A Special Class of Univalent Functions in Hele-Shaw Flow Problems

Abstract

We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection. Applying methods from the theory of univalent functions, we prove the invariance in time of $\Phi$-likeness property (a geometric property which includes starlikeness and spiral-likeness) for two basic cases: the inner problem and the outer problem. We study both zero and nonzero surface tension models. Certain particular cases are also presented.