Analytic Solutions to interfacial flows with nonlinear kinetic undercooling in a Hele-Shaw cell of a time-dependent gap

Abstract

This dissertation concerns the viscous fingering problem in Hele-Shaw flow with non-linear kinetic undercooling regulation. Hele-Shaw cells where the top plate is lifted uniformly at a prescribed speed and the bottom plate is fixed have been used to study interface related problems. This dissertation focuses on interfacial flows with nonlinear kinetic undercooling regularization in a radial Hele-Shaw cell with a time-dependent gap. The researcher obtained 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 Schwarz function approach is used to study the case where the initial shape is ellipse. Additionally, the researcher obtained the local existence of analytic solution of the moving boundary problem when the initial shape is a general analytic curve. In order to achieve this, the researcher first solve the Nonlinear Riemann-Hilbert Problem using techniques from PDE theory; then we reformulated the problem as an abstract Cauchy-Kowalewskaya evolution equation in some Banach spaces. Ultimately, the researcher applied the Nishida-Nirenberg theorem to obtain the existence and uniqueness of analytic solution to the nonlinear Hele-Shaw problem.