Qualitative properties and approximation of solutions of Bingham flows: On the stabilization for large time and the geometry of the support

Abstract

We study the transient flow of an isothermal and incompressible Bingham fluID. Similar models arise in completely different contexts as, for instance, in material science, image processing and differential geometry. For the two-dimensional flow in a bounded domain we show the extinction in a finite time even under suitable nonzero external forces. We also consIDer the special case of a threedimensional domain given as an infinitely long cylinder of bounded cross section. We give sufficient conditions leading to a scalar formulation on the cross section. We prove the stabilization of solutions, when t goes to infinity, to the solution $$ u_\infty $$ of the associated stationary problem, once we assume a suitable convergence on the right hand forcing term. We give some sufficient conditions for the extinction in a finite time of solutions of the scalar problem. We show that, at least under radially symmetric conditions, when the stationary state is not trivial, $$ u_\infty \ne 0 $$ , there are cases in which the stabilization to the stationary solution needs an infinite time to take place. We end the paper with some numerical experiences on the scalar formulation. In particular, some of those experiences exhibit an instantaneous change of topology of the support of the solution: when the support of the initial datum is formed by two disjoint balls, but closed enough, then, instantaneously, for any t > 0, the support of the solution u( ·, t) becomes a connected set. Some other numerical experiences are devoted to the study of the “profile” of the solution and its extinction time.