Planar Squeeze Flow of a Bingham Fluid

In this paper we study the planar squeeze flow of a Bingham plastic in the lubrication approximation. We assume that the domain occupied by the fluid is closed at one end and open at the other (planar geometry). We consider two cases: (i) planar walls approaching each other in a prescribed way; (ii) parallel walls whose shape depends on both time and longitudinal coordinate. The dynamics of the unyielded region is determined exploiting the integral formulation of the linear momentum balance. We prove that in proximity of the closed end the material is always yielded, so that the rigid part is always detached from it. When dealing with case (ii), we show that the dynamics of the rigid domain is governed by a very complex integral equation, whose qualitative analysis is beyond the aims of this paper. Conversely, in case (i) we obtain an almost explicit solution.