Abstract
The unsteady viscous flow over a continuously shrinking surface with mass suction is investigated using the optimal homotopy asymptotic method (OHAM). The nonlinear differential equation is obtained by means of the similarity transformation. The dual solutions exist for a certain range of mass suction and unsteadiness parameters. 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
The flow of the Newtonian and non-Newtonian fluids is important for engineers and applied mathematicians because of its several applications in engineering or industrial processes. In the last few decades, these fluids have attracted considerable attention from researchers in many branches of nonlinear dynamical systems in science and technology. The flow over a stretching/shrinking cylinder is an important problem in many engineering processes with applications in industries such as in plastic and metallurgy industries, glass-fiber production, and wire drawing. The pioneering works in the area of the flow inside a tube with time dependent diameter were [1, 2], where Uchida and Aoki and Skalak and Wang studied the internal flow velocity and pressure due to tube expansion or contraction. Miklavčič and Wang [3] investigated the flow over a shrinking sheet, obtaining an exact solution of the Navier-Stokes equations. Ishak et al. [4] reported that injection reduces the skin friction as well as the heat transfer rate at the surface while suction acts in the opposite manner. Fang et al. [5] obtained the exact solution of the unsteady state Navier-Stokes equations. Fang et al. [6] studied the viscous flow over a shrinking sheet by a newly proposed second order slip flow model. The exact solution of the full governing Navier-Stokes equation has two branches in a certain range of the parameters. The problem of unsteady viscous flow over a permeable shrinking cylinder was solved by Zaimi et al. [7] numerically using the shooting method. The effects of suction and unsteadiness parameters on the flow velocity and the skin friction coefficient have been analyzed and presented graphically and the same authors in [8] studied the effects of the unsteadiness parameter and the Brownian motion parameter on the flow field and heat transfer characteristics. Dual solutions are found to exist in certain conditions.
Analytical solutions to nonlinear differential equations play an important role in the study of the unsteady viscous flow over a shrinking cylinder, but it is difficult to find these solutions in the presence of strong nonlinearity. Many new approaches have been proposed to find approximate solutions of nonlinear differential equations. Perturbation methods have been applied to determine approximate solutions to weakly nonlinear problems [9]. But the use of perturbation theory in many problems is invalid for parameters beyond a certain specified range.
Homotopy perturbation method is employed to investigate steady-state heat conduction with temperature dependent thermal conductivity and heat generation in a hollow sphere by Khan et al. [10]. The same method is applied in the study of the effects of temperature distribution and heat transfer from solids of arbitrary shapes in [11]. Another procedure, the Adomian decomposition method, is used to compute the Sumudu transform of some typical functions in [12, 13]. Other methods have been proposed such as the various modified Lindstedt-Poincare method [14], some linearization methods [15], and the optimal homotopy perturbation method [16].
In this paper we consider the unsteady viscous flow over a shrinking cylinder. A version of the optimal homotopy asymptotic method is applied in this study to derive highly accurate analytical expressions of solutions. Our procedure does not depend upon any small or large parameters, contradistinguishing from other known methods. The main advantage of this approach is the control of the convergence of approximate solutions in a very rigorous way. A very good agreement was found between our approximate solutions and numerical solutions, which proves that our procedure is very efficient and accurate.
2. The Governing Equation
In what follows, we assume an unsteady laminar boundary layer flow of a nanofluid over an infinite cylinder or a tube with a time dependent diameter in shrinking motion as shown in Figure 1.
Also we consider the three-dimensional unsteady Navier-Stokes equations for incompressible fluids without body force such that based on the axisymmetric flow assumption and the fact that there is no azimuthal velocity component we have where is the velocity vector, is the fluid density, is the pressure, and is the kinematic viscosity. The diameter of the cylinder is assumed as a function of time with unsteady radius . For a positive value of , the cylinder radius becomes smaller with time, that is, contracting, while, for a negative value of , the diameter becomes larger with time, that is, expanding. In cylindrical polar coordinates and are measured in the radial and axial directions, respectively; (1) and (2) can be written as [5–8]
If we consider the constant mass transfer velocity () and a positive constant, then the boundary conditions are of the following form:
By means of the similarity variables [8] it is clear that , and, on the other hand, (3) is satisfied automatically. Based on the defined velocity components, it is straightforward to derive from (4) that the pressure gradient is a function of and and is independent on , such that, from (4), we obtain or using (7) the pressure may be written as where is the constant of the integration on and is the unsteadiness parameter for the expanding () or contraction () cylinder showing the strength of expansion or contraction. Substituting (7) into (5) and rearranging terms, this becomes with the boundary conditions transformed into the following: where prime denotes differentiation with respect to and is the dimensionless suction parameter.
3. Basic Ideas of the OHAM
Equation (10) can be written in a more general form where is a linear operator and is a nonlinear operator and the boundary conditions (11) in the form
Let be an initial approximation of such as
We point out that the linear operator from (12) and (14) is not unique.
Let us consider the function in the form where denotes an embedding parameter. It follows that the first-order approximate solution can be written as where are arbitrary parameters, which will be determined later. The boundary conditions are
We construct a family of equations [17–21]: with the properties where is an arbitrary auxiliary convergence-control function.
Now, equating only the coefficients of and into (18), we obtain the governing equation of given by (14) and the governing equation on ; that is,
In general, the nonlinear operator from (23) may be written as where the functions and are known and depend on the functions and also on the nonlinear operator, being a known integer number. It is known that the general solution of the nonhomogeneous linear equation (23) is equal to the sum of general solution of the corresponding homogeneous equation and some particular solutions of the nonhomogeneous equation. In what follows, we do not solve (23), but from the theory of differential equations it is more convenient to consider the unknown function in the form or where within expression of from (25) appear linear combinations of some functions , some of the terms which are given by corresponding homogeneous equation and a number of unknown parameters , , being an arbitrary integer number. The same considerations can be made for (26) where and are interchangeable.
4. The Convergence of the Approximate Solution (16)
The convergence of the approximate solution given by (16) depends upon the auxiliary functions , , which appear in (25). There are many possibilities to choose these functions . We try to choose such function so that within (25) the terms are of the same shape as the terms given by (24) [14–18]. The first-order approximate solution also depends on the parameters , . The values of these parameters can be optimally evaluated via various methods: the least-square method, minimization of the square residual error, the Galerkin method, collocation method or the Ritz method, and so on. In this way, it is clear that the first-order approximate solutions given by (16) are well determined. Because the auxiliary functions are not unique, we have freedom to determine multiple solutions for nonlinear differential equations (10) and (11). It should be emphasized that our procedure contains the auxiliary functions , , , which provides us with a simple way to adjust and control the convergence of the approximate solutions.
5. Multiple Approximate Solutions of the Unsteady Viscous Flow by OHAM
The linear operator can be chosen in the following forms: where is an unknown positive parameter and will be determined later.
The initial approximation can be obtained from (14), with boundary conditions
Equation (14) with the linear operators (28) or (29) has the solutions while (14) with the linear operators (30) or (31) has the solutions
The nonlinear operator corresponding to nonlinear differential equation (10) is defined as for linear operator defined by (28).
The same nonlinear operators for the linear operator defined by (29), (30), and (31) are, respectively,
Substituting (33) into (35) it holds that
Now, comparing (24) and (39), one gets
The first approximation can be written in the form where are arbitrary functions. Of course, we have freedom to choose such functions with conditions, obtained from (41):
For example are given by
Taking into consideration only the expression given by (43), from (33), (41), and (16) we obtain the first-order approximate solution of (10) and (11) in the form where are unknown parameters.
Many other approximate solutions can be obtained by means of combinations between initial approximations given by (33) and (34) and the nonlinear operators (36), (37), or (38).
6. Numerical Examples
In order to show the validity and accuracy of the OHAM, we compare previously obtained approximate solutions (46) with numerical integration results obtained by means of a fourth-order Runge-Kutta method in combination with shooting method and the Wolfram Mathematica 6.0 software. Using the least-square method for determination of the parameters and , we present the following four cases, for the different values of the coefficients and .
6.1. Case 1: and
We find dual solutions. (a) We have The first expression of the first-order approximate solution given by (46) can be written in the form (b) We have The second expression of the first-order approximate solution (48) is
6.2. Case 2: and
We obtain two dual solutions, respectively. (a) We have (b) We have
6.3. Case 3: and
We obtain the corresponding dual solutions, respectively. (a) We have (b) We have
6.4. Case 4: and
It holds that (a) (b)
In Table 1 we present a comparison between the skin friction coefficient obtained by means of OHAM and numerical results. The comparisons are found to be in very good agreement for the first and the second solutions.
In Tables 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 we present a comparison between all approximate solutions and and numerical results obtained by the Runge-Kutta method in combination with shooting method for different values of variable and different values of coefficients and .
It can be observed that the solutions obtained by OHAM are in excellent agreement with numerical results.
Figures 2 and 3 present the displacement for different values of unsteadiness , and , respectively. It is seen that for fixed value of the displacement decreases as increases for the first solutions. The opposite trend is observed for the second solutions.
Figures 4 and 5 depict the velocity profiles for fixed value of and some values of . It is observed that, in all cases, the velocity of fluid is damped faster as the magnitude of the unsteadiness parameter increases. The velocity boundary layer thickness decreases as decreases which implies the increase of the velocity gradient. For the first solution, the velocity gradient is positive, in contrast with the second solution. These conclusions are in concordance with results obtained in [8, 9].
From Table 1 it is seen that the magnitude of increases as the parameters increase in the case of the first solutions given by subcases 6.1(a), 6.2(a), 6.3(a), and 6.4(a). The opposite trend is observed for the variation of ; that is, increasing is to decrease the magnitude of the skin coefficient . In the case of the second solutions given by subcases 6.1(b), 6.2(b), 6.3(b), and 6.4(b) the variation of the skin friction coefficient is reverse.
7. Conclusions
The problem of unsteady viscous flow was solved by means of optimal homotopy asymptotic method and obtained results are compared with numerical results. The effects of the parameters and have been analyzed and presented graphically and in 13 tables. This problem admits a lot of solutions depending on some convergence-control parameters, and in certain conditions () every one of these solutions admits a dual solution. The magnitude of the skin friction coefficient decreases with the increasing of the unsteadiness parameter. The flow velocity and the skin friction coefficient are influenced by the parameters and . Our procedure is valid even if the nonlinear differential equation does not contain small or large parameters. In our construction of the homotopy appear some distinctive concepts such as the auxiliary convergence-control function , the linear operator , and several optimal convergence-control parameters which ensure a fast convergence of the solutions. The examples presented in this work lead to the conclusion that the obtained results are of the exceptional accuracy using only one iteration. The OHAM provides us with a rigorous way to control and adjust the convergence of the solutions through the auxiliary function involving several parameters which are optimally determined.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.