Método de Diferencias Finitas para un Problema de Bingham Unidimensional

Abstract

In this work, a numerical method of finite differences is proposed to solve an unidimensional Bingham problem, and some theoretical properties of the solution are proved, and corroborate numerical results. A Bingham fluid is a non-newtonian fluid, whose viscous behaviour makes his layers move only if shear stress is greater than a threshold value to. There are some previous theoretical results that allows
us to affirm existence and uniqueness of the solution under certain initial conditions.
The proposed method is a finite difference scheme with spatial variable step, that is, while we move a fixed time step, we adjust the grid in such a way that the spatial step represents the free boundary advance. Using an internal calculus of fixed point is possible to reduce the number of flops, making feasible the implementation of
the method in a computer. Besides that, a theorem of existence and uniqueness is proved for more general cases, and also a result about asymptotic behaviour is demonstrated.