[Paper Review] Weak solution of the Hele-Shaw problem: shocks and viscous fingering

Seung-Yeop Lee, Razvan Teodorescu+1 more|arXiv (Cornell University)|Dec 2, 2008

Cold Atom Physics and Bose-Einstein Condensates1 references3 citations

TL;DR

This paper proposes a weak solution to the Hele-Shaw problem that resolves finite-time cusp singularities via viscous shocks—discontinuities in pressure and vorticity that form a branching, growing tree structure. The solution preserves integrability and curl-free flow at macroscopic scales while allowing microscale relaxation, with a self-similar (2,3)-cusp singularity triggering a universal, scale-invariant shock branching event governed by transcendental equations and a unique transition ratio η ≈ 0.915.

ABSTRACT

In Hele-Shaw flows, boundaries between fluids develop unstable viscous fingers. At vanishing surface tension, the fingers further evolve to cusp-like singularities. We show that the problem admits a {\it weak solution} where shock fronts triggered by a singularity propagate together with a fluid. Shocks form a growing, branching tree of a mass deficit, and a line distribution of vorticity where pressure and velocity of the fluid have finite discontinuities. Imposing that the flow remain curl-free at macroscale determines the shock graph structure. We present a self-similar solution describing shocks emerging from a generic (2,3)-cusp singularity -- an elementary branching event.

Motivation & Objective

• To resolve finite-time cusp singularities in the Hele-Shaw problem that arise in viscous fingering at vanishing surface tension.
• To develop a weak solution framework that maintains macroscopic incompressibility and curl-free flow while allowing microscale relaxation of physical constraints.
• To characterize the emergence of viscous shocks as discontinuities in pressure and vorticity that form a growing, branching tree structure.
• To identify a universal, self-similar solution for the elementary branching event from a (2,3)-cusp singularity.
• To demonstrate that the regularization preserves the integrability of the Hele-Shaw flow, distinguishing it from other shock types in hydrodynamics.

Proposed method

• Formulate the Hele-Shaw problem using complex potential theory and the analytic height function Y(X,t), with Darcy’s law expressed in terms of the complex potential φ = ψ + ip.
• Introduce the integrated form of Darcy’s law via the functional Ω(X) = −i∫ρ₀Y dX, linking time evolution to the complex potential through ẌΩ = iφ.
• Identify shocks as anti-Stokes lines of Ω where Re Ω = 0, selected by the admissibility condition Ω > 0 on the upper side of the shock boundary.
• Use transcendental equations derived from elliptic functions (Weierstrass ℘-functions) to describe the shock geometry and endpoint motion.
• Apply Rankine-Hugoniot conditions to compute shock velocities, showing shocks propagate faster than the fluid absorption rate (t⁵/⁴ vs. t).
• Compute the universal transition ratio η = limₜ→₀ ẇCₜ>₀ / ẇCₜ<₀ ≈ 0.915, characterizing the abrupt change in capacity across the singularity.

Experimental results

Research questions

• RQ1 How can finite-time cusp singularities in the Hele-Shaw problem be regularized in the absence of surface tension?
• RQ2 What physical mechanism allows the flow to continue past a singularity while preserving macroscopic incompressibility and curl-free conditions?
• RQ3 What is the universal structure of the shock network formed by the branching of viscous shocks from a (2,3)-cusp singularity?
• RQ4 How does the shock structure relate to the complex potential and anti-Stokes lines of the integrated Darcy law?
• RQ5 What is the role of the capacity transition ratio η in characterizing the shock formation process?

Key findings

• The Hele-Shaw problem admits a weak solution where a cusp-like singularity triggers the formation of viscous shocks—lines of finite pressure discontinuity and vorticity distribution.
• Shocks form a growing, branching tree structure that propagates faster than the fluid is drained, with shock velocity scaling as V⊥ = X⊥ / t.
• The shock graph is uniquely determined by the condition that the flow remains curl-free at the macroscale, ensuring physical consistency.
• The elementary branching event from a (2,3)-cusp is self-similar and independent of flow details, described by transcendental equations involving Weierstrass elliptic functions.
• The transition in flow capacity is abrupt, characterized by a universal ratio η ≈ 0.91522, derived from the ratio g₃/g₂ at critical time.
• The mass deficit accumulated on shocks is quantified as ∫ₑ₃ᵉ₁₂ σ dℓ = (4/5)|ψ(e₁,₂)| ≈ (4/5)(6.34513) t⁵/⁴, showing shock-induced fluid retreat at the finger tip.