[Paper Review] Darcy's flow with prescribed contact angle -- Well-posedness and lubrication approximation

Hans Knüpfer, Nader Masmoudi|arXiv (Cornell University)|Apr 10, 2012

Fluid Dynamics and Thin Films31 references4 citations

TL;DR

This paper establishes local and global well-posedness for Darcy flow with a prescribed non-zero contact angle in a thin-film regime, proving uniform estimates across the contact angle parameter. It rigorously justifies the lubrication approximation by showing convergence to a degenerate fourth-order thin-film equation in the long-wave limit, resolving challenges in handling moving contact points and non-smooth domains via weighted Sobolev spaces and asymptotic analysis.

ABSTRACT

We consider the spreading of a thin two-dimensional droplet on a solid substrate. We use a model for viscous fluids where the evolution is governed by Darcy's Law. At the triple point where air and liquid meet the solid substrate, the liquid assumes a constant, non-zero contact angle ({\it partial wetting}). We show local and global well-posedness of this free boundary problem in the presence of the moving contact point. Our estimates are uniform in the contact angle assumed by the liquid at the contact point. In the so-called lubrication approximation (long-wave limit) we show that the solutions converge to the solution of a one-dimensional degenerate parabolic fourth order equation which belongs to a family of thin-film equations. The main technical difficulty is to describe the evolution of the non-smooth domain and to identify suitable spaces that capture the transition to the asymptotic model uniformly in the small parameter $\eps$.

Motivation & Objective

• To establish local and global well-posedness of Darcy flow with a prescribed, non-zero contact angle at the triple point where liquid, air, and solid meet.
• To develop a framework that allows for the movement of the contact point, overcoming limitations in prior Hölder space approaches.
• To rigorously justify the lubrication approximation (long-wave limit) by showing convergence to a degenerate fourth-order thin-film equation.
• To derive uniform estimates in the small parameter ε (representing the aspect ratio) that capture the transition to the asymptotic model without loss of regularity.

Proposed method

• Transforms the free boundary problem into a fixed domain using a geometric mapping to handle the moving contact point.
• Introduces weighted Sobolev spaces tailored to the geometry of the wedge-shaped domain near the contact line.
• Applies a localization argument using cutoff functions to reduce the problem to a half-space and analyze local behavior.
• Uses elliptic estimates in weighted spaces for the pressure and profile equations, including weighted L2 and H^k norms.
• Derives uniform estimates for the linearized operator in the half-space, proving boundedness and continuity of the solution operator in ε.
• Performs asymptotic analysis in the long-wave limit (ε → 0), showing convergence of solutions to the thin-film equation with contact angle conditions.

Experimental results

Research questions

• RQ1 Can a well-posedness theory be developed for Darcy flow with a prescribed, non-zero contact angle when the contact point is allowed to move?
• RQ2 How can uniform estimates be derived in the small parameter ε (aspect ratio) that remain valid as ε → 0, even when the contact angle is non-zero?
• RQ3 Does the lubrication approximation, which reduces the problem to a one-dimensional thin-film equation, accurately describe the long-wave limit of the Darcy flow with contact angle?
• RQ4 What is the precise asymptotic behavior of the solution as ε → 0, and how does the contact point motion influence the convergence to the reduced model?

Key findings

• The authors prove local and global well-posedness of the Darcy flow with prescribed contact angle in a class of weighted Sobolev spaces, allowing for moving contact points.
• Uniform estimates are established for the solution operator in the small parameter ε, with bounds independent of the contact angle.
• The solution of the Darcy flow problem converges to the solution of a degenerate fourth-order thin-film equation in the long-wave limit (ε → 0).
• The convergence is shown to hold in the sense of the asymptotic model, with the contact angle condition in the limit model matching the microscopic contact angle via Young's law.
• The analysis identifies the correct function spaces and norms that capture the transition to the lubrication approximation uniformly in ε.
• The method successfully handles the non-smooth domain and singular behavior at the contact line through careful localization and weighted estimates.