An Easy Computable Approximate Solution for a Squeezing Flow between Two Infinite Plates by using of Perturbation Method
Applied and Computational Mathematics
Volume 3, Issue 1, February 2014, Pages: 38-42
Received: Feb. 19, 2014; Published: Mar. 10, 2014
Views 3188      Downloads 192
Authors
U. Filobello-Nino, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
H. Vazquez-Leal, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
A. Perez-Sesma, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
J. Cervantes-Perez, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
V. M. Jimenez-Fernandez, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
L. Hernandez-Martinez, Electronics Department, National Institute for Astrophysics, Optics and Electronics, Sta. Maria Tonantzintla, Puebla, Mexico
D. Pereyra-Diaz, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
R. Castaneda-Sheissa, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
J. Sanchez-Orea, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
C. Hoyos-Reyes, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
S. F. Hernandez-Machuca, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
J. Huerta-Chua, Civil Engineering School, University of Veracruz, Poza Rica, Veracruz, Mexico
J. L. Rocha-Fernandez, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
A. D. Contreras-Hernandez, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
J. M. Mendez-Perez, Electronic Instrumentation Faculty, University of Veracruz, Circuito Gonzalo Aguirre Beltran s/n, Xalapa, Veracruz, Mexico
Article Tools
PDF
Follow on us
Abstract
This article proposes Perturbation Method (PM) to find an approximate solution for the problem of an axis symmetric 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 that PM is efficient.
Keywords
Mixed Boundary Conditions, Nonlinear Differential Equation, Perturbation Method, Approximate Solutions
To cite this article
U. Filobello-Nino, H. Vazquez-Leal, A. Perez-Sesma, J. Cervantes-Perez, V. M. Jimenez-Fernandez, L. Hernandez-Martinez, D. Pereyra-Diaz, R. Castaneda-Sheissa, J. Sanchez-Orea, C. Hoyos-Reyes, S. F. Hernandez-Machuca, J. Huerta-Chua, J. L. Rocha-Fernandez, A. D. Contreras-Hernandez, J. M. Mendez-Perez, An Easy Computable Approximate Solution for a Squeezing Flow between Two Infinite Plates by using of Perturbation Method, Applied and Computational Mathematics. Vol. 3, No. 1, 2014, pp. 38-42. doi: 10.11648/j.acm.20140301.16
References
[1]
X.J. Ran, Q.Y. Zhu, and Y. Li,"An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method", Communications in Nonlinear Science and Numerical Simulation, vol. 14, pp. 119-132, 2009.
[2]
W.E. Langlois, "Isothermal squeeze films", Applied Mathematics, vol. 20, p. 131, 1962.
[3]
E.O. Salbu, "Compressible squeeze films and squeeze bearings", Journal of Basic Engineering, vol. 86, p. 355, 1964.
[4]
J.F. Thorpe, Development in Theoretical and Applied Mathematics, W. A Shah, Ed., vol. 3, Pergamon Press, Oxford, UK, 1967.
[5]
R.L. Verma, "A numerical solution for squeezing flow between parallel channels", Wear, vol. 72, no. 1, pp. 89-95, 1981.
[6]
P. Singh., V. Radhakrishnan, and K. A. Narayan, "Squeezing flow between parallel plates", Ingenieur-Archiv, vol. 60, no. 4, pp. 274-281, 1990.
[7]
K.R. Rajagopal and A. S. Gupta, "On a class of exact solutions to the equations of motion of a second grade fluid", International Journal of Engineering Science, vol. 19, no. 7, pp. 1009-1014, 1981.
[8]
B.S. Dandapat and A. S. Gupta, "Stability of a thin layer of a second-grade fluid on a rotating disk", International Journal of Non-linear Mechanics, vo. 26, no. 3-4, pp. 409-417, 1991.
[9]
T.L.Chow, Classical Mechanics. John Wiley and Sons Inc., USA, 1995.
[10]
M.H. Holmes, Introduction to Perturbation Methods. Springer-Verlag, New York, 1995.
[11]
L.M.B.Assas, "Approximate solutions for the generalized K-dV-Burgers’ equation by He’s variational iteration method", Phys. Scr., vol. 76, pp. 161-164, DOI: 10.1088/0031-8949/76/2/008, 2007.
[12]
J.H. He, "Variational approach for nonlinear oscillators", Chaos, Solitons and Fractals, vol. 34, pp. 1430-1439. DOI: 10.1016/j.chaos.2006.10.026, 2007.
[13]
M. Kazemnia, S.A. Zahedi, M. Vaezi, and N. Tolou, "Assessment of modified variational iteration method in BVPs high-order differential equations", Journal of Applied Sciences, vol. 8, pp. 4192-4197, DOI:10.3923/jas.2008.4192.4197, 2008.
[14]
R. Noorzad, A. Tahmasebi Poor and M. Omidvar, "Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics", Journal of Applied Sciences, vol. 8, pp. 369-373, DOI:10.3923/jas.2008.369.373, 2008.
[15]
D.J. Evans and K.R. Raslan, "The Tanh function method for solving some important nonlinear partial differential", Int. J. Computat. Math., vol. 82, pp. 897-905. DOI: 10.1080/00207160412331336026, 2005.
[16]
F.Xu, "A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method", ZeitschriftNaturforschung - Section A Journal of Physical Sciences, vol. 62, no. 12, pp. 685-688, 2007.
[17]
J. Mahmoudi, N. Tolou, I. Khatami, A. Barari, and D.D. Ganji, "Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method", Journal of Applied Sciences, vo. 8, pp. 358-363.DOI:10.3923/jas.2008.358.363, 2008.
[18]
H. Naher, F.A. Abdullah, M.A. Akbar, "New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method", Journal of Applied Mathematics, vol. 2012, pp. 1--14, doi:10.1155/2012/575387, 2012.
[19]
G.Adomian, "A review of decomposition method in applied mathematics, Mathematical Analysis and Applications, vol.135, pp. 501-544, 1988.
[20]
E.Babolian, and J. Biazar, "On the order of convergence of Adomian method", Applied Mathematics and Computation, vol. 130, no. 2, pp. 383-387, DOI: 10.1016/S0096-3003(01)00103-5, 2002.
[21]
A.Kooch, and M. Abadyan, "Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method", Trends in Applied Sciences Research, vol. 7, pp. 57-67, DOI:10.3923/tasr.2012.57.67, 2012.
[22]
A.Kooch, and M. Abadyan, "Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method", Journal of Applied Sciences, vol. 11, pp. 3421-3428, DOI:10.3923/jas.2011.3421.3428, 2011.
[23]
S.K.Vanani, S. Heidari, and M. Avaji, "A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations", Journal of Applied Sciences, vol. 11, pp. 3504-3509, DOI:10.3923/jas.2011.3504.3509, 2011.
[24]
S. H. Chowdhury, "A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations", Journal of Applied Sciences, vol. 11, pp. 1416-1420, DOI:10.3923/jas.2011.1416.1420, 2011.
[25]
L.-N. Zhang and L. Xu, "Determination of the limit cycle by He’s parameter expansion for oscillators in a u3/1+u2potential",ZeitschriftfürNaturforschung - Section A Journal of Physical Sciences, vol. 62, no. 7-8, pp. 396-398, 2007.
[26]
V. Marinca and N.Herisanu, Nonlinear Dynamical Systems in Engineering, 1st edition,Springer-Verlag, Berlin Heidelberg, 2011.
[27]
J.H.He, "A coupling method of a homotopy technique and a perturbation technique for nonlinear problems", Int. J. Non-Linear Mech., vol. 351, pp. 37-43, DOI: 10.1016/S0020-7462(98)00085-7, 1998.
[28]
J.H.He, "Homotopy perturbation technique", Comput. Methods Applied Mech. Eng., vol. 178, pp. 257-262, DOI: 10.1016/S0045-7825(99)00018-3, 1999.
[29]
J.H. He, "Homotopy perturbation method for solving boundary value problems", Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006.
[30]
J.H.He, "Recent Development of the Homotopy Perturbation Method", Topological Methods in Nonlinear Analysis, vol. 31, no. 2, pp. 205-209, 2008.
[31]
A.Belendez, C. Pascual, M.L. Alvarez, D.I. Méndez, M.S. Yebra, and A. Hernández, "High order analytical approximate solutions to the nonlinear pendulum by He’s homotopy method", PhysicaScripta, vol. 79, no. 1, pp. 1-24, DOI: 10.1088/0031-8949/79/01/015009, 2009.
[32]
J.H.He, "A coupling method of a homotopy and a perturbation technique for nonlinear problems", International Journal of Nonlinear Mechanics, vol. 35, no. 1, pp. 37-43, 2000.
[33]
M. El-Shaed, "Application of He’s homotopy perturbation method to Volterra’sintegro differential equation", International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, pp. 163-168, 2005.
[34]
J.H.He, "Some Asymptotic Methods for Strongly Nonlinear Equations.International Journal of Modern Physics B", vol. 20, no. 10, pp. 1141-1199, DOI: 10.1142/S0217979206033796, 2006.
[35]
D.D.Ganji, H. Babazadeh, F Noori, M.M. Pirouz, and M. Janipour,"An Application of Homotopy Perturbation Method for Non linearBlasius Equation to Boundary Layer Flow Over a Flat Plate", International Journal of Nonlinear Science, vol.7, no.4, pp. 309-404, 2009.
[36]
D.D.Ganji, H. Mirgolbabaei, Me. Miansari, and Mo. Miansari, "Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three", Journal of Applied Sciences, vol. 8, pp. 1256-126, DOI:10.3923/jas.2008.1256.1261, 2008.
[37]
A.Fereidon, Y. Rostamiyan, M. Akbarzade, and D.D. Ganji, "Application of He’s homotopy perturbation method to nonlinear shock damper dynamics", Archive of Applied Mechanics, vol. 80, no. 6, pp. 641-649, DOI: 10.1007/s00419-009-0334-x, 2010.
[38]
P.R. Sharma and G. Methi, "Applications of homotopy perturbation method to partial differential equations", Asian Journal of Mathematics & Statistics, vol. 4, pp. 140-150, DOI:10.3923/ajms.2011.140.150, 2011.
[39]
A. Hossein, "Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation", LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, vol. 2012, Article ID 957473, 10 pages doi: 10.5402/2012/957473, 2011.
[40]
H. Vazquez-Leal, U. Filobello-Niño, R. Castañeda-Sheissa, L. Hernandez Martinez, and A. Sarmiento-Reyes, "Modified HPMs inspired by homotopy continuation methods", Mathematical Problems in Engineering, vol. 2012, Article ID 309123, DOI: 10.155/2012/309123, 20 pages, 2012.
[41]
H.Vazquez-Leal, R. Castañeda-Sheissa, U. Filobello-Niño, A. Sarmiento-Reyes, and J. Sánchez-Orea, "High accurate simple approximation of normal distribution related integrals", Mathematical Problems in Engineering, vol. 2012, Article ID 124029, DOI: 10.1155/2012/124029, 22 pages, 2012.
[42]
U.Filobello-Niño, H. Vazquez-Leal, R. Castañeda-Sheissa, A. Yildirim, L. HernandezMartinez, D. PereyraDíaz, A. Pérez Sesma, and C. Hoyos Reyes, "An approximate solution of Blasius equation by using HPM method", Asian Journal of Mathematics and Statistics, vol. 2012, 10 pages, DOI: 10.3923 /ajms.2012, ISSN 1994-5418, 2012.
[43]
J. Biazarand H. Aminikhan, "Study of convergence of homotopy perturbation method for systems of partial differential equations", Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2221-2230, 2009.
[44]
J. Biazar, and H. Ghazvini, "Convergence of the homotopy perturbation method for partial differential equations", Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2633-2640, 2009.
[45]
U.Filobello-Niño, H. Vazquez-Leal, D. PereyraDíaz, A. Pérez Sesma, R. Castañeda-Sheissa, Y. Khan, A. Yildirim, L. Hernandez Martinez, and F. RabagoBernal,"HPM Applied to Solve Nonlinear Circuits: A Study Case", Applied Mathematics Sciences, vol. 6, no. 87, pp. 4331-4344, 2012.
[46]
D.D. Ganji, A.R. Sahouli, M. Famouri, "A New modification of He’shomotopy perturbation method for rapid convergence of nonlinear undamped oscillators", Journal of Applied Mathematics and Computing, vol. 30, pp. 181-192, 2009.
[47]
U.Filobello-Nino, H. Vazquez-Leal, Y. Khan, A. Perez-Sesma, A. Diaz-Sanchez, V.M. Jimenez-Fernandez, A. Herrera-May, D. Pereyra-Diaz, J.M. Mendez-Perez,and J. Sanchez-Orea, "Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals", Computational and Applied Mathematics, ISSN: 0101-8205, DOI= 10.1007/s40314-013-0073-z, 2013.
[48]
T.Patel, M.N. Mehta, and V.H. Pradhan,"The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method", Asian Journal of Applied Sciences, Vol. 5, pp. 60-66, DOI:10.3923/ajaps.2012.60.66, 2012.
[49]
U.Filobello-Niño, H. Vazquez-Leal, Y. Khan, A. Yildirim, V.M. Jimenez-Fernandez, A. L Herrera May, R. Castañeda-Sheissa, andJ.CervantesPerez,"Perturbation Method and Laplace-Padé Approximation to solve nonlinear problems", Miskolc Mathematical Notes, vol. 14, no. 1, pp. 89-101, ISSN: 1787-2405, 2013.
[50]
U.Filobello-Niño, H. Vazquez-Leal, K.Boubaker, Y. Khan, A. Perez-Sesma, A.Sarmiento Reyes, V.M. Jimenez-Fernandez, A Diaz-Sanchez, A. Herrera-May, J. Sanchez-Orea and K. Pereyra-Castro, "Perturbation Method as a Powerful Tool to Solve Highly Nonlinear Problems: The Case of Gelfand'sEquation",Asian Journal of Mathematics and Statistics, vol. 2013, 7 pages, DOI: 10.3923 /ajms.2013, ISSN 1994-5418, 2013.
[51]
H. Naher and F. Abdullah, ''The Basic (G'/G)-Expansion Method for the Fourth Order Boussinesq Equation", Applied Mathematics, vol. 3, no. 10, pp. 1144-1152. doi: 10.4236/am.2012.310168, 2012.
[52]
H. Naher and F.A. Abdullah, "New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for nonlinear evolution equation", vol. 3, no. 3, doi: 10.1063/1.4794947, 2013.
[53]
U.Filobello-Niño, H. Vazquez-Leal, Y. Khan, A. Perez-Sesma, A. Diaz-Sanchez, A. Herrera-May, D. Pereyra-Diaz, R. Castañeda-Sheissa, V.M. Jimenez-Fernandez, and J. Cervantes-Perez, "A handy exact solution for flow due to a stretching boundary with partial slip", Revista Mexicana de Física E, vol. 59, pp. 51-55. ISSN 1870-3542, 2013.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186