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 $0 \leq \delta \ll 1$. 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 ($\delta=0$), in the case $\frac{\ell}{\pi} \notin \mathbb{Z}$, steady states are unique. For $0 < \delta \ll 1$, steady states are shown to be locally unique for any $\ell$. These steady states are stable for $\ell < \pi$, while they are unstable for $\ell > \pi$. 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 $\ell = \pi$, we study the dynamics of the solutions close to the manifold of periodic orbits in the critical case $\ell = \pi$ on the large time scale $\tau = \delta^2 t$. It turns out that in the time scale $\tau$ this dynamics can be approximated by a system of ordinary differential equations.