On the question of the preferred mode in cellular thermal convection

Publisher Website
Google Scholar
Abstract
This paper modifies and refines earlier work of Palm (1960) concerning the finiteamplitude steady state of cellular convective motion attained when a horizontal layer of fluid becomes unstable as a result of being heated from below. The two non-linear ordinary differential equations to which the problem was reduced by Palm (under certain conditions) are given in a corrected form, and are then analysed in some detail. The principal conclusions are that, for the model considered, hexagonal convection cell may be the stable equilibrium state only if the variation of kinematic viscosity with temperature is sufficiently great. Under the same circumstances a two-dimensional roll cell is also possible, the initial conditions determining which state actually occurs. Although further work is indicated, it seems probable that in an actual experiment with sufficiently large kinematic-viscosity variation, the hexagonal cells are more likely to appear. The analysis enables conclusions to be drawn concerning the flow direction at the cell centre, and also shows that a disturbance of sufficient magnitude may grow even though the situation is a stable one by linearized theory. Comparison with experiment is discussed.