Effect of high frequency vibration on convection initiation

Abstract

In [1] equations are derived which describe convection in a gravity force field and a criterion is introduced which determines the onset of convection.

In the present study we consider the case when, in addition to the gravity forces, there are vibrational forces acting on a liquid enclosed in a vessel. These forces arise as a result of vibration of the vessel along the vertical axis. In order to study the effect of vibrations we use the method of averaging with respect to small vibrations [2,3]. The unknowns are sought in the form of the sum of two terms, one which varies slowly with time and one of small amplitude which varies rapidly.

Averaged convsction equations are formulated (§1).

We consider the case of a plane horizontal strip and, under the assumption of satisfaction of the stability exchange principle, we introduce two parameters which define the onset of convection. One of these parameters is already known [1]—the product of the Grashof and Prandtl numbers. The second, arising as a result of the action of the vibrational forces, is apparently introduced here for the first time. The conclusions concerning the effect of vibrations on the initiation of convection (§3) are made for a model problem, on the assumption of spatially periodic disturbances (without accounting for the actual boundary conditions). In this case the vibrations stabilize the relative state of rest, and we can choose the velocity so that stability will exist for all temperature gradients A [see (3.7)].

It is found that in the presence of vibration (even small) the relative state of rest is stable with high temperature gradients. Moreover, if it is known that for given values of the vibration velocitya=a ₀and the temperature gradient A=A₀ the relative state of rest is stable, then we can, starting from the valuesa=0, A=0, by simultaneous variation ofa and A reach the indicated values (a ₀, A₀) while remaining in the region of stability. (We note that, generally speaking, the relative state of rest cannot be stabilized by simply increasing the gradient, since in the course of the increase the instability zone may be entered.) These conclusions are clearly valid only for vibrations with sufficiently high frequency which permit use of the averaging method. Justification of the method is not presented here.

The study of the model problem gives an idea of the qualitative picture of the phenomenon. We would expect that the qualitative conclusions drawn are also valid under actual boundary conditions. The authors hope to carry out these calculations in the near future and present a justification for the proposed method.