Evolution of waves in liquid films on moving substrates

TL;DR

This paper extends a classic liquid film flow model to include moving substrates, analyzing stability in linear and nonlinear regimes, and finds a nonlinear stabilization mechanism in the absence of surface tension.

Contribution

It introduces an extended IBL model for moving substrates and provides a comprehensive stability analysis including nonlinear effects, relevant for industrial coating processes.

Findings

Wide stability regions identified for moving and falling films.

Nonlinear effects can stabilize the film interface without surface tension.

Operational conditions for galvanizing lines are within stable regimes.

Abstract

Accurate and computationally accessible models of liquid film flows allow for optimizing coating processes such as hot-dip galvanization and vertical slot-die coating. This paper extends the classic three-dimensional integral boundary layer (IBL) model for falling liquid films (FF) to account for a moving substrate (MS). We analyze the stability of the liquid films on vertically moving substrates in a linear and a nonlinear setting. In the linear analysis, we derive the dispersion relation and the temporal growth rates of an infinitesimal disturbance using normal modes and linearized governing equations. In the nonlinear analysis, we consider disturbances of finite size and numerically compute their evolution using the set of nonlinear equations in which surface tension has been removed. We present the region of (linear) stability of both FF and MS configurations, and we place the operating conditions of an industrial galvanizing line in these maps. A wide range of flow conditions was analyzed and shown to be stable according to linear and nonlinear stability analyses. Moreover, the nonlinear analysis, carried out in the absence of surface tension, reveals a nonlinear stabilizing mechanism for the interface dynamics of a liquid film dragged by an upward-moving substrate.