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Dynamics of Nonlinear Systems

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Research Article | Open Access

Volume 2014 |Article ID 617453 | https://doi.org/10.1155/2014/617453

Vasile Marinca, Remus-Daniel Ene, Bogdan Marinca, "Analytic Approximate Solution for Falkner-Skan Equation", The Scientific World Journal, vol. 2014, Article ID 617453, 22 pages, 2014. https://doi.org/10.1155/2014/617453

Analytic Approximate Solution for Falkner-Skan Equation

Academic Editor: Y. Xia
Received14 Jan 2014
Accepted10 Mar 2014
Published30 Apr 2014

Abstract

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. A very good agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

1. Introduction

It is known that the word “viscoelastic” means the simultaneous existence of viscous and simultaneous elastic responses of a material. Some materials having a viscoelastic behavior are relevant in many fields of study for industrial and technological applications such as polymers, plastic processing, cosmetics, geology composites, paint flow, adhesives, towers generators, accelerators, electrostatic filters, droplet filters, and the design of heat exchanges [1].

Motivated by significant applications of viscoelastic materials, a substantial amount of research works has been invested in the study of nonlinear systems. In 1931, Falkner and Skan [2] have used some approximate procedures to solve boundary-layer equations. Hartree [3] found the numerical solution using a shooting method with (see (7)) as free parameter. The boundary conditions (8) arise in the study of viscous flow past a wedge of angle ; corresponds to flow toward the wedge and corresponds to flow away from the wedge. The special case is called the Blasius equation where the wedge reduced to a flat plate. In [4, 5], it is proved that if then the Falkner-Skan equation (7) with initial conditions (8) admits a unique smooth solution. For there exist two solutions, that is, one withand the other one with . Botta et al. [6] showed that the solution of Falkner-Skan equation is unique for under the restriction . Forced convection boundary-layer flow over a wedge with uniform suction or injection is analyzed by Yih [7]. Asaithambi [8] studied the Falkner-Skan equation using finite difference scheme. In [9], Zaturska and Banks presented a new solution branch in function of parameter . This solution branch is found to end singularity at ; its structure is analytically investigated and the principal characteristics are described. Also the spatial stability of such solutions is commented on. The differential transformation is adopted to investigate the velocity and shear-stress fields associated with Falkner-Skan boundary-layer problem in [10]. A group of transformations is used to reduce the boundary value problem into a pair of initial value problems, which are then solved by means of the differential transformation method. The nonlinear ordinary differential equation is solved using Adomian decomposition method (ADM) by Elgazery [11] such that the condition at infinity was applied to a related Padé approximation and Laplace transformation to the obtained solution. Also ADM is used in [12] by Alizadeh et al. to find an analytical solution in the form of infinite power series. Magnetohydrodynamic effects on the Falkner-Skan wedge flow are studied by Abbasbandy and Hayat in [13]. The same authors used Hankel-Padé and homotopy analysis method for the derivation of the solutions [14]. From a fluid mechanical point of view, the pathophysiological situation in myocardical bridges involves fluid flow in a time dependent flow geometry caused by contracting cardiac muscles overlying an intramural segment of the coronary artery. A boundary-layer model for the calculation of the pressure drop and flow separation is presented in [15] under the assumption that the idealized flow through a constriction is given by near equilibrium velocity profiles of the Falkner-Skan-Cooke family, the evolution of the boundary-layer is obtained by the simultaneous solution of the Falkner-Skan equation and the transient non-Kármán integral momentum equation.

Pirkhedri et al. [16] developed a numerical technique transforming the governing partial differential equation into a nonlinear third-order boundary value problem by similarity variables and then solved it by the rational Legendre collocation method. It used transformed Hermite-Gauss nodes as interpolation points. The steady Falkner-Skan solution for gravity-driven film flow of micropolar fluid is investigated in [17]. The ordinary differential equations are solved numerically using an implicit finite difference scheme known as the Keller-box method. In [18], Lakestani truncated the semi-infinite physical domain of the problem to a finite domain expanding the required approximate solution as the elements of Chebyshev cardinal functions. Yun proposed in [19] an iterative method for solving the Falkner-Skan equation in the form of polynomial series without requiring any differentiations or integrations of the previous iterate solutions. The author suggests a correction method which is compared with the successive differences of the iterations. In [20], Hendi and Hussain considered Falkner-Skan flow over a porous surface taking into account the case of uniform suction/blowing. Stream function formulation and suitable transformations reduce the arising problem to ordinary differential equation which has been solved by homotopy analysis method.

In science and engineering there exist a lot of nonlinear differential equations and even strongly nonlinear problems which are still very difficult to solve analytically by using traditional methods. Many methods exist for approximating the solutions of nonlinear problems, for example, the Adomian decomposition method [21], the modified Lindstedt-Poincare method [22], the parameter-expansion method [23], optimal variational method [24], optimal homotopy perturbation method [25], and so on [26].

The aim of the present paper is to propose an accurate approach to Falkner-Skan equation using an analytical technique, namely, optimal homotopy asymptotic method [2628].

The validity of our procedure, which does not imply the presence of a small parameter in the equation, is based on the construction and determination of the auxiliary functions combined with a convenient way to optimally control the convergence of the solution. The efficiency of the proposed procedure is proved while an accurate solution is explicitly analytically obtained in an iterative way after only one iteration.

2. The Governing Equation

The two-dimensional laminar boundary-layer equations of an incompressible fluid subject to a pressure gradient are [2, 3, 9, 12] where is the pressure gradient, , is the streamwise velocity in the direction of the fluid flow, is the velocity in the direction normal to , is the constant kinematic viscosity, and is the velocity at the edge of the boundary-layer which obeys the power-law relation , (), where is the mean stream velocity and is a constant. The relevant boundary conditions for fixed plate are

A stream function is introduced such that

Equation (2) of continuity is satisfied identically. The momentum equation (1) becomes

Integrating (5) and using similarity variable yield

Substituting (6) into (5) gives the equation of Falkner-Skan in the form with the initial and boundary conditions where is a measure of the pressure gradient and prime denotes derivative with respect to.

3. Fundamentals of the OHAM

In what follows, we consider nonlinear differential equation with boundary/initial condition

In (9),is a linear operator andis a nonlinear operator. In (10),is a boundary operator.

According to the basic ideas of OHAM [2628], one constructs a family of equations

The boundary condition is where,   is an unknown function, is an embedding parameter, and is an auxiliary function such that and for . When increases from 0 to 1, the solution  , changes from initial approximation to the solution . For and it holds that, respectively,

Expanding in series with respect to the parameter, one has

The series (14) contains the auxiliary functionwhich determines their convergence region. For the auxiliary functionwe propose that where and are functions of variable and of a number of unknown parameters . In this paper we consider the th-order approximation in the form

Inserting (14) into (11) we obtain where is the coefficient of in the expansion of about the embedding parameter .

Substituting (14) and (15) into (11) and equating the coefficients of like powers of , we obtain the following linear equations:

At this moment, themth-order approximate solution (16) depends on the functions , . The parameters which appear in the expression of , can be identified optimally via various methodologies such as the least square method, the Galerkin method, and the collocation method. The parameters can be determined, for example, if we substitute (16) into (9), such that the residual becomes

If and are two values from the domain of the problem and , , then the residual (21) must vanish with q—the number of parameters which appear in the expression of the functions ,.

We remark that our procedure contains the auxiliary functions which provides us with a simple but rigorous way to adjust and control convergence of the solution. It must be underlined that it is very important to properly choose the functions which appear in the th-order approximation (16). With these parameters known, the approximate solution is well determined. The parameters are, namely, convergence-control parameters.

4. Application of OHAM to Falkner-Skan Equation

To use the basic ideas of the proposed method, we choose the linear operator whereis the unknown parameter at this moment.

The nonlinear operator is

The boundary conditions are

Equation (18) can be written in the form and has the solution

From (17), (23), and (24) one obtain the expression

Substituting (27) into (28), we obtain

If we consider the first-order approximate solution , (16) becomes where is obtained from (20):

Substituting (27) and (29) into (31) we obtain the equation

There are many possibilities to choose the function which appears into (32). The convergence of the solution and consequently the convergence of the approximate solution given by (30) depend on the auxiliary function . Basically, the shape of should follow the term appearing in (29) which is the product of polynomial and exponential functions. In general, we try to choose the function so that the product from (31) and would be of the same form. In our paper, for example, we can consider only the possibilities and so on, where are unknown parameters. In the following we have four cases.

4.1. Case 1

If the auxiliary convergence-control function has the form then (32) can be written as

Finally, using (27) and solving (35), we determine the first-order approximate solution given by (30) in the form

4.2. Case 2

The auxiliary function has the form In this case, (32) becomes

The first-order approximate solution (30) in this case is obtained from (38) and (27) and can be written as

4.3. Case 3

If the auxiliary function has the form then the first-order approximate solution equation (30) has the form where is given by (36).

4.4. Case 4

In the last case, we consider such that the first-order approximate solution equation (30) becomes where is given by (39).

5. Numerical Examples

In order to prove the accuracy of the obtained results, we will determine the convergence-control parameters which appear in (36), (39), (41), and (43) by means of Galerkin method. Let be the residual within the approximate solution (or ) given by (36), (39), (41), and (43) which satisfies (7):

Since contains the parameters , . The parameters can be determined from the conditions where are linear independent functions, taken as weighting functions. Equations (36) and (39) contain nine unknown parameters: and , , and therefore we consider the following nine weighting functions :

For (41) and (43) which contain the unknown parameters: , , and , , we consider weighting functions ()

In this way, the convergence-control parameters , are optimally determined and the first-order approximate solutions are known for different values of the known parameter .

In what follows, we illustrate the accuracy of the OHAM comparing previously obtained approximate solutions with the numerical integration results computed by means of the shooting method combined with fourth-order Runge-Kutta method using Wolfram Mathematica 6.0 software. Also we will show that the error of the solutions decreases when the number of terms in the auxiliary convergence-control function increases. For some values of the parameter , we will determine the approximate solutions given by (36), (39), (41), and (43) and with the unknown parameters , , and obtained from the system given by (45).

Example 1. In the first case we consider that .
(a) For (36) and from the system (45), following the procedure described above the convergence-control parameters are obtained and consequently the first-order approximate solution (36) can be written in the form

In Tables 1 and 2 we present a comparison between the first-order approximate solution given by (49) and velocity obtained from (49), respectively, with numerical results for some values of variable and the corresponding relative errors.


( ) from (49) Relative error =

0
4/5 0.2543480764 0.2543149422 0.0000331341
8/5 0.8550267840 0.8550621314 0.0000353473
12/5 1.6045273996 1.6043588322 0.0001685673
16/5 2.3963133788 2.3962404320 0.0000729467
4 3.1955002598 3.1953781616 0.0001220981
24/5 3.9954529746 3.9949610905 0.0004918840
28/5 4.7954513976 4.7946972370 0.0007541605
32/5 5.5954513676 5.5946526226 0.0007987450
36/5 6.3954513670 6.3946242000 0.0008271670
8 7.1954513667 7.19442747097 0.0010238957


from (49) Relative error =

4/50.58330481770.58331495140.0000101337
8/50.87609756970.87596514540.0001324243
12/50.97606875610.97595195820.0001167979
16/50.99719207500.99737689780.0001848228
40.99980819860.99946930790.0003388907
24/50.99999257690.99952229750.0004702794
28/50.99999983980.99983315890.0001666808
36/50.99999999950.99988326480.0001167346
80.99999999950.99961749490.0003825046

(b) From (39), obtained by means of the auxiliary convergence-control function given by (37), we obtain the following results for the parameters:

The first-order approximate solution (39) becomes

In Tables 3 and 4 we present some values of stream function (51) and velocity obtained from (51), respectively, for different values of and the corresponding relative errors.


( ) from (51) Relative error =

0
4/50.25434807640.25433337460.0000147017
8/50.85502678400.85495612910.0000706549
12/51.60452739961.60427147790.0002559216
16/52.39631337882.39589021420.0004231645
43.19550025983.19488436750.0006158922
24/53.99545297463.99430629910.0011466754
28/54.79545139764.79369445320.0017569444
32/55.59545136765.59325627780.0021950898
36/56.39545136706.39301634520.0024350218
87.19545136677.19290609450.0025452721


from (51) Relative error =

0
4/50.58330481770.58319818490.0001066327
8/50.87609756970.87607694280.0000206268
12/50.97606875610.97572197530.0003467808
16/50.99719207500.99708769200.0001043829
40.99980819860.99934562220.0004625764
24/50.99999257690.99920257370.0007900032
28/50.99999983980.99931919190.0006806479
32/50.99999999780.99958682000.0004131778
36/50.99999999950.99979705960.0002029398
80.99999999950.99991364810.0000863514

(c) For (41), which depends on the auxiliary convergence-control function given by (40), we obtain

Therefore, the first-order approximate solution for stream function is

In Tables 5 and 6 we present some values of stream function (53) and velocity obtained from (53), respectively, for different values of variable and the corresponding relative errors.


( ) from (53) Relative error =

0