Flow of Bingham fluid through a three dimensional thin layer

Abstract

The paper is devoted to the study of asymptotic behavior of the solution of three dimensional steady flow of Bingham fluid through a three dimensional thin layer with Dirichlet boundary conditions. We are interested in the asymptotic behavior, to this aim we prove some convergence results concerning the velocity and pressure when the thickness tends to zero. The limit problem obtained after transforming the original problem into one posed over a fixed reference domain and the parameter representing the thickness of the layer tend to zero is studied. The lower-dimensional constitutive law and the differential equation satisfied by the limit variables in the non rigid zone are obtained. In addition, the uniqueness of limit solution has been also established. Existence and uniqueness results and a lower-dimensional constitutive law are obtained. An identical study of a two-dimensional problem yields a one dimensional model prevalent in engineering literature.