Regularization procedures of singular free boundary problems in rotational Bingham flows

The flow of a Bingham fluid in a concentric cylinder viscometer can be give rise to free boundary problems. In the region between the two rotating cylinders the fluid behaves like a viscous fluid if the shear stress, expressed as a linear function of the shear rate, exceeds a yield value, and like a rigid body otherwise. The surfaces dividing fluid and rigid zones are the free boundaries, where we assume that the shear rate is zero. Starting from a model proposed in a previous work, where just one fluid and one rigid zone were present, here we study the physical problem taking into account the possible formation of further rigid zones inside the fluid. We introduce a kind of "regularization" of the problem, pointing out the existence of a unilateral constraint to which the formation of new rigid zones is related. Each time the constraint comes into play, the problem must be reformulated by letting new free boundaries appear bounding the newly created rigid region. Moreover we analyse the possible collapsing of the rigid zones.