##
Conclusions

1. For natural convection flows where the Grashof number is larger, a boundary layer can be expected close to a solid surface.

2. For situations where a natural convection boundary-layer is occurring, heat transfer will be governed by coupled energy and momentum equations.

3. For transport processes that occur on a semi-infinite domain, where there is __no geometric length scale__, it is often possible to define a (dimensionless) similarity variable that contains the natural length scale.

4. It is often possible to reduce PDE's to ODE's through simplifications made possible by use of the similarity variable.

5. From the scaling identified by the similarity variable, it is often possible to predict the macroscopic behavior without solving the differential equations. In this case Nu ~ .

6. This prediction agrees with the recommended correlation for high Gr heat transfer.

7. The coupled, nonlinear ODE's can be readily solved with a shooting method.

8. Note the shape of the temperature profile and the location of the maximum velocity.

Converted by *Mathematica*
April 30, 2000