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In this paper, we study the onset and non-linear regimes of thermal buoyancy convection in an inclined porous layer saturated with fluid. The layer is subject to a gravitational field and a strictly vertical temperature gradient. This problem is important for geological applications. The linear stability of the heat-conducting regime to two-dimensional perturbations was previously studied by Kolesnikov and Lyubimov [J. Appl. Mech. Tech. Phys. 14, 400–404 (1973)]. In the first part of our work, we numerically, using the finite difference method, investigate two-dimensional nonlinear convection regimes that arise after the loss of stability of the heat-conducting regime. In the second part of the paper, the linear stability of the heat-conducting regime to three-dimensional perturbations is investigated. It has been found that for any layer inclination angle, three-dimensional perturbations are more dangerous than two-dimensional ones, and the most dangerous perturbations have the form of longitudinal rolls. For the layer inclination angle [Formula: see text], the wavenumber of critical perturbations is equal to zero, and for [Formula: see text], it differs from zero. Numerical calculations by the finite volume method within the framework of the full three-dimensional nonlinear approach confirm the conclusions of the linear stability analysis.

The effect of acoustic wave propagating in the direction of temperature gradient on a stability of stationary plane-parallel flow of binary fluid, generated by horizontal temperature and concentration gradients in a horizontal layer is investigated for two types of thermal boundary conditions: perfectly conductive and adiabatic boundaries. The study is performed analytically for the longwave perturbations and numerically for finite-wavelength perturbations. Stability maps for fixed Prandtl number value Pr = 0.01 and different values of Schmidt number and acoustic Reynolds number are obtained. The most dangerous instability modes and critical perturbation structure are discussed. Comparison with the case where acoustic wave is absent is performed. It is shown, that longwave perturbations are always stabilized by acoustic wave. Numerical investigation demonstrates, that acoustic wave also stabilizes the flow with respect to finite-wavelength perturbations when negative concentrational Grashof number is higher in modulus than certain value depending on acoustic wave intensity.

In the present work, we study the resonance oscillations of a single liquid drop or gas bubble relative to their equilibrium spherical shape, which arise as a result of harmonic, translational vibrations of a volume of liquid of different densities, containing such an inclusion. The parametric resonance of the forced translational mode with the main resonant frequency equal to the sum of two adjacent eigenfrequencies of capillary oscillations of the interface was previously discovered in an inviscid approximation [Lyubimov et al., “Resonance oscillations of a drop (bubble) in a vibrating fluid,” J. Fluid Mech. 909, A18 (2021)], also taking into account low viscosity using a phenomenological approach. Now, the effect of viscosity on the resonance is investigated by rigorous accounting for the viscosity in the dynamic boundary layer near the interface. The relations governing the evolution of the resonance modes near the threshold of instability excitation are determined. The equivalence of the two approaches taking viscosity into account is revealed: the rigorous one, built on a method of multiple scales, and the previously developed phenomenological approach, which is based on the artificial addition of the dissipative terms to the amplitude equations.

The effect of acoustic wave propagating in longitudinal direction on stationary convective flow in a horizontal channel of rectangular cross-section in the presence of horizontal temperature gradient and its stability is studied. It is shown that in the presence of sufficiently intensive acoustic wave stationary flow is of coaxial character, the fluid moves in the direction of temperature gradient near sidewalls and in the opposite direction in the interior of the channel. At Pr

The effect of evaporation and condensation on the Rayleigh-Taylor instability is studied with a rigorous approach, finding that phase change induces an additional flow that affects heating of the interphase boundary. This reduces the ratio of the phase change rate to the interface position deviation.

On the basis of the phase-field approach, we investigate the simultaneous diffusive and convective evolution of an isothermal binary mixture of two slowly miscible liquids that are confined in a horizontal plane layer. We assume that two miscible liquids are brought into contact, so the binary system is thermodynamically unstable and the heavier liquid is placed on top of the lighter liquid, so the system is gravitationally unstable. Our model takes into account the non-Fickian nature of the interfacial diffusion and the dynamic interfacial stresses at a boundary separating two miscible liquids. The numerical results demonstrate that the classical growth rates that characterise the initial development of the Rayleigh-Taylor instability can be retrieved in the limit of the higher Peclet numbers (weaker diffusion) and thinner interfaces. The further nonlinear development of the Rayleigh-Taylor instability, characterised, e.g., by the size of the mixing zone, is however limited by the height of the plane layer. On a longer time scale, the binary system approaches the state of thermodynamic and hydrodynamic equilibrium. In addition, a novel effect is found. It is commonly accepted that the interface between the miscible liquids slowly smears in time due to diffusion. We however found that when the binary system is subject to hydrodynamic transformations the interface boundary stretches, so its thickness changes (the interface becomes thinner) on a much faster convective time scale. The thickness of the interface is inversely proportional to the surface tension, and the stronger surface tension limits the development of the Rayleigh-Taylor instability.On the basis of the phase-field approach, we investigate the simultaneous diffusive and convective evolution of an isothermal binary mixture of two slowly miscible liquids that are confined in a horizontal plane layer. We assume that two miscible liquids are brought into contact, so the binary system is thermodynamically unstable and the heavier liquid is placed on top of the lighter liquid, so the system is gravitationally unstable. Our model takes into account the non-Fickian nature of the interfacial diffusion and the dynamic interfacial stresses at a boundary separating two miscible liquids. The numerical results demonstrate that the classical growth rates that characterise the initial development of the Rayleigh-Taylor instability can be retrieved in the limit of the higher Peclet numbers (weaker diffusion) and thinner interfaces. The further nonlinear development of the Rayleigh-Taylor instability, characterised, e.g., by the size of the mixing zone, is however limited by the height of the plane lay...

The oscillatory instability of advective conducting fluid flow in a horizontal circular cylinder in the presence of a rotating magnetic field is investigated. For Prandtl numbers Pr = 0 and 0.01 calculations showed that in both cases the monotonic instability observed in the absence of a field and in weak fields transforms into oscillatory instability at sufficiently large magnetic Taylor numbers. At Pr ≠ 0, oscillatory instability appears at substantially higher magnetic Taylor numbers.

The onset of convection in a horizontal fluid layer in the presence of density inversion is studied under given heat fluxes at the layer boundaries. The stability diagram is obtained and the nature of disturbances is studied in the linear approximation. Depending on the density maximum position within the layer the instability may be both long-wave and cellular in nature.

Three-dimensional advective flows in a horizontal cylinder of square section are numerically investigated under thermally insulated lateral boundaries and in the presence of a uniform longitudinal temperature gradient. The flow structure is shown to considerably depend on the Grashof number, the channel length, and the Prandtl number; depending on these parameters, the flow symmetrymode and its temporal behavior may be different. It is established that different cases of transition to oscillatory flow regimes are possible, namely, either with a preliminary violation of the flow symmetry (fork bifurcation) or without a change in the type of symmetry. The parameter range on which only the fork bifurcation is observable, while oscillatory flow patterns do not appear, is also determined.

Thermal convection of an elastoviscous fluid in a horizontal square cylinder heated from below is investigated. For the description of rheological properties of the fluid, a generalized Oldroyd model is used. The study is based on a weakly nonlinear analysis and the solution of the complete nonlinear equations by a finite-difference method. The boundaries dividing the plane of rheological parameters into regions with different convection excitation mechanisms are found. The parameters of nonlinear convection regimes are determined.

Numerical modeling of the temperature distribution, the velocity fields, and the dopant (Ge:Si) concentration during germanium crystal growth by the floating zone method under microgravity is carried out. The time-dependent three-dimensional problem is solved using the difference method. The deformations of the free surface of a liquid bridge and of crystallization and melting fronts are neglected. The relaxation of axisymmetric flow regimes to a steady state is observed, whereas the unsteadiness of three-dimensional flows develops with time.

The steady-state advective flow in a long horizontal rectangular channel with rigid adiabatic boundaries in the presence of a uniform longitudinal pressure gradient is investigated. The stability of this flow with respect to perturbations of various types is studied. The dependence of the critical Rayleigh number on the Prandtl number is found for various aspect ratios.

Within the framework of the fractal mobile-immobile medium model describing non-Fickian effects occurring in admixture seepage due to particle adhesion to the solid matrix, an expression for the admixture flux is derived. Flow discretization intended for finite-difference calculations is proposed and used as a basis for a conservation-law scheme for solving the model equations with account for admixture sources. Several one-dimensional test problems of admixture propagation in an imposed seepage flow are solved using the approach developed.

The noise effect on the stability of equilibrium in systems with slowly varying parameters is investigated. With reference to some model examples and for the case of convection in a closed region in the presence of gravity modulation it is shown that taking the noise into account is of key importance at low modulation frequencies, since it can even change the sign of the effect: modulation leads to equilibrium stabilization in the absence of the noise and to destabilization in its presence.

We investigate the oscillatory convective instability in a liquid layer with a deformable free surface. We provide linear stability analysis within a non-Boussinesq approach as our aim is to examine the influence of deformability of the free surface on the Rayleigh–Benard–Marangoni instability. Within this approach, fluid is assumed to be isothermally incompressible; the density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. The numerical results show significant differences in stability behavior as compared to results obtained using the Boussinesq approach. Moreover, a novel oscillatory mode of instability is revealed for the zero Marangoni number and zero gravity. This result could not be obtained within the framework of the conventional Boussinesq approximation. Thorough investigation of the novel oscillatory mode let us propose the mechanism of this mode. It is connected with the capillary wave that enforced with thermal expansion of fluid. Weakly nonlinear analysis shows that supercritical branching of standing rolls is possible.We investigate the oscillatory convective instability in a liquid layer with a deformable free surface. We provide linear stability analysis within a non-Boussinesq approach as our aim is to examine the influence of deformability of the free surface on the Rayleigh–Benard–Marangoni instability. Within this approach, fluid is assumed to be isothermally incompressible; the density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. The numerical results show significant differences in stability behavior as compared to results obtained using the Boussinesq approach. Moreover, a novel oscillatory mode of instability is revealed for the zero Marangoni number and zero gravity. This result could not be obtained within the framework of the conventional Boussinesq approximation. Thorough investigation of the novel oscillatory mode let us propose the mechanism of this mode. It is connected with the capillary wave that enforced with thermal expans...

The stability of a steady-state advective flow of a binary mixture in a plane horizontal layer with perfectly heat conducting solid boundaries in the presence of a uniform longitudinal temperature gradient is investigated. The problem is solved with account for thermodiffusion. The limits of the stability of the steady-state flow with respect to long-wave disturbances are found analytically. The stability of the main flow with respect to plane and spiral disturbances with finite wavelengths is studied numerically. The stability maps in the “Rayleigh number-Soret parameter” plane are constructed for a number of typical liquid and gaseous mixtures.

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...

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.

The stability of mechanical equilibrium of a horizontal layer of conducting fluid in the presence of a magnetic field rotating in a horizontal plane is considered. Both finite field rotation frequencies and the limiting case of high frequencies are investigated. It is shown that the magnetic field stabilizes the equilibrium. The dependence of the critical perturbation wavelength on the field strength is non-monotonic, and with increase in the magnetic field strength the mode of most dangerous perturbations changes from long-to short-wave type. Nonlinear three-dimensional convection regimes are calculated numerically. It is found that at finite supercriticalities and a sufficiently strong magnetic field the rolls and the hexagonal cells may be stable simultaneously.

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.

The effect of angular vibration on the flow, segregation, and interface morphology during vertical Bridgman crystal growth was investigated. Transparent experiments using SCN containing about 0.02 wt.% acetone were performed. To simulate the observed results, both direct numerical simulation (DNS) and Schlichting boundary layer approximation (SBLA) were considered in the computer model. The simulated morphological breakdown patterns, as a result of acetone accumulation (segregation), are consistent with the experimental observation. At high frequency with low amplitudes, both simulation approaches gave consistent results. However, care must be taken in using SBLA for low frequency vibration of several hertz.

The stability of the state of a two-layer system consisting of a homogeneous liquid and a solid particle suspension in the same liquid with a plane interface between them is investigated. The system performs both horizontal and vertical high-frequency vibration with an arbitrary phase shift. It is shown that a mean flow develops under the simultaneous action of both horizontal and vertical vibration; quantitative flow stability characteristics are determined numerically using the differential sweep method. It is shown that a traveling wave relief exists on the liquid-suspension interface. Transverse oscillations in phase with longitudinal oscillations destabilize the entire system. The presence of an oscillation phase shift can lead to an increase in the stability limit; the direction of motion of the wave relief can differ depending on the value of the phase shift. Instability of the system in the presence of strictly vertical vibration is observed, the crisis being associated with longwave monotonic perturbations.

Natural oscillations of a hemispherical drop on a solid substrate are considered. The Hocking condition is used to take into account the contact angle dynamics. The natural oscillations attenuate due to dissipation on the contact line and the degeneracy in the azimuthal number is eliminated. Forced oscillations of the drop are studied for tangential vibrations of the substrate. For the case of a fixed contact line, the amplitude of the forced oscillations grows without bound near the fundamental requency. In other cases, the amplitude is finite.

The effect of a standing acoustic wave on the development of long-wave convective perturbations in a horizontal layer with thermally insulated boundaries is investigated. The main two-dimensional flow is determined. A nonlinear amplitude equation with spatially-periodic coefficients is derived for investigating the stability of the main flow and secondary convection flows in the neighborhood of the stability threshold. The intensity of the acoustic field is assumed to be low. It is shown that the acoustic action leads to destabilization of the layer. Plane and three-dimensional perturbations are critical at large and small Prandtl numbers, respectively. Nonlinear one-dimensional steady-state solutions of the amplitude equation are obtained and their stability is investigated.

The two-dimensional thermal convection of a elasticoviscous fluid in a horizontal cylinder with a square cross-section heated from below is investigated. For describing the rheological properties of the fluid the Oldroyd model with the upper convective derivative is used. The investigation is carried out numerically using the finite-difference method. The instability limits of mechanical equilibrium with respect to monotonic and oscillatory perturbations are determined. Supercritical convection regimes are investigated numerically.

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.

Mean flows induced by a heated or cold sphere vibrating in a liquid are studied. The cases of a sphere immersed in an unbounded liquid and a liquid layer of finite thickness, bounded by two rigid coaxial spheres, are considered. An analytical solution is obtained in the creeping flow approximation for small values of the governing parameters. The vibrating flow structure is shown to depend to a considerable extent on the clearance and to be fairly complex for small clearances. For a fixed clearance, the creeping flow structure is determined by a single parameter, the ratio of the vibrating Grashof number to the pulsating Reynolds number (Schlichting parameter). For large values of the parameters, the vibrating flow structure is determined by the vibrating Grashof number and the Schlichting parameter separately. The changes in the flow structure with increase in the values of the governing parameters are studied numerically by a finite difference method.

Time-averaged flows induced by oscillations of a heated (or cooled) solid sphere immersed in a liquid are investigated. The cases of an infinite surrounding liquid and the liquid placed into the spherical rigid envelope are considered. The gravity is absent. The interaction of thermovibrational flows and vibrational Schlichting flows is studied. New generalized equations are used for the description of thermovibrational convection. Analytical solutions have been obtained for small values of the governing parameters, under restriction of creeping flows, both in the case of infinite surrounding liquid and liquid placed into the rigid envelope. It has been found that the vibrational flow essentially depends on the layer thickness and can be rather complicated in the case of small thicknesses. At a fixed layer thickness, the creeping flow structure is defined by only one parameter, which is the ratio of vibrational Grashof number and Schlichting parameter. In the case of arbitrary values of the parameters, vibrational flows have been studied by a finite-difference method. The evolution of the finite-amplitude vibrational flow structure with the increase of the vibrational Grashof number has been investigated.

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.

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.

International audience; In this paper, we study the effect of a homogeneous longitudinal through flow on the onset of convection in a horizontal porous layer saturated by a binary fluid and heated from below or above. The layer boundaries are subjected to a constant heat flux. The investigation is made by taking the Soret effect into account. It is found that in the case of positive separation ratio when the denser component moves toward the cooler wall, through flow has no effect on the stability threshold but exerts an orientating effect on the convective patterns. For negative separation ratio, a strong destabilization occurs of the spatially homogeneous state with respect to long-wave disturbances. The stability range for long-wavelength convective rolls is defined.

In the hydrodynamics of a Newtonian fluid, nonlinear effects are connected only with the presence of convective derivatives in the equations and therefore disappear when plane-parallel flows are considered. Non-Newtonian effects are usually taken into account either phenomenologically in the expression for the stress tensor or by explicitly considering additional degrees of freedom. A theory of the effective viscosity of a magnetic fluid is constructed in [1] by regarding a magnetic fluid as a medium with internal rotation. It was shown that the flow of fluid in a magnetic field is non-Newtonian. Later, many authors (see, for example, [2, 3]) studied one- and two-dimensional flows under the influence of a pressure difference. However, the study was usually limited to continuous and smooth solutions. In the present work, we study the plane-parallel flow of a magnetic fluid in a homogeneous magnetic field under the influence of a longitudinal pressure gradient. We also consider discontinuous solutions. It is shown that for large longitudinal pressure gradients and sufficiently great intensities of the magnetic field, the problem has an infinite number of steady solutions which differ in the number and position of discontinuities of the magnetization and the associated abrupt changes in the velocity profile. Steady regimes and their stability are studied numerically with allowance for weak diffusion of magnetization and internal angular momentum. It is shown that the degeneracy is then lifted; however, in a certain region of parameters several stable steady regimes still exist.

A study is made of plane-parallel convective motion of a viscoplastic liquid between parallel vertical planes on which different temperatures are maintained. In contrast to [1], the yield shear stress τ0 is not a constant but is assumed to be a function of the temperature; moreover, above a certain critical temperature T* the yield shear stress vanishes, so that for T > T* the liquid is purely Newtonian. The structure of the regions of quasirigid and viscoplastic flow is studied in its dependence on the Theological parameters. The velocity profiles corresponding to the different flow regimes are found, and the boundaries between the regimes and the longitudinal heat flux are determined.

Two-dimensional convective motion of a viscoplastic fluid in a long horizontal cylinder of square section heated on the side was studied numerically by the author in [1]. In the present paper, the problem of convection of a viscoplastic fluid in a rectangular region is solved numerically.