Optimal Auxiliary Functions Method for viscous flow due to a stretching surface with partial slip

1 Introduction

The study of boundary layer viscous flow due to a stretching surface is very important because of its several engineering applications such as: crystal growing, drawing of elastic films, the heat treated materials traveling between a feed roll and the wind-up roll or a conveyor belt poses the features of a moving continuous surface. There exist situations where there may be a partial slip between the fluid and the boundary. In the last few decades, nonlinear problems are widely used as models to describe complex physical phenomena in various scientific field of science. The boundary layer flow over a continuously stretching surface was first studied by Sakiadis [1, 2]. Crane [3] found a closed form exponential solution for the planar viscous flow of a linear stretching case. Sparrow et al. [4, 5] studied a number of flow problems taking the velocity slip into account. In [6] is given an algorithm for the moving boundaries and for the flow of a fluid past a rotating disk. Rajagopal et al. [7] obtained numerical solution for the flow of viscoelastic second order fluid past a stretching sheet. Ariel [8] and Anderson [9] have reported the analytical closed form solutions of the second grade fluid and of the fourth order nonlinear differential equations arising due to the MHD flow. Wang [10] studied axisymmetric cases for the viscous flow. Mirogolbabei et al. [11] considered a number of boundary layer flows induced by the axisymmetric stretching of a sheet. The unsteady axisymmetric flow and heat transfer of a viscous fluid is treated by Sajid et al. [12]. Also, the analytical solution for these problems has been considered by Sajid et al. in [13], Miklavčič and Wang [14] first analyzed properties of the flow to a shrinking sheet with suction. Many other researchers denoted themselves on investigating these problems [15, 16, 17, 18, 19].

In general in fluid mechanics of viscous fluids, by means of similarity transformations, the set of partial differential equations are reduced to that of ordinary differential equations. Much attention of the researchers was focused in obtaining the analytical solutions for the nonlinear problems. But, in general, the nonlinear problems cannot be solved analytically by means of traditional methods. For the weakly nonlinear problems, many methods exist for approximating the solutions. The most widely applied method by researches is perturbation technique [20] which is based on the existence of small parameters. Others methods for approximating the solutions of nonlinear problems are the modified Lindstedt-Poincare method [21], the Adomian decomposition method [22], the parameter-expansion method [23], the optimal homotopy perturbation method [24, 25, 26], the optimal homotopy asymptotic method [27, 28, 29].

Recently, several methods has been used for solving different nonlinear differential equations such as: traveling-wave transformation method [30], Cole-Hopf transformation method [31].

The objective of the present paper is to propose a new and accurate approach to nonlinear differential equation of viscous fluid due to a stretching surface with partial slip, using an analytical technique, namely the optimal auxiliary functions method. The motivation of our approach is to present a new procedure that is effective, explicit and very accurate for nonlinear approximations rapidly convergent to the exact solution using only one iteration. This procedure must provide a rigorous way to control and adjust the convergence of the approximate solutions using some convergence-control parameters. The proposed method is easy, concise and can be applied to other nonlinear problems. The validity of our procedure, which does not imply the presence of a small or large parameter in the equation or into the boundary / initial conditions, is based on the construction and determination of the auxiliary functions, combined with a convenient way to optimally control the convergence of the solutions. The efficiency of the proposed procedure is proven when an accurate solution is explicitly analytically obtained in an iterative way after only one iteration. Our procedure can be applied to a variety of engineering domains as [32, 33, 34, 35, 36, 37, 38, 39, 40]. The validity of this method is demonstrated by comparing the results obtained with the numerical solutions.

2 The governing equations

Consider the three dimensional flow of a viscous fluid bounded by a stretching surface. If (u, v, w) is the velocity components in the Cartesian directions (x, y, z) respectively, then the continuity and steady constant property Navier-Stokes equations for viscous fluid flow are [7], [10, 13]:

where p is the pressure, ρ is the density and ν is the kinematic viscosity. If a is the stretching constant, a > 0, W is the suction velocity and m is a parameter describing the type of stretching, then the velocity components on the stretching surface are:

It is known that for m = 1, we have planar stretching case, while for m = 2 we have axisymmetric stretching case. In order to simplify the governing equations we use the similarity transform [10]:

where prime denotes the differentiation with respect to η. Eq. (1) is identically satisfied and Eq. (4) becomes

with C a constant. From the Eqs. (2) and (3) we deduce that:

as there is no lateral pressure gradient at infinity. If N denotes a slip constant, then on the surface of the stretching sheet, the velocity slip is assumed to be proportional to the local sheer stress [41]:

From the similarity transform (6) we obtain

with λ=ρNaν > 0 a non-dimensional parameter indicating the relative importance of partial slip. If λ = 0 there is no slip. Given a suction velocity of − W on the stretching surface, we have the boundary condition

where α=Wmaν is a non-dimensional constant which determines the transpiration rate at the surface and α < 0 if injection from the surface occurs, α > 0 for suction and α = 0 for an impermeable sheet. Also, since there is no lateral velocity at infinity, we have yet the condition

3 Basic ideas of optimal auxiliary functions method

Eq. (8) with the initial / boundary conditions (10), (11) and (12) can be written in a more general form as [42, 43]:

where L is a linear operator, g is a known function and N is a given nonlinear operator, η denotes independent variable and Φ(η) is an unknown function. The initial / boundary conditions are

An exact solution for strongly nonlinear equation (13) and with initial / boundary conditions (14) are frequently scarce. To find an approximate analytic solution of Eqs. (13) and (14), we assume that the approximate solution can be written in the form with only two components:

where the initial approximation Φ₀(η) and the first approximation Φ₁(η, C_i) will be determined as follows. Substituting Eq. (15) into Eq. (13), it results in:

The initial approximation Φ₀(η) can be determined from the linear equation

and the first approximation Φ₁(η, C_i), from the following equation

Now, the nonlinear term from Eq. (18) is expanded in the form

In order to avoid the difficulties that appears in solving of the nonlinear differential equation (18) and to accelerate the rapid convergence of the first approximation Φ₁(η, C_i) and implicit of the approximate solution Φ(η), instead of the last term arising into Eq. (18), we propose another expression, such that the nonlinear differential equation (18) can be written as a linear differential equation taking into account the Eq. (19), in the form:

where A₁ and A₂ are two arbitrary auxiliary functions depending on the initial approximation Φ₀(η) and several unknown parameters C_i and C_j, i = 1, 2, …, p, i = p + 1, p + 2, …, s.

The auxiliary functions A₁ and A₂ (called optimal auxiliary functions) are not unique, and are of the same form like Φ₀(η) or of the form of the N[Φ₀(η)] or combinations of the forms of Φ₀(η) and N[Φ₀(η)].

For example, if Φ₀(η) or N[Φ₀(η)] contain polynomial functions, then A₁[Φ₀(η), C_i] and A₂[Φ₀(η), C_j] are sums of polynomial functions. If Φ₀(η) or N[Φ₀(η)] contain exponential functions, then A₁ and A₂ would be sums of the exponential functions. If Φ₀(η) or N[Φ₀(η)] contain trigonometric functions, then A₁ and A₂ would be sums of the trigonometric functions, and so on. In all these sums, the coefficients of the polynomial, exponential, trigonometric and so on functions, are the parameters C₁, C₂, …, C_s.

If in a special case N[Φ₀(η)] = 0, then it is clear that Φ₀(η) is an exact solution of Eqs. (13) and (14). The unknown parameters C_i and C_j can be optimally identified via different method such as: the Galerkin method, the least square method, the collocation method, the Ritz method, the Kantorovich method or by minimizing the square residual error, using:

where

a and b are two values depending on the given problem.

The unknown parameters C_i, C_j can be optimally identified from the equations

With these parameters C_i, C_j known, by this novel approach, the approximate solution (15) is well determined.

Our procedure proves to be a powerful tool for solving nonlinear problems not depending on small or large parameters. It should be emphasize that our method contains the optimal auxiliary functions A₁ and A₂ which provides us with a simple way to adjust and control the convergence of the approximate solutions after only one iteration. Also, it is remarkable that a nonlinear differential problem is transformed into two linear differential problems.

4 Application of OAFM to viscous fluid given by Eqs. (8), (10), (11) and (12)

We introduce the basic ideas of the OAFM by considering Eq. (8) with the initial / boundary conditions given by Eqs. (10), (11) and (12). We choose the linear operator L by the form [43]:

We can choose the linear operator in the form L(Φ(η)) = Φ″′ + K²Φ′, and so on, where K > 0 is an unknown parameter.

Eq. (17) becomes (g(η) = 0):

with the solution

The nonlinear operator N(Φ(η)) is obtained from Eqs. (8) and (25):

By means of the Eqs. (27) and (28) it is obtain

where

Having in view Eqs. (20), (25) and (29), the first approximation is obtained from the equation:

with the initial / boundary conditions

We have the freedom to choose the optimal auxiliary functions A₁ and A₂ in the following forms:

or

or yet

and so on.

If the auxiliary functions A₁ and A₂ are given by Eqs. (33) and (34) then Eq. (20) may be written as:

with the initial / boundary conditions given in Eq. (32), whose solution is:

where

If we consider m = 1 and β = 0 (the planar stretching case for impermeable sheet) into Eq. (29) then we have N[Φ₀(η)] = 0 and therefore we can obtain the exact solution of the equation:

which is:

where K is obtained from the equation:

Beyond this remarkable case, the approximate analytic solution of the Eqs. (8), (10), (11) and (12) can be obtained from Eqs. (15), (27) and (40).

5 Numerical examples

We illustrate the accuracy of OAFM by comparing obtained approximate solutions with the numerical integration results obtained by means of a fourth-order Runge-Kutta method in combination with the shooting method. In all cases, the unknown parameters are optimally identified via Galerkin method. For this, we use the following eleven weighted functions f_i given by [41]:

where γ, δ, K₁, K₂, K₃ and K₄ are unknown parameters.

The parameters K, K₁, K₂, K₃, K₄, γ, δ, C₁, …, C₆ are determined from equations (Galerkin method):

where the residual R(m, α, λ, η) is given by Eq. (23):

and Φ(η) is given by Eq. (15) with the initial / boundary conditions (10)-(12).

In Table 4 are presented the values of Φ″(0) and Φ(∞) for stretching flow with suction sheet.

From Tables 1-4 we deduced that there exist an excellent agreement between the numerical results and the results obtained by means of OAFM.

On the other hand, considering the effect of slip parameter on the velocity Φ(η) in both flows, Figures 1-4 have been displayed. Figure 1 shows the variation of Φ for planar flow and impermeable sheet. In Figure 2 the variation of Φ for planar flow and suction sheet is plotted.

Figure 1

Variation of Φ by increasing the slip parameter λ for planar flow and impermeable sheet (m = 1, α = 0): — numerical solution; …… OAFM solution

Figure 2

Variation of Φ by increasing the slip parameter λ for planar flow and suction sheet (m = 1, α = 3): — numerical solution; …… OAFM solution

Figure 3 shows the variation of Φ for axisymmetric flow and impermeable sheet. Figure 4 shows the variation of Φ for axisymmetric flow and suction sheet. It is clear that the velocity components decrease with an increase in the slip parameter for all cases.

Figure 3

Variation of Φ by increasing the slip parameter λ for axisymmetric flow and impermeable sheet (m = 2, α = 0): — numerical solution; …… OAFM solution

Figure 4

Variation of Φ by increasing the slip parameter λ for axisymmetric flow and suction sheet (m = 2, α = 3): — numerical solution; …… OAFM solution

In Figures 5-7 the planar cases for every value of slip parameter λ have been plotted and in Figures 8-10 the stretching cases for different values of λ have been plotted. It is evident that the velocity is less for the axisymmetric flow when compared with the planar case.

Figure 5

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 1): — numerical solution; …… OAFM solution

Figure 6

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 5): — numerical solution; …… OAFM solution

Figure 7

Variation of Φ by increasing the coefficient α for planar flow (m = 1, λ = 10): — numerical solution; …… OAFM solution

Figure 8

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 1): — numerical solution; …… OAFM solution

Figure 9

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 5): — numerical solution; …… OAFM solution

Figure 10

Variation of Φ by increasing the coefficient α for axisymmetric flow (m = 2, λ = 10): — numerical solution; …… OAFM solution

Finally, the residual functions obtained for the approximate analytic solutions given by Eqs. (47)-(58) are plotted in Figures 11-22.

Figure 11

The residual R(1, 1, 0, η) for Eq. (47) obtained by OAFM

Figure 12

The residual R(1, 5, 0, η) for Eq. (48) obtained by OAFM

Figure 13

The residual R(1, 10, 0, η) for Eq. (49) obtained by OAFM

Figure 14

The residual R(1, 1, 3, η) for Eq. (50) obtained by OAFM

Figure 15

The residual R(1, 5, 3, η) for Eq. (51) obtained by OAFM

Figure 16

The residual R(1, 10, 3, η) for Eq. (52) obtained by OAFM

Figure 17

The residual R(2, 1, 0, η) for Eq. (53) obtained by OAFM

Figure 18

The residual R(2, 5, 0, η) for Eq. (54) obtained by OAFM

Figure 19

The residual R(2, 10, 0, η) for Eq. (55) obtained by OAFM

Figure 20

The residual R(2, 1, 3, η) for Eq. (56) obtained by OAFM

Figure 21

The residual R(2, 5, 3, η) for Eq. (57) obtained by OAFM

Figure 22

The residual R(2, 10, 3, η) for Eq. (58) obtained by OAFM

6 Conclusions

In this work, the Optimal Auxiliary Functions Method (OAFM) is employed to propose an analytic approximate solution to the boundary nonlinear problem of the viscous flow induced by a stretching surface with partial slip. Our procedure is valid even if the nonlinear equations of motion do not contain any small or large parameters. The proposed approach is mainly based on a new construction of the solutions and especially on the auxiliary function. These auxiliary functions lead to an excellent agreement of the solutions with numerical results. This technique is very effective, explicit and accurate for non-linear approximations rapidly converging to the exact solution after only one iteration. Also OAFM provides a simple but rigorous way to control and adjust the convergence of the solution by means of some convergence-control parameters. Our construction is different from other approaches especially referring to the linear operator L depending of an optimal parameter K and to the auxiliary convergence-control function A₁ and A₂ which ensure a fast convergence of the solutions. In a special case OAFM lead to choice of the exact solution. Numerical results are obtained using the shooting method in combination with the fourth-order Runge-Kutta method, using Wolfram Mathematica 9.0 software.

Further important contributions in the area of this paper can be: OAFM allows us to obtain a very good results for both cases: planar and axisymmetric flow for any value of the slip parameter. We point out, also the influence of the coefficient α in the case of impermeable or suction sheet. The values Φ″(0) and Φ(∞) are given for both flow situations, and found in excellent agreement with numerical results.

The proposed method is straightforward, effective, concise and can be applied to other different nonlinear problems.