Dmitry V. Lyubimov

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.

Co-symmetry breakdown in problems of thermal convection in porous medium

Abstract We investigate two-dimensional thermal convection of saturating incompressible fluid in a horizontal cylinder filled with porous medium. The temperature distribution on the boundaries is time-independent and corresponds to the heating from below. At supercritical parameter values the problem has infinite number of stationary solutions for arbitrary shape of the region. This degeneracy is connected with the so-called co-symmetry property: the existence of the vector field which is orthogonal to the considered one. Non-coincidence of zeroes of these two fields leads generally speaking, to the degeneracy of the solutions. To destroy the degeneracy we add weak fluid seeping of the fluid through the boundaries either in vertical or in the horizontal direction. The breakdown of the family of the stationary solutions at high supercritical values of the Rayleigh number is studied in detail with the help of the corresponding normal form. Several limit cycles with the twisted leading manifolds appear as a result of the family destruction. To investigate the dynamical behavior the finite-dimensional models of the convection which maintain the breakdown of co-symmetry, are constructed on the base of the Galerkin approximation. The same scenario of the transition to chaos which seems to be connected with the co-symmetry breakdown, is recovered for both kinds of seeping. The quasi-periodic solution branches from the limit cycle. The further increase of the Peclet number leads to mode-locking, which is followed by the appearance of the homoclinic surface formed by the unstable manifold of the saddle periodic orbit; destruction of the latter surface leaves in the phase space the object with torus-like shape and non-integer fractal dimension.

Mechanisms of vibrational control of heat transfer in a liquid bridge

Convective flows in a liquid bridge subject to axial high frequency vibrations are studied on the base of a generalized Boussinesq approach. The generation of mean vorticity in the dynamical skin-layers near the rigid and free boundaries is taken into account, with the help of effective boundary conditions for the mean components of hydrodynamical fields. The role of viscous damping of surface waves in the generation of the mean flows is analyzed. Numerical calculations are carried out by a finite-difference method. The possibilities of the vibrational control of thermocapillary flow and of the orientation of the surfaces of constant temperature are demonstrated.

TWO MECHANISMS OF THE TRANSITION TO CHAOS IN FINITE-DIMENSIONAL MODELS OF CONVECTION

Abstract The onset of chaos in systems of ordinary differential equations possessing a stationary solution with a one-dimensional unstable manifold is studied both numerically and qualitatively with the help of an auxiliary piecewise monotomic discontinuous recursion relation. A connection is established between the route to chaos and the ratio of two leading eigenvalues of the vector field linearized near the fixed point. This connection is confirmed by numerical data obtained from the investigation of differential equations originating from a hydrodynamical problem. Two routes are considered — a well-known mechanism suggested by Lorenz and another one which is due to the accumulation of bifurcations corresponding to the emergence of homoclinic orbits of a saddle-point. The asymptotical properties of the latter route prove to be entirely determined by the above ratio.

The Onset and Nonlinear Regimes of Convection in a Two-Layer System of Fluid and Porous Medium Saturated by the Fluid

Abstract The onset of convection and its nonlinear regimes in a heated from below two-layer system consisting of a horizontal pure fluid layer and porous medium saturated by the same fluid is studied under the conditions of static gravitational field. The problem is solved numerically by the finite-difference method. The competition between the long-wave and short-wave convective modes at various ratios of the porous layer to the fluid layer thicknesses is analyzed. The data on the nature of convective motion excitation and flow structure transformation are obtained for the range of the Rayleigh numbers up to quintuple supercriticality. It has been found that in the case of a thick porous layer the steady-state convective regime occurring after the establishment of the mechanical equilibrium becomes unstable and gives way to the oscillatory regime at some value of the Rayleigh number. As the Rayleigh number grows further the oscillatory regime of convection is again replaced by the steady-state convective regime.

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.

Vibration Effect on the Nonlinear Regimes of Thermal Convection in a Two-Layer System of Fluid and Saturated Porous Medium

The effect of high-frequency vibrations on the nonlinear regimes of thermal convection in a two-layer system composed of a horizontal pure fluid layer and fluid-saturated porous layer heated from below is studied in the framework of the average approach. For large porous layer thicknesses it has been found, that at low vibration intensities the evolution of convective regimes with the growth of the Rayleigh number proceeds as follows: stationary regime—oscillatory regime—stationary regime. At high vibration intensities the stationary convective regimes take place at any values of the supercriticality used in the calculations. At close values of the fluid layer and porous layer thicknesses the interaction between the short-wave and long-wave instability modes is investigated. It has been found that at high vibration intensities the ambiguity of the stationary solutions is observed in a certain range of supercriticalities.

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.

Instability of a drop moving in a Brinkman porous medium

Sedimentation of a spherical drop of one fluid in the porous medium saturated by another is analyzed in the framework of the Brinkman model. The formula for the stationary velocity is obtained, which reproduces the results for the drop motion in a viscous fluid (Hadamard–Rybczynski formula) and for the Darcy model in the corresponding limiting cases. The steadily moving drop is found to be unconditionally unstable to interface perturbations. It is shown that the Brinkman model eliminates the unphysical features of the instability inherent to the Darcy model.

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.