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

Hans Knuepfer, Juan Velazquez

TL;DR

This work analyzes the spreading of thin liquid droplets in the lubrication regime under no-slip, and how slip regularization ε→0 yields three distinct limiting behaviors tied to the scaling of the microscopic contact angle. By formal matched asymptotics and inner-outer analyses, it connects type (a) fixed-support, type (b) moving-contact-line with logarithmic corrections, and type (c) fast-receding cases to specific θ_ε scalings, and shows how energy dissipation and interfacial energy balance permit these limits. It also derives Cox–Voinov-type relations for slow-contact-line motion in the weak-slip regime and Tanner-like results for complete wetting, clarifying how the no-slip equation can be interpreted as an asymptotic limit of regularized models. The findings illuminate the rich physical admissibility of the thin-film equation and provide a framework for understanding contact-line dynamics across wetting regimes from a unified lubrication perspective.

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.

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

TL;DR

This work analyzes the spreading of thin liquid droplets in the lubrication regime under no-slip, and how slip regularization ε→0 yields three distinct limiting behaviors tied to the scaling of the microscopic contact angle. By formal matched asymptotics and inner-outer analyses, it connects type (a) fixed-support, type (b) moving-contact-line with logarithmic corrections, and type (c) fast-receding cases to specific θ_ε scalings, and shows how energy dissipation and interfacial energy balance permit these limits. It also derives Cox–Voinov-type relations for slow-contact-line motion in the weak-slip regime and Tanner-like results for complete wetting, clarifying how the no-slip equation can be interpreted as an asymptotic limit of regularized models. The findings illuminate the rich physical admissibility of the thin-film equation and provide a framework for understanding contact-line dynamics across wetting regimes from a unified lubrication perspective.

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.

Paper Structure

This paper contains 19 sections, 2 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Discussion about results and previous literature
  4. Physical background
  5. Lubrication approximation
  6. Dissipation of energy
  7. Solutions in the case of no slip
  8. Sufficient conditions for non--spreading of solutions
  9. Type (a) solutions -- fixed support
  10. Type (b) solutions -- prescribed shrinking support
  11. Solutions in the case of weak slippage
  12. Well--posedness and Boundary conditions
  13. Solutions with prescribed contact line $s(t)$.
  14. Solutions with prescribed contact line angle.
  15. Approximating type (a) solutions -- Cox-Voinov Law
  16. ...and 4 more sections

Key Result

Lemma 2.4

Let $h$ be a solution of tfe-noslip with support $(s,\infty)$ for some function $s = s(t)$. We assume that $h$ is a type (b) in the sense that for $\xi :=x- s$ we have We also assume that the time derivative $h_t$ and up to three spatial derivatives of $h$ have the asymptotic behaviour, obtained by applying these derivatives on the right--hand side above. Furthermore, the solution and its derivat

Figures (1)

  • Figure 1: Thickness profile $h(t,x)$ of an evolving droplet.

Theorems & Definitions (10)

  • Remark 1.1: Time scales
  • Remark 2.1: Applicability and rescaled contact angle
  • Remark 2.2: Lubrication approximation and interfacial energies
  • Remark 2.3: Dissipation of energy for type (a) solutions
  • Lemma 2.4: Energy dissipation for type (b) solutions
  • proof
  • Conjecture 3.1: Finite dissipation implies fixed support
  • Theorem 3.2: Regularity implies fixed support
  • proof
  • Remark 4.1: Boundary conditions