- Open Access
A handy approximate solution for a squeezing flow between two infinite plates by using of Laplace transform-homotopy perturbation method
© Filobello-Nino et al.; licensee Springer. 2014
- Received: 12 March 2014
- Accepted: 18 July 2014
- Published: 10 August 2014
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.
- Laplace transform homotopy perturbation method
- Nonlinear fluid problems
- Power series
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.
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).
where p is a homotopy parameter, whose values are within range of 0 and 1, u0 is the first approximation for the solution of (3) that satisfies the boundary conditions.
Substituting (6) into (5) and equating identical powers of p terms, there can be found values for the sequence v0, v1, v2,…
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).
is the velocity vector, ρ the density, D represents the material time derivative, and T is the stress tensor, which is given by where μ is the dynamic viscosity of the fluid and P the pressure.
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).
We will assume that W is small enough so that, during the process, the gap 2l between the plates changes little and it can be considered approximately constant (see Figure 1).
where we have dropped * for simplicity.
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).
where, we have defined A = F′(0), B = F‴(0).
as the first approximation for the solution of (31) that satisfies the conditions F(0) = 0, F″(0) = 0.
and so on.
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).
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.
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.
- Adomian G: A review of decomposition method in applied mathematics. Math Anal Appl 1988, 135: 501-544.View ArticleGoogle Scholar
- Aminikhah H: The combined Laplace transform and new homotopy perturbation method for stiff systems of ODE s. Appl Math Model 2012, 36: 3638-3644.View ArticleGoogle Scholar
- Aminikhan H, Hemmatnezhad M: A novel effective approach for solving nonlinear heat transfer equations. Heat Transfer Asian Res 2012, 41(6):459-466.View ArticleGoogle Scholar
- Assas LMB: Approximate solutions for the generalized K-dV- Burgers’ equation by He’s variational iteration method. Phys Scr 2007 doi:10.1088/0031-8949/76/2/008, 76: 161-164.View ArticleGoogle Scholar
- Babolian E, Biazar J: On the order of convergence of Adomian method. Appl Math Comput 2002 doi: 10.1016/S0096-3003(01)00103-5, 130(2):383-387.View ArticleGoogle Scholar
- Belendez A, Pascual C, Alvarez ML, Méndez DI, Yebra MS, Hernández A: High order analytical approximate solutions to the nonlinear pendulum by He’s homotopy method. Phys Scr 2009 doi:10.1088/0031-8949/79/01/015009, 79(1):1-24.View ArticleGoogle Scholar
- Biazar J, Aminikhan H: Study of convergence of homotopy perturbation method for systems of partial differential equations. Comput Math Appl 2009, 58(No. 11–12):2221-2230.View ArticleGoogle Scholar
- Biazar J, Ghazvini H: Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Anal Real World Appl 2009, 10(No 5):2633-2640.View ArticleGoogle Scholar
- Chow TL: Classical Mechanics. John Wiley and Sons Inc, New York; 1995.Google Scholar
- Chowdhury SH: A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. J Appl Sci 2011 doi:10.3923/jas.2011.1416.1420, 11: 1416-1420.View ArticleGoogle Scholar
- Dandapat BS, Gupta AS: Stability of a thin layer of a second-grade fluid on a rotating disk. Int J Nonlinear Mech 1991, 26(no. 3–4):409-417.View ArticleGoogle Scholar
- El-Shaed M: Application of He’s homotopy perturbation method to Volterra’s integro differential equation. Int J Nonlinear Sci Numerical Simul 2005, 6: 163-168.Google Scholar
- Evans DJ, Raslan KR: The Tanh function method for solving some important nonlinear partial differential. Int J Comput Math 2005 doi:10.1080/00207160412331336026, 82: 897-905.View ArticleGoogle Scholar
- Fereidon A, Rostamiyan Y, Akbarzade M, Ganji DD: Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Arch Appl Mech 2010 doi: 10.1007/s00419-009-0334-x, 80(6):641-649.View ArticleGoogle Scholar
- Filobello-Niño U, Vazquez-Leal H, Castañeda-Sheissa R, Yildirim A, Hernandez Martinez L, Pereyra Díaz D, Pérez Sesma A, Hoyos Reyes C: An approximate solution of Blasius equation by using HPM method. Asian J Math Stat 2012 Doi: 10.3923 /ajms.2012, ISSN 1994–5418, 2012: 10.Google Scholar
- Filobello-Niño U, Vazquez-Leal H, Pereyra Díaz D, Pérez Sesma A, Castañeda-Sheissa R, Khan Y, Yildirim A, Hernandez Martinez L, Rabago Bernal F: HPM applied to solve nonlinear circuits: a study case. Appl Math Sci 2012, 6(2012, no. 85–88):4331-4344.Google Scholar
- Filobello-Niño U, Vazquez-Leal H, Khan Y, Yildirim A, Jimenez- Fernandez VM, Herrera May AL, Castañeda-Sheissa R, Cervantes-Perez J: Using perturbation methods and Laplace–Padé approximation to solve nonlinear problems. Miskolc Math Notes 2013, 14(1):89-101.Google Scholar
- Filobello-Niño U, Vazquez-Leal H, Khan Y, Perez-Sesma A, Diaz-Sanchez A, Herrera-May A, Pereyra-Diaz D, Castañeda-Sheissa R, Jimenez-Fernandez VM, Cervantes-Perez J: A handy exact solution for flow due to a stretching boundary with partial slip. Revista Mexicana de Física E 2013 ISSN 1870–3542, 59(2013):51-55. ISSN 1870–3542Google Scholar
- Filobello-Nino U, Vazquez-Leal H, Khan Y, Perez-Sesma A, Diaz-Sanchez A, Jimenez-Fernandez VM, Herrera-May A, Pereyra-Diaz D, Mendez-Perez JM, Sanchez-Orea J: Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals. Comput Appl Math 2013. ISSN: 0101–8205, 2013. doi:10.1007/s40314-013-0073-zGoogle Scholar
- Ganji DD, Mirgolbabaei H, Miansari M, Miansari M: Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three. J Appl Sci 2008 doi:10.3923/jas.2008.1256.1261, 8: 1256-1261.View ArticleGoogle Scholar
- Ganji DD, Babazadeh H, Noori F, Pirouz MM, Janipour M: An application of homotopy perturbation method for Non linear Blasius equation to boundary layer flow over a flat plate, ACADEMIC World Academic Union, ISNN 1749-3889(print), 1749–3897 (online). Int J Nonlinear Sci 2009, 7(4):309-404.Google Scholar
- He JH: A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 1998 doi:10.1016/S0020-7462(98)00085-7, 351: 37-43.Google Scholar
- He JH: Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999 doi:10.1016/S0045-7825(99)00018-3, 178: 257-262.View ArticleGoogle Scholar
- He JH: A coupling method of a homotopy and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 2000, 35(1):37-43.View ArticleGoogle Scholar
- He JH: Homotopy perturbation method for solving boundary value problems. Phys Lett A 2006, 350(1–2):87-88.View ArticleGoogle Scholar
- He JH: Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006 doi:10.1142/S0217979206033796, 20(10):1141-1199.View ArticleGoogle Scholar
- He JH: Variational approach for nonlinear oscillators. Chaos Solitons Fractals 2007 doi:10.1016/j.chaos.2006.10.026, 34: 1430-1439.View ArticleGoogle Scholar
- He JH: Recent development of the homotopy perturbation method. Topological Methods Nonlinear Anal 2008, 31(2):205-209.Google Scholar
- Holmes MH: Introduction to Perturbation Methods. Springer, New York; 1995.View ArticleGoogle Scholar
- Hossein A: Analytical approximation to the solution of nonlinear Blasius viscous flow equation by LTNHPM. Int Scholarly Res Netw ISRN Math Anal 2011 doi:10.5402/2012/957473, 2012(Article ID 957473):10.Google Scholar
- Hughes WF, Brighton JA: Dinámica De Los Fluidos. Mc Graw Hill, Sao Paulo; 1967.Google Scholar
- Kazemnia M, Zahedi SA, Vaezi M, Tolou N: Assessment of modified variational iteration method in BVPs high-order differential equations. J Appl Sci 2008 doi: 10.3923/jas.2008.4192.4197, 8: 4192-4197.View ArticleGoogle Scholar
- Khan Y, Wu Q: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 2011, 61(8):1963-1967.View ArticleGoogle Scholar
- Khan M, Gondal MA, Iqtadar Hussain S, Vanani K: A new study between homotopy analysis method and homotopy perturbation transform method on a semi infinite domain. Math Comput Model 2011, 55: 1143-1150.View ArticleGoogle Scholar
- Kooch A, Abadyan M: Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method. J Appl Sci 2011 doi:10.3923/jas.2011.3421.3428, 11: 3421-3428.View ArticleGoogle Scholar
- Kooch A, Abadyan M: Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends Appl Sci Res 2012 doi:10.3923/tasr.2012.57.67, 7: 57-67.View ArticleGoogle Scholar
- Langlois WE: Isothermal squeeze films. Appl Math 1962, 20: 131.Google Scholar
- Madani M, Fathizadeh M, Khan Y, Yildirim A: On the coupling of the homotopy perturbation method and Laplace transformation. Math Comput Model 2011, 53(9–10):1937-1945.View ArticleGoogle Scholar
- Mahmoudi J, Tolou N, Khatami I, Barari A, Ganji DD: Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method. J Appl Sci 2008 doi:10.3923/jas.2008.358.363, 8: 358-363.View ArticleGoogle Scholar
- Marinca V, Herisanu N: Nonlinear Dynamical Systems in Engineering. 1st edition. Springer, Berlin Heidelberg; 2011.View ArticleGoogle Scholar
- Murad U, Islam S, Zaman G, Khalid SN: A note on squeezing flow between two infinite parallel plates with slip boundary conditions. Int J Phys Sci 2011, 6(14):3296-3301.Google Scholar
- Noorzad R, Tahmasebi Poor A, Omidvar M: Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. J Appl Sci 2008 doi: 10.3923/jas.2008.369.373, 8: 369-373.View ArticleGoogle Scholar
- Papanastasiou TC, Georgiou GC, Alexandrou AN: Viscous Fluid Flow. CRC Press LLC, New York; 2000.Google Scholar
- Patel T, Mehta MN, Pradhan VH: The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method. Asian J Appl Sci 2012 doi: 10.3923/ajaps.2012.60.66, 5: 60-66.View ArticleGoogle Scholar
- Rajagopal KR, Gupta AS: On a class of exact solutions to the equations of motion of a second grade fluid. Int J Eng Sci 1981, 19(no. 7):1009-1014.View ArticleGoogle Scholar
- Ran XJ, Zhu QY, Li Y: An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method. Commun Nonlinear Sci Numer Simul 2009, 14: 119-132.View ArticleGoogle Scholar
- Salbu EO: Compressible squeeze films and squeeze bearings. J Basic Eng 1964, 86: 355.View ArticleGoogle Scholar
- Sharma PR, Methi G: Applications of homotopy perturbation method to partial differential equations. Asian J Math Stat 2011 doi:10.3923/ajms.2011.140.150, 4: 140-150.View ArticleGoogle Scholar
- Singh P, Radhakrishnan V, Narayan KA: Squeezing flow between parallel plates. Ingenieur Archiv 1990, 60(no. 4):274-281.View ArticleGoogle Scholar
- Spiegel MR: Teoría y Problemas de Transformadas de Laplace, Primera Edición. Serie de compendios Schaum, México; 1988.Google Scholar
- Stefan J: Versuche uber die scheinbare Adhasion, Sitzungsberichte der kaiserlichen Akademie der Wissenschaften, Mathematisch Naturwissenschaftliche Classe, 69, Band II. Abteilung Wien; 1874:713-735.Google Scholar
- Thorpe JF: Development in Theoretical and Applied Mathematics. Volume 3. Edited by: Shah WA. Oxford, UK: Pergamon Press; 1967.Google Scholar
- Vanani SK, Heidari S, Avaji M: A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations. J Appl Sci 2011 doi:10.3923/jas.2011.3504.3509, 11: 3504-3509.View ArticleGoogle Scholar
- Vazquez-Leal H: Generalized homotopy method for solving nonlinear differential equations”. Comput Appl Math 2014 doi:10.1007/s40314-013-0060-4, 33(1):275-288.View ArticleGoogle Scholar
- Vazquez-Leal H, Filobello-Niño U, Castañeda-Sheissa R, Hernandez Martinez L, Sarmiento-Reyes A: Modified HPMs inspired by homotopy continuation methods. Math Probl Eng 2012 doi:10.155/2012/309123, 2012(Article ID 309123):20.Google Scholar
- Vazquez-Leal H, Castañeda-Sheissa R, Filobello-Niño U, Sarmiento-Reyes A, Sánchez-Orea J: High accurate simple approximation of normal distribution related integrals. Math Probl Eng 2012 doi:10.1155/2012/124029, 2012(Article ID 124029):22.Google Scholar
- Verma RL: A numerical solution for squeezing flow between parallel channels. Wear 1981, 72(1):89-95.View ArticleGoogle Scholar
- Xu F: A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method. Zeitschrift Naturforschung A J Phys Sci 2007, 62(12):685-688.Google Scholar
- Zhang L-N, Xu L: Determination of the limit cycle by He’s parameter expansion for oscillators in a potential. Zeitschrift für Naturforschung A J Phys Sci 2007, 62(7–8):396-398.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.