Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Applications and Novel Regularization of the Thin-Film Equation

Khang Ee Pang

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.

Applications and Novel Regularization of the Thin-Film Equation

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.
Paper Structure (96 sections, 19 theorems, 459 equations, 59 figures, 3 tables, 1 algorithm)

This paper contains 96 sections, 19 theorems, 459 equations, 59 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation and Overview of Thesis
  4. Review of the Classical Thin-Film Equation
  5. Lubrication Approximation
  6. Similarity Solution
  7. Contact Line Singularity
  8. Classical Droplet Spreading Models
  9. Slip-length Model
  10. Precursor Film Model
  11. Preliminaries
  12. Gradient Flows
  13. Function Spaces
  14. Point Heated Droplets
  15. Overview
  16. ...and 81 more sections

Key Result

Lemma 3.3

If $h(x,t)$ is a moving contact line solution, then the area is conserved if and only if

Figures (59)

  • Figure 1: Radially symmetrical droplets on a horizontal surface. The liquid interface is marked in orange. (left) The droplet on the left has a contact angle greater than $90^\circ$, whereas (right) the droplet on the right has a contact angle less than $90^\circ$.
  • Figure 2: Schematic of a two-dimensional (a) thin-film flow and (b) droplet spreading setup. The orange line indicates the liquid-gas interface, and $h(x,t)$ describes the height of the interface measured from the substrate. $h_0$ and $\lambda_0$ are the typical vertical and horizontal length scales, respectively.
  • Figure 3: Similarity solution using the shooting method for case $n=1$ and $n=2$.
  • Figure 4: Geometrical interpretation of the boundary conditions. The slip-length parameter $\beta$ can be thought of as the extrapolation length of the velocity profile into the substrate.
  • Figure 5: Sketch of the droplet flow geometry with a precursor film of thickness $h_\infty$ in beyond the contact region.
  • ...and 54 more figures

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 3.5: Pointwise product
  • Definition 3.6
  • Proposition 3.7
  • ...and 34 more