Solutions of the thin film equation obtained in the limit of vanishing slip

TL;DR

This paper investigates the behavior of thin liquid droplets under different slip conditions at the solid interface, revealing multiple limiting solutions in the no-slip limit and clarifying their physical consistency.

Contribution

It demonstrates the existence of three distinct classes of solutions in the no-slip limit, linked to contact angle scaling, and clarifies their physical relevance within lubrication theory.

Findings

Three classes of limiting solutions identified in the no-slip limit.

Different contact angle scalings lead to distinct solution behaviors.

Refined analysis confirms physical consistency of solutions despite large contact angles.

Abstract

We analyze the evolution of thin liquid droplets in the lubrication approximation with different slip conditions at the liquid-solid interface. Motivated by the classical no-slip paradox which states that the Navier-Stokes equations with a no-slip boundary condition require unphysical infinite dissipation during droplet spreading, we focus on the limit of vanishing slip. We show that in the no-slip limit three fundamentally different classes of limiting solutions are approached, each of them corresponding to a different scaling of the microscopic contact angle as the regularization parameter vanishes. These findings suggest that the thin-film equation with no slip supports a rich family of physically admissible solutions, provided one interprets the no-slip thin film equation as the asymptotic limit of models which regularized slip conditions. Even though the large apparent contact angles in some of these solutions seem incompatible with the lubrication approximation, a refined analysis shows that the underlying physical variables remain consistent with the assumptions for the lubrication approximation.