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

TL;DR

This paper studies the well-posedness and Rayleigh-Taylor instability of nonhomogeneous incompressible fluids in an annular region with Naiver-slip boundary conditions, revealing conditions for linear and nonlinear instability.

Contribution

It provides a rigorous analysis of the well-posedness and instability criteria for nonhomogeneous fluids with Naiver-slip boundary conditions in an annular domain.

Findings

Established local existence of solutions using semi-Galerkin method.

Proved linear Rayleigh-Taylor instability for increasing density profiles.

Confirmed nonlinear instability through energy estimates.

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.