Symplectic Solution for Stokes Flow in the Thin Film Coating Applications

摘要

The symplectic method was introduced for solving the problem of the Stokes flow in the thin film coating applications. The Lagrangian function of the Stokes flow was established. Stresses was found to be the dual variables of velocities and the Hamiltonian function was achieved by the Legendre transformation. Taking velocities and stresses as the basic variables, the Hamiltonian formulation could be introduced into Stokes flow problems. In the symplectic space the problem could be solved via the method of separation of variables and expansion of eigenfunctions. This symplectic analysis required the solving of an eigenvalue equation which was completely rational without any guess functions. Using the lateral and two ends boundary conditions to determine the eigenvalues and the coefficients, the analytical solutions could be derived. When the driven lids of the square cavity was moving in same directions, there are two stagnation points on the centre-line between the free surfaces. When the driven lids was moving in opposite directions (including the case that the below lid was moving whilst the above was immobile), there is only one stagnation point in the cavity. Numerical examples show that the symplectic method was effective for Stokes flow problems in a rectangular cavity.