Local Invariant Structures in the Dynamics of Capillary Water Jet

TL;DR

This paper mathematically justifies the stability and instability of capillary water jets under different wave perturbations, confirming experimental observations and revealing invariant manifold structures in the jet's dynamics.

Contribution

It provides a rigorous mathematical analysis of water jet stability, demonstrating the existence of invariant manifolds and introducing a novel paradifferential propagator method for quasilinear systems.

Findings

Long wave perturbations lead to exponential instability consistent with Rayleigh-Plateau theory.

Short wave perturbations are stable and tangent to a center invariant set.

Infinite dimensional hyperbolic invariant manifolds exist without spectral gap.

Abstract

Physical experiments show that a capillary water jet is exponentially unstable under long wave perturbations, while remaining stable under short wave perturbations. Measurements indicate that the exponential growth rate in the long wave regime agrees quantitatively with the classical predictions of Rayleigh and Plateau, known as \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 axisymmetric 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 a (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. A notable finding is the existence of infinite dimensional hyperbolic invariant manifolds in the dynamics of infinitely long capillary water jet, in absence of spectral gap. The major methodological contribution is the construction of ``paradifferential propagator" corresponding to linear paradifferential hyperbolic systems, effectively balancing the loss of regularity due to quasilinearity. The method can be generalized to a broad class of quasilinear problems.