Exact solutions to interfacial flows with kinetic undercooling in a Hele-Shaw cell of time-dependent gap

Hele-Shaw cells where the top plate is moving uniformly at a prescribed speed and the bottom plate is fixed
have been used to study interface related problems. This paper focuses on interfacial flows with linear and nonlinear kinetic
undercooling regularization in a radial Hele-Shaw cell with a time dependent gap. We obtain some exact solutions of the
moving boundary problems when the initial shape is a circle, an ellipse or an annular domain. For the nonlinear case, a linear
stability analysis is also presented for the circular solutions. The methodology is to use complex analysis and PDE theory.