T. P. Lyubimova

Behavior of a drop on an oscillating solid plate

Oscillations of a nominally hemispherical, inviscid drop on a solid plate are considered accounting for the contact line dynamics. Hocking boundary conditions hold on the contact line: the velocity of the contact line is proportional to the deviation of the contact angle from its equilibrium value. Natural oscillations of a drop are studied, and both eigenfrequencies and damping ratios are determined for the axisymmetric modes. The linear oscillations caused by normal vibration of the substrate are considered. Well-pronounced resonant phenomena are revealed. The nonlinear oscillations of a drop are studied.

Motion of a Sphere Suspended in a Vibrating Liquid-Filled Container

The effects of small vibrations on the motion of a solid particle suspended in a fluid cell were investigated theoretically and experimentally. An inviscid model was developed to predict the amplitude of a solid particle suspended by a thin wire in the fluid cell which was vibrated horizontally. Both the model and experimental data showed that the particle amplitude is linearly proportional to the cell amplitude, and the existence of a resonance frequency. At higher cell vibration frequencies well above the resonance frequency, both the model and experiments showed that the particle amplitude becomes constant and independent of the wire length.

Convective instability of a system of horizontal layers of immiscible liquids with a deformable interface

The convective stability of equilibrium is considered for a system of two immiscible fluids which differ little in density. A generalized Boussinesq approximation is developed, making it possible to take the interface deformations properly into account. The stability of the equilibrium state of two fluids in a horizontal layer with a vertical temperature gradient is investigated. Several instability mechanisms are identified: long-wave and cellular monotonic disturbances and oscillatory disturbances. Increasing the deformability is shown to cause switching between instability mechanisms.

Stability of equilibrium of a double-layer system with a deformable interface and a prescribed heat flux on the external boundaries

The stability of mechanical equilibrium of a system of two horizontal immiscible-liquid layers with similar densities is studied. The problem is solved for a prescribed heat flux on the external boundaries. Within the framework of a generalized Boussinesq approximation, which takes the interface deformation correctly into account, the onset of convection caused by heating the system from above or below is considered. Two long-wave instability modes attributable to the presence of the deformable interface and the given heat flux on the external boundaries are detected. The system response to monotonic and oscillatory disturbances with finite wavelengths is investigated. A complete stability map is constructed.

Accumulation of solid particles in convective flows

Accumulation of solid particles suspended in unsteady convective flows is under theoretical investigation. The principal goal is to understand and interpret recent experiments by D. Schwabe [1,2]. Providing that volume particle concentration, nonisothermality, and relative size of particle are small, an effective single-fluid theoretical model is developed. The peculiarity of the obtained model is taking into account the distinction between fluid and particle inertia. This model is further applied to study particle accumulation in different flow setups: in a model oscillatory flow in a canal heated from below and subjected to the modulated gravity and in the Marangoni flow in a half-zone under microgravity conditions. These problems are investigated numerically by means of finite difference technique. We demonstrate, that the developed theoretical model properly describes generic features of particle accumulation in unsteady flows. Particularly, heavy particles tend to leave the centers of vortices, where the flow vorticity is maximal, and accumulate at their periphery. From numerical simulations in a floating zone, we try to clarify particle dynamics in Schwabe’s setup.

Rayleigh and parametric thermo-vibrational instabilities in supercritical fluids under weightlessness

Under the absence of gravity forces, the interaction of vibration with a thermal boundary layer (TBL) can lead to a rich variety of dynamics in a supercritical fluid (SCF). When subjected to vibration, a SCF can display different kinds of instabilities for different relative directions of the TBL and vibration. Rayleigh vibrational instability is formed when the vibration direction is tangential to the TBL. When the direction of vibration is perpendicular to the TBL, instabilities of parametric nature can develop. Two-dimensional finite volume numerical analysis of supercritical H2 filled in a square cell under vibration is carried out. The vibrational amplitudes range from 0.05 to 5 times the side of the cell and frequencies vary between 2.78 Hz and 25 Hz. Three different thermal boundary conditions (isothermal walls, adiabatic vertical/isothermal horizontal walls, and adiabatic horizontal/isothermal vertical walls) have been considered with various temperature proximities to the critical point (10 mK, 1...

The effects of vibrations on particle motion near a wall in a semi-infinite fluid cell

The effects of small vibrations on a particle-fluid system relevant to material processing such as crystal growth in space have been investigated experimentally and theoretically. An inviscid model for a spherical particle of radius, R 0 , suspended by a thin wire and moving normal to a cell wall in a semi-infinite liquid-filled cell subjected to external horizontal vibrations, was developed to predict the vibration-induced particle motion under normal gravity. The wall effects were studied by varying the distance between the equilibrium position of the particle and the nearest cell wall, H. The method of images was used to derive the equation of motion for the particle oscillating in an inviscid fluid normal to the nearest cell wall. The particle amplitude in a semi-infinite cell increased linearly with the cell vibration amplitude as expected from the results for an infinite cell, however, the particle amplitude also changed with the distance between the equilibrium position of the particle and the nearest wall. The particle amplitude was also found to increase or decrease depending on whether the cell vibration frequency was below or above the resonance frequency, respectively. The theoretical predictions of the particle amplitudes in the semi-infinite cell agreed well with the experimental data, where the effect of the wall proximity. on the particle amplitude was found to be significant for (H/R 0 < 2) especially near the resonance frequency. Experiments performed at high frequencies well above the resonance frequency showed that the particle amplitude reaches an asymptotic value independent of the wire length.

Vibration influence on morphological instability of a solidification front

The paper deals with the vibration influence on the stability of a planar solidification front moving with constant average velocity. The case of high frequency vibrations with small amplitude is studied. The density dependence on the solute concentration is accounted for in the framework of the Boussinesq approach. The equations for average velocity and concentration fields are obtained by multiple scale and averaging methods. A number of limit cases are studied analytically, by perturbation methods. For finite values of governing parameters the threshold of morphological instability is determined numerically. It is found that the vibrations normal to the solidification front exert destabilizing effect whereas tangential vibrations suppress the development of morphological instability.

Convective flows in a cylindrical fluid zone in a high-frequency vibration field

Convective flows of a nonuniformly heated fluid in a cylindrical fluid zone in a high-frequency longitudinal vibration field are studied. Vibration frequencies which are high as compared with dissipative decrements and capillary frequencies, but small as compared with acoustic frequencies are considered. The general method formulated earlier for describing the behavior of inhomogeneous fluids under the influence of high-frequency vibrations is used. The interaction between the vibrational flow mechanisms and thermocapillary effects on a free surface is analyzed.

Running interfacial waves in a two-layer fluid system subject to longitudinal vibrations.

We study the waves at the interface between two thin horizontal layers of immiscible fluids subject to high-frequency horizontal vibrations. Previously, the variational principle for energy functional, which can be adopted for treatment of quasistationary states of free interface in fluid dynamical systems subject to vibrations, revealed the existence of standing periodic waves and solitons in this system. However, this approach does not provide regular means for dealing with evolutionary problems: neither stability problems nor ones associated with propagating waves. In this work, we rigorously derive the evolution equations for long waves in the system, which turn out to be identical to the plus (or good) Boussinesq equation. With these equations one can find all the time-independent-profile solitary waves (standing solitons are a specific case of these propagating waves), which exist below the linear instability threshold; the standing and slow solitons are always unstable while fast solitons are stable. Depending on initial perturbations, unstable solitons either grow in an explosive manner, which means layer rupture in a finite time, or falls apart into stable solitons. The results are derived within the long-wave approximation as the linear stability analysis for the flat-interface state [D.V. Lyubimov and A.A. Cherepanov, Fluid Dynamics 21, 849 (1986)] reveals the instabilities of thin layers to be long wavelength.