D.V. Lyubimov

Vibrational control of crystal growth from liquid phase

Modeling and numerical simulations of the convective flows induced by the vibration of the monocrystal during crystal growth have been performed for two configurations simulating the Cz and FZ methods. This permitted to emphasize the role of different vibrational mechanisms in the formation of the average flows. It is shown that an appropriate combination of these mechanisms can be used to counteract the usual convective flows (buoyancy- and/or thermocapillary-driven) inherent to crystal growth processes from the liquid phase. While vibrational convection is rather complex due to these identified mechanisms, the new modeling used in the present paper opens up very promising perspectives to efficiently control heat and mass transfer during real industrial applications of crystal growth from the liquid phase.

Numerical Simulation of a Liquid Drop Freely Oscillating

Free oscillations of a viscous liquid drop surrounded by a dynamically inactive ambient gas, in zero gravity, are investigated numerically using FIDAP TM package in the axisymmetrical case. The full Navier-Stokes equations with appropriate interfacial conditions are solved by using Galerkin/Finite element technique along with the spine method for the advection of the free boundary are used. The aim of this preliminary study is to demonstrate the ability of the package to accurately solve nonlinear free surface problems. Oscillations of viscous drops released from an initial static deformation of small-to large-amplitude proportional to the second spherical harmonic, without initial internal circulation, are considered

Numerical investigation of meniscus deformation and flow in an isothermal liquid bridge subject to high-frequency vibrations under zero gravity conditions

This paper deals with meniscus deformation and flow in an isothermal liquid bridge maintained between two circular rods, when one rod is subject to axial monochromatic vibrations. It concerns a fundamental aspect of the problem of crystal growth from melt by the floating-zone technique which is often considered in weightlessness conditions. In the absence of vibrations the bridge is cylindrical; but due to vibration the mean shape of the meniscus is no more cylindrical and the meniscus oscillates around this mean shape. Two models are developed. First, we take into account the pulsating deformations of the meniscus (free surface), but we assume that the mean shape of meniscus remains cylindrical (i.e., we neglect the influence of vibration on this mean shape). For this simple case, a solution of the problem for the pulsating meniscus deformations and the pulsating velocity field is found in explicit form. For the mean flow, the problem is solved numerically by a finite-difference method. The calculations demonstrate the contribution of two basic mechanisms of mean flow generation due to vibrations, related to the generation of mean vorticity in the viscous boundary layer near the rigid boundaries and surface-wave propagation at a free surface. The intensity of the mean flow induced by surface waves is found to be sharply increasing when the vibration frequency approaches the resonance values that are determined from the explicit form of the solution of pulsation problem. In the second model, we take into account both pulsating and mean deformations of the meniscus. The governing equations for the potential of pulsating velocity and mean velocity, and for the pressure, are solved by using a finite-difference method and a boundary-fitted curvilinear coordinate system fitting the free surface.

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.

Influence of gravitational precipitation of solid particles on thermal buoyancy convection

Abstract We study a non-isothermal two-phase flow in a closed cavity heated from the sidewall, where one of the phase is a gas (or a liquid) and another phase consists of solid particles. The particles are subjected to the gravitational precipitation and the drag exerted by the flow. In the framework of the generalized Boussinesq approximation we derive a closed set of governing equations describing the dynamics of the suspensions. The linear stability of basic flow is analyzed. In addition, the transient processes of the thermal buoyancy convection of a fluid with solid inclusions is determined numerically.

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.

Influence of high-frequency vibration on the morphological instability in the directional crystallization of binary melts

The influence of various types of vibration on the morphological instability of the directional crystallization front in binary melts is investigated numerically under microgravity and terrestrial conditions. The vibration frequency is assumed to be high and the amplitude to be small and an averaged approach is used. It is shown that high-frequency rotational vibration generates an intense mean flow localized in the neighborhood of the crystallization front and the direction of this flow is opposite to the direction of gravity-convection flow. Under terrestrial conditions the interaction between vibration flow and gravity convection leads to the gravitational vortex being pushed away from the crystallization front. Under both terrestrial and microgravity conditions rotational vibration has a strong stabilizing action on the morphological instability and prevents the formation of an axial hollow.

Behavior of a drop (bubble) in a non-uniform pulsating flow

Abstract Behavior of a drop (bubble) in a non-uniform pulsating flow of a fluid with different density is studied. It is assumed that the pulsation frequency is high and amplitude is small. The gravity is absent. The drop (bubble) size is small in comparison with the characteristic scale of flow non-uniformity in the absence of inclusion and with the distance from the rigid walls. General formulas for the average force acting upon the drop (bubble) and for the average shape taken by the inclusion in a non-uniform pulsating flow of arbitrary kind are obtained. The average force is found to be determined by the velocity gradients of unperturbed flow and by the density ratio. This force vanishes for uniform pulsating flow. The deformation of average shape takes place for both uniform and non-uniform pulsating flows. For uniform pulsating flow, the drop (bubble) takes the shape of oblate spheroid. Non-uniformity of the pulsating flow results in more complex average shape of inclusion: the deviation of average shape from oblate spheroid is determined by velocity gradients of unperturbed flow. The calculations made for the pulsating flow generated by rotational vibrations in a narrow gap between two coaxial cylinders show that in this case the inclusion takes pear-shaped form extending to a surface of external cylinder. The average force acting upon inclusion from the pulsating flow is directed to the cylinder axis for the inclusion denser than the host fluid and in the opposite direction for less dense inclusion.

Rayleigh–Bénard–Marangoni convection in a weakly non-Boussinesq fluid layer with a deformable surface

The influence of surface deformations on the Rayleigh–Benard–Marangoni instability of a uniform layer of a non-Boussinesq fluid heated from below is investigated. In particular, the stability of the conductive state of a horizontal fluid layer with a deformable surface, a flat isothermal rigid lower boundary, and a convective heat transfer condition at the upper free surface is considered. The fluid is assumed to be isothermally incompressible. In contrast to the Boussinesq approximation, density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. Two different types of temperature dependence of the density are considered: linear and exponential. The longwave instability is studied analytically, and instability to perturbations with finite wavenumber is examined numerically. It is found that there is a decrease in stability of the system with respect to the onset of longwave Marangoni convection. This result could not be obtained within...

On the Boussinesq approximation for fluid systems with deformable interfaces

The assumptions underlying the Boussinesq approximation place restrictions on its applications to systems with free surfaces and interfaces. In this paper we reconsider the limits in which the Boussinesq approximation is valid and develop a generalization of the approximation which allows for self-consistent application to such systems. The Boussinesq limit is characterized two parameters, G = gL3ν2, and e = βθ, where G is a dimensionless measure of gravitational acceleration or system size and e is the product of the fluids coefficient of thermal expansion with a characteristic temperature difference. The Boussinesq limit, in general, corresponds to G → ∞ and e → 0 while the product Ge (equal to the familiar Grashof number) remains finite. We consider two problems involving deformable boundaries: the stability of a two-layer fluid system heated from above and below and the influence of buoyancy on long-wavelength Marangoni instability. In the first two examples, we examine the conditions required to consistently account for the effects of a deformable surface on thermal convection while simultaneously applying the Bousinesq approximation. In particular, the effect of the deformable surface can be included through a term proportional to the product of Gδ with the deflection, ζ, of the interface from planarity, but it is required to treat the density of the two fluids as equal in the equations of motion (i.e., δ → 0). In the second example, that of long-wavelength Marangoni instability, we find that to simultaneously consider the effects of thermal buoyancy together with a deformable surface we must either treat the surface as undeformable (G → ∞ and e → 0) or consider finite G and take Ge and e to be independent parameters in the equations of motion. This leads to a result which explicitly reveals the role of buoyancy as a destabilizing influence on long-wavelength instability.