Viscoplastic Fluids: Mathematical Modeling and Applications

Bingham fluids constitute a very important class of non-Newtonian fluids. The modeling of Bingham materials is of crucial importance in industrial applications, since a large variety of materials (e.g. foams, pastes, slurries, oils, ceramics, etc.) exhibit the fundamental character of viscoplasticity, that is the capability of flowing only if the stress is above some critical value. The flow of these materials is difficult to predict, because of the presence of unknown interfaces separating the yielded and the unyielded regions which are difficult to track. This is particularly evident when the flow occurs in complex geometries and when major simplifications, such as lubrication approximation, can be applied. Indeed, in some cases the Bingham model may even lead to a paradox, known as the "lubrication paradox". In this chapter we focus on some practical situations of Bingham flow which are the subject of a current mathematical research (lubrication flows, asymptotic expansions, etc.). Such issues and developments arise, for example, in the petroleum industry and in many natural contexts.