Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures

The two-dimensional convective motion generated by buoyancy forces on the fluid in a long rectangle, of which the two long sides are vertical boundaries held at different temperatures, is considered with a view to the determination of the rate of transfer of heat between the two vertical boundaries. The governing equations are set up; they reveal that the flow is determined uniquely by the Prandtl number $\sigma$, the Rayleigh number $A = g\left ( {{T_1} - {T_0}} \right ){d^3}/\left ( {{T_0}\kappa \nu } \right )$, and the ratio of the sides of the rectangle $l/d$. In the case of cavities used for thermal insulation of buildings, which is kept specially in mind throughout the paper, $A$ is usually about 1000 d$^{3}$ (where $d$ is in centimeters), and $l/d$ takes values between about 5 and 200.