# A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform-homotopy perturbation method

- Uriel Filobello-Nino
^{1}, - Hector Vazquez-Leal
^{1}Email author, - Juan Cervantes-Perez
^{1}, - Brahim Benhammouda
^{2}, - Agustin Perez-Sesma
^{1}, - Luis Hernandez-Martinez
^{3}, - Victor Manuel Jimenez-Fernandez
^{1}, - Agustin Leobardo Herrera-May
^{4}, - Domitilo Pereyra-Diaz
^{1}, - Antonio Marin-Hernandez
^{5}and - Jesus Huerta Chua
^{6}

**3**:421

https://doi.org/10.1186/2193-1801-3-421

© Filobello-Nino et al.; licensee Springer. 2014

**Received: **12 March 2014

**Accepted: **18 July 2014

**Published: **10 August 2014

## Abstract

This article proposes Laplace Transform Homotopy Perturbation Method (LT-HPM) to find an approximate solution for the problem of an axisymmetric Newtonian fluid squeezed between two large parallel plates. After comparing figures between approximate and exact solutions, we will see that the proposed solutions besides of handy, are highly accurate and therefore LT-HPM is extremely efficient.

## Keywords

## Introduction

Although the studies of squeezing flows have their origins in the 19 th century, at present time, it is an issue of considerable importance due to its practical applications in different areas such as physical, biophysical, chemical engineering, and food industry, also they are relevant in liquid metal lubrication theory, polymer processing, compression and injection molding, among many others.

The goal of this study is to find an approximate solution for the problem of squeezing flow between two infinite parallel plates slowly approaching each other. As mentioned in (Ran et al. 2009) these fluids are of paramount importance, in hydrodynamic lubrication theory. Thus, (Langlois 1962) and (Salbu 1964) analyzed isothermal compressible squeeze films neglecting inertial effects, while (Thorpe 1967), found an explicit solution, taking into account these effects. Also have been found some numerical solutions to these problems, such as those provided in (Verma 1981) and (Singh et al. 1990). Additionally, (Rajagopal & Gupta 1981) and (Dandapat & Gupta 1991) extended the previous investigations for the case of flow between rotating parallel plates.

Laplace Transform (L.T.) (or operational calculus) has played an important role in mathematics, not only for its theoretical interest, but also because its methods let to solve, in a simpler fashion, many problems in science and engineering, in comparison with other mathematical techniques (Spiegel 1988). In particular the Laplace Transform is useful for solving linear ordinary differential equations with constant coefficients, and initial conditions, but also can be used to solve some cases of differential equations with variable coefficients and partial differential equations (Spiegel 1988). On the other hand, applications of L.T. for nonlinear ordinary differential equations mainly focus to find approximate solutions, thus in reference (Aminikhan & Hemmatnezhad 2012) was reported a combination of Homotopy Perturbation (HPM) and L.T. methods (LT-HPM), in order to obtain highly accurate solutions for these equations. However, just as with L.T; LT-HPM method has been used mainly to find solutions to problems with initial conditions (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012), because it is directly related with them. Therefore this paper presents the application of LT-HPM, in the search for an approximate solution of the higher order nonlinear ordinary differential equation, which describes a squeezing flow between two infinite plates with, mixed boundary conditions defined on a finite interval (Filobello-Nino et al. 2013).

The case of equations with boundary conditions on infinite intervals, has been studied in some articles, and correspond often to problems defined on semi-infinite ranges (Hossein 2011; Khan et al. 2011). However the methods of solving these problems, are different from what will be presented in this paper (Filobello-Nino et al. 2013). The importance of research on nonlinear differential equations is that many phenomena, practical or theoretical, are of nonlinear nature. In recent years, several methods focused to find approximate solutions to nonlinear differential equations, as an alternative to classical methods, have been reported, such those based on: variational approaches (Assas 2007; He 2007; Kazemnia et al. 2008; Noorzad et al. 2008), tanh method (Evans & Raslan 2005), exp-function (Xu 2007; Mahmoudi et al. 2008), Adomian’s decomposition method (Adomian 1988; Babolian & Biazar 2002; Kooch & Abadyan 2012; Kooch & Abadyan 2011; Vanani et al. 2011; Chowdhury 2011), parameter expansion (Zhang & Xu 2007), homotopy perturbation method (Vazquez-Leal 2014; Marinca & Herisanu 2011; He 1998; He 1999; He 2006a; He 2008; Belendez et al. 2009; He 2000; El-Shaed 2005; He 2006b; Ganji et al. 2009; Ganji et al. 2008; Fereidon et al. 2010; Sharma & Methi 2011; Hossein 2011; Vazquez-Leal et al. 2012a; Vazquez-Leal et al. 2012b; Filobello-Niño et al. 2012a; Biazar & Aminikhan 2009; Biazar & Ghazvini 2009; Filobello-Niño et al. 2012b; Khan & Wu 2011; Madani et al. 2011; Aminikhan & Hemmatnezhad 2012; Aminikhah 2012; Khan et al. 2011; Filobello-Nino et al. 2013), homotopy analysis method (Patel et al. 2012), and perturbation method (Filobello-Niño et al. 2013a) among many others. Also, a few exact solutions to nonlinear differential equations have been reported occasionally (Filobello-Niño et al. 2013b).

The paper is organized as follows. In Standard HPM, we introduce the basic idea of standard HPM method. For Basic Idea of Laplace Transform Homotopy Perturbation Method (LT-HPM) we introduce Laplace transform homotopy perturbation method. Additionally in Governing equations the basic equations for the flow in study are derived. Case Study present the application of LT-HPM method, in the search for an approximate solution for the higher order nonlinear ordinary differential equation, which describes a squeezing flow between two infinite plates. Besides a discussion on the results is presented in Discussion. Finally, a brief conclusion is given in Conclusions.

## Standard HPM

The standard homotopy perturbation method (HPM) was proposed by Ji Huan He, it was introduced like a powerful tool to approach various kinds of nonlinear problems. The Homotopy Perturbation Method (HPM) is considered as a combination of the classical perturbation technique and the homotopy (whose origin is in the topology), but not restricted to small parameters as occur with traditional perturbation methods. For example, HPM method requires neither small parameter nor linearization, but only few iterations to obtain highly accurate solutions (He 1998; He 1999).

*A*is a general differential operator,

*B*is a boundary operator,

*f*(

*r*) a known analytical function and Γ is the domain boundary for Ω.

*A*can be divided into two operators

*L*and

*N*, where

*L*is linear and

*N*nonlinear; so that (1) can be rewritten as

where p is a homotopy parameter, whose values are within range of 0 and 1, *u*_{0} is the first approximation for the solution of (3) that satisfies the boundary conditions.

*p*as

Substituting (6) into (5) and equating identical powers of *p* terms, there can be found values for the sequence *v*_{0}, *v*_{1}, *v*_{2},…

*p*→ 1, it yields the approximate solution for (1) in the form

## Basic idea of Laplace Transform Homotopy Perturbation Method (LT-HPM)

The objective of this section is to show, how LT-HPM, can be employed to find analytical approximate solutions of Ordinary Differential Equations (ODE, s), as (3).

*p*

*p*, with the same power leads to

*U*(0) =

*u*

_{0}=

*α*

_{0}, =

*α*

_{n − 1};

*U*′(0) =

*α*

_{1},..,

*U*

^{n − 1}(0) therefore the exact solution may be obtained as follows

## Governing equations

*W*(see Figure 1).

where

$\overrightarrow{V}$ is the velocity vector, *ρ* the density, *D* represents the material time derivative, and *T* is the stress tensor, which is given by $T=-\mathit{PI}+\mu \left(\nabla \overrightarrow{V}+{\left(\nabla \overrightarrow{V}\right)}^{T}\right),$ where *μ* is the dynamic viscosity of the fluid and *P* the pressure.

*ψ*(

*r*,

*z*,

*t*) defined by

Thus, we have to determine only one unknown function *ψ*(*r*, *z*, *t*), rather than the two functions *u*_{
r
}(*r*, *z*, *t*) and *u*_{
z
}(*r*, *z*, *t*).

*E*

^{2}is given by

*ψ*(

*r*,

*z*,

*t*)

We will assume that W is small enough so that, during the process, the gap 2*l* between the plates changes little and it can be considered approximately constant (see Figure 1).

*ψ*=

*ψ*(

*r*,

*z*), so that (22) is rewritten as

*u*

_{ z }(

*r*,

*z*) = − 2

*F*(

*z*), in such a way that from (24) is obtained

*u*

_{ z }(

*r*, 0) = − 2

*F*(0) = 0, and so on). In order to facilitate the evaluation of (26) we introduce the following dimensionless parameters given by

where we have dropped * for simplicity.

## Case study

from (28), *ϵ* is a positive parameter.

It is possible to find a handy solution by applying the LT-HPM method.

where prime denotes differentiation respect to z.

where we have defined *Y*(*s*) = *ℑ*(*F*(*z*).

*F*(0) = 0,

*F*″(0) = 0, the last expression can be simplified as follows

*Y*(

*s*) and applying Laplace inverse transform

*ℑ*

^{− 1}

where, we have defined *A* = *F*′(0), *B* = *F*‴(0).

*F*(

*z*), in the form

as the first approximation for the solution of (31) that satisfies the conditions *F*(0) = 0, *F*″(0) = 0.

*P*we have

*v*

_{0}(

*z*),

*v*

_{1}(

*z*),

*v*

_{2}(

*z*),.. we obtain

and so on.

*p*→ 1, results in a second order approximation

*A*and

*B*, we require that equations (49) and (50) satisfy the boundary conditions

*F*(1) = 1,

*F*′(1) = 0, respectively. This gives rise to a system of equations for the unknowns

*A*and

*B*, above mentioned. Considering as cases study

*ϵ*=1 and

*ϵ*=2 we obtain the values

respectively.

## Discussion

^{− 12}). Since LT-HPM is expressed in terms of initial conditions for a given differential equation (see (14)), our procedure was aimed to express the approximate solutions in terms of two unknown quantities

*A*=

*F*′(0),

*B*=

*F*‴(0). We noted that these values can be determined requiring that approximate solution satisfies the couple of boundary conditions

*F*(1) = 1

*F*′(1) = 0, from which one obtain an algebraic system of equations for the unknowns

*A*and

*B*above mentioned, whose solution concludes the procedure.

Figure 2 shows the comparison between numerical solution and approximate solution (53) for *ϵ*=1. It can be noticed that curves are in good agreement, from which is clear the accuracy of our approximation, as a matter of fact Figure 3 shows that the biggest absolute error (A.E) of (53) is scarcely of 0.0003, which is remarkably precise, above all taking into account that (53) is just a second order approximate solution for (31).

Next, we found an approximate solution for the case of parameter *ϵ*=2 Figure 4 shows that (54) is an accurate analytical approximate solution for (31); from Figure 5 we deduce that the biggest absolute error (A.E) is of little more than 0.0025, whereby it is clear the reliability of LT-HPM method in the search for approximate solutions of nonlinear problems with finite boundary conditions. An important fact from LT-HPM follows from equations as (31), which can be written in the form *L*(*z*) + *ϵN*(*z*) = 0 where, *L*(*z*) is linear and *N*(*z*) nonlinear. It’s well known that classical methods of approximation as perturbation method PM (Holmes 1995; Chow 1995) provide in general, better results for small perturbation parameters *ϵ* < < 1 (for our case, the perturbation parameter would be small for small values of the distance between the plates and of the density of the fluid (see (28)). To be precise, *ϵ* can be visualized as a parameter of smallness, that measures how greater is the contribution of linear term *L*(*z*) than the one of *N*(*z*)_{.} In general it is easier to find analytical approximate solutions to equations as (31) for small values of *ϵ* than for big values of the same. Figures 2, 3, 4 and 5 show a noticeable fact, that (53) and (54) provide a good approximation as solutions of (31), despite of the fact that perturbation parameters *ϵ*=1 and *ϵ*=2 cannot be considered small.

From the above, it is evident that for values of *ϵ* ≤ 2, the LT-HPM solution will describe efficiently the nonlinear problem (31). On the other hand, as we take bigger values of *ϵ* it will be necessary to consider higher order approximations of (15), in order to keep the accuracy, but possibly losing the handy character of our approximations. In any case, LT-HPM, is not a restricted method, to small parameters (Filobello-Nino et al. 2013). A reason by which LT-HPM applied to problems with boundary conditions is as efficient and converges so rapidly (Filobello-Nino et al. 2013), is that unlike other methods (for instance HPM) which include the boundary conditions from the beginning of the problem at the lowest order approximation, LT-HPM estimates one of the initial conditions unknown at first, requiring that the whole proposed solution satisfies one of the boundary conditions (the other boundary condition is satisfied from the beginning of the procedure), thus is ensured that the approximate solution fits correctly on both boundaries of the interval.

Is expected to be possible to apply other methods to solve the nonlinear problem proposed (31), for example, HPM and HAM. Since HPM is a particular case of the parameters of HAM (*h* = − 1), it is expected that in general, the approximation obtained with HAM turns out to be more accurate, because its region of convergence is based on adjusting of that parameter, while HPM corresponds to a fixed value of the aforementioned parameter and therefore is limited. However, HAM requires sometimes longer expressions, for getting accurate results, such as was reported in (Ran et al. 2009; Murad et al. 2011), where homotopy analysis method was employed to provide an approximate solution of (31). Although the solutions reported to have good accuracy, they require of major order of approximation (in (Ran et al. 2009), for example, approximations were calculated up to fiftieth order), besides generally, HAM is more complicated to applications than LT-HPM, because their approximate expressions are too long and cumbersome, in contrast to expressions like (49), (53) and (54).

*ϵ*=1. It shows a sketch for several streamlines for various values of the distance

*r*, and therefore, provides a graphical representation of the velocity field, because as it is well known, streamlines are lines in the flow field that are everywhere tangent to the velocities (Hughes & Brighton 1967).

## Conclusions

In this paper LT-HPM was employed to provide an approximate analytical solution for the fourth order nonlinear differential equation which describes a squeezing flow between two infinite plates with, mixed boundary conditions defined on a finite interval. LT-HPM method expresses the problem of finding an approximate solution for a nonlinear ordinary differential equation, in terms of solving an algebraic system of equations for some unknowns initial conditions. Figures 2, 3, 4 and 5, show the efficiency of this method in the search for solutions of nonlinear boundary value problems.

The above is an additional advantage for the method, considering that LT-HPM does not need to solve several recurrence differential equations, by which is a tool efficient, useful and precise in practical applications.

## Declarations

### Acknowledgements

We gratefully acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACyT) through grant CB-2010-01 #157024. The authors would like to express their gratitude to Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution to this paper.

## Authors’ Affiliations

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