Nonlinear periodic wavetrains in thin liquid films falling on a uniformly heated horizontal plate

TL;DR

This paper investigates nonlinear wave patterns in thin liquid films on heated horizontal plates, deriving analytical solutions for localized and modulated wave structures using perturbation methods and the Benney model.

Contribution

It provides analytical descriptions of complex nonlinear wave patterns in heated thin films, linking experimental observations with theoretical models under specific conditions.

Findings

Localized periodic wave structures can be analytically derived.

Modulated wave trains are described by solutions to the complex Ginzburg-Landau equation.

Conditions for formation of these structures depend on surface tension and Biot number.

Abstract

A thin liquid film falling on a uniformly heated horizontal plate spreads into fingering ripples that can display a complex dynamics ranging from continuous waves, nonlinear spatially localized periodic wave patterns (i.e. rivulet structures) to modulated nonlinear wavetrain structures. Some of these structures have been observed experimentally, however conditions under which they form are still not well understood. In this work we examine profiles of nonlinear wave patterns formed by a thin liquid film falling on a uniformly heated horizonal plate. In this purpose, the Benney model is considered assuming a uniform temperature distribution along the film propagation on the horizontal surface. It is shown that for strong surface tension but relatively small Biot number, spatially localized periodic-wave structures can be analytically obtained by solving the governing equation under appropriate conditions. In the regime of weak nonlinearity, a multiple-scale expansion combined with the reductive perturbation method leads to a complex Ginzburg-Landau equation, the solutions of which are modulated periodic pulse trains which amplitude, width and period are expressed in terms of characteristic parameters of the model.