Dual Approximate Solutions of the Unsteady Viscous Flow over a Shrinking Cylinder with Optimal Homotopy Asymptotic Method

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 andnumerical solutions,which prove thatOHAMis very efficient in practice, ensuring a very rapid convergence after only one iteration.


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, glassfiber 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.

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 k 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 () =  0 √1 − .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  ( < 0) and  0 a positive constant, then the boundary conditions are of the following form: By means of the similarity variables [8] it is clear that  ≥ 1, 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  =  2 0 /4] is the unsteadiness parameter for the expanding ( < 0) or contraction ( > 0) cylinder showing the strength of expansion or contraction.Substituting (7) into (5) and rearranging terms, this becomes   () +   () +  ()   () −  2 () −  [  () −   ()] = 0 (10) Advances in Mathematical Physics 3 with the boundary conditions transformed into the following: where prime denotes differentiation with respect to  and  = − 0 /2] > 0 is the dimensionless suction parameter.

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  0 () 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  ∈ [0, 1] denotes an embedding parameter.It follows that the first-order approximate solution can be written as where  1 ,  2 , . . .,   are arbitrary parameters, which will be determined later.The boundary conditions are We construct a family of equations [17][18][19][20][21]: with the properties where (,   ) is an arbitrary auxiliary convergence-control function.
Now, equating only the coefficients of  0 and  1 into (18), we obtain the governing equation of  0 () given by ( 14) and the governing equation on  1 (,   ); that is, In general, the nonlinear operator from (23) may be written as where the functions ℎ  () and   () are known and depend on the functions  0 () 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  1 (,   ) in the form or where within expression of   (, ℎ  (),   ) from ( 25) appear linear combinations of some functions ℎ  , some of the terms which are given by corresponding homogeneous equation and a number of unknown parameters   ,  = 1, 2, . . ., ,  being an arbitrary integer number.The same considerations can be made for (26) where   and   are interchangeable.
so that within (25) the terms ∑  =1   (, ℎ  (),   )  () are of the same shape as the terms ∑  =1 ℎ  ()  () given by (24) [14][15][16][17][18].The first-order approximate solution (,   ) also depends on the parameters   ,  = 1, . . ., .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   (,   (),   ),  = 1, . . ., ,  = 1, . . ., , which provides us with a simple way to adjust and control the convergence of the approximate solutions.

Multiple Approximate Solutions of the Unsteady Viscous Flow by OHAM
The linear operator can be chosen in the following forms: where  > 0 is an unknown positive parameter and will be determined later.
The initial approximation  0 () 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).

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 .
The first expression of the first-order approximate solution given by ( 46) can be written in the form ( Advances in Mathematical Physics 7 In Table 1 we present a comparison between the skin friction coefficient   (1) obtained by means of OHAM and        solution.These conclusions are in concordance with results obtained in [8,9].
From Table 1 it is seen that the magnitude of   (1) 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   (1).In the case of the second solutions given

2 AdvancesFigure 1 :
Figure 1: A schematic model of flow in an expanding cylinder with time dependent radius.

Figure 4 :
Figure 4: Velocity profile for different values of  and  = 1.

Table 1 :
Comparison between the skin friction coefficient

Table 7 :
Comparison between the derivative It can be observed that the solutions obtained by OHAM are in excellent agreement with numerical results.Figures2 and 3present the displacement () for different values of unsteadiness ,  = 1 and  = 2, 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.
() 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