We investigated an axisymmetric unsteady two-dimensional flow of nonconducting, incompressible second grade fluid between two circular plates. The similarity transformation is applied to reduce governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE) in dimensionless form. The resulting nonlinear boundary value problem is solved using homotopy analysis method and numerical method. The effects of appropriate dimensionless parameters on the velocity profiles are studied. The total resistance to the upper plate has been calculated. 1. Introduction Squeezing flows are induced by externally normal stresses or vertical velocities by means of moving boundary. Squeezing flows have many applications in food industry, especially in chemical engineering. Some practical examples of squeezing flow include polymer processing, compression and injection molding. In addition, the lubrication system can also be modeled by squeezing flows. The study of squeezing flows has its origins in the 19th century and continues to receive considerable attention due to the practical applications in physical and biophysical areas. Stefan [1] published a classical paper on squeezing flow by using lubrication approximation. Such types of flow exist in lubrication when there is squeezing flow between two parallel plates. The tackiness of liquid adhesives also reflects squeeze film effects [2]. The squeeze film geometry has been studied extensively since 1947. Other applications in the biomechanics area relate to squeezing flow between parallel plates and the alternation between contraction and expansion of the blood vessels. In addition, polymer extrusion processes are modeled using squeezing flow of viscous fluids [3]. The squeezing flow between parallel plates when the confining walls have a transverse motion has great importance in hydrodynamic lubrication theory Langlois [4] and Salbu [5] have analyzed isothermal compressible squeeze films neglecting inertial effects. Thorpe [6] presented an explicit solution of the squeeze flow problem taking inertial terms into account. Later, P. S. Gupta and A. S. Gupta [7] showed that the solution given by [6] fails to satisfy the boundary conditions. Gupta solution, however, is restricted only to small Reynolds number. Squeeze film between two plane annuli with fluid inertia effects has been studied by Elkouh [8]. Some numerical solutions of squeezing flow between parallel plates has been conducted by Verma [9] and later by Singh et al. [10]. In addition, Hamza [11] has considered suction and injection
References
[1]
M. J. Stefan, “Versuch Uber die scheinbare adhesion,” Akademie der Wissenschaften in Wien. Mathematik-Naturwissen, vol. 69, p. 713, 1874.
[2]
J. J. Bickerman, Development in Theoretical and Applied Mathematics, vol. 3, Academic Press, New York, NY, USA, 1958.
[3]
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.
[4]
W. E. Langlois, “Isothermal squeeze films,” Applied Mathematics, vol. 20, p. 131, 1962.
[5]
E. O. Salbu, “Compressible squeeze films and squeeze bearings,” Journal of Basic Engineering, vol. 86, p. 355, 1964.
[6]
J. F. Thorpe, in Development in Theoretical and Applied Mathematics, W. A. Shah, Ed., vol. 3, Pergamon Press, Oxford, UK, 1967.
[7]
P. S. Gupta and A. S. Gupta, “Squeezing flow between parallel plates,” Wear, vol. 45, no. 2, pp. 177–185, 1977.
[8]
A. F. Elkouh, “Fluid inertia effects in a squeeze film between two plane annuli,” Journal of Tribology, vol. 106, no. 2, pp. 223–227, 1984.
[9]
R. L. Verma, “A numerical solution for squeezing flow between parallel channels,” Wear, vol. 72, no. 1, pp. 89–95, 1981.
[10]
P. Singh, V. Radhakrishnan, and K. A. Narayan, “Squeezing flow between parallel plates,” Ingenieur-Archiv, vol. 60, no. 4, pp. 274–281, 1990.
[11]
E. A. Hamza, “Suction and injection effects on a similar flow between parallel plates,” Journal of Physics D: Applied Physics, vol. 32, no. 6, pp. 656–663, 1999.
[12]
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.
[13]
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, vol. 26, no. 3-4, pp. 409–417, 1991.
[14]
R. M. Barron and J. T. Wiley, “Newtonian flow theory for slender bodies in a dusty gas,” Journal of Fluid Mechanics, vol. 108, pp. 147–157, 1981.
[15]
A. K. Ghosh and L. Debnath, “Hydromagnetic Stokes flow in a rotating fluid with suspended small particles,” Applied Scientific Research, vol. 43, no. 3, pp. 165–192, 1986.
[16]
Lokenath Debnath and A. K. Ghosh, “On unsteady hydromagnetic flows of a dusty fluid between two oscillating plates,” Applied Scientific Research, vol. 45, no. 4, pp. 353–365, 1988.
[17]
M. H. Hamdan and R. M. Barron, “Analysis of the squeezing flow of dusty fluids,” Applied Scientific Research, vol. 49, no. 4, pp. 345–354, 1992.
[18]
R. J. Grimm, “Squeezing flows of Newtonian liquid films an analysis including fluid inertia,” Applied Scientific Research, vol. 32, no. 2, pp. 149–166, 1976.
[19]
W. A. Wolfe, “Squeeze film pressures,” Applied Scientific Research, vol. 14, no. 1, pp. 77–90, 1965.
[20]
D. C. Kuzma, “Fluid inertia effects in squeeze films,” Applied Scientific Research, vol. 18, no. 1, pp. 15–20, 1968.
[21]
R. J. Grimm, “Squeezing flows of Newtonian liquid films an analysis including fluid inertia,” Applied Scientific Research, vol. 32, no. 2, pp. 149–166, 1976.
[22]
. TICHY JA and . WINER WO, “Inertial considerations in parallel circular squeeze film bearings,” Transactions of the ASME: Journal of Lubrication Technology, vol. 92, pp. 588–592, 1970.
[23]
C. Y. Wang and L. T. Watson, “Squeezing of a viscous fluid between elliptic plates,” Applied Scientific Research, vol. 35, no. 2-3, pp. 195–207, 1979.
[24]
R. Usha and R. Sridharan, “Arbitrary squeezing of a viscous fluid between elliptic plates,” Fluid Dynamics Research, vol. 18, no. 1, pp. 35–51, 1996.
[25]
H. M. Laun, M. Rady, and O. Hassager, “Analytical solutions for squeeze flow with partial wall slip,” Journal of Non-Newtonian Fluid Mechanics, vol. 81, no. 1-2, pp. 1–15, 1999.
[26]
M. H. Hamdan and R. M. Barron, “Analysis of the squeezing flow of dusty fluids,” Applied Scientific Research, vol. 49, no. 4, pp. 345–354, 1992.
[27]
P. T. Nhan, “Squeeze flow of a viscoelastic solid,” Journal of Non-Newtonian Fluid Mechanics, vol. 95, pp. 43–362, 2000.
[28]
S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992.
[29]
S.-J. Liao, “An explicit, totally analytic approximate solution for Blasius' viscous flow problems,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 759–778, 1999.
[30]
S.-J. Liao, “On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet,” Journal of Fluid Mechanics, vol. 488, pp. 189–212, 2003.
[31]
S. J. Liao, Beyond Perturbation, vol. 2 of Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, FL, 2004.
[32]
S. J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004.
[33]
S. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,” International Journal of Heat and Mass Transfer, vol. 48, no. 12, pp. 2529–2539, 2005.
[34]
S. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186–1194, 2005.
[35]
S. Liao, J. Su, and A. T. Chwang, “Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body,” International Journal of Heat and Mass Transfer, vol. 49, no. 15-16, pp. 2437–2445, 2006.
[36]
M. M. Rashidi and S. A. Mohimanian Pour, “Analytic approximate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis method,” Nonlinear Analysis. Modelling and Control, vol. 15, no. 1, pp. 83–95, 2010.
[37]
M. M. Rashidi, D. D. Ganji, and S. Dinarvand, “Approximate traveling wave solutions of coupled Whitham-Broer-Kaup shallow water equations by homotopy analysis method,” Differential Equations & Nonlinear Mechanics, vol. 2008, Article ID 243459, 8 pages, 2008.
[38]
M. M. Rashidi, G. Domairry, and S. Dinarvand, “Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 708–717, 2009.
[39]
G. Birkhoff, Hydrodynamics, Princeton University Press, Princeton, NJ, USA, 1960.
