Analysis of a three-dimensional fluid flow in rotating cylinders

Abstract

Subject of consideration is the modelling and analysis of a capillary-driven three-dimensional rimming-flow problem. We present the derivation of a fourth-order quasilinear degenerate-parabolic partial differential equation for the height $h>0$ of a fluid film coating the inner wall of a cylinder that rotates around a horizontal axis. The equation arises from a rescaled Navier-Stokes system for thin fluid films by means of a lubrication approximation and accounts for the physical effects of rotation, surface tension and gravity. The effect of the latter is measured by a non-dimensional parameter $0\le \hat{I}´\ll 1$. We characterise the structure of the steady states depending on the ratio $\ell$ of the cylinder length to its radius. In the absence of gravity ($\hat{I}´=0$), in the case $\frac{\ell }{\ddot{I}}€\notin \mathbb{Z}$, steady states are unique. For $0<\hat{I}´\ll 1$, steady states are shown to be locally unique for any $\ell$. These steady states are stable for $\ell <\ddot{I}€$, while they are unstable for $\ell >\ddot{I}€$. Furthermore, in the absence of gravity, for all $\ell >0$, we show that there exists a manifold of time-periodic solutions. In the critical case $\ell =\ddot{I}€$, we study the dynamics of the solutions close to the manifold of periodic orbits in the critical case $\ell =\ddot{I}€$ on the large time scale $\ddot{I}„=\hat{I}{´}^{2}t$. It turns out that in the time scale $\ddot{I}„$ this dynamics can be approximated by a system of ordinary differential equations.

43 pages, 8 figures