On the Initial-boundary Value Problem for a Bingham Fluid in a Three Dimensional Domain

The initial-boundary value problem associated with the motion of a Bingham fluid is considered. The existence and uniqueness of strong solution is proved under a certain assumption on the data. It is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time-periodic 0. Introduction. The purpose of this paper is to establish the existence of strong solutions to a variational inequality which describes the motion of a Bingham fluid in a bounded three dimensional domain. A Bingham fluid is a rigid viscoplastic fluid which is governed by a special constitutive law such that it moves like a rigid body if a certain function of the stresses does not reach the yield limit and it behaves like a viscous fluid when the yield limit is reached. Since the motion is governed by two entirely different stress-strain relations depending on the state of stresses, the conservation of momentum is expressed in terms of a variational inequality so that one can avoid the difficulty of separating the fluid zone and the rigid zone. The initial-boundary value problem we shall study is formulated as (—,w-u\ +a(u,w u) + b(u,u,w) + J(w) J{u)