The asymptotic behaviour of quasi‐Newtonian flow through a very thin or a very long curved pipe

Abstract

We consider an injection of incompressible viscous quasi‐Newtonian fluid in a curved pipe with a smooth central curve γ. Starting from the Navier–Stokes system with shear‐dependent viscosity, used to describe the polymeric flow or the blood flow, the 1‐dimensional model was obtained via singular perturbation as ε, the ratio between cross section area and the length of the pipe, tends to 0. The method of two‐scale convergence was applied. The limit depends only on the tangential injection along γ and the velocity, as well as the pressure drop have the tangential direction.