We consider in this paper the classical one-phase Stefan problem in dimensions two and three in the undercooled situation. By means of matched asymptotic expansions, a mechanism of cusp formation is presented for interfaces that are initially smooth.

The dynamics of blobs of viscous fluid in a non-planar Hele–Shaw cell is considered. The general approach developed by Richardson and some of the resulting analytic techniques are extended to flows in non-planar cells, including cells shaped as surfaces of revolution and helical surfaces. An example related to the development and coalescence of two initially separated blobs in a cell on a spherical surface is presented. Some applications to mathematically equivalent problems dealing with planar Hele–Shaw cells with a non-uniform gap and flows through porous media are also discussed.

A thin-film approximation is used in an analysis of the flow of a thin trickle of viscous fluid down a near-vertical plane. An approximate similarity solution is obtained, representing essentially a source (or sink) flow. Several interpretations of the solution are discussed.

The problem of steady-state propagation of a finger or a bubble of inviscid fluid through a Hele–Shaw cell filled by a viscous non-Newtonian, including visco-plastic (Bingham) fluid is addressed. Only flows symmetric relative to the cell axis are considered. It is shown that, using a hodograph transform, this non-linear free boundary problem can be reduced to the solution of an elliptic system of linear partial differential equations in a fixed domain with part of the boundary being curvilinear. The resulting boundary-value problem is solved numerically using the Finite Element Method. Finger shapes are calculated, and the approach is verified for one-parameter family of solutions which correspond to the well-known Saffman–Taylor solutions for the case of a Hele–Shaw cell filled by a Newtonian fluid. Results are also shown for fingers with non-Newtonian fluids. In the case of a cell filled by visco-plastic (Bingham) fluid, it is shown that stagnant zones propagate with the finger, and that the rear part of the finger has constant width. The same approach is applied to finding a two-parametric family of solutions for steady propagating bubbles. Results are shown for bubbles in Hele–Shaw cell filled by power-law and Bingham fluids.

A fully nonlinear, degenerate parabolic equation arising in the theory of damage mechanics is shown to be well-posed. Its solutions blow up in finite time and, under suitable conditions on the initial configuration, the blow-up set, corresponding to the portion of the material which breaks at the blow-up time, is an interval of nonzero measure. In a special but physically relevant case the problem reduces to the study of the blow-up set of solutions of the quasilinear equation