GDIM-droplet-spreading

Matlab code for solving the Geometric Diffuse Interface Method (GDIM) thin film equation in one spatial dimension. REF: A mathematical model and mesh-free numerical method for contact-line motion in lubrication theory (Open Access)

Introduction

The classical thin film equation in 1D is given by

      

In this work, we propose the regularisation

      

with a new potential

      

For derivations, please see ref [placeholder]. This repository presents two numerical methods for solving the regularised thin film equation, a fully implicit finite difference scheme and a mesh-free particle method. We also implemented a fast summation algorithm for the particle method that reduces the computational complexity of the particle method from O(N^2) to O(N), where N is the number of particles.

Usage

The directory /complete_wetting/ contains cases where (i.e. no potential energy) and /partial_wetting/ contains cases with . We recommend starting with /complete_wetting/ to get a feeling for the workings of the numerical methods. Both folder contain the following solvers:

• solve_particle.m solves the GDMI TFE with the particle method using direct summation.
• solve_sparse.m is the same as solve_particle.m but drop particles with zero weight since they do not contribute to the solution.
• solve_fast.m solves the GDMI TFE with the particle method with the fast summation algorithm.
• solve_newton.m solves the GDMI TFE using a fully implicit finite difference method with Newton's optimisation.

The folders also contain other utility functions, namely clamp.m, my_diag.m, and my_centered_array.m to enhance redability of the code. Dependencies are noted within the comment section of every solvers.