Well-posedness of the free surface problem for non-Newtonian fluids between cylinders rotating at different speeds: Weissenberg effect

Abstract

In this paper, we aim to provide a rigorous mathematical proof of the Weissenberg effect. We consider a liquid filling the semi-infinite annular region between two concentric cylinders rotating at different steady speeds. The behavior of the fluid leads to different shapes of the free surface. For a Newtonian fluid, the meniscus descends due to centrifugal forces. However, for a certain non-Newtonian fluid, the meniscus climbs up the inner cylinder phenomenon known as the Weissenberg effect. In a previous paper (2022), the author studied the case of a Newtonian fluid and developed a rigorous mathematical theory. In the present paper, we focus on the non-Newtonian case. By decomposing the problem into a system consisting of the Stokes problem and a transport equation, we prove the convergence of the formal perturbation series introduced by Joseph and Fosdick (1973), thereby providing a rigorous justification of the Weissenberg effect.