Modeling film flows down a rotating slippery cylinder

TL;DR

This paper develops a model for gravity-driven viscous films on a rotating slippery cylinder, analyzing how wall slip and rotation influence flow stability, nonlinear wave behavior, and film breakup.

Contribution

It introduces an asymptotic model incorporating rotation and wall slip effects, providing new insights into flow stability and nonlinear dynamics on a rotating cylinder.

Findings

Wall slip enhances flow instability on both surfaces.

Rotation amplifies instability for outer surface flow but reduces it for inner surface flow.

Increasing slip length promotes film breakup and droplet formation.

Abstract

This study investigates the nonlinear stability and dynamics of gravity-driven viscous films on a vertical rotating cylinder, considering both outer and inner surface flows with slip conditions at the cylinder wall. We develop an asymptotic model for the combined effects of rotation and wall slippage. Linear stability analysis indicates that wall slippage enhances instability on both surfaces, while rotation has differing impacts: it amplifies instability due to slip for outer surface flow but reduces it for inner surface flow. A weakly nonlinear stability analysis is then conducted to explore the combined impact of rotation and wall slip on flow stability beyond the linear regime, including the bifurcation of the nonlinear evolution equation for both surfaces. The traveling wave solution of the model is analyzed, showing how rotation affects nonlinear wave speed with a slippery wall. A stability analysis of the traveling wave solutions is also performed. Numerical simulations of the nonlinear evolution of the free surface reveal that increasing slip length enhances the choke phenomenon in inner surface flow, while rotation can delay this effect. Additionally, simulations show that for flow along the outer surface of a slippery rotating cylinder, the film tends to break up into droplets in the presence of rotation.