Oscillating flows in capillaries filled with immiscible liquids

Hydrodynamic flows generated by mechanical vibrations of a capillary filled with immiscible liquids are investigated. Air bubbles are contained at the hermetically sealed ends of the capillary. Equations for the change in the volumes of the air bubbles as functions of time and velocity distribution in the liquids are obtained for the case when the radius of the capillary is much less than the lengths of the liquid columns. Results of numerical calculations are given for a capillary filled with two liquids: water and mercury. Amplitude-frequency dependences of the change in volumes of the air bubbles are constructed which have a resonance nature. Graphs of the dependence of the velocity of the water and the mercury on the radial coordinate at different times are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 13–18, September–October, 1987.     

Electric effects of vibrations of capillaries filled with droplets of mercury and an electrolyte solution

The appearance of a time-dependent difference of the electric potentials due to mechanical vibrations of capillaries filled with droplets of mercury and an electrolyte solution was studied in [1–5]. The phenomenon is used in electrokinetic transducers [6, 7]. Equivalent electric and mechanical circuits of the transducers were proposed in [1–4]. The experimental dependences of the amplitude of the potential difference that arises between the mercury droplets on the vibration frequency (the amplitude-frequency characteristics of the transducers) were investigated in [2, 4]. In the present paper, expressions are obtained for the difference of the electric potentials between mercury droplets that arises in the case of harmonic vibrations of a capillary with small amplitudes. The theoretical and experimental amplitudefrequency characteristics of an electrocapillary transducer are constructed. Good agreement between the results of the calculations and the experimental data is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika-Zhidkosti i Gaza, No. 2, pp. 131–136, March–April, 1988.     

Relative Permeability Analysis of Tube Bundle Models

The analytical solution for calculating two-phase immiscible flow through a bundle of parallel capillary tubes of uniform diametral probability distribution is developed and employed to calculate the relative permeabilities of both phases. Also, expressions for calculating two-phase flow through bundles of serial tubes (tubes in which the diameter varies along the direction of flow) are obtained and utilized to study relative permeability characteristics using a lognormal tube diameter distribution. The effect of viscosity ratio on conventional relative permeability was investigated and it was found to have a significant effect for both the parallel and serial tube models. General agreement was observed between trends of relative permeability ratios found in this work and those from experimental results of Singhal et al. (1976) using porous media consisting of mixtures of Teflon powder and glass beads. It was concluded that neglecting the difference between the average pressure of the non-wetting phase and the average pressure of the wetting phase (the macro-scale capillary pressure) – a necessary assumption underlying the popular analysis methods of Johnson et al. (1959) and Jones and Roszelle (1978) – was responsible for the disparity in the relative permeability curves for various viscosity ratios. The methods therefore do not account for non-local viscous effects when applied to tube bundle models. It was contended that average pressure differences between two immiscible phases can arise from either capillary interfaces (micro-scale capillary pressures) or due to disparate pressure gradients that are maintained for a flow of two fluids of viscosity ratio that is different from unity.     

A model for film-forming with Newtonian and shear-thinning fluids

The formation of a thin film by (i) the slow penetration of a gas bubble into a liquid filled tube, (ii) the withdrawal of a planar substrate from a liquid filled gap, is investigated theoretically for the cases of both Newtonian and shear-thinning liquids; the latter conforming to either a power–law or Ellis model. Formulated as a boundary value problem underpinned by lubrication theory, the analysis gives rise to a system of ordinary differential equations which are solved numerically subject to appropriate boundary conditions. For Newtonian liquids comparison of the predicted residual film thickness for a wide range of capillary number, Ca  (10⁻⁴, 10), is made with others obtained using existing expressions, including the classical one of Bretherton, in the region of parameter space over which they apply. In the case of (i), prediction of the behaviour of the residual fluid fraction and gap-to-film thickness ratio, for a Newtonian liquid and one that is shear-thinning and modelled via a power–law, is found to be in particularly good agreement with experimental data for Ca < 0.2. For (ii), both shear-thinning models are utilized and contour plots of residual film thickness generated as a function of Ca and the defining parameters characteristic of each model.     

Asymptotic method for solving problems with radial laminar jets of immiscible liquids

Two problems involving radial laminar jets of immiscible liquids are considered: a free radial slit jet and a jet on a rotating disk. An asymptotic method of solution is proposed that makes it possible to determine the flow parameters far from the source. The difference between these flows and those of homogeneous liquids is demonstrated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 192–28, May–June, 1980.     

Filtration of a piecewise-homogeneous liquid

The nonsteady filtration of two immiscible liquids, in the absence of pressure, in a homogeneous and isotropic porous medium, is considered. A nonlinear hydrodynamic problem is formulated, and a number of its features are noted. The solution of the corresponding linearized problem is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 62–69, November–December, 1976.     

Percolation model of two-phase flow through porous media

A physical model of the process of two-phase flow of immiscible fluids through a porous medium is developed and used to make an analytical calculation of the dependence of the relative phase permeabilities on the saturation of the medium by one of the phases. The theory is compared qualitatively with experiment for a model capillary radius frequency function and quantitatively with numerical calculations made on a computer. In both cases good agreement is obtained. The pressure dependences of the phase permeabilities are analyzed. The question of residual saturation with the wetting fluid after completion of the displacement process is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 88–95, January–February, 1987.     

Two-dimensional problems of displacement in the case of nonlinear filtration laws

A method is proposed for calculating the multidimensional displacement of oil possessing anomalous properties from productive strata by water. Numerical solutions are obtained on the basis of the most generally accepted models of non-Newtonian oil within the framework of the theory of two-phase filtration of immiscible liquids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 70–75, September–October, 1976.     

Dynamics of viscoelastic liquid filaments: Low capillary number flows

Many applications of viscoelastic free surface flows requiring formation of drops from small nozzles, e.g., ink-jet printing, micro-arraying, and atomization, involve predominantly extensional deformations of liquid filaments. The capillary number, which represents the ratio of viscous to surface tension forces, is small in such processes when drops of water-like liquids are formed. The dynamics of extensional deformations of viscoelastic liquids that are weakly strain hardening, i.e., liquids for which the growth in the extensional viscosity is small and bounded, are here modeled by the Giesekus, FENE-P, and FENE-CR constitutive relations and studied at low capillary numbers using full 2D numerical computations. A new computational algorithm using the general conformation tensor based constitutive equation [M. Pasquali, L.E. Scriven, Theoretical modeling of microstructured liquids: a simple thermodynamic approach, J. Non-Newtonian Fluid Mech. 120 (2004) 101–135] to compute the time dependent viscoelastic free surface flows is presented. DEVSS-TG/SUPG mixed finite element method [M. Pasquali, L.E. Scriven, Free surface flows of polymer solutions with models based on conformation tensor, J. Non-Newtonian Fluid Mech. 108 (2002) 363–409] is used for the spatial discretization and a fully implicit second-order predictor–corrector scheme is used for the time integration. Inertia, capillarity, and viscoelasticity are incorporated in the computations and the free surface shapes are computed along with all the other field variables in a fully coupled way. Among the three models, Giesekus filaments show the most drastic thinning in the low capillary number regime. The dependence of the transient Trouton ratio on the capillary number in the Giesekus model is demonstrated. The elastic unloading near the end plates is investigated using both kinematic [M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469–501] and energy analyses. The magnitude of elastic unloading, which increases with growing elasticity, is shown to be the largest for Giesekus filaments, thereby suggesting that necking and elastic unloading are related.     

An Analysis of the Movement of Wetting and Nonwetting Fluids in Homogeneous Porous Media

The movement of wetting and nonwetting fluid flow in columns packed with glass beads is used to understand the more complicated flows in homogeneous porous media. The motion of two immiscible liquids (oil and water) is observed with different surfactants. Through dimensional analyses, fluid velocity is well correlated with interfacial tension and less dependent on particle size. In water–oil (W/O) experiments, finger pattern flows are observed if water is the displacing fluid that flows in an oil-filled porous media, whereas oil ganglia tend to form if oil is the displacing fluid in the water-wetted porous media. The results are well described by a simple model based on an earlier theory of flow in a tube.     

Numerical finding of the equilibrium interface of liquids in an axisymmetric vessel

A study is made of the problem of numerical determination of doubly connected axisymmetric equilibrium shapes of the interface of two immiscible liquids in an arbitrary axisymmetric vessel under conditions of reduced gravity. Problems of this kind are of practical interest in connection with the study of the behavior of fuel in the tanks of spacecraft [1]. It is assumed that the liquids are homogeneous and incompressible and have a temperature constant throughout the complete volume. In the investigation of the equilibrium of the interface of liquids under conditions of complete or partial weightlessness, it is necessary to take into account the forces of surface tension, which play a decisive part in such a situation. The equilibrium of liquids in a vessel with allowance for surface tension is described by a system of nonlinear equations [2]. In the present paper, the problem is solved by the Kantorovich— Newton iterative method [3], which makes it possible to reduce the nonlinear problem to a succession of linear problems solvable by the method of finite differences.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–137, September–October, 1981.I thank I. E. Tarapov and I. I. Ievlev for constant interest in the work and helpful comments.     

Development of thermogravitation convection in a two-layer system in the presence of a surface-active material on the boundary

Convective instability of equilibrium in a system of two horizontal layers of immiscible liquids, caused by the Rayleigh instability mechanism, has been studied within the framework of the linear theory in [1–5]. The present study will investigate the effect of a surface-active material (SAM), deposited on the boundary between the liquids, on the development of thermogravitation convection. Calculations were performed for two types of systems, which in the absence of a SAM show instability of a monotonic or an oscillatory character. A new type of oscillatory equilibrium instability was observed, produced by the effect of the SAM. In some region of parameter values the oscillatory instability may prove to be the more dangerous one. The action of the Marangoni effect on thermogravitation oscillations is considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 76–81, September–October, 1986.In conclusion, the authors express their gratitude to E. M. Zhukhovitskii for his helpful evaluation.     

Electrization of dielectric liquids flowing in tubes

The methods of the mechanics of continuous media [1] are used to consider the problem of electrization of dielectric liquids flowing in tubes [2–6]. According to modern ideas [2–6], there is always dissolved in such liquids a slight admixture of an electrolyte, whose molecules in such a dilute solution dissociate to a certain extent into positively and negatively charged ions. On the walls, oxidizing and reducing reactions take place, as a result of which the negative and positive ions, respectively, give up to the wall surplus electrons or take missing electrons from it. Thus, a positive (respectively, negative) total electric charge is induced in the liquid by the flow. We consider in this paper the electrization of a dielectric liquid in laminar flow in a circular cylindrical tube. We find the distribution of the electric charge in the liquid, the maximal electric current, and the dependence of the length over which the distribution of the electric charge in the tube is established on the tube radius, the Debye radius of the liquid, and the Péclet diffusion number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 41–47, November–December, 1979.We thank V. V. Gogosov for helpful comments made in a discussion of thwe work.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part II. Applications of Model and Comparison of Interacting and Non-Interacting Capillary Bundle Models

In the first part of this work (Dong et al., Transport Porous Media, 59, 1–18, 2005), an interacting capillary bundle model was developed for analysing immiscible displacement processes in porous media. In this paper, the second part of the work, the model is applied to analyse the fluid dynamics of immiscible displacements. The analysis includes: (1) free spontaneous imbibition, (2) the effects of injection rate and oil–water viscosity ratio on the displacement interface profile, and (3) the effect of oil–water viscosity ratio on the relative permeability curves. Analysis of a non-interacting tube bundle model is also presented for comparison. Because pressure equilibration between the capillaries is stipulated in the interacting capillary model, it is able to reproduce the behaviour of immiscible displacement observed in porous media which cannot be modelled by using non-interacting tube bundle models.     

Oscillations of an ideal liquid acted upon by subface-tension forces. Case of a doubly connected free surface

Many articles have appeared on the problems of small oscillations of an ideal liquid acted upon by surface-tension forces. Oscillations of a liquid with a single free surface are treated in [1, 2]. Oscillations of an arbitrary number of immiscible liquids bounded by equilibrium surfaces on which only zero volume oscillations are assumed possible are investigated in [3], We consider below the problem of the oscillations of an ideal liquid with two free surfaces on each of which nonzero volume disturbances are kinematically possible. The disturbances satisfy the condition of constant total volume. A method of solution is presented. The problem of axisymmetric oscillations of a liquid sphere in contact with the periphery of a circular opening is considered neglecting gravity. The first two eigenfrequencies and oscillatory modes are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 64–71, May–June, 1976.In conclusion, the author thanks F. L. Chernous'ko for posing the problem and for his attention to the work.     

Capillary instabilities of charged drops and electrical dispersion of liquids

The phenomenology of the instability of charged and uncharged drops and liquid menisci at the end of a capillary through which liquid is fed in constant and variable external electric fields, both uniform and nonuniform, is considered. Methods of investigating such instability are described. The steady-state deformation of drops in electric fields, the critical conditions under which their charged surface loses stability, and the laws of electrical dispersion of liquids in the final stage of instability are discussed.Yaroslavl'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 3–22, May–June, 1994.     

一类轴对称充液陀螺系统自旋运动稳定性研究

研究了一类复杂充液旋转对称陀螺（其圆柱形容腔充有互不相溶的双元液体，且带充液圆柱中心杆）的章动振荡与其所充双元液体自由振荡之耦合问题．利用留数方法研究了系统的自旋稳态运动的稳定性条件     

Thermocapillary instability of the interface between two immiscible liquids in the presence of volume heat sources

The problem of the stability of the interface between two infinite layers of different immiscible liquids is considered. It is assumed that within the liquid a distributed volume heat source, simulating Joule heating, is given. The stability of the rest state with respect to small unsteady disturbances is investigated. The investigation is carried out using the real boundary conditions at the interface between the two liquids rather than the model boundary conditions usually employed in such problems [5]. The problem considered is related to the practical question of the stability of electrolyzer processes. In the present case a possible threshold mechanism of development of oscillations of the electrolyte-aluminum interface is examined. A numerical example with liquid parameters that coincide with those of the electrolyte and aluminum shows that the thermocapillary instability mechanism can, in fact, be the source of surface waves at the electrolyte-aluminum interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1990.     

Pressure Waves in a Gas–Liquid Medium with a Stratified Liquid–Bubbly Mixture Structure

Evolution and decay of pressure waves of moderate amplitude in a vertical shock tube filled by a gas–liquid medium with a nonuniform (stepwise) distribution of bubbles over the tube cross section are studied experimentally. The gas–liquid layer has the form of a ring located near the tube wall or the form of a gas–liquid column located in the center of the tube. It is shown that the nonuniformity of bubble distribution over the tube cross section increases the attenuation rate of pressure waves.     

The motion of a closely-fitting sphere in a fluid-filled tube

Singular perturbation techniques are used to investigate the slow, asymmetric flow around a sphere positioned eccentrically within a long, circular, cylindrical tube filled with viscous fluid. The results apply to situations in which the sphere occupies virtually the entire cross section of the cylinder, so that the clearance between the particle and tube wall is everywhere small compared with both the sphere and tube radii. The technique is an improvement over conventional “lubrication-theory” analyses.Asymptotic expansions, valid for small dimensionless clearances, are obtained for the hydrodynamic force, torque and pressure drop for flow past a stationary sphere, as well as for the case of a sphere translating or rotating in an otherwise quiescent fluid. These expansions are employed to predict the macroscopic behavior of both a neutrally-buoyant sphere suspended in a Poiseuille flow, and a sedimenting sphere in a vertical tube.The results find application in capillary blood flow, pipeline transport of encapsulated materials, and falling-ball viscometers.