Thin Film Motion of an Ideal Fluid on the Rotating Cylinder Surface

TL;DR

This paper derives and analyzes shallow water equations for thin ideal fluid films on rotating cylinders, revealing solitary wave solutions and integrability properties through analytical and symbolic methods.

Contribution

It introduces a novel set of shallow water equations for rotating cylinders and demonstrates their integrability and solitary wave solutions using advanced analytical techniques.

Findings

Thin film precesses with velocity different from cylinder rotation

Exact solitary wave solutions are obtained

Equations are shown to be integrable via Painleve analysis

Abstract

The shallow water equations describing the motion of thin liquid film on the rotating cylinder surface are obtained. These equations are the analog of the modified Boussinesq equations for shallow water and the Korteweg-de Vries equation. It is clear that for rotating cylinder the centrifugal force plays the role of the gravity. For construction the shallow water equations (amplitude equations) usual depth-averaged and multi-scale asymptotic expansion methods are used. Preliminary analysis shows that a thin film of an ideal incompressible fluid precesses around the axis of the cylinder with velocity which differs from the angular velocity of rotating cylinder. For the mathematical model of the liquid film motion the analytical solutions are obtained by the Tanh-Function method. To illustrate the integrability of the equations the Painleve analysis is used. The truncated expansion method and symbolic computation allows to present an auto-Backlund transformation. The results of analysis show that the exact solutions of the model correspond to the solitary waves of different types.