Unstable Interface Dynamics for Gravity Stokes Flow

TL;DR

The paper addresses instability mechanisms in a 2D two-fluid, gravity-driven Stokes flow with a sharp interface, formulating the problem via contour dynamics and a weak Stokes-Transport framework. It develops a monotone potential energy and uses symmetry reductions to prove that, in the unstable stratification, either the interface length $L(t)$ or the maximal curvature $\mathcal{K}(t)$ must grow without bound, with explicit sublinear growth rates; it also proves finite-time turning in the Rayleigh-Taylor-stable regime and provides a priori bounds on finger growth through analyticity-based arguments. The methodological core combines a weak-strong equivalence to contour dynamics, symmetry preservation, and energy methods, yielding quantitative growth and breakdown criteria supported by numerical simulations. These results illuminate how viscous fingering and RT-like instabilities manifest in viscous two-phase flows at low Reynolds number and offer rigorous insights into long-time interface dynamics and singularity formation. The findings have implications for understanding interface complexity and pattern formation in porous-media analogues and sedimentation-type problems under gravity.

Abstract

We investigate some unstable behavior of the interface given by two incompressible fluids of different densities evolving by the regular Stokes law with gravity force. In the unstable scenario, where the denser fluid lies above the lighter fluid, we prove infinite-in-time growth of the length or the curvature of the interface. We support these analytical results with numerical simulations that confirm the predicted growth phenomena. In the stable configuration, where the denser fluid lies below the lighter fluid, we show that certain initial configurations evolve into the unstable regime in finite time.

Figures (9)

• Figure 1: An example of an unstable, un-stratified configuration satisfying our symmetry assumptions in Theorem \ref{['thm:main']}. Note that $\Gamma_0$ is centrally symmetric about the point $(0,0)$, and evenly symmetric with respect to the orange dashed lines $x_1 = \pm\frac{\pi}{2}$.
• Figure 2: Sketch of the turning instability at $\alpha = 0$. The blue curve represents the initial interface and the red curve represents its evolution into a partially unstable regime.
• Figure 3: Illustration of the definitions of $D_\epsilon^+$ (red region), $D_\epsilon^-$ (blue region), and $\Gamma_\epsilon^+$ (yellow region), and $h_\epsilon(t)$ ($x_2$-coordinate of the purple line in the example on the left). Note that $h_\epsilon(t)$ might not be well-defined if $D_\epsilon^+ \cap \{x_2<0\}$ is empty, as illustrated in the example on the right.
• Figure 4: Illustration of Proof of Proposition \ref{['prop:largeslope']}. In the worst case scenario, the tangent line of the curve $z$ at $(y_1,y_2)$ is such that the tangent circle with radius $\epsilon_0^{1/3}$ is also tangent to the vertical line $x_1=-\frac{\pi}{2}$.
• Figure 5: Illustration of the worst case scenario for Proposition \ref{['prop:budget']}. When the curve $z$ coincides with the purple curve, it takes arclength $\sim \epsilon_0^{1/3}$ from the point $z(\beta)$ for the tangent to be horizontal.

Theorems & Definitions (48)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1
• Proposition 2.2
• Remark 2.3
• Proposition 2.4: Conservation of Central Symmetry