Thin film flow dynamics on fiber nets

TL;DR

This paper studies the mathematical behavior of thin film flows on fiber nets, proving existence, qualitative properties, and convergence to uniform coating distribution relevant for industrial water and oil separation applications.

Contribution

It introduces a rigorous analysis of fourth order degenerate parabolic equations on graph domains with specific boundary conditions, addressing a novel problem in thin film flow modeling.

Findings

Existence of non-negative weak solutions established.

Convergence to a uniform steady state proven for certain parameters.

Qualitative behavior of solutions characterized.

Abstract

We analyze existence and qualitative behavior of non-negative weak solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff's boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.