Circular oscillations of a cylinder in a viscous fluid

Summary The steady streaming velocity induced by the circular motion of a cylinder of elliptic cross-section in a viscous fluid is considered. The amplitude of this circular motion is supposed small compared with a typical diameter of the cylinder, which maintains a fixed orientation throughout the motion. Outside the Stokes shear-wave layer Reynolds stresses contribute to the induced steady streaming. The outer flow is calculated in the case of large streaming Reynolds numbers for two particular cylinders.

---

Zusammenfassung Es wird die stationäre Strömung untersucht, die durch die kreisförmige Bewegung eines Zylinders mit elliptischem Querschnitt in einer zähen Flüssigkeit induziert wird. Die Amplitude dieser Kreisbewegung wird als klein angenommen gegenüber den Halbachsen der Ellipse, deren Orientierung während der Bewegung unverändert bleibt. Ausserhalb der Stokes-Schicht tragen die Reynolds-Spannungen zur Induzierten stationären Strömung bei. Die äussere Strömung wird im Falle von grossen Reynolds-Zahlen der Strömung für zwei besondere Zylinder berechnet.

     

Anisotropic transient wave motions in a conducting fluid under uniform magnetic and current fields

An initial value investigation into the development of two-dimensional anisotropic surface waves generated by a harmonically oscillating pressure distribution acting on the undisturbed free surface of an inviscid, incompressible homogeneous and electrically conducting fluid is made in this paper in considerable detail. The problem is solved by the use of generalized function treatment in conjunction with asymptotic methods. An asymptotic solution of the problem related to some physically realistic pressure distributions is presented. It is shown that an ultimate steady state is set up in the limit. Two limiting cases such as (i) very deep fluid and (ii) very shallow fluid, which are of particular interest have been examined with some emphasis. Finally, the effects of the imposed magnetic and current fields as well as the surface tension on the wave motions has been examined in some detail. Additionally, it is shown that the present method of solution provides an interesting example of the applicability of the generalized function method in problems of magnetohydrodynamics     

Unsteady Couette flow of viscous fluid under a non-uniform magnetic field

The aim of this letter is to construct the analytic solution for unsteady Couette flow in the presence of an arbitrary non-uniform applied magnetic field. The flow is induced by a generalized velocity given to the lower plate. The perturbed eigenfunction expansion method is employed to develop a series solution for small magnetic field.     

Asymptotics of homogeneous oscillations in a compressible viscous fluid

The large amplitude high frequency oscillations of the flow of a compressible viscous fluid have been shown to obey to an integro-differential system. We consider here the case of homogeneous oscillations, with both gas-like pressure law and a van der Waals one. We show that the solution admits a limit as the time increases. This limit is constant in the former case, but may take up to three distinct values in the latter.     

Nonlinear oscillations of an electrically conducting layer in a magnetic field with periodic electromagnetic excitation

We study the mechanical, thermal, and electromagnetic effects accompanying forced oscillations of a non-ferromagnetic electrically conducting layer in a transverse magnetic field.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 26–30.     

Flow and heat transfer for a power-law electrically conducting fluid flowing between parallel plates under transverse magnetic field with viscous dissipation

The flow and heat transfer problem with viscous dissipation for electrically conducting non-Newtonian fluids with power-law model in the thermal entrance region of two parallel plates with magnetic field under constant heat flux and constant wall temperature conditions has been studied. The governing equations have been solved numerically using quasilinearization technique and implicit finite-difference scheme. It has been found that the effect of viscous dissipation on heat transfer is quite significant for heating and cooling conditions at the wall.     

Magnetic field in a turbulent conducting fluid

The problem of magnetic field in conducting turbulent, incompressible fluid is considered. The velocity of the fluid is taken to be independent of the magnetic field and is described by a Gaussian field, ‘white noise’ in time with smooth space correlation. The main result is that no fast dynamo (by which is meant almost sure exponential growth of magnetic field) can exist for an incompressible fluid when the magnetic viscosity is positive. For d = 2, sharper results are obtained; the magnetic field dies out when the magnetic viscosity is strictly positive. Furthermore, when d = 2, existence and characterization of invariant measure are given for d = 2 when the magnetic viscosity is zero. The results are compared to those discussed by Baxendale and Rosovskii in [2]     

Weak fluctuation modes of an electrically and thermally conducting viscous compressible flow in a homogeneous magnetic field

Zusammenfassung Eine Perturbationstheorie erster Ordnung zeigt, daß die Gleichungen, die die Bewegung eines elektrisch und thermisch leitenden zähen kompressiblen Strömungsfeldes bestimmen, welches endliche elektrische Leitfähigkeit hat, vier verschiedene Typen von Störungsfeldern enthalten kann; diese werden von vier unabhängigen partiellen Differentialgleichungen bestimmt. Die drei ersten Typen sind: Wirbel-Stromdichte- und magnetisch-akustische Wellen-Typen. Sie sind hyperbolischer Art und pflanzen sich nicht isotrop als Wellen fort. Der vierte Typ ist ein Entropie-Typ, der sich nicht ausbreitet und der von einer parabolischen Gleichung bestimmt wird. Jeder dieser verschiedenen Typen kann nicht weiter zerlegt werden wie im Falle inkompressibler Strömung. Zusätzlich wird gezeigt, daß die magnetisch-akustischen Wellen im Falle endlicher elektrischer Leitfähigkeit neue und interessante Eigenschaften besitzen.     

Radial flow of a viscous,incompressible, electrically conducting fluid between two stationary,porous discs in the presence of transverse magnetic field

The problem of incompressible, electrically conducting viscous laminar flow between two non-conducting coaxial discs is studied, in the presence of transverse magnetic field of uniform intensity, when there is constant injection or suction at the discs. The problem is interesting both mathematically and physically. The solution, for small suction or injection, has been obtained for highly viscous fluid. The velocity profiles have also been drawn graphically.     

Secondary flow about a magnetized sphere rotating in viscous conducting fluid

The problem of secondary motion induced by the steady rotation of a magnetized sphere in an infinite incompressible viscous conducting fluid is considered. It is found that the secondary flow adds nothing to the couple required tomaintain the motion and the effect of the magnetic field is to damp the secondary velocity field.     

Rayleigh-Taylor instability of compressible finitely conducting rotating fluid in the presence of a magnetic field

The character of the equilibrium of a non-viscous, compressible finitely conducting rotating fluid in the presence of a vertical magnetic field along the direction of gravitational field has been investigated. It is shown that the solution is characterised by a variational principle. Based on the existence of variational principle, an approximate solution has been derived for the case of a fluid having exponentially varying density in the vertical direction. Due to finite resistivity of the medium it is found that potentially stable or unstable configuration retains its character. Further the growth rate of disturbance has been obtained corresponding to short and long wavelengths and it is found that electrical resistivity suppresses the growth rate for large wavelengths but it increases the same for small wavelengths. It is further shown that magnetic field has a destabilizing influence for large wavelengths and a stabilizing influence for small wavelengths.     

Conservation theorems for magnetic field and vorticity in two-dimensional conducting fluid flows

The classical conservation theorems for magnetic force lines, magnetic flux through a fluid surface, and intensity of magnetic vector tubes are generalized to plane flows of a finitely conducting fluid in an orthogonal magnetic field. The Helmholtz and Kelvin vorticity conservation theorems are generalized for plane motion of a viscous conducting fluid in an orthogonal magnetic field and the Bernoulli integral is derived. The Bernoulli integral is also generalized for plane motion of viscous ideally conducting fluid in a longitudinal magnetic field. Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 46–49, 1994.     

Localization of the spectrum of oscillations of a viscous capillary fluid

Translated from Matematicheskie Zametki, Vol. 47, No. 5, pp. 116–126, May, 1990.     

Exact solutions for equilibrium configurations of the surface of a conducting fluid in a nonuniform magnetic field

We study the two-dimensional magnetic shaping problem for the situation where the free surface of a perfectly conducting fluid is deformed by the magnetic field of a system of linear current-carrying conductors. Equilibrium is achieved due to the balance of capillary and magnetic pressures. We obtain exact solutions of the problem using conformal map techniques. These solutions describe a system of two-dimensional dimples that appear on the initially flat surface of a liquid conductor under the action of a nonuniform magnetic field. We consider the case of two symmetrically located dimples in detail.     

Asymptotic model for free surface flow of an electrically conducting fluid in a high-frequency magnetic field

We study the behaviour of a layer of an electrically conducting inviscid incompressible fluid in a high-frequency alternating magnetic field. We derive nonlinear asymptotic equations governing the evolution of the fluid layer in the high-frequency limit. As a test for the model, we consider the linearised stability problem for an infinite planar free surface of a layer of finite depth.     

Damping of the free oscillations of a neutral buoyancy sphere in a viscous stratified fluid

A technique is developed for calculating the oscillations of balanced spheres at neutral buoyancy levels based on the linearization of the equations of the mechanics of a viscous, continuously stratified fluid. A self-consistent system of integro-differential equations is obtained and analysed using perturbation theory methods. The results of calculations of the displacements of the centres of the spheres are reduced to a form which a permits direct comparison with a laboratory experiment and they agree with the data of measurements. A comparison is made with calculations of the free oscillations of a sphere in an ideal fluid.