The One Step Optimal Homotopy Analysis Method to Circular Porous Slider

An incompressible Newtonian fluid is forced through the porous of a circular slider which is moving laterally on a horizontal plan. In this paper, we introduce and apply the one step Optimal HomotopyAnalysisMethod one step OHAM to the problem of the circular porous slider where a fluid is injected through the porous bottom. The effects of mass injection and lateral velocity on the heat generated by viscous dissipation are investigated by solving the governing boundary layer equations using one step optimal homotopy technique. The approximate solution for the coupled nonlinear ordinary differential equations resulting from the momentum equation is obtained and discussed for different values of the Reynolds number of the velocity field. The solution obtained is also displayed graphically for various values of the Reynolds number and it is shown that the one step OHAM is capable of finding the approximate solution of circular porous slider.


Introduction
An interesting subject in mathematical physics is the study and analysis of flow between plates 1-6 .An analytical overview of study of porous bearing has been carried out by Morgan and Cameron in 3 .Gorla 7 discussed the fluid dynamical and heat transfer of the circular porous slider bearing.The study of the effects of the Reynolds number on circular porous slider has been investigated in 8 by using the Variational Iteration Method VIM which is one of the semi analytical methods.The fluid dynamics in a slider bearing have been discussed in 9 by using the series expansion and asymptotic expansion.Wang 9 , in fact, discussed the numerical solution for the porous slider for the large Reynolds number.As it is well known the numerical methods such as finite difference and finite element are time consuming and may be difficult due to stability constraints.Toward this end, in this paper, we introduce and apply an effective method so-called one step Homotopy Analysis Method Our paper is organized as follows.In Section 2, we present description of conservation mass and momentum density Navier-Stokes equations and also transformation.In Section 3, we have introduced the one step OHAM to nonlinear system of equations.In Section 4, solutions are given to illustrate capability of one step OHAM.Finally, in Section 5, we give the conclusion of this study.

Formulation
In this paper, the flow field due to a circular porous slider Figure 1 is calculated by using one step OHAM.A fluid of constant density is forced through the porous bottom of the slider and thus separates the slider from the ground.An incompressible fluid is forced through the porous wall of the slider with a velocity W. Figure 2 shows the slider which is fixed at the plane z d, with a viscous fluid injected through it.The base is the plane at z 0, which is moving in the x-direction with velocity U.For detail, please see 7, 24 .
As the gap d is small, it can be assumed that both planes are extended to infinity 24 .Considering the u, v, and w to be the velocity components in the direction x, y, and z, respectively, the conservation mass and conservation momentum density Navier-Stokes equations are as follows: where ρ is density of fluid, ν is kinematic viscosity, and P is pressure.The boundary conditions are as follows: where U is velocity of the slider in lateral and longitudinal direction and W is velocity of fluid injected through the porous bottom of the slider.For transforming 2.2 -2.4 , the following equations are defined 7, 8, 24 : w −2Wh η .

2.9
By substituting 2.7 -2.9 into Navier-Stoks equations 2.2 -2.4 , it can be obtained that 7, 8, 24 where R Wd/ν is the cross-flow Reynolds number, A and K are constants which will have to be determined.The boundary conditions for the transformation are as follows:

2.13
The series solution for small values of R was obtained by Wang in 9 .Gorla 7 solved nonlinear equation 2.10 and 2.11 by using the fourth-order Runge-Kutta method together with the shooting method.

One Step Optimal Homotopy Analysis Method
To illustrate the basic idea of the one step Optimal Homotopy Analysis Method, we consider the system of nonlinear differential equation where L i are linear operators and N i are nonlinear operators.The optimal φ i x; q : Ω × 0, 1 → R which satisfies

3.4
Here η ∈ Ω and q ∈ 0, 1 is an embedding parameter, H i q are nonzero auxiliary functions for q / 0, and H i 0 0. For q 0 and q 1

3.5
When q 0 and q 1, φ i η; 0 u i,0 η and φ i η; 1 u i η , respectively.The zeroth-order problem u i,0 η is obtained from 3.2 and 3.4 with q 0 giving The auxiliary functions H i q are chosen in the form where C i,j are constants.The 3.7 , in fact, is a simple case of auxiliary function H τ, q in 25, 26 .Marinca and Heris ¸anu 25, 26 proposed the auxiliary function H τ, q that has the following form: where C i,1 , C i,2 , . .., C i,m−1 can be constants and the last value C i,m τ can be a function depending on the variable τ.To get an approximate solution, φ i η; q, C i,j , i, j 1, 2, . . ., n are expanded in a Taylor's series about q as u i,j η; C i,j q j , j 1, 2, 3, . . . .

3.9
Substituting 3.9 into 3.4 and equating the coefficient of like powers of q, the first-and second-order problems are given as 21 and the general governing equations for u i,j η are given as 21 where N i,j−k u i,0 η , u i,1 η , . . ., u i,j−k η is the coefficient of q j−i in the expansion of N φ x; q about the embedding parameter q and N i φ i η; q, C i,j N i,0 u i,0 η j≥1 N i,j u i,0 , u i,1 , . . ., u i,j q j .

3.12
It can be seen in the number of papers that the convergence of the series 3.9 depends upon the auxiliary constants C i,1 , C i,2 , C i,3 , . . .17-21 .If the series is convergent at q 1, then which m denotes the mth order of approximation.Substituting 3.13 into 3.1 gives the following expression for the residual:

3.14
If R i η; C i,j 0, then u i η; C i,j are the exact solutions of nonlinear system differential equations.For the determination of auxiliary constants C i,j , the least squares can be used.Consider It is to be noted that, at the first order of approximation, the square residual error Δ i,1 only depends on C i,1 .To obtain the optimal value of Δ i,1 , we need to solve the following system of nonlinear algebraic equation: 0, i 1, 2, . . ., n.

3.16
For the second-order approximation, the square residual error Δ i,2 are functions with respect to C i,1 and C i,2 .The values of C i,1 have been obtained.To obtain the optimal value of Δ i,2 , we need to solve the following system of nonlinear algebraic equations: dΔ i,2 dC i,2 0, i 1, 2, . . ., n.

3.17
By repeating the above process, the square residual error Δ i,n will contain only the unknown convergence-control parameter C i,n .To obtain the optimal value of square residual error Δ i,n , we should solve the following system of nonlinear algebraic equations: dΔ i,n dC i,n 0, i 1, 2, . . ., n.

3.18
Contrary to Marinca's approach which requires the solution of a set of m nonlinear algebraic equation for m unknown convergence-control parameters C 1 , C 2 ,. .., C m , in one step OHAM the square residual error is minimized at each equation of system and each order so as to obtain the optimal convergence-control parameter only one by one.In fact, it is needed to solve only one nonlinear algebraic equation to obtain the C i,n at each order of approximation.An advantage of one step OHAM is that it is easy to implement and obtain high order of approximation with less CPU time 22 .The disadvantage of the OHAM is the need to solve a set of coupled nonlinear algebraic equations for the unknown convergencecontrol parameters C 1 , C 2 , C 3 , . ..,C m which will be obtained from relation 3.9 .For low order of m, the nonlinear algebraic system can be easily solved but for large m it is more difficult.Therefore, the necessary CPU time increases exponentially 22 .

Application of One Step OHAM
According to OHAM formulation described in Section 3, we start with

4.1
We can easily choose the initial approximation as By applying the OHAM, we can obtain components of OHAM series solution 3.9 .By substituting the solution obtained into 2.10 and 2.11 , we can obtain the following residual function as

4.3
The square residual errors at the m order of approximation are defined by 22 where For m 1, it is found that  In Tables 1, 2, and 3, we have calculated the convergence-control parameters C i,j i, j 1, . . ., n and square residual error Δ i,j for various orders of n at R 0.01, R 0.1, and R 1 by using one step OHAM introduced in Section 3. It is to be noted that the square residual error decreases quickly as the order of approximation increases.In Table 4, we have calculated the residual functions Eh and Ef for various t ∈ 0, 1 and R 0.01 and R 0.1.It can be seen that the errors are very small.

4.5
In Tables 5 and 6, we have compared the values of h 0 , h 1 , h 0 and h 1 between the solution obtained by using one step OHAM and solution obtained by using fourth-order Rung-Kutta method in 7 .
In Table 7, we have compared the values of f 0 and f 1 between the solution obtained by using one step OHAM and solution obtained by using fourth-order Runge-Kutta method in 7 .
In Figure 3, we have displayed the vertical velocity h η for various Reynolds number.It can be seen that h η increases within the gap width as the Reynolds number increases.In Figure 4, we have shown the lateral velocity h η .It is clear from Figure 4 that for increasing the Reynolds number R, the h η increases whereas the position of maximum velocity tends to move closer to the moving wall.In Figure 5, it can be seen that in the case of R 0.01, the lateral velocity f η is linear.These results are consistent with those obtained in 7 .

Conclusion
In this paper, the one step Optimal Homotopy Analysis Method one step OHAM has been successfully introduced and applied for solving the problem of circular porous slider.The influence of the Reynolds number has been discussed through graphs.The graphical behavior of f, h, and h for different values of the Reynolds number small and big Reynolds number is presented graphically for fifth-order approximation solution using one step OHAM.In optimal homotopy asymptotic method OHAM , the control and adjustment of the convergence of the series solution using the control parameters C i 's are achieved in a simple way.A disadvantage of OHAM is that it is necessary to solve a set of nonlinear algebraic equations with m unknown convergence-control parameters C 1 , . . ., C m and this is time consuming, specially for large m.In contrast to OHAM, in one step OHAM introduced in this paper, algebraic equations with only one unknown convergence-control parameter at each level should be solved.In fact, the one step OHAM is easy to implement and obtain high order of approximation with less CPU time.

Figure 3 :η 1 R = 5 R = 10
Figure 3: Effect of the various Reynolds number on vertical velocity profile h η .

Figure 4 :
Figure 4: Effect of the various Reynolds number on lateral velocity profile h η .

Figure 5 :
Figure 5: Effect of the various Reynolds number on lateral velocity profile f η .
η Vertical velocity profile h(η) for various Reynolds number

Table 4 :
The values of residual functions given by 6 terms of one step OHAM.

Table 5 :
Wall gradients of vertical functions for various Reynolds number.
For m 2, the square residual error Δ 2,1 and Δ 2,2 are only dependent C 1,2 and C 2,2 since C 1,1 and C 1,2 are known.Thus, the optimal values of C 1,2 and C 2,2 are obtained by solving the following system of equations: In this approach, the optimal values of convergence-control parameters C 1,1 , C 2,1 , C 1,2 , . . .are obtained one by one until an accurate enough approximation 22 .

Table 6 :
Wall gradients of vertical functions for various Reynolds number.

Table 7 :
Wall gradients of lateral functions for various Reynolds number.