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On the dynamical Rayleigh-Taylor instability of non-homogeneous fluid in annular region with Naiver-slip boundary

Liang Li, Quan Wang

TL;DR

The paper analyzes the dynamical Rayleigh–Taylor instability for a 2D, nonhomogeneous incompressible Navier–Stokes system in an annulus with mixed boundary conditions, using polar coordinates to exploit the geometry. It establishes local well-posedness via a semi-Galerkin approach and Stokes estimates, and then demonstrates linear RT instability by separating variables and a modified variational method. Building on this, it proves nonlinear instability in both Lipschitz and Hadamard senses through detailed energy estimates and a variational construction of the maximum growth rate, valid under reasonably broad density profiles. The results illuminate the transition from linear growth to nonlinear breakdown in annular geophysical/astrophysical flows and provide rigorous instability criteria valuable for understanding stratified viscous fluids with slip boundaries.

Abstract

This paper investigates the well-posedness and Rayleigh-Taylor (R-T) instability for a system of two-dimensional nonhomogeneous incompressible fluid, subject to the non-slip and Naiver-slip boundary conditions at the outer and inner boundaries, respectively, in an annular region. In order to effectively utilize the domain shape, we analyze this system in polar coordinates. First, for the well-posedness to this system, based on the spectral properties of Stokes operator, Sobolev embedding inequalities and Stokes' estimate in the context of the specified boundary conditions, etc, we obtain the local existence of weak and strong solutions using semi-Galerkin method and prior estimates. Second, for the density profile with the property that it is increasing along radial radius in certain region, we demonstrate that it is linear instability (R-T instability) through Fourier series and the settlement of a family of modified variational problems. Furthermore, based on the existence of the linear solutions exhibiting exponential growth over time, we confirm the nonlinear instability of this steady state in both Lipschitz and Hadamard senses by nonlinear energy estimates.

On the dynamical Rayleigh-Taylor instability of non-homogeneous fluid in annular region with Naiver-slip boundary

TL;DR

The paper analyzes the dynamical Rayleigh–Taylor instability for a 2D, nonhomogeneous incompressible Navier–Stokes system in an annulus with mixed boundary conditions, using polar coordinates to exploit the geometry. It establishes local well-posedness via a semi-Galerkin approach and Stokes estimates, and then demonstrates linear RT instability by separating variables and a modified variational method. Building on this, it proves nonlinear instability in both Lipschitz and Hadamard senses through detailed energy estimates and a variational construction of the maximum growth rate, valid under reasonably broad density profiles. The results illuminate the transition from linear growth to nonlinear breakdown in annular geophysical/astrophysical flows and provide rigorous instability criteria valuable for understanding stratified viscous fluids with slip boundaries.

Abstract

This paper investigates the well-posedness and Rayleigh-Taylor (R-T) instability for a system of two-dimensional nonhomogeneous incompressible fluid, subject to the non-slip and Naiver-slip boundary conditions at the outer and inner boundaries, respectively, in an annular region. In order to effectively utilize the domain shape, we analyze this system in polar coordinates. First, for the well-posedness to this system, based on the spectral properties of Stokes operator, Sobolev embedding inequalities and Stokes' estimate in the context of the specified boundary conditions, etc, we obtain the local existence of weak and strong solutions using semi-Galerkin method and prior estimates. Second, for the density profile with the property that it is increasing along radial radius in certain region, we demonstrate that it is linear instability (R-T instability) through Fourier series and the settlement of a family of modified variational problems. Furthermore, based on the existence of the linear solutions exhibiting exponential growth over time, we confirm the nonlinear instability of this steady state in both Lipschitz and Hadamard senses by nonlinear energy estimates.

Paper Structure

This paper contains 21 sections, 43 theorems, 370 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proofs of main results on well-posedness
  4. Proof of Theorem \ref{['jubucunzai0605']}
  5. Proof of Theorem \ref{['qiangjie0613']}
  6. Proof of Theorem \ref{['qiangjiele0614']}
  7. Proof of Theorem \ref{['weiyixing0615']}
  8. Linear instability
  9. linear unstable solution
  10. Construction of a solution to system \ref{['bianlianfenli0403']}
  11. Exponential growth rate
  12. Nonlinear instability in Lipschitz sense
  13. Uniqueness of solution to linearized equation
  14. Nonlinear estimates
  15. Construction of the solution to linearized problem
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose $\mathbf{V}_{0}\in\mathbf{G}^{1,2}$, $\rho_{0}\in L^{\infty} \left(\widetilde{\Omega}\right)$ and $\rho_{0}\left(r,\theta\right)>\delta>0$ ($\delta$ is a constant fixed) in $\widetilde{\Omega}$. Then, there exist a number $T^{*}>0$ and a weak solution $\left(\mathbf{V},\rho\right)$ to the sy where $C=C\left(\mu,R_{1},R_{2},\rho_{0},\mathbf{V}_{0}\right)>0$.

Figures (1)

  • Figure 1: $\mathbf{g}=-g\left(\frac{x}{r}, \frac{y}{r}\right)^T$.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.1
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.2
  • Lemma 2.1
  • ...and 77 more