Finite element methods for PDEs with moving boundaries

Abstract : Fluid flows with moving boundaries are encountered in many applications such as spray cooling, spray coating, ink-jet printing, fuel injecting, fluid-structure interactions, aerodynamics, ship hydrodynamics, etc. The fluid flow and/or scalar quantities (temperature, concentration, etc) in these applications are described by parabolic PDEs (Navier-Stokes equations, energy equation, etc) in a time-dependent domain. Apart from other challenges associate with the numerical solution of these nonlinear PDEs, presence of the moving boundaries make the computations more challenging.

In this talk, a finite element scheme based on arbitrary Lagrangian-Eulerian (ALE) approach will be presented for PDEs in time-dependent domain. After a brief discussion on the numerical challenges, the stability estimates for the continuous and discrete forms of the model will be provided. In addition, the algorithms and the implementations will be discussed. Finally, an array of numerical results for impinging droplets, rising bubble, flows with surfactants on free surface, flow over oscillating aerofoil, etc, will be presented.