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—It is known that linearly polarized translational vibrations can multiply the capture cross section of deposited solid particles by a single gas bubble suspended in a liquid. This fact is of great theoretical and practical importance for using vibrations to intensify the flotation process, which, however, involves multiple rising bubbles. The present work studied the capture of solid particles by a gas bubble ensemble in liquid exposed to high-frequency small amplitude vibrations in two-dimensional formulation numerically. The numerical solution was obtained using ANSYS Fluent software. The obtained fields of the averaged and pulsating components of the flow velocity were then processed by our own code in order to calculate the trajectories of small particles from a “cloud” located in the computational domain. The description of forces acting on the particle took the added mass of the fluid, gravity, the Archimedean force, Stokes force, Basset force, and the vibrational force due to the inhomogeneity of the pulsating field into account. The calculations were performed for millimeter air bubbles in water with particle characteristics significant for the flotation process. The influence of vibrations is shown to be significantly attenuated within the bubble ensemble due to the “screening” effect, which peaks at the highest vibration frequency and depends weakly on the distance between the bubbles. However, vibrations of a lesser frequency, at which the gas in bubbles is compressible, are still able to penetrate deep into the liquid volume at a sufficient excitation intensity, and thereby increase the particle capture region.

The onset and nonlinear regimes of the Soret-induced convection in a mixture of dodecane, isobutylbenzene, and tetralin, taken in equal mass fractions and placed in a rectangular porous cavity with rigid impermeable boundaries elongated in the horizontal direction and heated from below, have been numerically investigated. The vertical boundaries of the cavity are thermally insulated. The components of the mixture under consideration are representatives of the main groups of chemical compounds that are oil components. The values of the porosity and permeability of the medium were chosen to be close to the values of real media, such as sand, sandstone, or limestone. The area of such a configuration simulates a hydrocarbon field. Due to the Soret effect (thermal diffusion), the light components of the mixture with positive separation ratios (dodecane and isobutylbenzene) are accumulated in the hot region, whereas the heavy component (tetralin) is accumulated in the cold region. In the case of heating from below (provided by the presence of geothermal gradient under natural conditions), this process may induce convection. It was found that a steady flow occurs in the cavity at a certain value of Rayleigh number; this flow becomes oscillatory with an increase in the Rayleigh number. With a further increase in the Rayleigh number, irregular oscillations arise. Several monotonic and oscillatory regimes, characterized by different spatial scales (from one to ten vortices), were found in the range of Rayleigh numbers under consideration. The oscillations of instantaneous flow characteristics in oscillatory regimes have a complex shape. A steady-state flow regime with an asymmetric structure was found. The ranges of Rayleigh numbers in which the found regimes may exist and the time dependences of the integral characteristics during the formation of steady flows were determined. The effect of supercriticality on the flow structure and concentration distributions of mixture components were analyzed.

The issues of formation and assessment of diffuse pollution of water objects in the zones near large mining and chemical industrial complexes are discussed. The study object is the Kama River (Kama Reservoir) in the zone of influence of the Solikamsk–Berezniki Industrial Hub. The major flows of pollution are due to seepage recharges from the operating and removed from service sludge storage sites and storages of liquid and solid wastes. The anthropogenic impact was identified, and its qualitative and quantitative characteristics were evaluated based on expedition data, state monitoring data, and mathematical simulation. The formation mechanisms are considered for the local pollution zones in the Kama River segment (the Kama Reservoir), in the context of considerable vertical heterogeneity―the concentrations of pollutants in bottom zones can be an order of magnitude higher than their concentration in the surface layer. More than 90% of pollution in this part of the Kama Reservoir were found to arrive in diffuse manner. Six types of measures were proposed for reducing the diffuse runoff in the zone of influence of the Solikamsk–Berezniki Industrial Hub.

In this work, secondary convection regimes in a. fluid with a. viscosity linearly dependent on temperature, enclosed between two vertical parallel planes heated to different temperatures, are studied. The boundaries of the layer were considered hard and perfectly heat-conducting. The problem was solved numerically by the finite difference method. Calculations were carried out for Prandtl numbers equal to one and twenty. In the first case, in a. fluid with constant viscosity, the loss of stability of the main flow is related to the development of hydrodynamic perturbations, which comprise motionless vortices at the boundary of the counter flows. In the second case, the instability of the main flow is caused by oscillatory perturbations, which comprise thermal waves. Dependences of the Nus-selt number on the Grashof number and data on the structure of the secondary flows are obtained. It is found that, if the Prandtl number is equal to unity, the Nusselt number monotonically increases with the Grashof number and, near the instability threshold of the main flow, it increases according to the root law; i.e., the secondary flow arises softly. The secondary structures look like drifting vortices at the boundary of counter flows, which, after a. transient process, leads to the establishment of steady oscillations of the heat flux. At a. Prandtl number of twenty, the relationship of the Nusselt number with the Grashof number is nonmonotonic; the curve contains sections in which the Nusselt number is equal to unity and sections in which the Nusselt number increases/decreases with increasing Grashof number. This behavior is explained by the fact that, for the Prandtl number equal to twenty, there are two instability modes; oscillatory and monotonic, and the region of increasing oscillatory perturbations with a. fixed wavenumber is bounded both above and below.

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.

Gravitational–capillary waves, arising in a system of thick layers of two aggregate states of a substance, are studied using the example of water and its vapour. Their initially flat interface is in hydrostatic and thermodynamic equilibrium at atmospheric pressure and the corresponding saturation temperature $100\,^{\circ }\text{C}$, provided that a balance is maintained for heat fluxes transported through immovable phases in the process of heat conduction. From the problem of linear stability for small perturbations, estimation relations for their damping coefficient and eigenfrequency shift are obtained, including the factors of viscosities of the media and phase transition. The first two contributions, proportional to the kinematic viscosity of the liquid and the square root of the kinematic viscosity of its vapour, respectively, are consistent, accurate to additionally accounted for capillary forces, with the result of Dore’s analysis, which is a development of the weakly viscous Lamb theory for ‘deep water’. Numerical calculations have shown that, with increasing wavelength, the accuracy of the proposed approximation increases, especially for the wave damping coefficient, and the contribution of the viscosity of the light phase increases in percentage terms. As for the phase transition effect, which was previously overestimated using the quasi-equilibrium approximation, it remains insignificant at the level of heat fluxes acceptable in the model of thick layers of homogeneous media. The factor of thermocapillarity is found to be very weak; it becomes qualitatively noticeable only on the background of disappearing heating. Here, together with the factor of dependence of the saturation temperature on the phase pressure, it determines a non-zero correction to the wave damping coefficient.

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.

Vibrational effect on heterogeneous systems is one of the mechanisms for controlling the processes occurred in these systems. For hydrodynamical systems, vibrations can greatly affect the type of flows and the shape of the interfaces, leading to the behavior that is significantly different from that in static fields. In this paper, the flows and interface deformations are studied numerically for a cylindrical liquid zone surrounded by a coaxial layer of gas. In vertical direction the system is bounded by the parallel rigid plates subjected to the axial vibrations of finite frequency and amplitude. The aim of the work is to study and explain the nature of new vibrational phenomena observed experimentally. The study is performed in the framework of a full non-average approach using the fluid volume method. The data are obtained on the instantaneous and average velocity fields and the instantaneous and average shape of the interface at various frequencies and amplitudes of vibrations. It is shown that the axial vibrations of the rigid plates induce the waves at the interface which propagate from the plates to the zone center. These waves generate an average flow with the direction near interface from the oscillating plates to the zone center. Additionally, vibrations generate an average flow near rigid plates in the form of toroidal vortices with the direction near rigid plates from interface to the zone axis.

The influence of differences in the hydrochemical characteristics of the waters of merging rivers on the formation of density currents and the behavior of the density jump layer is studied numerically. The problem is solved for the velocity regimes of the confluent Chusovaya and Sylva rivers (Russian Federation), which are characterized by significantly different salinity values. To study the specific features of layered structures, numerical simulation is performed in the framework of a two-dimensional approach. The calculations are carried out for a section of the Chusovaya River with a length of 300 m, located downstream from confluence of the Chusovaya and Sylva rivers. The Chusovskoy water intake of the city of Perm is located on this site. In the winter-spring season, in this area a regime is established in which a stable layer of density and velocity jumps is observed. With the confluence parameters typical of the winter low water season in the area of the Chusovskoy water intake, 8 km below the confluence point, field measurements detected fluctuations in the concentration and flow velocity in the density jump layer. The two-dimensional numerical simulations performed in the present work also showed the presence of velocity and concentration fluctuations in the jump layer. We found that vortex structures are formed in the jump layer, leading to fluctuations in the concentration in time at a given point and to the formation of a wave structure of the concentration field at different times.

Based on the thermovibrational convection equations, we have investigated the structure of the averaged plane-parallel convective flow in a plane vertical layer of Williamson fluid executing high-frequency linearly polarized vibrations along the layer. We show that as the vibrations are intensified, the nonlinear viscous properties of a pseudoplastic fluid cease to affect the structure and intensity of its main flow, and it becomes similar to a flow of ordinary Newtonian fluid. The linear problem of stability of an averaged plane-parallel flow of pseudoplastic Williamson fluid has been formulated and solved for the case of longitudinal high-frequency linearly polarized vibrations for small periodic perturbations along the layer. Numerical calculations have shown that, as in a Newtonian fluid, the monotonic hydrodynamic perturbations are most dangerous at low Prandtl numbers. As the Prandtl number increases, the thermal instability modes begin to exert an undesirable effect. An enhancement of pseudoplastic fluid properties leads to destabilization of the main flow for both types of perturbations. Similarly to a Newtonian fluid, an additional vibrational instability mode to which small Grashof numbers correspond appears in the presence of vibrations. The influence of this vibrational mode on the stability of the main flow is determined by the vibration frequency and the temperature gradient. An intensification of the vibrations destabilizes the flow for all of the investigated instability modes. For a given set of rheological parameters of the Williamson model, there are critical values of the modified and vibrational Grashof numbers at which the averaged flow completely loses its stability with respect to the types of perturbations under consideration. Absolute destabilization of the main flow in a pseudoplastic fluid occurs at higher values of the vibrational Grashof number than those in a Newtonian fluid.

In this work, we numerically investigate the capture of deposited solid particles by an ensemble of floating gas bubbles in a liquid that is subjected to the action of low-amplitude acoustic vibrations. The calculations were carried out for millimeter air bubbles in water with particle characteristics corresponding to the flotation process. It is shown that the pulsating field is significantly weakened inside the bubble ensemble as a result of the "screening" effect, which is fully manifested in the limit of a very high vibration frequency. However, pulsations of a high but finite frequency, taking into account the compressibility of the gas in the bubbles and with a sufficient intensity of their excitation, are able to penetrate deep into the volume of the liquid and thereby increase the area of particle capture.

The paper deals with the resonance oscillations of a drop (bubble) surrounded by a fluid of different density in a container subjected to small amplitude vibrations in zero gravity conditions. The drop size is considered to be large in comparison with both the vibration amplitude and the thickness of viscous Stokes layers. The calculations for parametrically excited oscillations of the drop are carried out in the linear approximation, for inviscid and low viscous media, neglecting compressibility effects. The resonant oscillation is a doublet of neighbouring modes of eigen-oscillations of the drop, for which the sum of frequencies coincides with the frequency of the forced vibrations. This means that the basic state becomes unstable against quasi-periodic oscillations. The finite viscosity implies a finite threshold for the excitation of resonance. On the other hand, the viscosity plays a destabilizing role; at non-zero (even infinitesimal) viscosity the width of the instability frequency range turns out to be greater than in the case of inviscid fluids.

The liquid/liquid displacement through a 2D uniform network of capillaries is numerically modelled with the use of the phase-field approach. The detailed structure of the flow fields within the uniform matrices of different sizes (with the different number of pores) is examined with the aim to reveal the asymptotic behaviour, pertinent for a sufficiently large matrix, that could be used for representation of a porous medium. The integral characteristics of the flow that do not depend on the matrix size can be used for calculation of the parameters of a macroscopic (Darcy) approach. We demonstrate that qualitatively the displacement occurs very similarly in the matrices with the different number of structural elements. In particular, we show that the capillary pressure remains nearly constant during the displacement run until the break-through time (with some minor variations related to a particular shape of a matrix). Upon increase of the matrix size the magnitude of the capillary pressure (and all other integral characteristics of the two-phase flow) quickly converges to the limiting value (so that the limiting results are already reached for a matrix with 6 × 6 elements), giving the direct procedure for calculation of the capillary pressure in a porous medium.

International audience; A rapid downstream weakening of the processes that drive the intensity of transverse mixing at the confluence of large rivers has been identified in the literature and attributed to the progressive reduction in channel scale secondary circulation and shear-driven mixing with distance downstream from the junction. These processes are investigated in this paper using a three-dimensional computation of the Reynolds averaged Navier Stokes equations combined with a Reynolds stress turbulence model for the confluence of the Kama and Vishera rivers in the Russian Urals. Simulations were carried out for three different configurations: an idealized planform with a rectangular cross-section (R), the natural planform with a rectangular cross-section (P), and the natural planform with the measured bathymetry (N), each one for three different discharge ratios. Results show that in the idealized configuration (R), the initial vortices that form due to channel-scale pressure gradients decline rapidly with distance downstream. Mixing is slow and incomplete at more than 10 multiples of channel width downstream from the junction corner. However, when the natural planform and bathymetry are introduced (N), rates of mixing increase dramatically at the junction corner and are maintained with distance downstream. Comparison with the P case suggests that it is the bathymetry that drives the most rapid mixing and notably when the discharge ratio is such that a single channel-scale vortex develops aided by curvature in the post junction channel. This effect is strongest when the discharge of the tributary that has the same direction of curvature as the post junction channel is greatest. A comprehensive set of field data are required to test this conclusion. If it holds, theoretical models of mixing processes in rivers will need to take into account the effects of bathymetry upon the interaction between river discharge ratio, secondary circulation development, and mixing rates.

International audience; At present, a significant weakening of the intensity of transverse mixing at the confluence of large rivers, which is observed in a number of cases, is widely discussed. Since the observed features of the confluence of large watercourses are not only of research interest but also of significant economic importance associated with the characteristics of water management at these water bodies, a large number of works are devoted to their study. Water resources management requires measures for the organization of water use which can be rational only under the understanding of processes occurring in water basins. To explain the phenomenon of suppression of the transverse mixing, which is interesting and important from the point of view of ecology, a wide range of hypotheses is proposed, up to the negation of turbulence in rivers. One of the possible mechanisms for explaining the suppression of transversal mixing can be the presence of transverse circulation manifesting itself as Prandtl’s secondary flows of the second kind. The characteristic velocity of these circulation flows is very small and difficult to measure directly by instruments; however, in our opinion, they can significantly complicate the transverse mixing at the confluence. The proposed hypothesis is tested in computational experiments in the framework of the three-dimensional formulation for dimensions of a real water object at the mouth of the Vishera River where it meets the Kama. Calculations demonstrate that, at sufficiently large flow rates, the two waters practically do not mix in the horizontal direction throughout the depth over long distances from the confluence. It has been found that a two-vortex flow is formed downstream the confluence, which just attenuates the mixing; the fluid motion in the vortices is such that, near the free surface, the fluid moves from the banks to the middle of the riverbed.

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.

In this paper we present the results of numerical simulation of nonlinear regimes of a NaCl aqueous solution in a square cavity with rigid and impermeable for substance boundaries. The vertical boundaries are thermally insulated, there are different constant temperatures on horizontal ones, it corresponds to the heat from below. The calculations are carried out within the non-stationary approach, with using of Boussinesq approximation and taking into account the polynomial dependence of the thermal diffusion coefficient on temperature. According to experimental data, the sign of thermal diffusion coefficient of the mixture under study changes at a temperature 285.4 K. In this work we consider a temperature range such that the sign of the thermal diffusion coefficient changes inside the simulated region. Other transport coefficients factors are considered as constant. The calculations were carried out for the cases of Earth and reduced gravity. A comparison is made with the case of the thermal diffusion coefficient constant value, which is negative at the considered average temperature.

The paper is devoted to the numerical study of two-dimensional convective motion of a viscoplastic fluid in a closed region under lateral heating. The problem is solved numerically using the ANSYS Fluent software. The Bulkeley-Herschel model describing viscoplastic behavior was chosen as a non-Newtonian fluid model. Under certain rheological parameters, this model is reduced to a Newtonian fluid model, the behavior of which is also investigated as a limit case. Based on the results of the calculations, the dependence of the maximum value of the stream function in the cavity on the Rayleigh number is plotted and the evolution of the unyielded zones with an increase in the Rayleigh number is traced. The threshold value of the Rayleigh number, at which a sharp growth of the flow intensity is observed, has been determined. The calculated threshold value of the Rayleigh number for the Bulkley-Herschel liquid was found to be very close to the threshold value of the Rayleigh number at which the flow of the Bingham liquid arises, which was obtained analytically.

The stability of water supply systems is one of the main factors that determine the reliability of the functioning of large industrial complexes. This problem is considered on the example of the “Azot” branch of URALCHEM JSC in the city of Berezniki, which takes fresh industrial water from the Kama river (Kama reservoir). A specific feature of this water body is the significant variability of its hydrological regime. During the summer dry season, it is characterized by low flow velocities. In winter, with a decrease in the water level in the reservoir, a typical river regime is observed. In addition, there is a strong technogenic impact on the water body. The impact is formed largely due to non-declared dispersed (diffuse) sources. Their characteristic feature is high mineralization and, accordingly, the density of the formed wastewater. Their behavior is fundamentally different from the processes of dilution and migration of effluents with neutral buoyancy. A two-layer structure is formed, when the content of the main ions in the surface horizon is more than an order of magnitude less than in the bottom layer. This circumstance must be taken into account when assessing the reliability of the functioning of water intakes. The most effective tool for solving such problems is the conjugation of a complex of field monitoring observations with computational experiments using hydrodynamic models in one-, two- and three-dimensional formulations.

The paper deals with the investigation of interaction of thermocapillary, parametric and frozen wave instability mechanisms in a two-layer system of immiscible incompressible viscous fluids of different densities (more dense fluid is located below). External boundaries are rigid. They are maintained at constant different temperatures. The system is subjected to the gravity force and horizontal vibrations of finite frequency and amplitude. The study is performed taking into account the interface deformations. The thermal expansion of fluids is neglected. It is found that vibrations make strong destabilizing effect on the longwave instability mode related to the interface deformations and stabilizing effect on Pearson instability mode.

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

This study focuses on the infiltration of saturated brine from liquid-waste storage facilities into the surrounding groundwater and surface-water bodies. The storage facilities are situated at the Verkhnekamsk Potassium and Magnesium Salt Deposit (Perm krai, Russian Federation); they store highly mineralized brine of potassium, sodium, and magnesium chlorides. An analytical solution of a one-dimensional equation of contaminant transport by a homogeneous steady-state groundwater flow is used to estimate the time it takes the contaminant to travel from the storage facilities to the nearest surface water body, as well as to evaluate the formation time of a stationary concentration profile, with contaminant adsorption in the porous matrix either neglected or taken into account. The contaminant concentration at the point of brine entry into the surface water body is calculated. The ANSYS Fluent software package is used for direct 3D simulation of brine infiltration into the surrounding medium. The simulation results revealed different stages of contaminant propagation in the porous medium. The contaminant was found to spread over a wide area with an almost uniform high brine concentration close to the saturation value. The contaminant reaches the nearest riverbed approximately 20 days after the start of infiltration. The estimates of the time required for the contaminant front to reach the surface water body obtained by three-dimensional simulation agree with the analytical estimates derived from a one-dimensional model. The proposed system of physical models adequately describes the hydrodynamic processes accompanying the operation of large storage facilities and can be used to predict contaminant-front propagation in the groundwater near storage facilities.

The stability of a shear flow imposed along a diffusive interface that separates two miscible liquids (a heavier liquid lies underneath) is studied using direct numerical simulations. The phase-field approach is employed for description of a thermo- and hydrodynamic evolution of a heterogeneous binary mixture. The approach takes into account the dynamic interfacial stresses at a miscible interface and uses the extended Fick's law for setting the diffusion transport (the diffusion flux is proportional to the gradient of chemical potential). The shear flow is unstable to two kinds of instabilities: (i) the Kelvin-Helmholtz instability, with an immovable vortex formed in the middle of an interface (in the vertical direction), and (ii) the Holmboe instability, with travelling waves along the interfacial boundary. The development of the Holmboe instability results in a stronger enhancement of molecular mixing between the mixture components. Earlier, the boundaries of these instabilities were determined using the linear stability analysis and employing the concept of a 'frozen interface'. In the current work, through the solution of full equations, we obtain the stability boundaries for several sets of governing parameters, showing a greater variety of the possible shapes of the stability diagrams. The Kelvin-Helmholtz instability always occurs at lower gravity effects (lower density contrasts), while the Holmboe instability occurs when gravity is stronger. We show that for some parameters these two instabilities are separated by a zone where the shear flow is stable, and this zone disappears for the other sets of parameters.

Nonlinear regimes of the Soret-induced convection of a ternary mixture in a rectangular cavity with the vertical heat flux at the boundaries are studied numerically. Linear stability analysis performed earlier for a plane horizontal layer has shown that in a wide range of parameters the longwave instability is dominant. Nonlinear calculations performed in the present work for the parameter ranges where the monotonic longwave instability is dominant according to the linear stability analysis, show that at large supercriticalities the multi-vortex stationary flows could be realized. With the decrease of absolute value of the Rayleigh number Ra the series of flow structure transformations takes place. This transformation is accompanied by the decrease of a number of the vortices such that near the threshold the flow becomes two-vortex, i.e. the flow with maximum possible wavelength (minimum possible wave number) is realized. Thus, the nonlinear calculations confirm the conclusions of the linear stability analysis that the longwave instability is dominant. For the parameter ranges where according to the linear stability analysis the oscillatory longwave instability is dominant, the nonlinear calculations show that at small values of the net separation ratio the oscillatory regimes could be observed just in a narrow range of Ra close to the convection threshold and with the growth of net separation ratio the range of Ra where they can be observed is extended.

We develop a numerical model for a two-phase flow of a pair of immiscible liquids within a capillary tube. We assume that a capillary is initially saturated with one liquid and the other liquid is injected via one of the capillary's ends. The governing equations are solved in the velocity-pressure formulation, so the pressure levels are imposed at the capillary inlet and outlet ends. We model the structure of the flow and the shape of the liquid/liquid interface. We are able to reproduce the flow for a wide range of capillary numbers, when the meniscus preserves its shape moving together with the flow, and when the meniscus constantly stretches resembling the transport of a passive impurity. We demonstrate that the phase-field approach is capable of reproducing all features of the liquid/liquid displacement, including the motion of a contact line, the dynamic changes of the capillary pressure, and the dynamic changes of the apparent contact angle.

In the formation of the hydrochemical regime of water bodies and adjacent territories, a significant role is played by the density stratification effects caused by the heterogeneity of the distribution of mineralization fields. When the concentration of the heavy solute is larger than one ppm under terrestrial conditions, these effects have a great influence on the nature of the flow. Such effects can not be described using hydrodynamic models within the framework of shallow water equations; for their correct description, it is necessary to use hydrodynamic models in a full three-dimensional non-hydrostatic formulation. The present paper presents the results of numerical modeling of the formation of diffuse pollution of the Vyatka River during washing of floodplain water bodies (quarries, lakes) in the region of the Kirovo-Chepetsk industrial complex. The assessment of the scale and intensity of the supply of nitrogen-containing compounds was made. The calculation results form the basis for the development of a number of possible measures aimed at both reducing diffuse runoff and ensuring the standard water quality at the main drinking water intake in Kirov, Russia.

In this work, it is shown numerically in COMSOL Multiphysics and experimentally that a change in the NaCl concentration in water significantly affects the distribution of acoustic pressure in a laboratory sonochemical reactor. Thus, in distilled water under the action of ultrasound, two areas of increased pressure were ob-served, one of which was located directly above the ultrasound source, and the second was near the surface of the liquid, this effect is associated with the reflection of sound waves from the surface of the liquid. With the increase in the salt content, the maximum value of the acoustic pressure in the liquid decreases, which is associated with the dependence of the acoustic impedance of the liquid on the salt concentration and the peculiarities of the dynamics of vapor-gas bubbles in such solutions.

In this paper we present the results of numerical simulation of nonlinear convection regimes of a NaCl aqueous solution heated from below. The calculations are carried out for square and horizontally elongated rectangular cavities, both in the approximation of a constant thermal diffusion coefficient and in case of taking into account its temperature dependence. Other transport coefficients are considered as constant. We observe the local and integral characteristics of the realized flows at constant and variable thermal diffusion coefficients in the conditions of the Earth and reduced gravity.

This paper describes the experimental study of the mechanism of selective fixation of bubbles formed in a liquid under the influence of ultrasound on the surface of plates of various degrees of wettability when the liquid is degassed as a result of ultrasonic action. The experiments used solutions of NaCl and KCl of various concentrations, as well as distilled water. The plates are made of organic glass (wetting angle 51°) and Teflon (wetting angle 118°). As a result of the study, the experimental data on the surface area occupied by bubbles on the surface of plates were obtained, and the mechanism of selective fixation of bubbles under the influence of ultrasound in the conditions of emerging degassing was shown.

The paper deals with the numerical investigation of the possibilities to control convective flows in the liquid bridge in zero gravity conditions applying axial vibrations. The surface tension is assumed to be dependent both on the temperature and on the solute concentration. The free surface deformations and the curvature of the phase change surfaces are neglected but pulsational deformations of the free surface are accounted for. The first part of the paper concerns axisymmetric steady flows. The calculations show that the evolution of convective flow with the variation of thermal Marangoni number at a fixed value of the solutal Marangoni number is accompanied by the hysteresis phenomenon, which is related to the existence of two stable steady regimes in a certain parameter range. One of these regimes is thermocapillary dominated, it corresponds to the two-vortex flow, and the other is solutocapillary dominated, it corresponds to the single-vortex flow. Under vibrations, the range of the Marangoni numbers where hysteresis is observed becomes narrower and is shifted to the area of larger values. The second part of the paper concerns the stability of axisymmetric thermo-and solutocapillary flows and the transition to three-dimensional regimes. Significant mutual influence of flows generated by each process on the stability of the other is discovered. Stability maps in the parametric plane for the thermal Marangoni number, the solutal Marangoni number, are obtained for different values of vibration parameters. It is shown, that vibrations exert a stabilizing effect, increasing critical Marangoni numbers for all modes of instability. However, this effect is different for different modes and at high vibration intensity destabilization is possible. Consequently, vibrations can modify the scenario of the transition to the three-dimensional mode.

The stability of the horizontal interface of two immiscible viscous fluids in a Hele-Shaw cell subject to gravity and horizontal vibrations is studied. The problem is reduced to the generalized Hill equation, which is solved analytically by the multiple scale method and numerically. The long-wave instability, the resonance (parametric resonance) excitation of waves at finite frequencies of vibrations (for the first three resonances), and the limit of high-frequency vibrations are studied analytically under the assumption of small amplitudes of vibrations and small viscosity. For finite amplitudes of vibrations, finite wave numbers, and finite viscosity, the study is performed numerically. The influence of the specific natural control parameters and physical parameters of the system on its instability threshold is discussed. The results provide extension to other results [J. Bouchgl, S. Aniss, and M. Souhar, Phys. Rev. E 88, 023027 (2013)10.1103/PhysRevE.88.023027], where the authors considered a similar problem but took into account viscosity in the basic state and did not consider it in the equations for perturbations.

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.

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 cavity's size, as $\ensuremath{\sim}{L}^{1/2}$.

The paper deals with the investigation of the onset and weakly nonlinear regimes of the Soret-driven convection of ternary liquid mixture in a horizontal layer with rigid impermeable boundaries subjected to the prescribed constant vertical heat flux. It is found that there are monotonous and oscillatory longwave instability modes. The boundary of the monotonous longwave instability in the parameter plane Rayleigh number Ra - net separation ratio $\Psi$ at fixed separation ratio of one of solutes consists of two branches of hyperbolic type. One of the branches is located at ${\rm Ra} >0$ , the other one at ${\rm Ra} 0$ only for the heating from below and at $\Psi 0$ and the other at ${\rm Ra}

The paper deals with the numerical investigation of convective flows and heat and mass transfer in the directional solidification of binary melts in the presence of rotational vibrations of finite amplitude and frequency. The study is performed in the framework of the Boussinesq approximation and unsteady axisymmetric approach, taking into account the existence of the two-phase zone. The simulation is conducted for binary melt with low phase change temperature (succinonitrile with ethanol), using ANSYS Fluent package which realizes the finite volume method. The data on the temporal evolution of velocity, temperature and solute concentration fields in the melt and on the solute distribution in the grown crystal with and without vibrations are obtained. It is demonstrated that the vibrations make strong stabilizing effect; they reduce the radial segregation and prevent the pit formation and solidification front breakdown, which leads to a substantial increase of grown crystal homogeneity.

Summary The peculiarities of the formation of density currents in the zone of confluence of two rivers with strongly different hydrochemical regimes are studied numerically and experimentally. The three-dimensional numerical simulation shows that the water of the river of higher mineralization and density flows under the water of the river of lower mineralization and density and vice versa. And besides, such overlapping of the water streams is observed both upstream and downstream of the confluence of two rivers. The results of numerical simulation are supported by the data of expedition observations and in situ measurements. A similar phenomenon, namely, a flow of two overlapped oppositely directed water streams was previously discovered in the mouth zone of the rivers flowing in the sea. Our study reveals the existence of a new type of the hydrological systems, in which such a phenomenon occurs.

The solution of a wide range of tasks of protection and rational use of water bodies requires, first of all, an assessment of the levels of impact of industrial complexes. The solution of such problems for extreme technological and hydrometeorological conditions is carried out on the basis of appropriate computational experiments. However, verification of the underlying mathematical models can be performed only on the basis of correct input data. The paper presents materials of field studies of the temperature distribution of water both in the water area and in the depth of the Magnitogorsk reservoir, in the zone of influence of thermal effluents of PJSC “Magnitogorsk Iron and Steel Works” (PJSC MMK). It is shown that in the warmest limiting period, a significant vertical temperature inhomogeneity is formed.

Thermal convection in a flat horizontal layer of a porous medium with solid impermeable boundaries on which the heat flow is given is considered. The porous medium is saturated with a viscous incompressible fluid pumped along the layer. In the system under discussion, with a vertical heat flow inhomogeneous along the layer, localized convective structures may occur in the region where the heat flow exceeds the critical value corresponding to homogeneous heating from below and corresponding to the beginning of convection in the layer. With an increase in the rate of longitudinal pumping of the liquid through the layer, a transition from a state in which the localized convective structures are stable to a state in which the localized convective flow is completely washed out of the region of its excitation occurs. Calculations were performed in the framework of Darcy-Boussinesq. Results of the numerical calculation of the process removal of localized convective structures from the zone of its excitation with an increase in the rate of longitudinal pumping of liquid through the layer are presented. The map of the system state modes is obtained.

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 paper deals with the investigation of a behavior of a system of particles suspended in a fluid in a container subjected to high frequency translational vibrations of linear polarization. Pair interaction forces act on the particles under these conditions. These forces decrease with the distance and depend on the interacting particles orientation with respect to the vibration direction. The presence of these forces leads to the formation of structures in space. The problem is solved numerically using molecular dynamics method with pair interaction approximation. It is shown that the process of the structures formation consists of the fast stage of compact cluster formation and slow evolution of these clusters. It is found that for vibrations of linear polarization the particles form the chains oriented perpendicular to the direction of vibrations. At long time-scales these chains form the layers perpendicular to the direction of the vibrations and located almost periodically all over the fluid volume.

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.