Hydrostatics in weak force fields. On annular figures of equilibrium of a rotating liquid and their stability

We consider the problem of annular equilibrium figures of a rotating weightless liquid, having surface tension, and their stability. This question has been studied by Charreaux (for a discussion of his results see [1]), who examined the evolution of the forms of annular equilibrium figures and showed that there exists a family of stable equilibrium shapes. However, these studies of Charreaux are incomplete, and the conclusion on the existence of stable forms is valid only for axisymmetric disturbances.In the following we examine the properties of annular equilibrium figures of a rotating liquid. From the results of numerical integration on a digital computer we construct a family of equilibrium forms and present data which permit finding the corresponding equilibrium form from the ensemble of physical parameters which define the equilibrium state. In studying the stability we use the technique of [2, 3].The results of the numerical calculation and the asymptotic representat ons show that stable annular equilibrium figures of a rotating liquid do not exist.The author wishes to thank M. A. Belyaev for compiling the program for the numerical calculation, and also N. D. Kopachevskii, A. D. Myshkis, and A. D. Tyuptsov for discussions of the results and helpful remarks.     

Existence of axisymmetric equilibrium figures

Archive for Rational Mechanics and Analysis -     

Convective instability of a rotating fluid

A physical system may be in thermodynamic equilibrium when participating as a whole in uniform rotational motion [1]. In particular, mechanical equilibrium of a liquid in a cavity rotating about a stationary axis with the constant angular velocity (solid-body rotation of the liquid) is possible. If the liquid is uniform in composition and isothermal, then such equilibrium, as shown in [2], is stable for all . However, in the case of a nonuniformly heated liquid, stability of the solid-state rotation is, generally speaking, impossible.The appearance of two steady-state force fields is associated with uniform rotation: the centrifugal field and the Coriolis force field. The former field forces the liquid elements which are less heated and therefore more dense to move away from the axis of rotation, displacing the less dense liquid layers (centrifugation). If we maintain in the liquid a temperature gradient which prevents the establishment of equilibrium stratification of the liquid, then with a suitable value of this gradient (the magnitude obviously depending on ) undamped flows—convection—will develop in the liquid. Thus, while in conventional gravitational convection the gravity field is the reason for the appearance of the Archimedes buoyant forces, in the rotating cavity the mixing of the nonuniformly heated liquid is caused by the centrifugal field. As soon as the convective flows arise the Coriolis forces come into play. Account for the latter, as is shown below, prevents reducing in a trivial fashion the study of convective stability of rotating liquid to the well-studied problems of gravitational convection.     

Fluctuating flow in a rotating fluid

Summary Fluctuating flow of a viscous fluid rotating over a disk whose angular velocity oscillates about a nonzero mean is investigated. Initially the disk and the fluid rotate in the same sense with different angular velocities ₁ and ₂ ( ₂> ₁) and at a particular instant of time, the angular velocity of the disk becomes ₁[1+ sin( )]. The problem is solved as an initial boundary value problem and it is found that for small values of the results of analytical and numerical methods are in excellent agreement. The effect of frequency parameter on surface skin frictions has been analysed for various values of angular velocity ratio s and amplitude parameter .

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Fluktuierende Strömung in einer rotierenden Flüssigkeit

Übersicht Untersucht wird die fluktuierende Strömung einer viskosen Flüssigkeit, die über einer Scheibe, deren Winkelgeschwindigkeit um einen von Null verschiedenen Mittelwert schwankt, rotiert. Anfangs drehen sich die Scheibe und die Flüssigkeit gleichsinnig, aber mit verschiedenen Winkelgeschwindigkeiten ₁ und ₂ ( ₂> ₁). Zu einem Anfangszeitpunkt geht die Winkelgeschwindigkeit der Scheibe über in ₁[1+ sin ( )]. Die Aufgabe wird als Anfangs-/Randwertproblem gelöst. Für kleine Werte stimmen die analytischen und numerischen Ergebnisse hervorragend überein. Für verschiedene Werte des Winkelgeschwindigkeitsverhältnisses und des Amplitudenparameters wurde der Einfluß des Frequenzparameters auf die Reibspannungen an der Scheibe untersucht.

     

Self-gravitating stability of a rotating fluid layer

The self-gravitating instability of the present model is discussed by using a simple linear theory. The problem is formulated for a rotating fluid layer, and a dispersion relation valid for all kinds of perturbations is derived and discussed. The self-gravitating force is found to be a destabilizing factor for a small range of wavenumbers, while it is stabilizing in other ranges, depending on the density ratio of the fluids. For high values of the angular velocity, the rotational force produces a stabilizing effect and can suppress the self-gravitating instability. In the absence of the self-gravitating force, the model of a rotating fluid layer is marginally stable.     

Motion of a fluid between rotating cones

A study is made of the steady axisymmetric flow of a viscous fluid between two cones rotating in opposite ways round a common axis. It is shown that as in the case of the flow of fluid swirled by plane disks rotating at different speeds [1], there can be two regimes of motion in the system: a Batchelor regime with quasirigid rotation of the fluid outside the boundary layers [2] and a Stewartson regime in which the azimuthal flow is concentrated only in the boundary layers [3]. In the Stewartson regime, a boundary layer analogous to that in the single disk problem (see, for example, [4–6]) forms in the region of each cone far from the apex. For the flows outside the boundary layers, simple expressions are found which make it possible to obtain a conception of the circulation of the fluid as a whole. With minor alterations, the results can be applied to the case of the rotation of other curved surfaces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–64, March–April, 1985.The author thanks A. M. Obukhov for suggesting the subject and for his interest in the work, and A. V. Danilov and S. V. Nesterov for useful discussions.     

Drift of spheres in a rotating fluid

The drift of spheres in a rotating fluid is investigated. The problem is studied experimentally and numerically using the Galerkin method. It is shown that for small angular velocities of the fluid Ω the drift velocity of the spheres is almost independent of Ω, but once a certain threshold value Ω_* is attained the drift velocity rapidly decreases. The experimental dependence of the translational velocity of the sphere on the fluid angular velocity is explained on the basis of a theoretical analysis.     

Barotropic elliptical dipoles in a rotating fluid

Barotropic f-plane dipolar vortices were generated in a rotating fluid and a comparison was made with the so-called supersmooth f-plane solution which—in contrast to the classical Lamb–Chaplygin solution—is marked by an elliptical separatrix and a doubly continuously differentiable vorticity field. Dye-visualization and high-resolution particle-tracking techniques revealed that the observed dipole characteristics (separatrix aspect ratio, cross-sectional vorticity distribution and vorticity versus streamfunction relationship) are in close agreement with those of the supersmooth f-plane solution for the entire lifespan of the dipolar vortex.     

Diffusion stratification in a rotating linearly-stratified fluid

The velocity characteristics of the circulation developing in a homogeneously-rotating linearly-density-stratified fluid are investigated experimentally. It is shown that various wave and vortex regimes of motion in which the fluid velocities can reach 1 cm/s are possible. A steady-state rigid-body rotation regime exists for certain parameters of the system. Moscow, Cambridge (Great Britain). Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–67, March–April, 1998. The work was carried out in the Department of Applied Mathematics and Theoretical Physics of Cambridge University (Great Britain) and in the Obukhov Atmospheric Physics Institute of the RAS.     

Point force in a MHB rotating fluid

We consider the steady flow of an inviscid, rotating fluid confined in a cylinder under the influence of a uniform axial magnetic field, in the presence of an isolated point force. It is found that, the wave-like terms, infinite in number, occurring on the downstream side are not persistent and decay at far distances, in contrast to the non-magnetic case. Also the presence of a uniform magnetic field induces an additional finite number of decaying modes on the upstream side and an infinite number of similar modes on the downstream side, which reduce to the geostrophic terms, in the corresponding non-magnetic case. The downstream solution reveals that discontinuities occur in the velocity gradients inside the fluid region.     

Flow of a Viscoplastic Fluid on a Rotating Disk

ＦＬＯＷＯＦＡＶＩＳＣＯＰＬＡＳＴＩＣＦＬＵＩＤＯＮＡＲＯＴＡＴＩＮＧＤＩＳＫＦａｎＣｈｕｎ（范椿）（ＩｎｓｔｉｕｉｅｏｆＭｅｃｈａｎｉｃｓ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ）（ＲｅｃｅｉｖｅｄＮｏｖ．２０，１９９２；Ｃｏｍｍｕｎｉｃａｔ...     

Flow of a Casson fluid on a rotating disk

The flow of a thin layer of a Casson fluid on a fast rotating disk is considered. The film thickness distribution at various times for various initial thickness distribution is calculated. The stability of the flow is examined.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Evolution of solitary vortices in a rotating fluid

The evolution of cyclones and anticyclones with a characteristic scale substantially exceeding the radius of deformation is investigated numerically. An equation obtained by an asymptotic method for small values of the Kibel'-Rossby number is employed. The model ensures conservation of the potential vorticity in the fluid particles for motions of finite amplitude (when the thickness of the layer H deviates considerably from the undisturbed value H₀). It is shown that anticyclones of a certain type adapt themselves to a steady shape and travel in a direction opposite to the direction of rotation of the sphere (westwards). It is found that the existence of a steadily migrating anticyclone requires the presence of a region of closed isolines of the potential vorticity (trapping region), in which the fluid is transported together with the anticyclone. Cyclones drift westwards more slowly than anticyclones and migrate towards the poles, losing energy by emitting Rossby waves. It is found that in cyclones the trapping region retains an almost circular shape, the decay of the eddy, associated with wave emission, showing as its initial intensity increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 52–59, July–August, 1986.     

Stability of the equilibrium of rotating drillstrings

The quasistatic stability of a rotating drillstring under longitudinal force and torque is analyzed. Constitutive equations are derived, and a technique to solve them is proposed. It is shown that the buckling mode of the drillstring is helical within a section subjected to compressive forces __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 101–109, June 2006.     

On permanent nonlinear waves in a rotating fluid

The governing equation for long nonlinear gravity waves in a rotating fluid changes with the value of the Coriolis parameter f. (1) When f is large, i.e. in the strong rotation case, in an infinite ocean, there are only Sverdrup waves; in a semi-infinite ocean or in a channel, there are either solitary Kelvin waves, for which the governing equation is a KdV equation, or Poincaré waves, which can be obtained by superposition of two Sverdrup waves. (2) When f is small, i.e. in the weak rotation case, in an infinite ocean there are solitary or cnoidal waves governed by the Ostrovskiy equation, and we provide an explicit solution for both solitary and cnoidal Ostrovskiy progressive waves; and in a semi-infinite ocean or a channel, there are Sverdrup waves, which are governed either by Ostrovskiy equations or by the Grimshaw-Melville equation. (3) When f is very small, i.e. in the very weak rotation case, in an infinite ocean, or in a channel, there are solitary waves with a horizontal crest, but with a velocity component or a pressure gradient, which are governed by KdV equations as in the non-rotating case. Physically, that means that the most determining factor is the ratio of the Rossby radius of deformation over a characteristic length of the wave.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

Coherent vortices in a turbulent and rotating fluid

The emergence of coherent, intense and quasi-steady vortices in a rotating fluid under the action of turbulence is a phenomenon which is not yet well accounted for. A first deterministic approach by Maxworthy, Hopfinger and Redekopp (1985) related it to the occurence of mixed density fluid intrusions in a linearly stratified fluid. In the first part we present visualizations of the vortex genesis verifying qualitatively Maxworthy et al.'s ideas. In the second part, a linear instability model is proposed, very similar to the thermal convection instability in a rotating fluid. For marginal stability conditions, the model shows the occurrence of a regular spatial distribution of steady vortices. This new approach does not contradict the visualizations presented in the first part.