluids are considered.The aim of the present work is to solve the problem of the steady boundary layer flow of an upper-convected Maxwell fluid in a porous channel with wall transpiration.We have solved the governing nonlinear equation of present problem using the homotopy perturbation method, HPM, He [2005] and optimal homotopy asymptotic method, OHAM, Marinca and Heris ¸anu [2008].We noticed that the solution obtain from the two methods are in a very good agreement for suction case.It is noted from the solution series of homotopy perturbation method that the results in the case of injection fails to be in accordance with available analytical data.

Statement of the problem
Consider the steady MHD two-dimensional flow of an upperconvected Maxwell fluid in a channel with permeable walls.The distance between the channel width, is 2l apart.Porous medium fills the space between the walls of the channel.The fluid motion is generated by suction/injection of the channel walls.The complete set of governing equations of the upper-convected Maxwell fluid consists of the incompressibility conditions ∂u ∂x a e-mail: rozaini@uthm.edu.my where u and v are the velocity components in the x and y−axis, respectively, ρ is the fluid density, λ is the relaxation time, k is the permeability of the porous medium, φ is the porosity, ν is the kinematic viscosity, σ is the electrical conductivity and B 0 is the constant magnetic field in y direction.
The appropriate boundary conditions are: where V 0 is the transpiration velocity.Hayat et al. [2011] has demonstrated that by using the similarity variables: The momentum equation (2) now can be reduced to the following similarity equation: The corresponding boundary conditions (3) reduce to where M is the Hartman number, K is the permeability of the porous medium, Re is the Reynolds number and De is the Deborah number.Must be noted that for Re > 0 corresponds to suction case, Re < 0 for injection and De = 0 for Newtonian fluid.

Solution using the OHAM
According to OHAM by Marinca and Heris ¸anu [2008] the differential equation ( 5) satisfied by the velocity f (η) is decomposed into a linear part L ( f ) and a non-linear part N( f ) and is written in the form: where We now construct a homotopy ψ (η, p) : R × [0, 1] → R which satisfies the family of equations: where p ∈ [0, 1] is an embedding parameter.Expand ψ (η, p) in a Taylor series with respect to p, we obtain (12) where f 0 (η) is the initial approximation.Using the boundary conditions (6) , we choose the initial approximation f 0 (η) as Substituting equation ( 12) into equation ( 11) and equating the coefficients of like powers of p, we obtain the following sets of problems for the first-order and second order problem respectively, as follows: dη 2 = 0, and ( 14) The second order approximate solution f : Hence, the approximate analytical solution (16) yields the following residual R and the functional J, respectively 17) For calculation of constants C 1 and C 2 the method of least squares is employed.

Solution using the HPM
According to the homotopy perturbation method by He [2005], the differential equation ( 5) satisfied by the velocity field f (η) is decomposed into a linear part L( f ) and a non-linear part N( f ) similar to (7)-( 9).We construct a homotopy class h (η, p) : R × [0, 1] → R which satisfies the following equation: where f 0 is an initial approximation to the solution f (η) .Assuming: Taking p as small parameter and taking a power series solution of equation ( 26) in the form: where h k are unknown function of η.Now setting p → 1, equation ( 28) yield the approximate analytical solution of f (η) Substituting equation ( 28) into equation ( 26) and the boundary conditions ( 6) and equate the coefficients of like powers of p we obtain the following sets of problems for the first-order and second order problem respectively, as follows: dη 2 = 0 , and ( 27) In view of equation ( 29) , the second order approximate solution is given in the form The case of Newtonian fluid (De = 0), by fixing K = 0.5 and M = 1, we obtained the approximate solutions of the second order for suction, Re = 20 and injection, Re = −20 cases respectively as follows: The case of Newtonian fluid (De = 0.5), by fixing K = 0.5 and M = 1, we obtained the approximate solutions of the second order for suction, Re = 20 and injection, Re = −20 cases respectively as follows: 261667η 5 − 0.308506η 6 + 0.000695406η 7 −0.109331η 8 − 0.239972η 9 + 0.00497605η 10 +0.00015907η 11 + 0.017362η 12 − 0.0790405η 13 −0.00417921η 14− 0.00692374η 15 + 0.00741998η 16 −0.000278536η 18+ 0.000797702η 20 , (32)  .The result show similar behaviour as that for Newtonian fluid (Fig. 1) and also the results are contrast with the available analytical data (Hayat et al. [2011] Fig. 16 and Fig. 22).Fig. 4 shows the variation of Hartman number H on the non-dimensional velocity f using OHAM with suction case (Fig. 4 (a)) and injection (Fig. 4 (b)).The result show similar behaviour as that for Newtonian fluid (Fig. 2) and also the results are in good agrement with the available analytical data (Hayat et al. [2011]).

Conclusion remark
Approximate analytical solutions for velocity upper-convected Maxwell fluid in a porous channel with wall transpiration dance with the available analytical data.

Fig. 1 .
Fig. 1.Effects of Hartman number M on f in the case of suction Re > 0 (a) and injection Re < 0 (b) using HPM