Penalty method and problems of liquid-solid interaction

Some applications of the so-called solidification method are presented. The first one deals with the classical problem on the motion of rigid bodies in viscous incompressible fluids. The solidification method consists in considering the bodies as fluids whose viscosity tends to infinity. This technique enables us to solve the problem on an arbitrary time interval and take into account possible collisions of the bodies. The behavior of the bodies before and after the collisions and during the time when they are touching one another is also described. As a second application, a problem on a solid-liquid phase transition in a movable substance is considered. Dedicated to the memory of V. N. Monakhov