| Abstract: | When the surface temperature of a liquid is a harmonic function of time with a frequency , a temperature wave propagates into the liquid. The amplitude of this wave decreases exponentially with distance from the surface. The temperature oscillation is essentially concentrated in a layer of the order of (2 /)1/2, where x is the thermal conductivity of the liquid (thermal boundary layer). Depending on the phase, at certain positions below the surface the temperature gradient is directed downwards and if its magnitude is sufficiently large (the magnitude is a function of the amplitude and frequency of the surface oscillations) the liquid can become unstable with respect to the onset of convection. In that case the convective motion may spread beyond the initial unstable layer. For low frequencies the stability condition can be derived from the usual static Rayleigh criterion, on the basis of the Rayleigh number and the average temperature gradient of the unstable layer. This quasi-static approach, used by Sal'nikov 1], is appropriate to those cases in which the period of the temperature oscillations is much larger than the characteristic time of the perturbations. But when these times are of the same order, the problem must be analyzed in dynamic terms. The stability problem must then be formulated as a problem of parametricresonance excitation of velocity oscillations due to the action of a variable parameter-the temperature gradient.In an earlier work 2] we considered the problem of the stability of a horizontal layer of liquid with a periodically varying temperature gradient. It was assumed that the thickness of the layer was much smaller than the penetration depth of the thermal wave, so that the temperature gradient could be assumed to be independent of position. In the present work we consider the opposite case, in which the liquid layer is assumed to be much larger than the penetration depth, i. e., a thermal boundary layer can be defined. The temperature gradient at equilibrium, which is a parameter in the equations determining the onset of perturbations, is here a periodic function of time and a relatively complicated function of the depth coordinate z. The periodic oscillations are solved by the Fourier method; the equations for the amplitudes are solved by the approximate method of KarmanPohlhausen.The authors are grateful to L. G. Loitsyanskii for helpful criticism. |
