The accurate and comprehensive model of thin fluid flows with inertia on curved substrates

TL;DR

This paper develops a comprehensive mathematical model for thin viscous Newtonian fluid flows over curved substrates, accurately capturing effects like curvature, gravity, inertia, and dissipation, applicable to complex geometries and phenomena.

Contribution

It introduces the most complete model for thin film Newtonian fluid flow on curved surfaces, derived using centre manifold theory, encompassing various physical effects and geometries.

Findings

Model accurately includes curvature, gravity, inertia, and dissipation effects.

Applicable to wave phenomena, drop formation, and vortices on curved substrates.

Provides a basis for deriving simpler models through truncation.

Abstract

Consider the 3D flow of a viscous Newtonian fluid upon a curved 2D substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We derive a comprehensive model of the dynamics of the film, the model being expressed in terms of the film thickness and the average lateral velocity. Based upon centre manifold theory, we are assured that the model accurately includes the effects of the curvature of substrate, gravitational body force, fluid inertia and dissipation. The model may be used to resolve wave-like phenomena in the dynamics of viscous fluid flows over arbitrarily curved substrates such as cylinders, tubes and spheres. We briefly illustrate its use in simulating drop formation on cylindrical fibres, wave transitions, Faraday waves, viscous hydraulic jumps, and flow vortices in a compound channel. These models are the most complete models for thin film flow of a Newtonian fluid; many other thin film models can be obtained by different truncations of the dynamical equations given herein.