Weakly nonlinear instability of the flow of two immiscible liquids with different viscosities in a pipe

The weakly nonlinear stability due to an axisymmetric disturbance of two immiscible liquids with different viscosities in a pipe is worked out. The most important conclusions drawn from this work are the independence of the Reynolds number in the stability criterion, the existence of stable waves with sawtooth profile when the more viscous fluid is located at the core and it occupies most of the pipe, and the destabilizing effect of the surface tension, effect that can be overtaken by the stabilizing effect due to the difference in viscosity of the two fluids, when the viscosity of the ring fluid is much smaller than the viscosity of the core fluid and the thickness of the ring flow is smaller than the thickness of the core flow.     

Thermocapillary Instability in a Two-Layer System with a Deformable Interface

The stability of equilibrium of a system consisting of two plane immiscible viscous liquid layers under conditions of weightlessness is considered. The surface tension force, which depends linearly on the temperature, acts on the deformable interface between the media. Three model systems are considered. In each system the development of instability is due to a particular asymmetry of the properties of the liquids or the layer thicknesses. The conditions of excitation of longitudinal thermocapillary and transverse capillary waves supported by the thermocapillary effect on the interface are discussed.     

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     

Stability of a steady-state discontinuity of a fluid layer on the surface of an immiscible fluid

A new stable structure of the three-phase system formed by a gas, a horizontal liquid layer with a free upper surface and an underlying immiscible liquid substrate is investigated experimentally and theoretically. When the upper layer has a greater surface tension than the lower layer and its thickness is fairly small, a local deformation of its surface can lead to the development of a steady-state concentric discontinuity within whose limits the lower layer os in contact with the gas. The conditions of stability of such a phase system with a steady-state discontinuity are studied and the dependences of the discontinuity parameters on the vessel diameter, the upper layer thickness, and the liquid surface tensions are obtained for various pairs of liquids. The formulation of the analytic problem of the layer discontinuity is discussed. The experimental data are compared with the results of calculations carried out for a model of a discontinuity in an infinite layer.     

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.     

Bénard-Marangoni convection in a rotating fluid with and without surface deformation

This paper deals with Bénard-Marangoni convection in a thin horizontal layer heated from below and rotating uniformly about a vertical axis. The analysis is linear and its objective is to determine the critical temperature drop and the critical wavenumber at the onset of convection. The lower surface of the layer is in contact with an uniformly heated rigid plate and the upper face is deformed and subjected to a temperature-dependent surface tension. Exchange of stability and Boussinesq's approximation are taken for granted. It is shown that rotation has a stabilizing effect while surface deflection plays a stabilizing role at large angular velocity and has an opposite influence at low angular velocity. By increasing the angular velocity, one reinforces the stability, whatever the surface deformation.     

Flutter of infinite elastic plates in the boundary-layer flow at finite Reynolds numbers

The stability of an infinite elastic plate in supersonic gas flow is investigated taking into account the presence of the boundary layer formed on the plate surface. The effect of viscous and temperature disturbances of the boundary layer on the behavior of traveling waves is studied at large but finite Reynolds numbers. It is shown that in the case of the small boundary layer thickness viscosity can have both stabilizing and destabilizing effect depending on the phase velocity of disturbance propagation.     

Einfluß der Dichteanomalie von Wasser auf das Einsetzen der Konvektion einer von unten beheizten bzw. gekühlten halbunendlichen Schicht

The influence of the density anomaly of water on the onset of convection is analytically studied for a horizontal semiinfinit water layer heated or cooled, respectively from below. The water temperature varied in the range 0 to 20°C. Besides of the Rayleigh-number the onset of convection, i.e. the limits of stability can be described with two additional parameters the Non-Linearity N and the Startparameter γ. The calculated values for the stability limits using linear stability analysis and the Galerkin approach are in excellent agreement with those found in the literature. As a general result, it can be stated that the density anomaly has a stabilizing effect.     

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.     

Calculation of the wave motion on the uniformly charged interface between two immiscible viscous liquids on the basis of the boundary layer theory

A modified theory of the boundary layer associated with a periodic capillary-gravity wave on the uniformly charged interface between two immiscible viscous incompressible liquids is proposed. A model problem simpler than the exact problem is proposed for describing the boundary layer flows in the upper and lower liquids. The structure of the solution of this problem reflects the principal features of the exact asymptotic solution, namely, the rapidity of the decrease in the vortex part of the flow as a function of the depth and the insignificance of certain components. Estimates of the boundary layer thicknesses for which the difference between the exact asymptotic solution and the solution of the simplified model problem (formulated within the framework of the theory proposed) can be specified with a predetermined accuracy are obtained.     

Effect of surface tension on the onset of convection in a layer of liquid with a free surface

The problem of stability of a plane horizontal layer of liquid heated from below is considered with surface tension at the upper surface taken into account. The problem is stated in section 1, proof of the existence of stability threshold is given in section 2, while section 3 concerns the construction of neutral curves by numerical methods and with the stabilizing effect of surface tension on the state of equilibrium.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 10, No. 3, pp. 89–92, May–June, 1969.     

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.     

Stability of the interface in two-layer couette flow of upper convected maxwell liquids

The linear stability of a two-layer Couette flow of upper convected Maxwell liquids is considered. The fluids have different densities, viscosities, and elasticities, with surface tension at the interface. At low speeds, the interfacial mode may become unstable, while other modes stay stable. The shortwave asymptotics of the interfacial mode is analyzed. It is found that an elasticity difference can stabilize or destabilize the flow even in the absence of a viscosity difference. As the viscosity difference increases, the range of elasticities for which there is shortwave stability widens. A linearly stable arrangement can be achieved by placing the less viscous fluid in a thin layer to stabilize longwaves and using elasticities to stabilize shortwaves. Such an arrangement can be stable even when the density stratification is adverse.     

Stability of resonant interfacial waves in the presence of uniform magnetic field

The aim of this work is to study the instability of interacting waves between two immiscible magnetic liquids. The effects of gravitation and a uniform normal magnetic field are taken into account. The method of multiple scales is used to determine the stability criteria of the considered problem. The various stability criteria are discussed both analytically and graphically. According to the numerical examples, we have remarked that the increase of the ratio of the permeability of the liquids appears to be the destabilizing effect of the magnetic field. The short waves below the critical wavenumbers are stable whereas a number of long waves are unstable. The viscosity effect on the stability criteria is a dual-role one, depending on the strength of the applied magnetic field.     

Gravitational instability of thin gas layer between two thick liquid layers

We consider the problem of gravitational instability (Rayleigh–Taylor instability) of a horizontal thin gas layer between two liquid half-spaces (or thick layers), where the light liquid overlies the heavy one. This study is motivated by the phenomenon of boiling at the surface of direct contact between two immiscible liquids, where the rate of the “break-away” of the vapor layer growing at the contact interface due to development of the Rayleigh–Taylor instability on the upper liquid–gas interface is of interest. The problem is solved analytically under the assumptions of inviscid liquids and viscous weightless vapor. These assumptions correspond well to the processes in real systems, e.g., they are relevant for the case of interfacial boiling in the system water-n-heptane. In order to verify the results, the limiting cases of infinitely thin and infinitely thick gas layers were considered, for which the results can be obviously deduced from the classical problem of the Rayleigh–Taylor instability. These limiting cases are completely identical to the well-studied cases of gravity waves at the liquidliquid and liquid–gas interfaces. When the horizontal extent of the system is long enough, the wavenumber of perturbations is not limited from below, and the system is always unstable. The wavelength of the most dangerous perturbations and the rate of their exponential growth are derived as a function of the layer thickness. The dependence of the exponential growth rate on the gas layer thickness is cubic.     

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.     

Optimally growing boundary layer disturbances in a convergent nozzle preceded by a circular pipe

We report the findings from a theoretical analysis of optimally growing disturbances in an initially turbulent boundary layer. The motivation behind this study originates from the desire to generate organized structures in an initially turbulent boundary layer via excitation by disturbances that are tailored to be preferentially amplified. Such optimally growing disturbances are of interest for implementation in an active flow control strategy that is investigated for effective jet noise control. Details of the optimal perturbation theory implemented in this study are discussed. The relevant stability equations are derived using both the standard decomposition and the triple decomposition. The chosen test case geometry contains a convergent nozzle, which generates a Mach 0.9 round jet, preceded by a circular pipe. Optimally growing disturbances are introduced at various stations within the circular pipe section to facilitate disturbance energy amplification upstream of the favorable pressure gradient zone within the convergent nozzle, which has a stabilizing effect on disturbance growth. Effects of temporal frequency, disturbance input and output plane locations as well as separation distance between output and input planes are investigated. The results indicate that optimally growing disturbances appear in the form of longitudinal counter-rotating vortex pairs, whose size can be on the order of several times the input plane mean boundary layer thickness. The azimuthal wavenumber, which represents the number of counter-rotating vortex pairs, is found to generally decrease with increasing separation distance. Compared to the standard decomposition, the triple decomposition analysis generally predicts relatively lower azimuthal wavenumbers and significantly reduced energy amplification ratios for the optimal disturbances.     

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.