Finite Element Approximation of a Moving Boundary Problem Arising in the Modeling of the Spin Coating Process for Thin Films

Abstract: Using the Navier-Stokes equations along with a continuity equation, a one-dimensional model is developed to describe the spin coating process of thin polymeric films. The resulting model is a system of a parabolic partial differential equation coupled with an integral equation as well as with an ordinary differential equation describing the motion of a moving boundary. Viscosity and diffusivity are allowed to be varied in the model. To be able to perform the finite element approximation of the model equations, the moving boundary is fixed. Then the finite element method is applied along with the so called Method of Lines resulting in a semi-discrete problem, a large system of ordinary differential equations which is then solved with MATLAB. We present an existence and uniqueness result what concerns the semi-discrete solutions. Finally, we illustrate numerically the behavior of the solutions to our model.