FANDOM

137 Pages

The RT instability occurs when a dense liquid is laid on top of a tenuous fluid. The more exact mathematical condition is that the density gradient is opposite to the gravitational force.

$\mathbf{g} \cdot \nabla \rho < 0$

In the next sections we will develop the growth rate in the linear stage.

Equation of continuity

$\nabla \mathbf{v} = 0$

Conservation of momentum

$\rho \frac{\partial \mathbf{v}}{\partial t} + \rho \mathbf{v} \cdot \nabla \mathbf{v} + \nabla P = \mathbf{g}$

We can simplify the equation by replacing the velocity by a flow potential $\mathbf{v} = \nabla \varphi$. Substituting in the equations of motion yields

$\nabla^2 \varphi = 0$

$\nabla \left(\rho \frac{\partial \varphi}{\partial t} + \frac{1}{2} \rho \left(\nabla \varphi \right )^2 + P - \rho \mathbf{g} \cdot \mathbf{r} \right ) = 0$

We assume an initial configuration where a heavy fluid of density $\rho_u$ is on top $y>0$ of a lighter fluid of density $\rho_d$ (which occupies the region $y<0$.

Next, we introduce a perturbed flow potential

$\varphi = \delta \varphi_{u,d} \exp \left[ i \left(t \omega - k x + i q |y| \right) \right]$

From the continuity of the normal velocity on the interface between the two fluids $\delta \varphi_u = - \delta \varphi_d$. The Bernouli invariant should also be continuous across the interface

$\left[\rho \frac{\partial \varphi}{\partial t} + \frac{1}{2} \rho \left(\nabla \varphi \right )^2 + P - \rho g y \right] = 0$

where the square brackets denote difference across the interface. The pressure is continuous across the interface, so it vanishes. The kinetic energy term also vanish because it is of a higher order in the perturbation. The position of the interface is

$y = \frac{1}{i \omega} \frac{\partial \varphi}{\partial y}$

We are thus left with a dispersion equation

$\omega^2 = -k g \frac{\rho_u - \rho_d}{\rho_u + \rho_d}$

Since the frequency is complex, the amplitude of this mode grows exponentially if a heavy fluid is on top, and decays if the lighter fluid is on top.

There is a more intuitive way to understand the this growth rate. Let us focus on a single cell of size $1/k$ near the interface between the two fluids. Let us denote by $z$ the position of the interface. The entire cell can be thought of as moving as a single body whose mass is $\left(\rho_u + \rho_d \right) k^{-2} L$, where $L$ is in dimension along the z axis. The force driving this cells is given by Archimedes' buoyancy force $g \left(\rho_u - \rho_d\right) k^{-1} L z$, so the equation of motion for the cell is $\left(\rho_u + \rho_d\right) k^{-2} L \ddot{z} = g \left(\rho_u-\rho_d\right) k^{-1} L z$

and the growth rate is $\frac{z}{\ddot{z}} = \frac{g k \left(\rho_u - \rho_d\right)}{\rho_u+\rho_d}$