The flow field induced by an oscillating fluid drop immersed in another fluid

The flow field induced by a translatory oscillating spherical drop immersed in another fluid is considered. It is assumed that the amplitude of the oscillation is small compared with the radius of the drop. We are concerned, for the most part, with the case of a small frequency parameter M. Of particular interest is the steady streaming induced both inside and outside of the drop. The problem has been solved on the basis of the Navier-Stokes equations by the method of matched asymptotic expansions.     

The torsional oscillations of a rigid spherical inclusion embedded in an elastic stratum

In this paper we investigate the axisymmetric harmonic torsional vibrations of a rigid spherical inclusion, embedded centrally in an elastic stratum of finite thickness. For an infinite solid the problem has a simple closed form solution. Here the stratum problem is formulated as a Fredholm integral equation of the first kind, and a perturbation method employed to solve this equation approximately for large values of the ratio of sphere radius to stratum depth. Approximations are given for quantities of physical interest such as the stress distribution at the inclusion-elastic medium interface, and the stress couple on the sphere.

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Résumé Dans cette étude on recherche les vibrations axisymmétriques harmoniques et torsionales d'une inclusion sphérique et raide, située centralement dans une couche d'une épaisseur bornée. Pour un solide infini le problème a une solution élémentaire.On formule le problème de couche comme une équation intégrale de Fredholm de la première sorte. On se sert d'une méthode de perturbation pour résoudre cette équation d'une façon approximative pour les grandes valeurs de la proportion du rayon de la sphère contre la profondeur de la couche. Les approximations sont calculées pour la distribution d'effort employée sur l'inclusion et pour la couple de torsion employée sur la sphère.

     

Longitudinal and torsional oscillations of a rod in a viscoelastic fluid

The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed. Received: 14 March 1997 Accepted: 14 May 1998     

Tangential oscillations of a differentially rotating spherical fluid layer

The natural harmonic oscillations of a differentially rotating fluid layer under the action of a potential force are considered. The rise in the layer level is assumed to be negligible. The oscillations satisfy an ordinary second-order differential equation with singular coefficients that depend on the spatial coordinate. This equation is solved by the method of local separation of singularities based on the use of the properties of the Fuchs series for a bounded solution. Various laws for the latitude dependence of the angular rate of ocean rotation and the effect of these laws on the problem spectrum are considered. An equation is obtained for the streamlines of the oscillations investigated. Two cases in which the latitude dependence of the base flow velocity coincides with the real dependence for a celestial body are considered and the corresponding modes are found.     

Non-linear oscillations of a spherical particle in a rotating fluid

The motion of a spherical particle contained within a rotating fluid is analysed using equations of classical mechanics. Conditions are established under which the second order differential system simplifies to the well known Svedberg equation which describes the radial motion in centrifugation theory. The complete particle trajectories are obtained by solving the second order system numerically and approximations to these are derived using perturbation methods.     

Compliance effects on the torsional flow of a viscoelastic fluid

The effects of transducer compliance on transient stress measurements in torsional flows of a viscoelastic fluid are investigated theoretically. The analysis is based on the torsional flow of an upper-convected Maxwell fluid between a rotating and ‘stationary’ disk, which is allowed to twist and displace axially as a result of the stresses exerted on the disk by the fluid. An approximate analytical solution to the governing equations is obtained using a standard perturbation method. Results of the analysis are used to examine how the fluid velocity is altered by the motion of the stationary disk and to gain insight on how transient stress measurements are affected by transducer compliance. The analysis shows that compliance effects increase with applied shear rate and that the effects of torsional and axial compliance are coupled in measurements of the shear stress and first normal stress difference.     

Airfoil flow instabilities induced by background flow oscillations

 The effect of background flow oscillations on transonic airfoil (NACA 0012) flow was investigated experimentally. The oscillations were generated by means of a rotating plate placed downstream of the airfoil. Owing to oscillating chocking of the flow caused by the plate, the airfoil flow periodically accelerated and decelerated. This led to strong variations in the surface pressure and the airfoil loading. The results are presented for two angles of attack, α=4° and α=8.5°, which correspond to the attached and separated steady airfoil flows, respectively. Received: 6 June 2000 / Accepted: 18 October 2001     

Axisymmetric oscillations of two spherical shells in a compressible fluid

To find the interaction between spherical shells at the frequency of their free oscillations in a fluid, we examine the problem of axisymmetric oscillations of two identical spherical shells under the assumption that the shell centers of curvature do not coincide. The solution is found for the cases of a compressible and an incompressible fluid by the series method with reduction to an infinite system of linear equations. A mathematical justification of the method used is presented.     

Natural oscillations of a rotating spherical fluid layer of variable depth

Small harmonic oscillations of the free surface of a thin fluid layer covering a rotating sphere are considered. The fluid is in the central field of sphere gravity and is exposed to the centrifugal and Coriolis forces. It is assumed that the fluid layer depth is independent of the longitude. In this formulation the problem is governed by a differential equation with singular coefficients that generalizes the Laplace tidal equation. The method of local separation of singularities is applied to integrate this equation. The solutions obtained are compared with the corresponding modes of the Laplace tidal equation, that is, the solutions of the problem for a fluid layer of constant depth.     

Non-axisymmetric oscillations of a hemispherical drop

Natural oscillations of a hemispherical drop on a solid substrate are considered. The Hocking condition is used to take into account the contact angle dynamics. The natural oscillations attenuate due to dissipation on the contact line and the degeneracy in the azimuthal number is eliminated. Forced oscillations of the drop are studied for tangential vibrations of the substrate. For the case of a fixed contact line, the amplitude of the forced oscillations grows without bound near the fundamental requency. In other cases, the amplitude is finite.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 8–20. Original Russian Text Copyright © 2004 by Lyubimov, Lyubimova, Shklyaev.     

The EHD-driven fluid flow and deformation of a liquid jet by a transverse electric field

The aim of this study is to investigate the effect of a uniform transverse electric field on the steady-state behavior of a liquid cylinder surrounded by another liquid of infinite extent. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and in the limit of small electric field and fluid inertia. A detailed analysis of the electrical and hydrodynamic stresses acting on the interface separating the two fluids is presented, and an expression is found for the interface deformation for small distortions from a circular shape. The electrical stresses acting on the interface of two leaky-dielectric liquids are compared with those acting on an interface separating a perfect dielectric or infinitely conducting core fluid cylinder from a surrounding perfect dielectric fluid. A comparison is made between the results of this study and those of a similar study for fluids with permeable interfaces and the classical results for liquid drops.     

Effect of surface electric current on the electrohydrodynamic flow inside and outside a spherical drop

The problem of electrohydrodynamic flow of a viscous, low-conducting, polarizable liquid inside and outside a spherical drop in an applied homogeneous constant electric field is analytically solved with account for the effect of both surface conduction current and surface convection current. The influence of the drop deformation on the field and the flow is neglected. The solution is obtained in the form of asymptotic expansions in a small parameter corresponding to weak surface convection electric currents.