  We look at a homogeneous incompressible fluid with a time and space variable
rheology of Bingham type, which is governed by a coupling equation. Such a
system is a simplified model for a gas-particle mixture that flows under the
influence of gravity. The main application of this kind of model is pyroclastic
flows in the context of volcanology. In order to prove long time existence of
weak solutions, classical Galerkin approximation method coupled with a priori
estimates allows us to partially solve the problem. A difficulty remains with
the stress tensor, which must satisfy an implicit constitutive relation. Some
numerical simulations of a flow of this type are given in the last section.
These numerical experiments highlight the influence of the fluidization
phenomenon in the flow.
