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Mathematical Problems in Engineering
Volume 2014, Article ID 151470, 12 pages
Research Article

Symplectic Exact Solution for Stokes Flow in the Thin Film Coating Applications

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Received 6 December 2013; Revised 27 February 2014; Accepted 27 February 2014; Published 24 April 2014

Academic Editor: Balaji Raghavan

Copyright © 2014 Yan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The symplectic analytical method is introduced to solve the problem of the stokes flow in the thin film coating applications. Based on the variational principle, the Lagrangian function of the stokes flow is established. By using the Legendre transformation, the dual variables of velocities and the Hamiltonian function are derived. Considering velocities and stresses as the basic variables, the equations of stokes flow problems are transformed into Hamiltonian system. The method of separation of variables and expansion of eigenfunctions are developed to solve the governing equations in Hamiltonian system, and the analytical solutions of the stokes flow are obtained. Several numerical simulations are carried out to verify the analytical solutions in the present study and discuss the effects of the driven lids of the square cavity on the dynamic behavior of the flow structure.