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Free boundary problem for two-dimensional ElectroHydroDynamic Equations with a gravity field

Lili Du, Yuanhong Zhao

TL;DR

This work studies a two-phase free boundary problem for the 2D ElectroHydroDynamic equations with gravity, focusing on singularities at stagnation points. It adopts a variational formulation with the EHD functional and uses Weiss-type monotonicity, a frequency formula, and concentration-compactness to analyze blowups of the combined potential $u$ near stagnation. A key finding is the identification of a critical decay rate $|x-x^{st}|^{1/2}$ for the electric field that determines the local interface geometry: faster decay yields the classical Stokes corner, critical decay yields Stokes-type or symmetric/asymmetric corners depending on boundary interaction, and slower decay leads to a cusp; the blowups are homogeneous of degree $3/2$ on cones of opening $2\pi/3$. The results extend the Stokes cusp/corner framework to two-phase EHD flows with gravity, clarifying how electric stresses and gravity shape stagnation-point singularities and connecting with prior GVW and VW analyses.

Abstract

This paper studies a two-phase free boundary problem governed by the ElectroHydroDynamic equations, which describes a perfectly conducting, incompressible, irrotational fluid with gravity, surrounded by a dielectric gas. The interface separating fluid and gas is referred to as the free boundary. It is known that the free surface remains smooth away from the stagnation points, where the relative velocity of the incompressible fluid vanishes. In the presence of gravity, the Stokes conjecture, proved by Varvaruca and Weiss [Acta. Math. 206, 363-403, (2011)], implies that the corner type singularity will occur in the one-phase incompressible fluid. It is natural to ask whether this conjecture still holds in the two-phase flow problem. As a consequence, the primary objective of this work is to characterize the possible singular profiles of the free interface near the stagnation points in the presence of an electric field. Our main result is the discovery of a critical decay rate of the electric field near the stagnation points which indicates the classification of the singular profiles of the free surface. More precisely, we showed that when the decay rate of the electric field is faster than the critical decay rate, its negligible effect implies that the singular profile must be the well-known Stokes corner. When the electric field decays as the critical decay rate, the symmetry of the corner region may be broken, giving rise to either a Stokes corner or an asymmetric corner as the possible singular profile. If the decay rate is slower than the critical decay rate, the electric field dominates and completely destroys the corner structure, resulting in a cusp singularity. The analysis of these singularities relies on variational principles and geometric methods. Key technical tools include a Weiss-type monotonicity formula, a frequency formula, and a concentration-compactness argument.

Free boundary problem for two-dimensional ElectroHydroDynamic Equations with a gravity field

TL;DR

This work studies a two-phase free boundary problem for the 2D ElectroHydroDynamic equations with gravity, focusing on singularities at stagnation points. It adopts a variational formulation with the EHD functional and uses Weiss-type monotonicity, a frequency formula, and concentration-compactness to analyze blowups of the combined potential near stagnation. A key finding is the identification of a critical decay rate for the electric field that determines the local interface geometry: faster decay yields the classical Stokes corner, critical decay yields Stokes-type or symmetric/asymmetric corners depending on boundary interaction, and slower decay leads to a cusp; the blowups are homogeneous of degree on cones of opening . The results extend the Stokes cusp/corner framework to two-phase EHD flows with gravity, clarifying how electric stresses and gravity shape stagnation-point singularities and connecting with prior GVW and VW analyses.

Abstract

This paper studies a two-phase free boundary problem governed by the ElectroHydroDynamic equations, which describes a perfectly conducting, incompressible, irrotational fluid with gravity, surrounded by a dielectric gas. The interface separating fluid and gas is referred to as the free boundary. It is known that the free surface remains smooth away from the stagnation points, where the relative velocity of the incompressible fluid vanishes. In the presence of gravity, the Stokes conjecture, proved by Varvaruca and Weiss [Acta. Math. 206, 363-403, (2011)], implies that the corner type singularity will occur in the one-phase incompressible fluid. It is natural to ask whether this conjecture still holds in the two-phase flow problem. As a consequence, the primary objective of this work is to characterize the possible singular profiles of the free interface near the stagnation points in the presence of an electric field. Our main result is the discovery of a critical decay rate of the electric field near the stagnation points which indicates the classification of the singular profiles of the free surface. More precisely, we showed that when the decay rate of the electric field is faster than the critical decay rate, its negligible effect implies that the singular profile must be the well-known Stokes corner. When the electric field decays as the critical decay rate, the symmetry of the corner region may be broken, giving rise to either a Stokes corner or an asymmetric corner as the possible singular profile. If the decay rate is slower than the critical decay rate, the electric field dominates and completely destroys the corner structure, resulting in a cusp singularity. The analysis of these singularities relies on variational principles and geometric methods. Key technical tools include a Weiss-type monotonicity formula, a frequency formula, and a concentration-compactness argument.
Paper Structure (12 sections, 7 theorems, 308 equations, 18 figures, 2 tables)

This paper contains 12 sections, 7 theorems, 308 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Description of the mathematical problem
  4. A variational approach
  5. The analysis of the transition condition
  6. Main results
  7. Preliminaries
  8. Singular profiles near the stagnation points
  9. The possible singular asymptotics for Case 1.1
  10. The possible singular asymptotics for Case 1.2
  11. The possible singular asymptotics for Case 1.3
  12. The possible singular profiles for Case 1

Key Result

Theorem 1

Let $u$ be a weak solution of problem 1.20 (as defined by Definition A in Appendix A). For $r_0>0$ small enough, we assume the following statements hold Then the following asymptotic behaviors hold

Figures (18)

  • Figure 1: Two-phase EHD flow
  • Figure 2: EHD inkjet printing apparatus, as described in JB
  • Figure 3: Stokes conjecture for 2D water wave
  • Figure 4: The physical model of two-phase EHD flow
  • Figure 5: The two-phase free boundary problem
  • ...and 13 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.10
  • Theorem 3.17