Dmitriy Lyubimov

Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number

The paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the cross-section aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleigh-type steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized.

Rotating magnetic field effect on convection and its stability in a horizontal cylinder subjected to a longitudinal temperature gradient

A rotating magnetic field (RMF) is used in crystal growth applications during the solidification process in order to improve the crystal quality. Its influence on the convective flows in molten metals and on their stability is studied here in the case of a horizontal infinite cylindrical channel subjected to a longitudinal temperature gradient. The steady convective flows, which correspond to the usual longitudinal counterflow structure, with four vortices in the cross-section for non-zero Prandtl number, Pr, are modified by the RMF (parametrized by the magnetic Taylor number Ta m ). For zero Prandtl number, the flow in the cross-section corresponds to circular streamlines and the longitudinal flow structure is moved in the direction of the magnetic field rotation, with a decrease in its intensity and an asymptotic variation as 1/Ta m . For non-zero Prandtl numbers, depending on the respective values of Ta m on one side and Prandtl and Grashof numbers on the other side, different structures ranging from the circular streamlines with transport by rotation of the longitudinal velocity and the temperature field, to the more usual counterflow structure almost insensitive to the RMF with four cross-section vortices, can be obtained. The decrease in the flow intensity with increasing Ta m is also delayed for non-zero Pr, but the same asymptotic limit is eventually reached. The stability analysis of these convective flows for Ta m = 0 shows a steep increase of the thresholds around Pr = Pr t,0 ≈ 3 x 10 -4 , corresponding to the transition between the usual counterflow shear mode and a new sidewall shear mode. This transition is still present with an RMF, but it occurs for smaller Pr values as Ta m is increased. Strong stabilizing effects of the rotating magnetic field are found for Pr Pr t,0 (i.e. in the domain where the sidewall instability is dominant), in contrast, the stabilization by the RMF is weak.

Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 2. Effect of a magnetic field

The stabilization of buoyant flows by a magnetic field is an important matter for crystal growth applications. It is studied here when the cavity is an infinite channel with rectangular cross-section typical of horizontal Bridgman configurations and when the magnetic field is applied in the vertical and transverse directions. The steady basic flow solution is first calculated: the usual counter flow structure is modified by the magnetic field and evolves towards jets in the cross-section corners when the magnetic field is vertical and towards a more uniform structure in the transverse direction when the magnetic field is transverse. The stability results show a very good stabilization of the convective flows for a vertical magnetic field with exponential increases of the thresholds for any width of the channel and for various Prandtl numbers Pr . The results for a transverse magnetic field are more surprising as a destabilizing effect corresponding to an initial decrease of the thresholds is obtained at Pr =0 and for small channel widths. A kinetic energy budget at the thresholds reveals that the main destabilizing factor is connected to the vertical shear of the longitudinal basic flow and that it is the modifications affecting this shear energy which are mainly responsible for the variation of the thresholds when a magnetic field is applied.

Effect of the contact-line dynamics on the oscillations of a compressed droplet

This paper studies the natural and forced oscillations of a deformed droplet of an inviscid liquid surrounded by a different liquid and bounded in the axial direction by solid planes. In equilibrium, the droplet is a figure of revolution and the ratio of its radius to height is significant. The equilibrium contact angle between the side surface of the droplet and the solid surface is different from a right angle. The motion of the contact line is taken into account by setting an effective boundary condition. It is shown that three characteristic ranges of natural frequencies exist.

Thermal vibrational convection in near-critical fluids. Part 2. Weakly non-uniform heating

A theoretical model describing the response of a single-phase near-critical fluid to small-amplitude, high-frequency translational vibrations is developed on the basis of the multiple-scale and averaging methods. Additionally to the usual terms of thermal convection, the equations and boundary conditions contain new terms responsible for the generation of pulsating and average flows due to vibrational forcing and fluid compressibility. The effect of compressibility is taken into account both in the boundary layers and in the bulk. A number of classical problems of convection and thermoacoustics are considered on the basis of the new model.

Two-dimensional thermal convection in porous enclosure subjected to the horizontal seepage and gravity modulation

Coupled effect of horizontal seepage and gravity modulation on the onset and nonlinear regimes of two-dimensional thermal buoyancy convection in horizontal fluid-saturated porous cylinder of rectangular cross section with perfectly conductive boundaries is studied. It is shown that gravity modulation makes destabilizing effect. Null-dimensional dynamical system describing supercritical convective regimes is derived. Conditions for existence of stable periodical regimes are defined. It is found that system demonstrates dynamics on a torus which can be either resonant or non-resonant depending on the parameters. Synchronization domains which correspond to the resonant torus existence in the parameter space are determined by the rotation number technique. It is found that at certain values of the ratio of the cross section height to width, degeneracy takes place. In this case different stable periodical regimes forming one-parametric family coexist. Linear stability and nonlinear dynamics of the system at fi...

The Rayleigh–Taylor instability of the externally cooled liquid lying over a thin vapor film coating the wall of a horizontal plane heater

The linear instability of a vapor film formed at the surface of a flat horizontal heater surrounded by an externally cooled liquid is investigated in the presence of a gravitational field. Consideration is given to the case when the stationary base state is characterized by the heat fluxes balanced at the interface between the two media. The critical value of the heat flux required for the complete suppression of the Rayleigh–Taylor instability by the phase transition has been evaluated mainly in the absence of the natural convection in the liquid layer and is found to be different from the known data obtained by approximate approaches. The case of the instability suppression in the system when long-wave disturbances have the longest lifetime is described. It has been shown that the media pressure influence on the phase transition, revealed in thin vapor films, can markedly increase the growth rate of long-wave disturbances and prevent their suppression.

Dynamic detachment of a gas bubble from a solid flat substrate performing normal harmonic vibrations

This paper presents an investigation of the detachment of a bubble from a solid flat substrate according to a dynamic scenario, i. e., due to strong vibrations of the surface shape of the bubble caused by normal nonacoustic harmonic vibrations of the substrate. The Layzer’s model based on an analysis of single-mode solutions near the bubble top was used to study its detachment in microgravity, where the dynamics of the bubble surface is due to competition between liquid inertia forces and surface tension forces. Detachment of the bubble from the substrate was determined from the condition of its elongation during vibrations by a magnitude comparable to the radius of the bubble in equilibrium. The dependence of the vibration intensity required for the detachment of the bubble on the problem parameters was determined using a number of empirical assumptions. The volume of the detached bubbles was estimated.

Phase diagram of a binary mixture in a closed cavity

Normally, the phase diagram is reported as a property of the binary mixture. We show that the phase diagram (that is, the zones of thermodynamic stability of the states of the binary mixture) is also affected by the size of the container. We investigate the thermodynamic stability of the binary mixture in a closed cavity, and identify the zone in parameters where the binary mixture is heterogeneous in equilibrium (the zone of spinodal decomposition), the zone where the mixture is always homogeneous in equilibrium, and the zone where the transition between these two states is possible (the metastable nucleation zone). In addition, we investigate the properties of the smallest single droplet that may be in equilibrium in the closed cavity (for the given average concentration, all smaller droplets would always dissolve). We show that the size of such droplets depends on the cavitys size, as ∼L^{1/2}.

Numerical modeling of frozen wave instability in fluids with high viscosity contrast

This paper deals with the direct numerical simulation of quasi-stationary (frozen) wave formation at the interface of two immiscible fluids with large viscosity contrast, in a rectangular container subjected to the horizontal vibrations of finite frequency and amplitude. The critical conditions for the origination of a frozen wave as well as the dependences of the frozen wave height and wavelength on the vibration intensity are obtained. The time-evolution of the interface shape during the vibration period is analyzed. Numerical results are found to be in a good agreement with known experimental and linear stability results. The average deformation of the interface and the structure of average flows are calculated for different vibration intensities. It is shown that a change in the dependencies of the frozen wave characteristics on the vibration intensity follows a change in average flow structure.