Application of HPM to Solve Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates

Authors

  • H. Vazquez-Leal Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • U. Filobello-Nino Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • A. Sarmiento-Reyes National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla. 72840 Puebla, México.
  • M. Sandoval-Hernandez Doctorado en Ciencia, Cultura y Tecnología, Universidad de Xalapa, Km 2 Carretera Xalapa-Veracruz, Xalapa 91190, Veracruz, México,Centro de Bachillerato Tecnológico Industrial y de Servicios No. 268, Av. La Bamba, Geovillas del Puerto, Veracruz, 91777, Veracruz, México.
  • J. A. A. Perez-Sesma Facultad de Ingeniería Electrónica y Comunicaciones, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolución, 93390, Poza Rica, Veracruz, México.
  • A. Perez-Sesma Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • V. M. Jimenez-Fernandez Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • J. Huerta-Chua Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • F. Castro-Gonzalez Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • J. Sanchez-Orea Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • S. F. Hernández Machuca Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • L. Cuellar-Hernández Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • J. E. Pretelin Canela Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • A. E. Gasca-Herrera Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • C. E. Sampieri-Gonzalez Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • B. E. Palma-Grayeb Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • A. D. Contreras-Hernandez Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • O. Alvarez-Gasca Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.
  • F. J. Gonzalez-Martinez Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 91000, México.

Keywords:

boundary layer, fluid mechanics, kinematics viscosity, density, homotopy perturbation method.

Abstract

In this article, Homotopy Perturbation Method (HPM) is used to provide two approximate solutions to the nonlinear differential equation that describes the behaviour for the unsteady squeezing flow of a second grade fluid between circular plates. Comparing results between approximate and numerical solutions shows that our results are capable to provide an accurate solution and are extremely efficient.

References

[1] Hughes W F and Brighton J A, Dinámica De Los Fluidos, Mc Graw Hill. 1967.
[2] Resnick R and Halliday D, Física Vol. 1, John Wiley and Sons. 1977.
[3] Landau L D and Lifschitz E M, Fluid Mechanics Vol. 6 2nd Edition, Pergamon Press. 1987.
[4] Mohammad Mehdi Rashidi, Abdul Majid Siddiqui, and Mostafa Asadi,
Application of Homotopy Analysis Method to the Unsteady Squeezing Flow of a Second Grade Fluid between Circular Plates. Mathematical Problems in Engineering Hindawi Publishing Corporation, vol. 2010, pp. 1-18, 2010.
[5] He, J.H. A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Non-Linear Mech., 351: 37-43. DOI: 10.1016/S0020-7462(98)00085-7, 1998.
[6] He, J.H. Homotopy perturbation technique. Comput. Methods Applied Mech. Eng., 178: 257-262. DOI: 10.1016/S0045-7825(99)00018-3, 1999.
[7] Assas, L.M.B. Approximate solutions for the generalized K-dV- Burgers’ equation by He’s Variational iteration method. Phys. Scr., 76: 161-164. DOI: 10.1088/0031-8949/76/2/008, 2007.
[8] He, J.H. Variational approach for nonlinear oscillators. Chaos, Solitons and Fractals, 34: 1430-1439. DOI: 10.1016/j.chaos.2006.10.026, 2007.
[9] Kazemnia, M., S.A. Zahedi, M. Vaezi and N. Tolou, Assessment of modified variational iteration method in BVPs high-order differential equations. Journal of Applied Sciences, 8: 4192-4197.DOI:10.3923/jas.2008.4192.4197, 2008.
[10] Evans, D.J. and K.R. Raslan. The Tanh function method for solving some important nonlinear partial differential. Int. J. Computat. Math., 82: 897-905. DOI: 10.1080/00207160412331336026, 2005.
[11] Xu, F. A generalized soliton solution of the Konopelchenko-Dubrovsky equation using Exp-function method. Zeitschrift Naturforschung - Section A Journal of Physical Sciences, 62(12): 685-688, 2007.
[12] Mahmoudi, J., 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, 8: 358-363.DOI:10.3923/jas.2008.358.363, 2008.
[13] Adomian, G. A review of decomposition method in applied mathematics. Mathematical Analysis and Applications. 135: 501-544, 1998.
[14] Babolian, E. and J. Biazar. On the order of convergence of Adomian method. Applied Mathematics and Computation, 130(2): 383-387. DOI: 10.1016/S0096-3003(01)00103-5, 2002.
[15] Kooch, A. 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, 7: 57-67.DOI:10.3923/tasr.2012.57.67, 2012.
[16] Kooch, A. 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, 11: 3421-3428. DOI:10.3923/jas.2011.3421.3428, 2011.
[17] Vanani, S. K., S. Heidari and M. Avaji. A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations. Journal of Applied Sciences, 11: 3504-3509. DOI:10.3923/jas.2011.3504.3509, 2011.
[18] Chowdhury, S. H. A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. Journal of Applied Sciences, 11: 1416-1420. DOI:10.3923/jas.2011.1416.1420, 2011.
[19] Zhang, L.-N. and L. Xu. Determination of the limit cycle by He’s parameter expansion for oscillators in a potential. Zeitschrift für Naturforschung - Section A Journal of Physical Sciences, 62(7-8): 396-398, 2007.
[20] He, J.H. Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(1-2): 87-88, 2006.
[21] He, J.H. Recent Development of the Homotopy Perturbation Method. Topological Methods in Nonlinear Analysis, 31.2: 205-209, 2008.
[22] Belendez, A., 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. Physica Scripta, 79(1): 1-24. DOI: 10.1088/0031-8949/79/01/015009, 2009.
[23] He, J.H. A coupling method of a homotopy and a perturbation technique for nonlinear problems. International Journal of Nonlinear Mechanics, 35(1): 37-43, 2000.
[24] El-Shaed, M. Application of He’s homotopy perturbation method to Volterra’s integro differential equation. International Journal of Nonlinear Sciences and Numerical Simulation, 6: 163-168, 2005.
[25] He, J.H. Some Asymptotic Methods for Strongly Nonlinear Equations. International Journal of Modern Physics B, 20(10): 1141-1199. DOI: 10.1142/S0217979206033796, 2006.
[26] Ganji, D.D, H. Babazadeh, F Noori, M.M. Pirouz, M Janipour. 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). International Journal of Nonlinear Science Vol.7 No.4, pp. 309-404, 2009.
[27] Ganji, D.D., 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, 8: 1256-1261.DOI:10.3923/jas.2008.1256.1261, 2008.
[28] Fereidon, A., Y. Rostamiyan, M. Akbarzade and D.D. Ganji. Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Archive of Applied Mechanics, 80(6): 641-649. DOI: 10.1007/s00419-009-0334-x, 2010.
[29] Sharma, P.R. and G. Methi. Applications of homotopy perturbation method to partial differential equations. Asian Journal of Mathematics & Statistics, 4: 140-150. DOI:10.3923/ajms.2011.140.150, 2011.
[30] Aminikhah Hossein. Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation by LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, Volume 2012, Article ID 957473, 10 pages doi: 10.5402/2012/957473, 2011.
[31] Noorzad, R., A. Tahmasebi Poor and M. Omidvar, 2008. Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. Journal of Applied Sciences, 8: 369-373. DOI:10.3923/jas.2008.369.373.
[32] Patel, T., 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, 5: 60-66.DOI:10.3923/ajaps.2012.60.66, 2012.
[33] Vazquez-Leal H, 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.
[34] Vazquez-Leal H., 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.
[35] U. Filobello-Niño, H. Vazquez-Leal, R. Castañeda-Sheissa, A. Yildirim, L. Hernandez Martinez, D. Pereyra Díaz, 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.
[36] Biazar, J. and 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, (2221-2230), 2009.
[37] Biazar, J. and H. Ghazvini. Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Analysis: Real World Applications, Vol. 10, No 5, (2633-2640), 2009.
[38] U. Filobello-Niño, H. Vazquez-Leal, D. Pereyra Díaz, A. Pérez Sesma, J. Sanchez Orea, R. Castañeda-Sheissa, Y. Khan, A. Yildirim, L. Hernandez Martinez, and F. Rabago Bernal. HPM Applied to Solve Nonlinear Circuits: A Study Case. Applied Mathematics Sciences, 6 (85-88) 4331-4344, 2012.
[39] Fernandez, Francisco. M. Rational approximation to the Thomas-Fermi equations. Applied mathematics and computation. Vol 217, 4 pages, DOI: 10.1016/ j.amc. 2011.01.049, 2011.
[40] Yao, Baoheng A series solution to the Thomas-Fermi equation. Applied mathematics and computation. Vol 203, 6 pages, DOI: 10.1016/ j.amc. 2008.04.050, 2008.
[41] Noor Muhammad Aslam, Syed Tauseef Mohyud- Din. Homotopy approach for perturbation method for solving Thomas- Fermi equation using Pade Approximants. International of nonlinear science, Vol 8, 5 pages, 2009.
[42] Y.C. Jiao, Y. Yamamoto, et al. An Aftertreatment Technique for Improving the Accuracy of Adomian,s Decomposition Method. An international Journal computers and mathematics with applications. Vol 43, 16 pages, 783-798, 2001.
[43] M Merdan, A Gökdogan and A. Yildirim. On the numerical solution of the model for hiv infection of cd t cells. Computers and Mathematics with Applications. 62:118-123, 2011.
[44] Utku Erdogan and Turgut Ozis. A smart nonstandard finite difference scheme for second order nonlinear boundary value problems. Journal of Computational Phyciscs, 230(17): 6464-6474, 2011.
[45] G.A. Baker. Essentials of Padé approximations. Academic Express, London, Klebano, 1975.
[46] Elias Deeba, S.A. Khuri, and Shishen Xie. An algorithm for solving boundary value problems. Journal of Computational Physics, 159: 125-138, 2000.
[47] Xinlong Feng, Liquan Mei, and Guoliang He. An efficient algorithm for solving Troesch, s problem. Applied Mathematics and Computation, 189(1): 500-507, 2007.
[48] S. H Mirmoradia, I. Hosseinpoura, S. Ghanbarpour, and A. Barari. Application of an approximate analytical method to nonlinear Troesch, s problem. Applied Mathematical Sciences, 3(32):1579-1585, 2009.
[49] Hany N. Hassana and Magdy A. El-Tawil. An efficient analytic approach for solving two point nonlinear boundary value problems by homotopy analysis method. Mathematical methods in the applied sciences, 34:977-989, 2011.
[50] Mohammad Madani, Mahdi Fathizadeh, Yasir Khan, and Ahmet Yildirim. On the coupling of the homotopy perturbation method and laplace transformation. Mathematical and Computer Modelling, 53 (910): 1937-1945, 2011.
[51] M Fathizadeh, M Madani, Yasir Khan, Naeem Faraz, Ahmet Yildirim and Serap Tutkun. An effective modification of the homotopy perturbation method for mhd viscous flow over a stretching sheet. Journal of King Saud University-Science, 2011.
[52] Yasir Khan, Qingbiao Wu, Naeem Faraz, and Ahmet Yildirim. The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet. Computers and Mathematics with Applications, 61(11):3391-3399, 2011.
[53] Naeem Faraz and Yasir Khan. Analytical solution of electrically conducted rotating flow of a second grade fluid over a shrinking surface. Ain Shams Engineering Journal, 2(34): 221 -226, 2011.
[54] Yasir Khan, Hector Vazquez Leal and Q.Wu. An efficient iterated method for mathematical biology model. Neural Computing and Applications, pages 1-6, 2012.
[55] S. Thiagarajan, A. Meena, S. Anitha, and L. Rajendran. Analytical expression of the steady–state catalytic current of mediated bioelectrocatalysis and the application of He,s Homotopy perturbation method. J Math Chem Springer, DOI 10.1007/s10910-011-9854-z, 2011.
[56] U. Filobello-Niño, H. Vazquez-Leal, Y. Khan, A. Yildirim, V.M. Jimenez- Fernandez, A.L. Herrera May, R. Castañeda-Sheissa, and J.Cervantes-Perez. Using Perturbation methods and Laplace-Padé approximation to solve nonlinear problems. Miskolc Mathematical Notes, 14 (1) 2013 89-101 e-ISSN: 1787-2413. 2013
[57] J. Vleggar, Laminar boundary-layer behaviour on continuous accelerated surfaces. Chemical Engineering Science, Vol. 32, 1517-1525, 1977.
[58] M.M. Rashidi and D.D. Gangi J. Homotopy perturbation method for solving flow in the extrusion processes. IJE Transactions A: Basics Vol. 23, 267-272, 2010.
[59] C.Y.Wang. Flow due to a stretching boundary with partial slip-an exact solution of the Navier-Stokes equations. Chemical Engineering Science Vol. 23, 267-272, 2002.
[60] Crane, L.J Flow past a stretching plate. Zeitschrift fuer Angewandte Mathematik und Physik, 21, 645-647, 1970.
[61] D. Yao, V.L. Virupaksha, and B. Kim, “Study on squeezing flow during nonisothermal embossing of polymer microstructures,” Polymer Engineering and Science, Vol. 45, no. 5, pp. 652-660, 2005.
[62] W. E. Langlois, “Isothermal squeeze films,” Applied Mathematics, vol. 20, p. 131, 1962.
[63] R. L. Verma, “A numerical solution for squeezing flow between parallel channels” Wear, vol 72, no. 1, pp. 89-95, 1981.
[64] P Singh, V. Radhakrishnan and K. A. Narayan, “Squeezing flow between parallel plates” Ingenieur-Archin, vol. 60, no 4, pp. 274-281, 1990.
[65] W.H. Enright, K.R. Jackson, S.P. abd Norsett, and P.G. Thomsen. Interpolants for runge-kutta formulas. ACM TOMS, 12:193–218, 1986.
[66] E. Fehlberg. Klassische runge-kutta-formeln vierter und niedrigerer ordnung mit schrittweiten-kontrolle und ihre anwendung auf waermeleitungsprobleme. Computing, 6:61–71, 1970.

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Published

2017-01-18

How to Cite

Vazquez-Leal, H., Filobello-Nino, U., Sarmiento-Reyes, A., Sandoval-Hernandez, M., Perez-Sesma, J. A. A., Perez-Sesma, A., Jimenez-Fernandez, V. M., Huerta-Chua, J., Castro-Gonzalez, F., Sanchez-Orea, J., Hernández Machuca, S. F., Cuellar-Hernández, L., Pretelin Canela, J. E., Gasca-Herrera, A. E., Sampieri-Gonzalez, C. E., Palma-Grayeb, B. E., Contreras-Hernandez, A. D., Alvarez-Gasca, O., & Gonzalez-Martinez, F. J. (2017). Application of HPM to Solve Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates. American Scientific Research Journal for Engineering, Technology, and Sciences, 27(1), 161–178. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/1421

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