Influence of high-frequency vibrations on the onset of convection in a two-layer system

Abstract

This Note deals with the influence of high-frequency translational oscillations on the onset of convection in a two-layer system of weakly heterogeneous immiscible fluids with deformable interface. The averaging method is applied to the generalized Oberbeck-Boussinesq equations. Vibration-generated forces and tensions appear as the result. A transition to the Oberbeck-Boussinesq approximation is made in the averaged equations. Analysis of averaged equations leads to the following conclusions. Horizontal vibrations are obtained not influencing the onset of convection, and in the cases of other directions the influence of vibration is determined by a single parameter, depending on velocity amplitude and direction. Vibration is shown to generate effective surface tension, smoothing the interface. Critical parameters are calculated for the case of homogeneous fluids. To cite this article: S.M.

Many Faces of Boussinesq Approximations

“…(D4) The asymptotic for gravity Φ * ∼ ε 2k−q at ε → 0 in (2.24) is 'singular' for 2k − q < 0 however it does not prevent one against of using it. For example, the 'singular' asymptotic Φ * ∼ 1/ε has been actively used in the vibrational convection, see Zenkovskaya & Simonenko (1966), Simonenko (1972), Gershuni & Lyubimov (1997), Levenshtam (1996). It is apparent that, without the use of this 'singular' asymptotic, the theory of vibrational convection can not be constructed.…”

Mixing Under Vibrations in Reduced Gravity

“…Then all physical quantities may be presented as the superposition of a slow (the characteristic time τ vis is large with respect to the vibration period) and fast (the characteristic time 1/ω is of the order of the vibration period) parts (Zenkovskaya and Simonenko 1965;Gershuni and Lyubimov 1998). The variables (V, P, C) take form:…”

“…Let us look in the principal of averaging approximation (Zenkovskaya and Simonenko 1965;Gershuni and Lyubimov 1998). Taking into account the relations (2), the viscous terms in the equations for the fast velocity can be neglected and the fast flow is considered as potential.…”

Effects of vibrations on dynamics of miscible liquids

“…In the limit of high frequency and small amplitude of periodical vibration the averaging method can be applied effectively to study the property of vibrational convection, see, e.g. [5,6]. The averaging procedure is rather confusing and for better understanding we will give some details.…”

“…One can speak about low or high frequencies depending on whether the period is comparable with or much smaller than the reference viscous and heat/mass diffusion times. The high frequency limit is of special interest: here the flow can be represented as a superposition of`fast' part, which oscillates with the frequency of vibration, and`slow' time-average part (mean flow), which describes the non-linear response of the fluid to a periodic excitation [5,6].…”