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Local Invariant Structures in the Dynamics of Capillary Water Jet

Chengyang Shao, Haocheng Yang

Abstract

Physical experiments show that a capillary water jet is exponentially unstable under long-wave perturbations, while remaining stable under short-wave perturbations. Measurements further indicate that the exponential growth rate in the long-wave regime agrees quantitatively with the classical predictions of Rayleigh and Plateau. This phenomenon is known as the \emph{Rayleigh-Plateau instability}. In this paper, we provide a mathematical justification of these experimental observations. The motion of the water jet is modeled by an irrotational Eulerian free-boundary system governed by surface tension. We prove that the (un)stable directions in the linearized system, corresponding to long-wave perturbations, are indeed tangent to an (un)stable invariant manifold of the full nonlinear system. On the other hand, the elliptic directions, corresponding to short-wave perturbations, are indeed tangent to a center invariant set in a generalized sense. These results give a positive answer to the question raised by Lin-Zeng concerning the existence of invariant manifolds for Eulerian free-boundary systems. The major methodological contribution is the construction of ``paradifferential propagator" corresponding to linear paradifferential hyperbolic systems along elliptic directions, making Lyapunov-Perron type arguments applicable. The method effectively balances the loss of regularity in quasilinear problems and can be generalized to a broader class of PDEs.

Local Invariant Structures in the Dynamics of Capillary Water Jet

Abstract

Physical experiments show that a capillary water jet is exponentially unstable under long-wave perturbations, while remaining stable under short-wave perturbations. Measurements further indicate that the exponential growth rate in the long-wave regime agrees quantitatively with the classical predictions of Rayleigh and Plateau. This phenomenon is known as the \emph{Rayleigh-Plateau instability}. In this paper, we provide a mathematical justification of these experimental observations. The motion of the water jet is modeled by an irrotational Eulerian free-boundary system governed by surface tension. We prove that the (un)stable directions in the linearized system, corresponding to long-wave perturbations, are indeed tangent to an (un)stable invariant manifold of the full nonlinear system. On the other hand, the elliptic directions, corresponding to short-wave perturbations, are indeed tangent to a center invariant set in a generalized sense. These results give a positive answer to the question raised by Lin-Zeng concerning the existence of invariant manifolds for Eulerian free-boundary systems. The major methodological contribution is the construction of ``paradifferential propagator" corresponding to linear paradifferential hyperbolic systems along elliptic directions, making Lyapunov-Perron type arguments applicable. The method effectively balances the loss of regularity in quasilinear problems and can be generalized to a broader class of PDEs.
Paper Structure (42 sections, 41 theorems, 389 equations, 4 figures)

This paper contains 42 sections, 41 theorems, 389 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Equations Governing a Capillary Fluid Jet
  3. Experimental Facts and Formal Explanation
  4. The Main Result
  5. Classical Lyapunov-Perron Method and its Limitations
  6. Idea of the Proof
  7. Paralinearization
  8. Paradifferential Propagator
  9. Twisted Duhamel Formula, Lyapunov-Perron Method
  10. Future Perspectives
  11. Toolset of Paradifferential Calculus
  12. Notations and conventions
  13. Preliminaries from Fourier Analysis
  14. Functional Spaces and Littlewood-Paley Decomposition
  15. Regular Mappings
  16. ...and 27 more sections

Key Result

Theorem 1.1

Fix the unperturbed radius $0<\rho<1$ with $\rho^{-1}\notin\mathbb{N}$. For any $s_0>5$, there exists a small real number $\varepsilon>0$, and a unique finite-dimensional $C^\infty$ submanifold $M_\mathtt{u}$ (resp. $M_\mathtt{s}$) of $H^{s_0+1}\times \dot{H}^{s_0+1/2}$ (see Notation note:ZeroMean), Moreover, the manifold $M_\mathtt{u}$ (resp. $M_\mathtt{s}$) satisfies the following properties:

Figures (4)

  • Figure 1: Shape of the water jet. Code adapted from HK2023.
  • Figure 2: Above: amplifying disturbance near the most unstable mode leading to finite time break-up, corresponding to $\rho\xi\simeq0.678$. Below: non-amplifying disturbance, corresponding to $\rho\xi\simeq1.07$. Reprint from figure 8 and 10 of DG1966.
  • Figure 3: Dependence of the exponential growth rate $\omega_e$ on the wave number $k=\rho\xi$. Solid line: Rayleigh's theoretical prediction Rayleigh1878. $\square$: data from Donnelly-Glaberson DG1966. $\triangle$: data from Goede-Yuen GY1970. Reprint from Figure 3 of Lafrance1975.
  • Figure :

Theorems & Definitions (76)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5: Classical Invariant Manifold Theorem
  • proof
  • Definition 2.5: Decreasing Family
  • Definition 2.6: Regular Mappings
  • Remark 2.7
  • Lemma 2.8
  • ...and 66 more