Applications and Novel Regularization of the Thin-Film Equation

TL;DR

This paper introduces a new regularization of the thin-film equation to better model droplet spreading, along with numerical methods and theoretical analysis of the solution's properties.

Contribution

It presents the Geometric Thin-Film Equation as a novel regularization and develops robust numerical schemes with proven regularity and convergence.

Findings

Proven regularity and convergence of the numerical solutions

Existence and uniqueness of solutions for a wide range of initial conditions

Development of a mesh-free, fast numerical scheme for 2D and 3D simulations

Abstract

The classical no-slip boundary condition of the Navier-Stokes equations fails to describe the spreading motion of a droplet on a substrate due to the missing small-scale physics near the contact line. In this thesis, we introduce a novel regularization of the thin-film equation to model droplet spreading. The solution of the regularized thin-film equation -- the Geometric Thin-Film Equation is studied and characterized. Two robust numerical solvers are discussed, notably, a fast and mesh-free numerical scheme for simulating thin-film flows in two and three spatial dimensions. Moreover, we prove the regularity and convergence of the numerical solutions. The existence and uniqueness of the solution of the Geometric Thin-Film Equation with respect to a wide range of measure-valued initial conditions are also discussed.