Abstract

The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient.

1. Introduction

According to the classification of Prandtl, the fluid motion is divided into two regions. The first region is near the object where the effect of friction is important and is known as the boundary layer, while, for the second type, these effects can be neglected [1–3]. It is common to define the boundary layer as the region where the fluid velocity parallel to the surface is less than of the free stream velocity [1].

The boundary layer thickness increases from the edge along the surface on which the fluid moves. Even for the case of a laminar flow, the exact solution of equations describing the laminar boundary layer is very difficult to calculate and only few simple problems can straightforward be analysed [1, 3].

An interesting case is the one where a flow is induced into a viscoelastic fluid by a linearly stretched sheet [4–6] (see Figure 1). Extrusion of molten polymers through a slit die for the production of plastic sheets is an important process in polymer industry [4]. The process is normally complicated from the physical point of view, because it requires significant heat transfer between the sheet and a surrounding fluid that plays the role of a cooling medium. An important aspect of the flow is the extensibility of the sheet which can be employed to improve its mechanical properties along the sheet. To obtain better results is necessary to improve the cooling rate, whereby it is common to add some polymeric additives into water (which is one of the most employed fluids as cooling medium) in order to have a better control on the cooling rate. The flow due to a stretching boundary is also important in other engineering processes of interest, such as the glass fibre drawing and crystal growing among many others. Detail discussion about this topic can be found in [4]. This work assumes that the boundary conditions for the problem under study are adequately described by Navier’s condition, which states that the amount of relative slip is proportional to local shear stress. Unlike what happens with fluids like water, mercury, and glycerine, which do not require slip boundary conditions [7], there are cases where partial slip between the fluid and the moving surface may occur. Some known cases include emulsions, as mustard and paints and polymer solutions and clay [7].

Figure 1 

Schematic showing a stretching boundary.

He [8, 9] proposed the standard HPM; it was introduced as a powerful tool to approach several kinds of nonlinear problems. The HPM can be considered as a combination between the classical perturbation technique and the homotopy (whose origin is in the topology), but not restricted to the limitations found in traditional perturbation methods. For instance, HPM method does need neither small parameter nor linearisation, just few iterations to obtain accurate results [5, 8–35]. The fundamentals of HPM convergence can be found in [21, 24, 25].

There are other modern alternatives to find approximate solutions to the differential equations that describe some nonlinear problems such as those based on variational approaches [36–39], tanh method [40], exp-function [41, 42], Adomian decomposition method [43–49], parameter expansion [50], homotopy analysis method [4, 51, 52], and perturbation method [53] among many others.

To figure out how HPM method works, consider a general nonlinear equation in the form with the following boundary conditions: where is a general differential operator, is a boundary operator, is a known analytical function, and is the domain boundary for .

Also, can be divided into two parts and , where is linear and nonlinear; from this last statement, (1) can be rewritten as

In a broad sense, a homotopy can be constructed in the following form [8, 9]: or where is a homotopy parameter, whose values are within range of and , and is the first approximation to the solution of (3) that satisfies the boundary conditions. Assuming that solution for (4) or (5) can be written as a power series of

Substituting (6) into (5) and equating identical powers for terms, it is possible to obtain the values for the sequence .

When , it yields in the approximate solution for (1) in the form

Another way to build a homotopy, which is relevant for this paper, is by considering the following general equation: where and are the linear and nonlinear operators, respectively. It is desired that solution for describes, accurately, the original nonlinear system.

By the homotopy technique, a homotopy is constructed as follows [18]:

Again, it is assumed that solution for (9) can be written in the form (6); thus, taking the limit when results in the approximate solution for (8).

The variation of homotopic parameter within the range amounts to a deformation that begins from an initial equation with known solution until it becomes the equation to be solved. From a practical point of view, taking the limit is just setting .

2. Padé Approximant

Let be an analytical function with the Maclaurin’s expansion Then the Padé approximant to of order which we denote by is defined by [54–57] where we considered , and the numerator and denominator have no common factors.

The numerator and the denominator in (11) are constructed so that and and their derivatives agree at up to . That is, From (12), we have From (13), we get the following algebraic linear systems: From (14), we calculate first all the coefficients , . Then, we determine the coefficients , from (15).

Note that for a fixed value of , the error (12) is the smallest when the numerator and denominator of (11) have the same degree or when the numerator has one degree higher than the denominator.

3. Laplace-Padé Resummation Method

Several approximate methods provide power series solutions (polynomial). Nevertheless, sometimes, this type of solutions lacks large domains of convergence. Therefore, Laplace-Padé resummation method [54] is used in literature to enlarge the domain of convergence of solutions or inclusive to find exact solutions.

The Laplace-Padé method can be explained as follows.(1)First, Laplace transform is applied to power series.(2)Next, is substituted by in the resulting equation.(3)After that, we convert the transformed series into a meromorphic function by forming its Padé approximant of order . and are arbitrarily chosen, but they should be of smaller values than the order of the power series. In this step, the Padé approximant extends the domain of the truncated series solution to obtain better accuracy and convergence.(4)Then, is substituted by .(5)Finally, by using the inverse Laplace transform, we obtain the exact or approximate solution.

4. Formulation

Consider a two-dimensional stretching boundary (see Figure 1). Experiments show that the velocity of the boundary is approximately proportional to the distance from the orifice [7, 58], so that where is a proportionality constant.

Let be the fluid velocities for the directions, respectively. In this case the boundary condition is adequately described by Navier’s condition, which states that the amount of relative slip is proportional to local shear stress: where is a constant of proportionality and is the kinematic viscosity of the bulk fluid. The relevant expressions for this case are the Navier-Stokes equations: and continuity where and are density and pressure, respectively.

To solve equations (18)–(20), we have to consider boundary conditions (16) and (17), besides the fact that there is no lateral velocity or pressure gradient away from the stretching surface.

Next, we will show that it is possible to get an ordinary nonlinear differential equation from (18).

With this end, we note that component of velocity is negative () and by symmetry arguments it only depends on (see Figure 1). Therefore, it is possible to define a function , , such that [7]: where

From (21) and continuity equation (20) we obtain

Clearly (20) is satisfied under transformations (21), (22), and (23), while (18) adopts the simpler form;

To deduce the boundary conditions of (24), we see from Figure 1 that and and therefore

Finally, substituting (16) and (23) into (17) we obtain

5. Approximate Solution for a Two-Dimensional Viscous Flow Equation

Next, we present some solution methods to the study problem.

5.1. HPM Method

In this section, HPM is used to find approximate solutions for (24). Identifying the linear part as and the nonlinear as we initiate the HPM method by constructing a homotopy based on (9), in the form

Assuming that the solution has the form [8, 9]:

Then, substituting (30) into (29) and equating terms having identical powers of we obtain

In order to fulfil the boundary conditions from (24) given by (25)-(26), we find that , , , , , , , , , , , , and so on. We have assumed the value for as some adequate constant , which adopts the following values for the corresponding as follows: , , , , , and [6]. Thus, the results obtained from above equations are and so on.

To exemplify, we will consider the seventh order approximation by substituting solutions (32) into (30) and calculating the limit when :

The above approximation is valid for all values of and corresponds to the results reported in [5] for the cases when and . Clearly, if higher order approximations are considered, better accuracy is obtained but the resulting expressions could be too long.

5.2. HPM Laplace-Padé Scheme (LPHPM)

Next, we study the case ; it means that (see (26)). This is an interesting case because the following exact solution was reported in [59]:

Thus, substituting in (33) we obtain the following series solution: