Convective instability of a system of horizontal layers of slightly compressible liquids

The convective stability of a system of two immiscible liquids with close densities is studied. The densities of the liquids depend nonlinearly on temperature and pressure. It is shown that the state of mechanical equilibrium is unstable. Neutral curves are plotted, and the critical values of the Rayleigh number are found. The calculations are performed for physical parameters characteristic of various northern, central, and southern zones of lake Baikal. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 15–22, July–August, 2007.     

Rayleigh-Benard problem for an anomalous fluid

The stability of the state of rest of a heated infinite horizontal layer of a viscous heat-conducting fluid (the Rayleigh-Benard problem) is considered. The equation of state for the fluid takes into account the nonmonotonic temperature and pressure dependence of water density. Instability of the mechanical equilibrium with respect to small monotonic perturbations is studied. The effect of the problem parameters on the Rayleigh numbers and their corresponding critical motions is investigated numerically using linear theory. Numerical investigation of the spectral problem is based on the Godunov-Abramov orthogonalization method. The calculation results are compared with the well-known results for the limiting case where the density is considered a quadratic function of temperature and does not depend on pressure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 27–38, March–April, 2007.     

Dynamics and radiation of a single bubble under conditions of abnormal compressibility of a bubbly liquid

An equation is proposed for the pulsation of a single cavity in an abnormally compressible bubbly liquid which is in pressure equilibrium and whose state is described by the Lyakhov equation. In the equilibrium case, this equation is significantly simplified. Numerical analysis is performed of the bubble dynamics and acoustic losses (the profile and amplitude of the radiation wave generated on the bubble wall from the side of the liquid). It is shown that as the volumetric gas concentration k₀ in the equilibrium bubbly medium increases, the degree of compression of the cavity by stationary shock wave decreases and its pulsations decrease considerably and disappear already at k₀ = 3%. In the compression process, the cavity asymptotically reaches an equilibrium state that does not depend on the value of k₀ and is determined only by the shock-wave amplitude. The radiation wave takes the shape of a soliton whose amplitude is much smaller and whose width is considerably greater than the corresponding parameters in a single-phase liquid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 51–57, May–June, 2007.     

Steady-state conditions of a nonisothermal film with a heat-insulated free boundary

The equilibrium shapes of a nonisothermal liquid film with a heat-insulated free surface for large Marangoni numbers are investigated in the long-wave approximation using a combination of analytical and numerical methods. It is proved that the two-dimensional problem of the equilibrium of a strip-shaped film has a steady-state solution for an arbitrary large temperature gradient on the boundaries of the strip. An increase in this gradient leads to an abrupt thinning of the film near the heated boundary, which can result in instability and rupture of the film. In the equilibrium problem for a film fixed on a circular contour, the nonuniform distribution of the heat flux on the contour was found to have a significant influence on the free-surface shape. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 59–73, July–August, 2008.     

Equations of the shallow water model on a rotating attracting sphere. 1. Derivation and general properties

A shallow water model on a rotating attracting sphere is proposed to describe large-scale motions of the gas in planetary atmospheres and of the liquid in the world ocean. The equations of the model coincide with the equations of gas-dynamic of a polytropic gas in the case of spherical gas motions on the surface of a rotating sphere. The range of applicability of the model is discussed, and the conservation of potential vorticity along the trajectories is proved. The equations of stationary shallow water motions are presented in the form of Bernoulli and potential vorticity integrals, which relate the liquid depth to the stream function. The simplest stationary solutions that describe the equilibrium state differing from the spherically symmetric state and the zonal flows along the parallels are found. It is demonstrated that the stationary equations of the model admit the infinitely dimensional Lie group of equivalence. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 24–36, March–April, 2009.     

Equilibrium of a free nonisothermal liquid film

The equilibrium of a free weightless liquid film fixed over a planar contour and acted upon by thermocapillary forces is studied. Trends in the behavior of free liquid films are important for understanding the processes occurring in foams. The equilibrium equations for a nonisothermal weightless free film are derived for the two limiting cases: the temperature of the film is considered a known function of the coordinates; the free surface of the film is thermally insulated. For the plane and axisymmetric cases, the existence conditions for the solutions of the resulting nonlinear boundary-value problems are found and their properties are studied. For the general case, an approximate solution of the equilibrium problem is obtained provided that the analogue of the Marangoni number is small. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 16–29, May–June, 2007.     

Pressure-dependent viscosity and free volume of atactic and syndiotactic polystyrene

The effect of pressure on viscosity is an important but often overlooked aspect of the flow properties of polymeric materials. In this work, two polymers (an atactic and a syndiotactic Polystyrene) were characterized to determine the effect of pressure on viscosity. In particular, a device was adopted to increase the exit pressure of a standard capillary rheometer, thus obtaining data of viscosity under high pressure and high shear rates. The Simha-Somcynsky equation of state was applied to the pressure–volume–temperature experimental data of both materials to obtain the dependence of free volume on temperature and pressure. The Doolittle equation was eventually employed to verify the dependence of viscosity on free volume. It was found that, for both materials, a linear relationship holds between the logarithm of zero-shear-rate viscosity (at several temperatures and pressures) and the inverse of free volume.     

Numerical analysis of the stability of nonisothermal couette flow

The stability of convective motion of a liquid between two rotating heated cylinders is investigated in the absence of external forces. The mathematical model for describing the convection is obtained from the general equations [1, 2] on the assumption that the density of the liquid, the thermal conductivity, the specific heat and the viscosity coefficients depend only on temperature, and that the work done by the pressure forces and the viscous dissipation are negligibly small. The thermal expansion coefficient of the liquid is not assumed to be small, which distinguishes the models in question from the classical Oberbeck-Boussinesq model [1, 3, 4]. Rostov-on-Don. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 70–76, September–October, 1988.     

Multivelocity effects in high-pressure-gradient flows of dilute bubbly media

The multivelocity effects associated with the behavior of gas or vapor bubbles in a region with high pressure gradients typical of the flows around a cavity in which the pressure is higher than that in the surrounding space are considered. For a low volume bubble concentration, the problem of fluid flow perturbation by the bubbles is examined. For gas bubbles, it is shown that taking multivelocity effects into account considerably reduces the additional jet momentum. It is found that, with time, the temperature distribution in the wake behind a vapor bubble becomes nonmonotonic and the maximum temperature may even exceed the initial bubble temperature. It is demonstrated that the bubbles may accumulate and a flow regime with a sharply pronounced two-phase jet extending to the outer edge of the main liquid jet may develop. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–100, January–February, 1998. The work received financial support from the Russian Foundation for Fundamental Research (project No.96-01-01442).     

New analysis of contact melting of phase change material around a hot sphere

The process of contact melting of the solid phase change material (PCM) around a hot sphere, which is driven by the temperature difference between the PCM and the sphere, is analyzed in this paper. Considering the difference of the normal angle between the sphere surface and the solid–liquid interface of the melting PCM, the fundamental equations of the melting process are derived with the film theory. The new film thickness and pressure distribution inside the liquid film and the variation law of the normal angle of the solid–liquid interface and the melting velocity of the sphere are also obtained. It is found that (1) while normal angle at sphere surface φ is within a certain value φ₀, which is related to Ste number and the outside force F, it has no obvious effect on the pressure distribution inside the liquid film and the numerical results by the present model are in accordance with the analytical results in the published literature, (2) the film thickness at φ = ±90° is constringent to a certain value and not the infinity, (3) the analytical results can be employed approximately to analyze the contact melting process except for the film thickness at φ = ±90°.     

Analysis of axisymmetric phase strains in plates and shells

The axisymmetric strain problem for a shell in the direct phase transformation interval is formulated approximately as a nonlinear boundary-value thermoelastic problem with an implicit temperature dependence (through a phase parameter simulating the volume fraction of the new-phase crystals). The buckling problems for a circular plate and a shallow spherical dome of TiNi alloy loaded by normal pressure in the direct phase transformation interval are solved numerically. The branches of buckled equilibrium states are obtained for various values of the loading and phase parameters. It is found that the deflections increase abruptly with an increase in the phase parameter for a fixed value of the loading parameter. The evolution of the buckling modes and the phase-strain distribution along the meridian are studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 163–171, March–April, 2007.     

Stability of a multilayered composite conical shell under uniform external pressure

The stability of equilibrium of a layered composite circular conical truncated shell loaded with uniform external pressure is investigated. A parametric analysis of the critical pressure intensities is carried out with allowance for the transverse shear, the moment character of the subcritical state of equilibrium, and the subcritical strains. Kemerovo State University, Kemerovo 650043. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 198–207, July–August, 1999.     

Propagation of nonlinear waves in an inhomogeneous gas-liquid medium. Derivation of wave equations in the Korteweg-de Vries approximation

Equations describing the propagation of waves of small but finite amplitude in a liquid with gas bubbles are derived. The bubble distribution density is a continuous function of bubble size and spatial coordinates. It is found that, for a uniform bubble distribution, the obtained equations become the Korteweg-de Vries, Kadomtsev-Petviashvili and Khokhlov-Zabolotskaya equations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 188–197, March–April, 2009.     

Diffusion stability of bubbles in a cluster

The diffusion stability of gas bubbles in one-fraction and two-fraction clusters subjected to an acoustic field is studied. For a one-fraction cluster, numerical values were obtained for the initial gas concentrations in the liquid at which the bubble tends to one of two equilibrium states because of diffusion processes between the bubble and the ambient liquid. It is found that a two-fraction cluster tends to become a one-fraction cluster. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 40–48, July–August, 2007.     

Direct numerical simulation of a liquid sheet in a compressible gas stream in axisymmetric and planar configurations

A thin liquid sheet present in the shear layer of a compressible gas jet is investigated using an Eulerian approach with mixed-fluid treatment for the governing equations describing the gas–liquid two-phase flow system, where the gas is treated as fully compressible and the liquid as incompressible. The effects of different topological configurations, surface tension, gas pressure and liquid sheet thickness on the flow development of the gas–liquid two-phase flow system have been examined by direct solution of the compressible Navier–Stokes equations using highly accurate numerical schemes. The interface dynamics are captured using volume of fluid and continuum surface force models. The simulations show that the dispersion of the liquid sheet is dominated by vortical structures formed at the jet shear layer due to the Kelvin–Helmholtz instability. The axisymmetric case is less vortical than its planar counterpart that exhibits formation of larger vortical structures and larger liquid dispersion. It has been identified that the vorticity development and the liquid dispersion in a planar configuration are increased at the absence of surface tension, which when present, tends to oppose the development of the Kelvin–Helmholtz instability. An opposite trend was observed for an axisymmetric configuration where surface tension tends to promote the development of vorticity. An increase in vorticity development and liquid dispersion was observed for increased liquid sheet thickness, while a decreasing trend was observed for higher gas pressure. Therefore surface tension, liquid sheet thickness and gas pressure factors all affect the flow vorticity which consequently affects the dispersion of the liquid.      

A Numerical Investigation of Transpiration Cooling with Liquid Coolant Phase Change

This article presents a numerical approach to investigate the transpiration cooling problems with coolant phase change within porous matrix. A new model is based on the coupling of the two-phase mixture model (TPMM) with the local thermal non- equilibrium (LTNE), and used to describe the liquid coolant phase change and heat exchange processes in this article. The effects of thermal conductivity, porosity, and sphere diameter of the porous matrix on the temperature and saturation distributions within the matrix are studied. The results indicate that an increase in the porosity or sphere diameter can lead to an area dilation of two-phase region and a rise of liquid temperature; whereas an increase in the thermal conductivity of the porous matrix results only in a rise of liquid temperature, but drops of solid temperature and temperature gradient on the hot surface. The influence of hot surface pressure on cooling effect is discussed by numerical simulations, and numerical results show that the effect of the transpiration cooling will be worse under higher pressure. The investigation also discovers an inverse phenomenon to the past investigations on the transpiration cooling without coolant phase change, namely in two-phase region, coolant temperature may be higher than solid temperature. This inversion can be captured only by the new LTNE–TPMM.     

Ice-water convection in an inclined rectangular cavity filled with a porous medium

This paper reports on the results of a numerical study on the equilibrium state of the convection of water in the presence of ice in an inclined rectangular cavity filled with a porous medium. One side of the cavity is maintained at a temperature higher than the fusion temperature while the opposite side is cooled to a temperature lower than the fusion temperature. The two remaining sides are insulated. Results are analysed in terms of the density inversion parameter, the tilt angle, and the cooling temperature. It appears that the phenomenon of density inversion plays an important role in the equilibrium of an ice-water system when the heating temperature is below 20°. In a vertical cavity, the density inversion causes the formation of two counterrotating vortices leading to a water volume which is wider at the bottom than at the top. When the cavity is inclined, there exist two branches of solutions which exhibit the bottom heating and the side heating characteristics, respectively (the Bénard and side heating branches). Due to the inversion of density, the solution on the Bénard branch may fail to converge to a steady state at small tilt angles and exhibits an oscillating behavior. On the side heating branch, a maximum heat transfer rate is obtained at a tilt angle of about 70° but the water volume was found to depend very weakly on the inclination of the cavity. Under the effect of subcooling, the interplay between conduction in the solid phase and convection in the liquid leads to an equilibrium ice-water interface which is most distorted at some intermediate cooling temperature.     

An Analytical Model for Capillary Pressure–Saturation Relation for Gas–Liquid System in a Packed-Bed of Spherical Particles

Capillary pressure is considered in packed-beds of spherical particles. In the case of gas–liquid flows in packed-bed reactors, capillary pressure gradients can have a significant influence on liquid distribution and, consequently, on the overall reactor performance. In particular, capillary pressure is important for non-uniform liquid distribution, causing liquid spreading as it flows down the packing. An analytical model for capillary pressure–saturation relation is developed for the pendular and funicular regions and the factors affecting capillary pressure in the capillary region are discussed. The present model is compared to the capillary pressure models of Grosser et al. (AIChE J., 34:1850–1860, 1988) and Attou and Ferschneider (Chem. Eng. Sci., 55:491–511, 2000) and to the experiments of Dodds and Srivastava (Part Part Syst. Charact., 23:29–39, 2006) and Dullien et al. (J. Colloid Interface Sci., 127:362–372, 1989). The non-homogeneity of real packings is considered through particle size and porosity distributions. The model is based on the assumption that the particles are covered with a liquid film, which provides hydrodynamic continuity. This makes the model more suitable for porous or rough particles than for non-porous smooth particles. The main improvements of the present model are found in the pendular region, where the liquid dispersion due to capillary pressure gradients is most significant. The model can be used to improve the hydrodynamic models (e.g., CFD and cellular automata models) for packed-bed reactors, such as trickle-bed reactors, where gas, liquid, and solid phases are present. Models for such reactors have become quite common lately (Sáez and Carbonell, AIChE J., 31:52–62, 1985; Holub et al., Chem. Eng. Sci, 47, 2343–2348, 1992; Attou et al., Chem. Eng. Sci., 54:785–802, 1999; Iliuta and Larachi, Chem. Eng. Sci., 54:5039–5045, 1999, IJCRE 3:R4, 2005; Narasimhan et al., AIChE J., 48:2459–2474, 2002), but they still lack proper terms causing liquid dispersion.     

Two-dimensional problem of the impact of a vertical wall on a layer of a partially aerated liquid

The two-dimensional unsteady problem of the impact of a vertical wall on a layer of a liquid which is mixed with air near the wall and does not contain air bubbles away from the wall is solved in a linear approximation. The gas-liquid mixture is modeled by a homogeneous, ideal, and weakly compressible medium with a reduced sound velocity dependent on the air concentration in the gas-liquid mixture. Outside the gas-liquid layer, the liquid is considered ideal and incompressible. During the initial stage of the impact, the liquid flow and the hydrodynamic pressure are determined using the linear theory of the potential motion of an inhomogeneous liquid. The dependence of the amplitude of the impact pressure along the wall on the air concentration in the gas-liquid layer and on the thickness of this layer is investigated. For a small relative thickness of the layer, the thin-layer approximation is used. It is shown that the solution of the original problem tends to the approximate solution as the thickness of the layer decreases. It is shown that the presence of the gas-liquid layer leads to wall pressure oscillations. Estimates are obtained for the pressure amplitude and the oscillation period. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 34–46, September–October, 2006.