The Rayleigh number is generally described as distinguishing between the convective and coductive regime of heat flow in a fluid. I'm looking at a specific geometry. and it's not clear to me precisely what
the characteristic length scale would be, and
what the critical Rayleigh number for the problem is.
The geometry consists of a large reservior of fluid at a temperature $T_1$. In one of the walls, there is a slit of thickness $d$ and length $L$, and the end of the slit is cooled to a temperature $T_0<T_1$, as indicated in the sketch:
Gravity is acting downwards (indicated by the green arrow). The box is very long in the third direction (into the screen). The boundary colours indicate the higher temperature (red) , the lower temperature (blue) and insulated boundaries (black). We can assume $L/d\gg1$ if necessary (at least $L/d>5$, more likely $>10$).
The reservoir is much larger than the slit, and I'm really only interested in the flow in the slit (and the heat flow at the cooled tip), in a stationary situation. I assume that, depending on the parameters, there will be a "conduction" regime without fluid flow, and heat flow by diffusion, and a "convection" regime, where the higher density of the cooled fluid at the tip leads to a steady flow into the (top of the) slit, cooling at the tip and flow out of the (bottom of the) slit. Intuitively, convection would be favoured by a large temperature difference (or large coefficient of thermal expansion), small $L$ and large $d$.
As far as I understand, the Rayleigh number depends on the difference in density (which in turn depends on thermal expandsion and temperature difference), viscosity $\eta$ and diffusivity $\alpha$, $$\text{Ra}\propto \frac{\Delta \rho g}{\eta \alpha}\cdot \left(\text{characteristic length scale}\right)^3\,,$$ and it contains the "characteristic scale" and possiby a numerical suppression factor.
The characteristic scale presumably is a function of $L$ and $d$ and encodes the fact that the viscous fluid needs to move along a long narrow slit.
So my questions are:
- Is my general intuition correct?
- If so, is there a formulation of the Reynolds number for this situation?
- What would be the critical value of the Rayleigh number here?
- Additionally, what happens if the slit is filled with a porous medium of permability $k$?