A theoretical analysis of the onset of surface-tension-driven convection in a horizontal liquid layer cooled suddenly from above

Abstract

The onset of surface-tension-driven convection in a horizontal liquid layer cooled suddenly from above is analyzed by using linear stability theory. To obtain the critical condition to mark the onset of instability in the form of regular cellular motion the time domain is divided into two. With small time the basic temperature profile is strongly nonlinear and therefore the propagation theory we have developed is employed. Based on the propagation theory a new set of stability equations are derived and their scale analysis is discussed. It is found that the fluid layer is more stable with decreasing the Biot number and the Prandtl number. With large time the frozen-time model is applied. In this time domain the stability criteria are independent of the Prandtl number. By connecting the predictions from the above two models the overall stability criteria are constructed. The interesting role of the Biot number on the critical condition is discussed in detail.