Convection development in a liquid layer with a free boundary

In certain calculations of the critical Rayleigh number for a liquid layer with free boundary which is heated from below, the linearization method has been used and it has been assumed that the temperature perturbations disappear at the undisturbed free boundary.Proper linearization shows that the temperature perturbation is proportional to the free surface perturbation, and the latter is proportional to the normal stress perturbation with the proportionality factor F=²/gh³ (g is the free-fall acceleration, is the kinematic viscosity, h is the liquid layer thickness). In §1 we present a formulation of the problem with account for the parameter F; in §2 we consider the linearized equations and the existence of a stability threshold is proved-a positive eigenvalue-and it is established that with an increase in the parameter F/P (P is the Prandtl number) the value of the critical Rayleigh number Ra_* decreases; §3 presents the results of a numerical calculation of Ra as a function of the parameter F/P.Convection development in a liquid layer with a free surface on which a given temperature is maintained was studied in [1, 2]. The value R_*=1100 found for the critical Rayleigh number agrees well with the experimental value. In the calculations made in [1, 2] the linearization method is used, and it is assumed that the temperature perturbations disappear at the undisturbed free boundary. Strictly speaking, this assumption is not correct.Correct linearization shows that the temperature perturbation is proportional to the perturbation of the free boundary, and the latter is proportional to the normal stress perturbation (see below (2.3)).The problem formulation is presented in §1; §2 deals with the linearized equations and the existence (Theorem 2.1) is demonstrated of a stability threshold—which is a simple positive eigenvalue; §3 presents the results of a numerical calculation of R_* as a function of the parameter =F/P.