[Paper Review] Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium

Francisco Gancedo, Rafael Granero-Belinchón+1 more|LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas)|Nov 8, 2019

Navier-Stokes equation solutions61 references4 citations

TL;DR

This paper establishes global existence and instantaneous analyticity of solutions to the Rayleigh-Taylor unstable Muskat problem in a porous medium with surface tension, proving that fluid interfaces with sufficiently small slope remain stable and do not form drops. Using a novel framework of Wiener-Sobolev anisotropic spaces, the authors show that capillary forces stabilize the system even in the presence of gravity-driven instability, without assuming irrotational flow.

ABSTRACT

This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top of a dry region. An important feature considered here is that the layer of fluid is under an impervious wall. This physical situation have been widely study by mean of thin film approximations in the case of small characteristic high of the fluid considering its strong interaction with the fixed boundary. Here, instead of considering any simplification leading to asymptotic models, we deal with the complete free boundary problem. We prove that, if the fluid interface is smaller than an explicit constant, the solution is global in time and it becomes instantly analytic. In particular, the fluid does not form drops in finite time. Our results are stated in terms of Wiener spaces for the interface together with some non-standard Wiener-Sobolev anisotropic spaces required to describe the regularity of the fluid pressure and velocity. These Wiener-Sobolev spaces are of independent interest as they can be useful in other problems. Finally, let us remark that our techniques do not rely on the irrotational character of the fluid in the bulk and they can be applied to other free boundary problems.

Motivation & Objective

• To analyze the stabilization of the Rayleigh-Taylor instability in a fluid layer confined under an impervious wall in a porous medium.
• To determine explicit conditions under which surface tension prevents drop formation or fingering in gravity-driven unstable configurations.
• To establish global regularity for the free boundary problem without relying on irrotationality assumptions or asymptotic approximations.
• To develop and apply a new class of non-standard Wiener-Sobolev anisotropic function spaces to describe the regularity of the interface, velocity, and pressure.
• To prove that solutions become instantaneously analytic in space, even for initial data with moderate slope in a suitable functional space.

Proposed method

• Formulates the problem in Eulerian coordinates using Darcy's law with gravity and surface tension, avoiding thin-film approximations.
• Reformulates the free boundary problem into a fixed domain via an Arbitrary Lagrangian-Eulerian (ALE) transformation.
• Introduces a new class of anisotropic Wiener-Sobolev spaces to capture the regularity of the interface, velocity, and pressure, with distinct smoothness in time and space.
• Derives elliptic estimates for the pressure and velocity fields in these anisotropic spaces, leveraging the structure of the ALE transformation.
• Establishes parabolic-type energy estimates for the interface height function $ h $, including bounds on nonlinear terms involving the interface curvature and surface tension.
• Uses Fatou’s lemma and convergence arguments in Wiener spaces to prove that the solution becomes analytic for all positive times, even if the initial data is only in a Wiener space.

Experimental results

Research questions

• RQ1 Under what conditions does surface tension prevent drop formation in a Rayleigh-Taylor unstable fluid layer in a porous medium?
• RQ2 Can global regularity be established for the full free boundary problem without assuming irrotational flow or asymptotic simplifications?
• RQ3 What functional space framework is necessary to capture the regularity of the interface and fluid fields in the presence of surface tension and gravity?
• RQ4 Does the solution become instantaneously analytic in space, even for non-smooth initial data?
• RQ5 Can the analysis be extended to systems that do not admit a contour dynamic formulation?

Key findings

• If the initial interface slope is smaller than an explicit threshold in a Wiener space, the solution exists globally in time.
• The solution becomes instantaneously analytic in space for all $ t > 0 $, even if the initial data is only in a Wiener space.
• The interface does not form drops or singularities in finite time due to surface tension stabilization.
• The analysis does not require the fluid to be irrotational, making the method applicable to broader classes of free boundary problems.
• The authors introduce a new class of Wiener-Sobolev anisotropic spaces that are of independent interest and may be useful in other fluid dynamics problems.
• Energy estimates and convergence in Wiener spaces are used to prove that the solution remains in a space of analytic functions for all positive times.