A theoretical and numerical study of gravity driven coating flow on cylinder and sphere: fingering instability

TL;DR

This paper develops a mathematical model and analytical formulas to understand fingering instability patterns in gravity-driven coating flows on cylinders and spheres, validated by numerical analysis.

Contribution

It introduces a unified asymptotic and modal analysis framework for fingering instability on curved surfaces, including new formulas for the most unstable wave number.

Findings

Derived analytical formulas for fingering patterns on cylinders and spheres.

Validated asymptotic theory with transient growth analysis.

Disjoining pressure influences growth rate but not the wave number.

Abstract

To find the regularities of formed fingers in gravity driven coating flows on upper cylinder and sphere, the mathematical formulation to model the fingering instability on cylindrical or spherical surface which consists of a capillary wave equation and a linear perturbation equation is constructed, based on the leading order governing equation in standard cylindrical or spherical coordinate system. A disjoining pressure model is introduced to simulate the partial wetting process near moving contact line. The fingering number in high $Bo$ coating flow is focused on from a linear perspective. Using an asymptotic theory, the high $Bo$ limits of the linear perturbation equations for both the cylindrical and spherical problems are proved to degenerate into a common eigenvalue problem. Two analytical formulae concerning the most unstable wave number which can be described as two power laws are derived via a method of modal analysis for the cylindrical and spherical problem, respectively. These formulae provide a succinct method to estimate the fingering number if the onset of the fingering instability is known. A method of transient growth analysis which makes no high $Bo$ assumptions is used to verify the accuracy of the asymptotic theory and modal analysis. In particular, an asymptotic behavior of the most unstable wave number as $Bo$ increases is highlighted. We show that the linear growth rate may be improved by the disjoining pressure, which can be attributed to the effect of partial wetting, but the most unstable wave number is insensitive to the wetting model.