The shape of axisymmetric rotating fluid

The equations for axisymmetric self-gravitating rotating fluid have been studied extensively since Poincaré. The model derives its primary interest from celestial mechanics, where it can be used to study the geometry of stars and planets. Existence of a solution for both the compressible and the incompressible cases is known. The smoothness of the boundary of the fluid is studied, and, in particular, it is proved that the rotating fluid has at most a finite number of rings.     

On the modelling of bubble plumes in a liquid pool

Turbulent, bubble plumes are investigated numerically using the commercial, Computational Fluid Dynamics (CFD) code CFX-F3D. A six-equation, two-fluid model approach is adopted, in which interphase momentum exchange models include buoyancy, drag, added mass, lift and turbulent dispersion effects. Particular attention is paid to turbulence modelling, in which generation and dissipation resulting from interaction between bubbles and liquid are specifically taken into account within the context of an extended k − ϵ turbulence model. Results from a number of calculations are presented and compared against published, experimental bubble plume data. It is suggested that existing bubble/liquid interaction models for plumes may be grouped into three categories: those which produce lateral bubble spreading, those which diffuse the ambient liquid velocity field, and those which couple the plume to the surrounding liquid and thereby ultimately govern the pool mixing behaviour.     

Weak oscillations of a gas bubble in a spherical volume of compressible liquid

The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.     

On the existence of stationary solitary waves in a rotating fluid

A mathematical proof that there are no stationary solutions of the soliton type is given for a number of equations related to Ostrovskii's equation which, in particular, describes the surface and internal waves in a rotating fluid. A physical interpretation of this fact is presented. It is shown that, in the case of a different character of the high frequency dispersion which corresponds, for example, to capillary waves on a shallow rotating fluid, the conditions of the theorem are not satisfied as a result of which the prohibition on the existence of solitons is lifted. In this case, both single solitons as well as stationary formations consisting of solitons, that is, multisolitons, are constructed using numerical calculations.     

On the Rayleigh problem for a Sisko fluid in a rotating frame

The unsteady rotating flow of a Sisko fluid bounded by a suddenly moved infinite flat plate is investigated. The fluid is electrically conducting in the presence of a transverse applied time-dependent magnetic field. A highly non-linear differential equation resulting from the balance of momentum and mass, coupled with appropriate boundary and initial conditions is solved numerically. The numerical solutions for different values of the parameters are compared and discussed.     

On pulsatile flow of a viscous fluid in a rotating channel

A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK ²=ΩL ²/v which is reciprocal of Ekmann Number and frequency parameter σ=αL ²/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK ² the flow near the plates has a multiple boundary layer character.     

On the stability of uniformly rotating liquid in a weak magnetic field

We analyze the simplest free boundary problem of magnetohydrodynamics governing the evolution of an isolated mass of a viscous incompressible liquid in the presence of the magnetic field. The motion of the liquid is governed by the Navier–Stokes equations, and for the magnetic field we have the Maxwell equations with an excluded displacement current. The magnetic field should be determined not only in the domain filled with the liquid, but also in the surrounding vacuum region. On the free boundary of the liquid standard jump conditions for the magnetic field are prescribed, as well as kinematic and dynamic boundary conditions, where the magnetic stress tensor is taken into account. We prove that the solution corresponding to a rigid rotation of the fluid and to zero magnetic field is stable if the functional of potential energy has a positive second variation. Bibliography: 11 titles.     

Analytical solution for a steady flow of enclosed rotating disks

Low-Reynolds-number flow plays an important role in the centrifugal separation of fluid particles under microgravity conditions and also in micromechanics due to the miniaturization of fluid mechanical parts. In this situation, the governing equations may be simplified. Here an analytical solution is presented for the steady flow of an incompressible viscous fluid between two finite disks enclosed by a cylindrical container for small Reynolds number (Re 10). The general solution is valid for all choices of the aspect ratio () and different cases of disk to cylinder rotation rates (s). An expression for the torque acting on the disk is obtained. The tangential velocity distribution is calculated and presented graphically for different values of ands. Known results in the literature for a single rotating disk and similar problems follow as a particular case of the general solution presented.

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Zusammenfassung Zahlreiche hydrodynamische Vorgänge unter der Bedingung verminderter Schwerkraft aber auch Vorgänge in der Mikromechanik finden im Bereich kleiner Reynoldszahlen statt. In solchen Situationen können die Bewegungsgleichungen vereinfacht und eventuell analytische Lösungen gefunden werden. In dieser Arbeit wird die stationäre Strömung einer viskosen, inkompressiblen Flüssigkeit für kleine Reynolds- und unterschiedliche Aspektzahlen untersucht. Die Flüssigkeit ist zwischen zwei rotierenden Scheiben und einem zylindrischen Behälter eingeschlossen. Eine analytische Lösung für die Tangentialkomponente des Geschwindigkeitsvektors ist für den allgemeinen Fall, dass die Scheiben und der Behälter unterschiedliche Winkelgeschwindigkeiten besitzen können, dargestellt. Des weiteren wurde eine Beziehung für das Widerstandsmoment der rotierenden Scheibe angegeben. Der Verlauf der Tangentialgeschwindigkeiten für verschiedene Rotations- und Aspektverhältnisse wird graphisch dargestellt und diskutiert. Bereits angegebene Lösungen in der Literatur bezüglich dieser Geometrie können als Sonderfall der hier dargestellten Lösung entwickelt werden.

     

On the theory of countercurrent flow in a rotating viscous heat-conducting gas

The countercurrent flow in a gas centrifuge is simulated. Mechanical and thermal methods for its excitation are discussed; thermal restructuring, the thermal control of the velocity field, and a shift in the inversion point are analyzed; and the formation of overtone flows in the rarefaction zone is studied.     

On the stability of a uniformly rotating viscous incompressible self-gravitating liquid

The paper is devoted to justification of the potential energy minimum principle in the problem of stability of a uniformly rotating viscous incompressible self-gravitating liquid. The capillary forces on the free boundary of the liquid are not taken into account. It is proved that the regime of rigid rotation is stable if the second variation of the energy functional is positive. The proof is based on the analysis of the evolution free boundary problem for perturbations in the velocity and pressure of the rotating liquid. Bibliography: 15 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 165–208.     

Bifurcation of the shape of a rotating string

We study the problem of the shape of an inextensible string rotating in a gravitational field. We discuss a method of solution and present some results: bifurcation diagrams and equilibrium shapes of the string for certain values of the angular velocity ω. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

On the flow of a non-Newtonian fluid between rotating,coaxial discs

In recent times, workers in numerical methods for non-Newtonian flows have been concerned with the high Weissenberg number problem. This problem which may be caused by a number of different things, manifests itself in the failure of numerical methods at some finite and often small Weissenberg number.This paper is concerned with the torsional flow of an Oldroyd-B fluid. The kinematics is restricted to that commonly referred to as Von Kármán kinematics. These restrictions allow the reduction of the problem to a set of ordinary differential equations. The problem is then solved with finite differences using well-known branch following and jumping techniques.The solution of this set of equations is discussed, and it is found that the solutions either lose stability at subcritical bifurcation points, or fold back on themselves at limit points. Either of these will cause a high Weissenberg number problem.Comparisons are also made with the known solutions to the Newtonian problem by considering small values of the Weissenberg number.     

Effect of a viscous fluid flow past a spherical gas bubble on the growth of its radius

A nearly spherical gas bubble expands adiabatically in a viscous incompressible fluid flowing past it. The Rayleigh-Plesset formula for the growth of the bubble radius is modified due to the flow of the viscous fluid.     

Instability of a rotating fluid

The paper is concerned with the problem of stability of a uniformly rotating viscous incompressible self-gravitating liquid bounded by a rotationally symmetric free surface. It is proved that it is unstable if the second variation of the energy functional is not positive. Bibliography: 11 titles.     

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

We consider the free boundary problem for the Navier–Stokes equations governing a nonstationary motion of a layer of a viscous incompressible liquid that covers the surface of a rigid ball rotating around a fixed axis with constant angular velocity ω. The liquid is subject to the gravitation force generated by the mass of the ball. The self-gravitation forces between the liquid particles and capillary forces on the free surface are not taken into account. We consider the problem of stability of the regime of the rigid rotation of the liquid with the same angular velocity and prove that it is stable if |ω| is less than a certain constant. Bibliography: 10 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 91–145.     

Stretching a surface in a rotating fluid

Summary A surface is stretched in a rotating fluid. The solution to the governing set of nonlinear differential equations depends on a parameter which is the ratio of the rotation rate to the stretching rate. Perturbation solutions for small and large compare well with exact numerical integration.     

Hydromagnetic penetration in a rotating fluid

Summary Fluid motion induced by the penetration of an oscillating axial magnetic field in a semi-infinite rotating fluid is considered. The redistribution of forces due to the generation of Maxwell stresses produce fluid fluctuations superposed on a steady motion. Estimates of the mean flow are obtained for different values of the mean applied magnetic pressure.

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Zusammenfassung Die Flüssigkeitsbewegung, die durch das Eindringen eines oszillierenden achsialen Magnetfeldes in eine halb-unendliche rotierende Fluid-Masse induziert wird, wird untersucht. Die Neuverteilung der Kräfte, bedingt durch die Erzeugung von Maxwell-Spannungen, erzeugt Fluktuationen des Fluids, die der stationären Bewegung überlagert sind. Schätzungen der mittleren Geschwindigkeit für verschiedene Werte des mittleren magnetischen Druckes wurden erhalten.