Thermal stability of a fluid layer under variable body forces (Q1087413)

The instability of a layer of fluid confined between two horizontal parallel planes and heated from above or below under the action of a gravitational field varying with height is investigated analytically. It is found that if gravity remains downward (upward) throughout the flow domain, neutral modes do not exist. A sufficient condition for stability of a layer heated from above is that gravity remains directed downward over a sufficiently large part of the flow domain. A circle theorem limiting the growth rate of an arbitrary oscillatory mode (whether stable, neutral or unstable) is established. A sufficient condition for instability of a layer heated from below is found to be that gravity remains directed downward and the gravity profile have concave curvature throughout most of the flow domain. For a linear varying gravity field with the downward gravitational acceleration increasing with height or decreasing with height slowly, (i) a layer heated from above is stable, (ii) a layer heated from below is unstable, and (iii) the circle theorem is modified depending on whether the gravitational acceleration is increasing or decreasing with height.