A note on co-current laminar flows of two immiscible elastico-viscous liquids

Summary An analysis is presented of steady (isothermal) co-current laminar flows of two immiscible elasticoviscous liquids in cylindrical channels to include (i) unidirectional stratified flow with ripple-free, plane liquid interface, and (ii) concentric-layered swirling flow with ripple-free cylindrical liquid interface. The general conditions are derived for such two-phase channel flows to be physically realizable. It is shown that, whereas (under certain circumstances)single-phase laminar flows are physically possible,two-phase flows, on the other hand, of liquids of the same class may not be. But liquids of theRoberts type (Roberts 1953), with a normal stress difference equivalent to an extra simple tension along the streamlines in simple shearing, are capable of steady unidirectional flowin all circumstances (whether in single or two-phase flow), though they are not in a privileged position so far astwo-phase swirling flows are concerned.     

Dynamics of two-phase bubble-droplets in immiscible liquids

This paper describes an experimental study of two-phase bubble-droplets of butane in distilled water. In the process of direct-contact heat transfer between two immiscible liquids, and with change of phase, evaporation occurs within a liquid droplet to form an interior bubble which ultimately may expand to absorb the entire droplet. The so-called stopped-evaporation two-phase bubble-droplets were formed by such evaporating droplets of butane rising in a column of water by application of pressure on the surface of the water to stop the evaporation. The configurations of such bubble-droplets, which depend on the ratio of mass of the vapor to mass of the liquid, are discussed. Results are also given for the rise velocity, and comparisons are made between the rise velocity of stopped-evaporation droplets and those of evaporating droplets.

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Dynamik zweiphasiger Blasen-Tropfen in unmischbaren Flüssigkeiten

Zusammenfassung Die experimentelle Untersuchung von zweiphasigen Blasen-Tropfen aus Butan in destilliertem Wasser wird beschrieben. Bei direktem Wärmeaustausch mit Phasenänderung zwischen zwei unmischbaren Flüssigkeiten beginnt die Verdampfung innerhalb eines Flüssigkeitstropfens und bildet eine innere Blase, die sich schließlich ausdehnt und den Tropfen absorbiert. Ein Ende der Verdampfung konnte beim Aufsteigen von Butantropfen in eine Wassersäule durch Anlegen von Druck an der Wasseroberfläche erreicht werden. Die Struktur dieser Blasen-Tropfen, die abhängt vom Verhältnis der Massen des Dampfes und der Flüssigkeit, wird diskutiert. Außerdem werden Ergebnisse für die Aufstiegsgeschwindigkeit gegeben und die Aufstiegsgeschwindigkeiten der verdampfenden Tropfen mit denen bei beendeter Verdampfung verglichen.

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Nomenclature d Equivalent spherical diameter of initial droplet (mm) - D Equivalent spherical diameter of bubble-droplet (mm) - m₀₀ Initial mass of droplet (kg) - m _v Mass of vapour (kg) - Re _c Reynolds number based on water properties (Re_c=UD/) - t Time (s) - T _c Temperature of water (°C) - U Rise velocity (m/s) - Z Position of bubble-droplet (mm) - t ₀ Overall temperature difference (C deg) - Density (kg/m³) - Viscosity (kg/m s)     

Oscillating flows of miscible fluids

A theoretical analysis is presented for the flows of two miscible, viscous, incompressible fluids, subject to oscillatory pressure gradients in a cylindrical tube. The extended, time-dependent Navier-Stokes equations with inter-component interaction for the fluids are reduced to ordinary, inhomogeneous differential equations of fourth order by separation and decoupling, and solved in closed form. As distinguished from the classical one-fluid Richardson flow, the dynamics of the two-component, oscillatory flow is determined by two eigenvalues which are complex. Since the coupling force between the fluids increases with their velocity difference, deviations from the one-fluid Richardson effect exist.On sabbatical leave from Department of Electrical Engineering, Colorado State University, U.S.A.     

Influence of contact-perimeter slip and heat transfer processes on oscillating hydrodynamic flows of immiscible liquids in capillary tubes

The hydrodynamic flows which arise in the presence of mechanical vibrations of a capillary tube filled with immiscible liquids are investigated. At the hermetically sealed ends of the tube there are air bubbles. It is assumed that the interfacial contact perimeters of the immiscible liquids can slip relative to the walls of the tube. The results of numerical calculations are given for a mercury electrocapillary transducer [1–4], which is a capillary tube filled with water and mercury. The calculated and experimental amplitude—frequency characteristics (the dependences of the amplitude of the electric potential difference on the vibration frequency) are compared.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 36–45, May–June, 1993.     

The behaviour of large drops in immiscible liquids

The motion of large drops with Eötvos numbers as high as 1000 has been investigated in large tanks where wall effects are negligible. Shapes observed include ellipsoidal- and spherical-caps with and without skirts, crescents, biconcave disks, toroids and wobbling irregular forms. The ellipsoidal-cap drops are shown to obey an equation based upon an extension of the Davies and Taylor theory for bubbles in liquids, regardless of whether skirts are being formed and shed at the rear.     

Unsteady flows in elastico-viscous liquids

Summary Consideration is given to two problems involving the unsteady motion of a sphere in aMaxwell liquid. In both problems the full solutions are obtained and compared in each case to the solution obtained using an approximation procedure. Good agreement is obtained for large values of the elastic parameter. The approximation reduces the mathematics considerably and thus allows experimentalists to determine the elastic parameter involved.With 2 figures     

Theory of nonsteady nonequilibrium filtration of immiscible liquids

A modification to the model developed in [1] for the nonequilibrium filtration of an inhomogeneous liquid is proposed. The structure of the saturation discontinuity is investigated, taking into account the nonequilibrium conditions; it is shown to be stable in a linear approximation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 51–58, May–June, 1978.It remains to thank G. I. Berenblatt for his constant interest in the work.     

Developing laminar flows with visco-elastic and non-Newtonian liquids

Summary Developing laminar flows have been studied experimentally in straight pipes of circular cross-section. Velocity profiles have been measured and entry lengths determined for the transition from an initially flat profile to the equilibrium form: flow visualisation using fine tracer particles was the technique used. It has been found that non-Newtonian (pseudo-plastic) fluids have significantly longer development regions than those corresponding to dynamically similar Newtonian conditions. Viscoelastic liquids depart even more from the simple case. The experimental evidence seems to indicate that elastic effects influence the flow field at two differing time scales, not only where the process time is of the same order of magnitude as the natural (relaxation) time of the liquid but also in much slower processes.

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Zusammenfassung Die Entwicklung von laminaren Strömungen wurde experimentell in geraden Rohren mit Kreisquerschnitt untersucht. Die Geschwindigkeitsprofile wurden gemessen und die Eintrittslängen bestimmt, die für den Übergang von dem ursprünglich flachen Profil zur Gleichgewichtsform nötig sind. Die Strömung wurde mit Hilfe von feinen Spurenteilchen sichtbar gemacht. Es zeigte sich dabei, daß nicht-Newtonsche Flüssigkeiten ein deutlich größeres Übergangsgebiet haben als solche, die sich auf dynamisch ähnliche Newtonsche Bedingungen beziehen. Viskoelastische Flüssigkeiten weichen stärker von dem einfachen Fall ab.Das Experiment scheint zu zeigen, daß elastische Effekte das Strömungsfeld in zwei unterschiedlichen Zeitmaßstäben beeinflussen: nicht nur in dem Bereich, in dem die Prozeßzeit von derselben Größenordnung ist wie die Relaxationszeit der Flüssigkeit, sondern auch bei wesentlich langsameren Prozessen.

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Heat transfer solutions in laminar co-current flow of immiscible liquids

Thermal entry region solutions are analytically determined for horizontal, co-current laminar flow of immiscible liquids in direct contact, inside circular tubes and parallel plate channels. The related eigenvalue problem for such a composite media is readily solved by extending the ideas in the recently advanced sign-count method. It is assumed a core-annular flow configuration for circular tubes and sheat-core flow for the parallel plates channel, without consideration of interface instabilities and stratified flow. First, the velocity problem is solved for fully developed flow and pumping power expressions established for different operating conditions. Then, the temperature problem is analytically handled to yield expressions for quantities of practical interest such as total heat exchange rates, along the duct length and, again, for different flow rates and pressure drop requirements. The analysis is illustrated through consideration of an application dealing with pumping of a very viscous oil with the addition of an external thin layer of a less viscous fluid (water). Pumping power and total heat exchange are then evaluated for both geometries and critically compared to the single fluid flow problem.Hier sind Lösungen für die thermische Eintritts-strecke von horizontalen, laminaren Gleichströmungen in unvermischbaren Flüssigkeiten, die in direktem Kontakt untereinander sind, analytisch bestimmt worden. Die Lösungen gelten für Gleichströmungen in Röhren und in parallelen flachen Kanälen. Das betreffende Eigenwertproblem für solch ein zusammengesetztes Medium ist vollkommen mit dem Gedanken des kürzlich weiterentwickelten Zeichenzählverfahrens gelöst worden. Für die Rohre sind kreisring- und kernförmige Strömungen und für die parallelen Plattenkanäle Schichtkernströmungen angenommen worden. Hierbei ist die Grenzflächeninstabilität nicht in Betracht gezogen worden. Als erstes ist das Geschwindigkeitsproblem für eine vollkommen entwickelte Strömung gelöst und es sind Ausdrücke für die Pumpleistung für verschiedene Betriebsbedingungen ermittelt worden. Dann ist das Temperaturproblem analytisch behandelt worden, um Ausdrücke für die Größen von praktischem Interesse zu erzielen, wie die gesamte Wärmeaustauschrate über die Kanallänge für verschiedene Strömungsgeschwindigkeiten und Druckverlustanforderungen. Die Berechnung ist durch die Betrachtung eines Anwendungsbeispiels veranschaulicht worden, bei dem sehr zähflüssiges Öl mit einer zusätzlichen äußeren, dünnen Schicht, die weniger zähflüssig ist, gepumpt wurde. Die Pumpleistung und der gesamte Wärmeaustausch sind für beide Geometrien ausgewertet und kritisch mit dem einfachen Strömungsproblem von Fluiden verglichen worden.     

One-dimensional filtration of immiscible liquids in a nonuniform porous medium

As is known, the differential equation for two-phase filtration with account for capillarity was obtained in [1], and later integrated numerically for the case of a uniform stratum of finite length in [2]. Other versions of the solution of the Rapoport-Leas equation or the system which is equivalent to it are known [3, 4]. This article presents the results of a numerical solution of an analogous problem with account for nonuniform permeability of the stratum.The authors wish to thank T. V. Startsev and L. Kh. Aminov for assistance in performing the calculations on the Ural-3 computer.     

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.     

Modeling electrohydrodynamic flows in slightly conductive liquids

Mechanical Mathematics Department, Moscow State University, Moscow 117234. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 36, No. 4, pp. 36–44, July–August, 1995.     

Heat transfer in vertical annular laminar flow of two immiscible liquids

The paper investigates heat transfer in annular laminar undisturbed flow of two immiscible liquids, with constant heat-flux generated at the wall of the tube. It presents an analytical solution for the fully developed temperature field. This is used to obtain a more general solution from a model, describing the temperature field as a superposition of the fully developed and the developing fields. This superposition model is solved by an orthogonal collocation method. An asymptotic model for short entry lengths is also described. Calculations for a kerosene-water system, show that the superposition solution converges to the entrance solution below 100 diameters and converges asymptotically to the solution of the fully developed temperature field beyond 5000 diameters. The effect of the wavy interface is assessed experimentally for annular kerosene-water flow, by comparing predicted and measured temperature profiles. It is found that experimental profiles are considerably flatter and measured Nusselt numbers for the kerosene phase are accordingly higher by 40–320% as compared to the undisturbed flow analyses.     

Motion of two viscous immiscible incompressible liquids in a rotating cylinder

The shape of the interface is determined using the boundary layer method and the position of the center of mass of the liquid and the value of the braking torque exerted by the liquid on the walls of the rotating cylinder are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 92–99. January–February, 1990.     

“Hanging” discontinuities in the saturation with the filtration of immiscible liquids in inhomogenous media

The article discusses the flow of two incompressible immiscible liquids in a porous medium. A study is made of “hanging” discontinuities of the saturation arising at the points of the breakaway of the absolute permeability, taking account of capillary forces and of the force of gravity.     

Shock-induced two-phase flows in an aligned baffle system filled with suspended particles

This paper describes the shock propagation through a dilute gas-particle suspension in an aligned baffle system. Numerical solution to two-phase flows induced by a planar shock wave is given based on the two-continuum model with interphase coupling. The governing equations are numerically solved by using high-resolution schemes. The computational results show the shock reflection and diffraction patterns, and the shock-induced flow fields in the 4-baffle system filled with the dusty gas.