On a Shallow Water Model for Non-Newtonian Fluids

Abstract

The aim of this work is to modelize the evolution of a viscoelastic fluid through a Shallow Water system. The fluid hydrodynamic in this situation comes from the Navier-Stokes equations but the difficulty lies in the definition of the stress tensor for this non-Newtonian fluid. In order to get an expression for it we focus on the microscopic properties of the fluid by considering a diluted solution of polymer liquids. A kinetic theory for this type of solutions gives us “constitutive equations” that relate the stress tensor to the velocity. They are known as the Fokker–Planck equations. Once the stress tensor is defined we shall derive the model by developing the asymptotic analysis of the joined system of equations to obtain a Shallow Water type model following [6]. Finally we show a numerical test to check the influence of the polymers in the behavior of the flow.