Approximate equations for the slow spreading of a thin sheet of Bingham plastic fluid

A model that can approximately describe a non-Newtonian fluid such as paint and fluid mud is a Bingham plastic with a yield stress. To facilitate the study of slow but transient spreading of a thin sheet of fluid mud, we need the approximate equations governing the nonlinear motion. Beginning with a modified Bingham model with two viscosities, the approximate equations are derived by a systematic perturbation analysis. The controlling parameters are found to be the Reynolds number R, the shallowness ratio h̄/L̄=δ1/2, and the ratio of viscosities β=ν/ν1 [as defined in (45)]. Results valid for R,β,δ≪1 but δ/β2≤O(1) are obtained. In the special case when δ/β2≪1, the limits can also be obtained by heuristic arguments similar to those in the lubrication theory. Two examples are discussed.