# Rayleigh-Taylor instability

The Rayleigh – Taylor instability (RTI) is a hydrodynamic instability which causes a disturbance at the interface of two liquids of different weight to grow exponentially. It is named after the two physicists John William Strutt, 3rd Baron Rayleigh , and Geoffrey Ingram Taylor .

## Description and occurrence

hydrodynamic simulation of a Rayleigh-Taylor instability

The RTI is a two-phase instability (like the Kelvin-Helmholtz instability ) that occurs when two liquids of different density are accelerated against each other. It doesn't matter what type of acceleration is. A heavy liquid is Rayleigh-Taylor unstable on a light one in the gravitational field , as is the shell of a star exploding as a supernova , which is accelerated against the thinner interstellar medium . The fringed appearance of the Crab Nebula , for example, is a result of the RTI. Typical of the RTI are the mushroom-shaped protrusions of the liquids into one another, which can be observed, for example, when a little milk is added to a cup of tea.

## theory

From the linear stability analysis of the fluid dynamics equations, the following dispersion relation is obtained for two adjoining, non-moving liquids of different density :

${\ displaystyle \ omega = {\ sqrt {gk \ left ({\ frac {1-a} {1 + a}} \ right)}}.}$

Here is the angular frequency of the disturbance, its wave number , the acceleration (e.g. gravitational ) and the ratio of the densities of the liquid layers . ${\ displaystyle \ omega}$${\ displaystyle k}$${\ displaystyle g}$${\ displaystyle a}$${\ displaystyle \ rho _ {\ text {above}} / \ rho _ {\ text {below}}}$

Is , d. H. the overhead liquid is the heavier so is imaginary , i.e. H. inserted into the wave equation of the perturbation one obtains an exponential increase of the perturbation. The configuration is therefore unstable to the smallest disturbances. In the opposite case (light liquid to heavier) you get the dispersion relation for surface waves. ${\ displaystyle a> 1}$${\ displaystyle \ omega}$ ${\ displaystyle \ rho \ sim e ^ {i (\ mathbf {kx} - \ omega t)}}$${\ displaystyle a <1}$

## Individual evidence

1. Perturbations at a two-fluid interface ( Perturbations at a two-fluid interface ( Memento of February 8, 2006 in the Internet Archive ))