Simplified diffusion-convection model w/ Rayleigh-Taylor unstability

[TboneWalker]

I am attempting to make a simple phenomenological model of convection in a fluid with a negative concentration gradient. The heavier fluid overlaying light fluid will under many conditions cause a so called Rayleigh-Taylor unstability, and the denser fluid will move downwards as a result. I've been trying to make a one-dimensional approximation of this effect that can predict concentration distribution over time. My first thought was to have a velocity vector that was in some way related to the density gradient, such as

v = constant * (dc/dx)^n
or
v = f(mu) * (dc/dx)^n
mu: viscosity

Im stuggling to find litterature on this, except for numerical solutions of 2D and 3D problems. My hope was to in some way get an average of 2D and 3D effects respresented in the 1D solution. Any thoughts or helpful advice?

 

[Andy Resnick]

I don't see how you can have a 1-D system where this occurs, since as the (say) heavy fluid falls, the light fluid rises. It's a fundamentally 3D problem.

Chandrasekhar's book "Hydrodynamic and Hydromagentic Stability" is the essential book for this problem. The Rayleigh-Taylor instability is solved analytically in section 10.

 

[TboneWalker]

I know that this convection can not physically happen in one dimension,
I just though it would be possible to approximate a formulation through a 1D equation, that in some way would give us an area-averaged density as a function of height and time...

Thanks for the book tip btw :)