Stability of a Horizontal Fluid Layer with Unsteady Heating from Below and Time-dependent Body Force

The stability of a horizontal layer of fluid heated unsteadily from below and subjected to a time-dependent body force field was investigated theoretically, assuming an incompressible fluid with small density changes resulting from heating. A stability criterion is developed and used to calculate critical Rayleigh numbers, which are found to be much higher than for the static case, dependent only on the shape of the density profile, and independent of heating rate and Prandtl number. The initial motion corresponds to approximately the same cell shape as for the static case. Velocity growth increases from zero at the critical time at a rate proportional to the Prandtl number.