A dynamical systems approach for the contact-line singularity in thin-film flows

TL;DR

This paper uses dynamical systems theory to analyze the contact-line singularity in thin-film flows, providing regularity results, extending to higher dimensions, and proving existence and uniqueness of self-similar solutions.

Contribution

It introduces a dynamical systems framework to characterize contact-line singularities, including regularity results and existence proofs for self-similar solutions in thin-film flows.

Findings

Regularity results for singular expansions near the contact line.

Extension of results to higher-dimensional radially-symmetric solutions.

Proof of existence and uniqueness of self-similar droplet solutions.

Abstract

We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this singularity can be understood by the study of invariant manifolds of a suitable dynamical system. In particular, we prove regularity results for singular expansions near the contact line for a wide class of mobility exponents and for zero and nonzero dynamic contact angles. Key points are the reduction to center manifolds and identifying resonance conditions at equilibrium points. The results are extended to radially-symmetric source-type solutions in higher dimensions. Furthermore, we give dynamical systems proofs for the existence and uniqueness of self-similar droplet solutions in the nonzero dynamic contact-angle case.