Modeling and dynamics of axisymmetric thin liquid film flow along a conical surface

TL;DR

This paper develops models to analyze the stability and wave dynamics of thin liquid films flowing along conical surfaces, revealing the influence of curvature and flow rate on wave behavior and stability thresholds.

Contribution

It introduces a Benney-type equation and a low-dimensional model specifically for conical surfaces, providing accurate, low-cost analysis of linear and nonlinear wave phenomena.

Findings

Streamwise curvature significantly affects stability thresholds.

A second-order model accurately captures wave dynamics with reduced computational cost.

Flow behavior on conical surfaces closely resembles flat-plate flow when local parameters are used.

Abstract

This study focuses on the modeling and dynamics of gravity-driven, axisymmetric thin liquid film flow along a conical surface. Spatial linear stability analysis is performed on the basis of a Benney-type equation derived for the present configuration. In particular, streamwise curvature of the free surface is found to exert a crucial influence on the stability threshold. For simulations of surface waves, a second-order low-dimensional model is developed under the long-wave assumption, achieving accuracy comparable to direct numerical simulations at far lower cost. With this model, the characteristics of both linear and nonlinear waves are examined. A key difference from the flow over a flat plate is the dependence of wave dynamics on radial distance from the cone apex. At relatively high flow rates, a transition from solitary to sinusoidal waves is observed, with the transition position correlating closely with the linear stability threshold. Within the parameter range investigated, quantitative results of the conical film flow are almost identical to those in the flat-plate case when local parameters are substituted, indicating that inertial effects of the conical geometry are negligible. The models and findings presented in this paper may aid the design and optimization of industrial processes such as film coating and liquid-film-based heat and mass transfer on conical surfaces.