A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.     

Stability of Filtration of a Gas—Liquid Mixture

The stability of steady regimes of filtration of a gas—liquid mixture at pressure lower than the saturation pressure is studied for the case of a nonmonotonic dependence of the relative phase permeability of the liquid on the gas saturation. It is shown that periodic self–oscillations can appear, and their evolution leads to deterministic chaos due to the appearance and destruction of quasiperiodic motions.     

Joule–Thomson Effects on the Flow of Liquid Water

We present a revised form of the energy balance for the coupled thermodynamics of liquid water flowing in porous media and give examples of situations where a commonly used formulation based on transport of enthalpy leads to erroneous results. Assuming negligible contribution from kinetic energy as well as sources and sinks such as energy from radioactive decay, total energy conservation is reduced to a balance between changes in internal energy, enthalpy, conductive heat flux, and gravitational potential energy. The Joule–Thomson coefficient is defined as the change in temperature with respect to an increase in pressure at constant enthalpy. Because liquid water has a negative Joule–Thomson coefficient at low temperatures, at a constant gravitational potential water cools as it compresses and heats as it expands. If one ignores the gravitational energy, transport of enthalpy alone leads to water heating by 2 \(^\circ \) C per kilometer as it is brought up from depth. The corrected energy balance transports methalpy, which is enthalpy plus gravitational potential energy. Although the simpler form leads to small changes in the temperature profile for typical simulations, there are several instances where this effect may prove to be important. The most important impact of the erroneous form is probably in the field of geothermal energy production, where the creation of a few degrees of heat in a simulation could lead to miscalculation of power plant efficiencies.     

Paradox of “Self-Outflow” of a Free Liquid Jet

The effect of accumulation of reversible strains on the pressure head is analyzed theoretically for elastic and viscoelastic liquid flow from convergent channels. It is shown that, depending on the rheological features of the liquids, the pressure head can both increase and decrease as compared with the pressure determined by the Bernoulli formula. In particular, a situation in which a liquid flows out without any pressure head applied (paradox of self-outflow) is possible within the framework of the model. Transition to different viscoelastic liquid flow regimes as a function of the constitutive parameters is considered with reference to a channel with a sharp bottleneck (abrupt decrease in the cross-section).     

Resonance Waves in a Model of a Two‐Layered Liquid

With allowance for surface interaction between phases, the behavior of longwave perturbations at the interface between two layers of dissimilar liquids, which form resonance triplets described by a pseudodifferential equation, is studied.     

Liquid suspensions of single and binary component solid particles—An overview

In this paper theoretical approaches and experimental findings relative to the hydrodynamics of liquid suspensions of solid particles by liquids are reported and discussed. For the single particle specie systems, advantages and possible faults of well known empirical correlations are discussed. For binary-solid mixture suspensions, experimental evidence are reviewed and approaches capable of successfully describing observed behaviour are reported.     

Steady Flow of a Navier–Stokes Liquid Past an Elastic Body

We perform a mathematical analysis of the steady flow of a viscous liquid, L{\mathcal{L}} , past a three-dimensional elastic body, B{\mathcal{B}} . We assume that L{\mathcal{L}} fills the whole space exterior to B{\mathcal{B}} , and that its motion is governed by the Navier–Stokes equations corresponding to non-zero velocity at infinity, v _∞. As for B{\mathcal{B}} , we suppose that it is a St. Venant–Kirchhoff material, held in equilibrium either by keeping an interior portion of it attached to a rigid body or by means of appropriate control body force and surface traction. We treat the problem as a coupled steady state fluid-structure problem with the surface of B{\mathcal{B}} as a free boundary. Our main goal is to show existence and uniqueness for the coupled system liquid-body, for sufficiently small |v _∞|. This goal is reached by a fixed point approach based upon a suitable reformulation of the Navier–Stokes equation in the reference configuration, along with appropriate a priori estimates of solutions to the corresponding Oseen linearization and to the elasticity equations.     

The Steady Motion of a Navier–Stokes Liquid Around a Rigid Body

Let be a body moving by prescribed rigid motion in a Navier–Stokes liquid that fills the whole space and is subject to given boundary conditions and body force. Under the assumptions that, with respect to a frame , attached to , these data are time independent, and that their magnitude is not “too large”, we show the existence of one and only one corresponding steady motion of , with respect to , such that the velocity field, at the generic point x in space, decays like |x|⁻¹. These solutions are “physically reasonable” in the sense of FINN [10]. In particular, they are unique and satisfy the energy equation. Among other things, this result is relevant in engineering applications involving orientation of particles in viscous liquid [14].     

Dispersion of Dynamically Passive Impurities in Non‐One‐Dimensional Liquid Flow

Equations that describe dispersion of a substance in a nononedimensional incompressible liquid flow through a plane channel are derived. The model under consideration extends the traditional Taylor model to the case where sources of the substance are present in the flow and relaxation transfer processes are taken into account. Additional conditions for the dispersion equations are obtained. The relation between the proposed model and the Taylor model is analyzed. Based on the equations obtained, the mass transfer between circulation regions in the flow is calculated and a system of cellularmodel equations for stagnant cavities is constructed.     

Liquid Filtration in an Unbounded Water‐Bearing Stratum with an Inclined Confining Bed

PolubarinovaKochina, Numerov, and other authors paid much attention to filtration problems of a heavy incompressible liquid in inclined waterbearing strata. In this work, therefore, classical schemes of liquid filtration on inclined confining beds are considered along with the general problem of filtration for arbitrary polygonal impermeable walls of a waterbearing stratum. In doing so, we also consider direct problems of physical and geometrical parameters of filtration flows.     

Liquid phase heterogeneous photocatalytic ozonation of phenol in liquid-solid fluidized bed:Simplified kinetic modeling

The isotherms of original AC (activated carbon) and photocatalysts (TiO2-AC) calcined at 500℃ for phenol were measured. The results showed a reversible adsorption of phenol onto both kinds of particles at 25℃, and could be fitted well to the Freundlich adsorption equation for the dilute solution. Five oxidation processes, namely O3 , O3 /UV, O3 /UV/AC, O2 /UV/TiO2 and O3 /UV/TiO2 , for phenol degradation in fluidized bed were evaluated and compared, and the photocatalytic ozonation was found to give the hig...     

The Motion of Viscous Liquid Column with Finite Length in a Vertical Straight Capillary Tube

This paper deals with the motion of viscous liquid column with finitelength and two free surfaces in a vertical straight capillary tube.It isassumed that fluid is Newtonian.Linearizing the boundary conditions,analytic expressions in the form of infinite series have been obtainedfor velocity,plessure and free surface at low Reynolds number.Thenumerical calculation is carried out for a set of cylinder’s length ofwater and blood.It has been revealed that there are considerable cir-culating currents at the upper and lower meniscuses.Its maximum ve-locity is about57%of the average velocity of the mainstream.Iner-tial effect is also studied in this paper.Using the time-dependent methodin finite difference techniques,numerical solution of the correspondiugnonlinear equation at Re≤24.5 is computed.Comparing it with analyticexact solution at low Reynolds number shows that inertial effect isnegligible provided Re≤24.5.     

Vapor–Liquid Jump Conditions within a Porous Medium: Results for Mass and Energy

In this paper, we develop the mass and energy jump conditions for a vapor–liquid boundary within a porous medium. The analysis is restricted to a single fluid component and leads to the appropriate jump conditions for an evaporation front. The condition of local thermal equilibrium is assumed to exist in the homogeneous liquid and vapor regions, but not in the boundary region where rapid changes in the saturation occur.     

Effect of Liquid Viscosity on Dispersion of Quasi-Lamb Waves in an Elastic-Layer–Viscous-Liquid-Layer System

The dispersion curves are constructed and propagation of quasi-Lamb waves are studied for wide range of frequencies based on the Navier–Stokes three-dimensional linearized equations for a viscous liquid and linear equations of the classical theory of elasticity for an elastic layer. For a thick liquid layer, the effect of the viscosity of the liquid and the thickness of elastic and liquid layers on the phase velocities and attenuation coefficients of quasi-Lamb modes is analyzed. It is shown that in the case of a thick liquid layer for all modes, there are elastic layers of certain thickness with minimal effect of liquid viscosity on the phase velocities and attenuation coefficients of modes. It is also discovered that for some modes, there are both certain thicknesses and certain ranges of thickness where the effect of liquid viscosity on the phase velocities and attenuation coefficients of these modes is considerable. We ascertain that liquid viscosity promotes decrease of the penetration depth of the lowest quasi-Lamb mode into the liquid. The developed approach and the obtained results make it possible to ascertain for wave processes the limits of applicability of the model of ideal compressible fluid. Numerical results in the form of graphs are adduced and analyzed.     

Stability of Thermal Vibrational Flow in an Inclined Liquid Layer Against Finite‐Frequency Vibrations

Stability of the flow that arises under the action of a gravity force and streamwise finitefrequency vibrations in a nonuniformly heated inclined liquid layer is studied. By the Floquet method, linearized convection equations in the Boussinesq approximation are analyzed. Stability of the flow against planar, spiral, and threedimensional perturbations is examined. It is shown that, at finite frequencies, there are parametricinstability regions induced by planar perturbations. Depending on their amplitude and frequency, vibrations may either stabilize the unstable ground state or destabilize the liquid flow. The stability boundary for spiral perturbations is independent of vibration amplitude and frequency.     

Stability and Self‐Oscillations of a Rotor Containing a Conducting Liquid in a Magnetic Field

This paper considers the problem of the stability in the small of the steadystate spinning of a rotor with a cylindrical cavity partly filled with a viscous, incompressible, conducting liquid in a magnetic field. The responses of the buttend boundary layers and the resultant force exerted by the liquid on the rotor performing circular precession of small radius are determined. The plane of the viscoelastic restraint parameters of the rotor axis was Dpartitioned into regions with different degrees of instability is constructed. Steadystate spinning near the boundary of the region of stability in the space of parameters is studied assuming nonlinear responses of the supports. It is shown that passage through the boundary of the region of stability leads to bifurcation of the steadystate spinning regime, resulting in periodic motion of the type of circular precession. The origin ofperiodic motion from steadystate spinning can be subcritical or supercritical.     

Stationary Shear Flows of Nematic Liquid Crystals: A Comprehensive Study via Ericksen–Leslie Model

Journal of Dynamics and Differential Equations - We apply the Ericksen–Leslie dynamic model to investigate stationary solutions of planar shear flows for nematic liquid crystals. Nematic...     

Calculation of Linear Stability of a Stratified Gas–Liquid Flow in an Inclined Plane Channel

Linear stability of liquid and gas counterflows in an inclined channel is considered. The full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode (corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase. The wave length and the phase velocity of neutral disturbances of both modes are calculated as functions of the Reynolds number for the liquid. It is shown that these dependences for the surface mode are significantly affected by the gas velocity.     

Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond

We study global minimizers of a continuum Landau–De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W ^1,2, to a global minimizer predicted by the Oseen–Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen–Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau–De Gennes global minimizer. We also study the interplay between biaxiality and uniaxiality in Landau–De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions.     

High‐Frequency Changes in the Orientation of Nematic Liquid Crystals in Radio Frequency Crossed Electric Fields

The orientational dynamics of the director of a nematic liquid crystal located in radio frequency crossed electric fields were studied by numerical calculations and experimentally. This system is shown to be a physical object of nonlinear dynamics. Depending on the parameters of the problem, the following types of states of the director were observed: stationary (an analog of the nonthreshold Freedericksz transition), periodic, quasiperiodic (multimode), and stochastic of the strange attractor type. In the calculations, all states were obtained by solving a deterministic system of two timedependent nonlinear differential equations of the first order with no electrohydrodynamic terms. All types of solutions obtained, including stochastic ones, were observed experimentally.