Simplified diffusion-convection model w/ Rayleigh-Taylor unstability

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In summary, the conversation revolved around creating a simple phenomenological model for convection in a fluid with a negative concentration gradient, specifically the Rayleigh-Taylor instability. The speaker's initial thought was to use a velocity vector related to the density gradient, but is struggling to find literature on this topic. They were hoping to find a 1D solution that could represent the 2D and 3D effects, but it seems to be a fundamentally 3D problem. The book "Hydrodynamic and Hydromagnetic Stability" by Chandrasekhar was recommended as a helpful resource.
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TboneWalker
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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?
 
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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.
 
  • #3
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 :)
 

1. What is a simplified diffusion-convection model with Rayleigh-Taylor instability?

A simplified diffusion-convection model with Rayleigh-Taylor instability is a mathematical model used to study the mixing of fluids with different densities due to gravitational forces. It takes into account diffusion, which is the gradual mixing of fluids due to random molecular motion, and convection, which is the mixing caused by bulk fluid motion. The Rayleigh-Taylor instability refers to the instability that arises when a heavier fluid sits on top of a lighter fluid, causing the heavier fluid to sink and the lighter fluid to rise, leading to mixing.

2. How is the Rayleigh-Taylor instability incorporated into the model?

The Rayleigh-Taylor instability is incorporated into the model through the inclusion of the Rayleigh-Taylor parameter, which is a measure of the relative strength of the gravitational force to the diffusive force. As this parameter increases, the instability becomes more prominent and leads to greater mixing.

3. What is the significance of the Rayleigh-Taylor instability in fluid dynamics?

The Rayleigh-Taylor instability plays a crucial role in many natural and industrial processes, such as ocean currents, atmospheric phenomena, and mixing in nuclear fusion reactors. Understanding and modeling this instability is important for predicting and controlling these processes.

4. How does the simplified diffusion-convection model with Rayleigh-Taylor instability differ from other models?

This model differs from other models by simplifying the complex physics of fluid mixing into two main processes: diffusion and convection. It also takes into account the Rayleigh-Taylor instability, which is not always included in other models. This allows for a more efficient and simplified approach to studying fluid mixing.

5. What are the limitations of the simplified diffusion-convection model with Rayleigh-Taylor instability?

One limitation of this model is that it assumes ideal conditions and does not account for factors such as turbulence or viscosity. It also does not consider the effects of other forces, such as surface tension, which may play a role in certain systems. Additionally, the model may not accurately describe highly complex or non-linear systems.

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