Analysis of a three-dimensional fluid flow in rotating cylinders

TL;DR

This paper derives and analyzes a PDE model for a thin fluid film in a rotating cylinder, exploring steady states, stability, and periodic solutions influenced by rotation, surface tension, and gravity effects.

Contribution

It introduces a new PDE model for the fluid film in rotating cylinders and characterizes the stability and periodicity of steady states under various physical parameters.

Findings

Steady states are unique when gravity is absent and the ratio of length to radius is not an integer multiple of pi.

Steady states are locally unique and stable for cylinder length ratios less than pi, unstable otherwise.

Existence of a manifold of time-periodic solutions in the absence of gravity for all cylinder length ratios.

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.