Small Angular Velocity Research Articles

An effect of sudden circumferential strain on the laminar boundary layer on a rotating cylinder in an axial flow

The laminar boundary layer which develops on a rotating thin cylinder fitted with an aft-section rotating with an angular velocity different from that of a fore-section is examined numerically. This problem concerns a relaxation process of the boundary layer subjected to a sudden circumferential rate of strain. Two methods are adopted: one is to regard the flow on the aft-cylinder as perturbations of that on the fore-section, and the other is to approximate the discontinuous change in the angular velocity by the cumulative normal distribution function. It is shown that the flow fields are largely influenced by the degree of favorable pressure gradient produced in the boundary layer. Especially, in case of flow passing onto the aft-section with a smaller angular velocity, adverse pressure gradient is induced immediately after the junction between two cylinders, and when the degree of the discontinuity is increased, flow separation can be provoken.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

Kelvin–Helmholtz problem in a rotating Hall plasma

The Kelvin–Helmholtz problem in a Hall plasma, including the effects of rotation, is studied. In contrast to previous results, it is shown that if the angular velocity is normal to the wave vector that describes perturbations of the interface, the rotation does not affect the stability of a shear plane. The special case of very small angular velocity is studied and it is shown that the rotation has either a stabilizing or a destabilizing effect, according as the gravitational speed is greater or less than the Alfvén speed.

Scattering by a rotating dielectric sphere

The scattering of a plane wave by a rotating dielectric sphere of arbitrary radius, \epsilon_{r} and \mu_{r} is investigated. A relativistically correct solution is found for the scattered fields and the fields inside the sphere. This solution is valid in the limit of small angular velocities. Attention is focused on the fields at large distances. The depolarization of the backscattered signal and the scattering in the low frequency limit are treated in some detail.

Slowly rotating configurations of an incompressible fluid in the bimetric theory of gravitation

In [I, 2], the most general solution to Einstein's equations was studied in the case of small angular velocities (the ~ approximation), which is equivalent to the gravtational energy of the star being many times greater than the rotational energy: B = ~2/8~GPc << i (Pc is the density at the center of the configuration). In this approximation, the diagonal components of the metric tensor outside the mass distribution keep their Schwarzschild form, and there is also a nondiagonal component g03 = (iJ/r) sin 2 8, where J is the total angular momentum. This approximation corresponds to considering rotation of a sphere with allowance for Coriolis forces without change in its shape. The solution depends on the two mass and angular momentum parameters and can obviously be obtained from the Kerr solution [3] by expanding the latter in a series in the angular momentum and retaining the terms linear in J.

New special standard to ensure unity of measurements of small angular velocities

New special standard to ensure unity of measurements of small angular velocities

Methods and means of reproducing small angular velocities

Methods and means of reproducing small angular velocities

Dynamical Friction Effects on the Motion of Stars in Rotating Spherical Clusters

Effects of dynamical friction on star orbits in a spherical cluster uniformly rotating with small angular velocity about a fixed axis are considered, deformations of the cluster due to the rotation being neglected. The test star is supposed to move in a noncircular restricted orbit under the influence of both the attraction of the cluster with the smoothed-out distribution of stellar matter and dynamical friction due to random encounters of the test star with other stars of the cluster.The approximate formula for dynamical friction has been deduced, the encounters being supposed to be the binary ones. The differential equations for the osculating elements of the star orbit have been obtained for the two cases of the density distribution - the uniform and the exponential ones. The numerical results demonstrate the complicated character of dynamical friction effects on the evolution of the orbit. The orbit tends to become circular, and its inclination decreases. These effects are proportional to the mass of the test star. This leads to the conclusion that dynamical friction contributes noticeably to the concentration of massive stars near the center of the cluster.

Stationary-platform Maneuvers Of Gyrostat Satellites

In this paper we present some novel results regarding the dynamics of gyrostats containing N axisymmetric rotors subject to control torques applied by the platform. The particular class of maneuvers we are interested in are termed maneuvers, because they are based on the notion of keeping the platform almost stationary throughout the maneuver. The basic idea behind this class of maneuvers is to control the rotors in such a way that the maneuver remains near a branch of equilibrium motions for which the platform angular velocity, w, is zero. Of course, u> is not actually zero, but it does remain small provided that the internal torques are small. We develop the equations of motion, and then discuss the stationary-platform conditions, which lead to the development of stationaryplatform control laws for the control torques. We give a simple proof that all stationary-platform equilibria are nonlinearly stable in the absence of energy dissipation. Numerical results are given which include the dissipative effects of a viscously damped rotor. The results confirm the effectiveness of the stationary-platform maneuver for a large-angle rotation with small angular velocity throughout the maneuver.

Synchronous Analysis for Diagnostics of Rolling Bearings in the Turbojet Engine

Designed in the Aviation Institute, the K-15 turbojet engine has got rolling bearings, which answers with frequency5.87×ωrthe unbalance excitation. The signal with such frequency indicates a fault of the outer race of the rolling bearing. A set of the digital synchronous filters was used for the K-15 vibration spectrum analysis. A procedure of filtration was performed by the computer software. The sychronous summation of the measured signals was carried out before the spectrum analysis. Two cases were considered: the engine with a small force due to unbalance (a small angular velocity of the rotor), and the engine with a big unbalance force (high angular velocity). In the first case, the outer race frequency was not observed, despite the existence of the vibration amplitude (caused by unknown disturbances) with such frequency before the synchronous summation. In the second case, the outer race frequency after synchronous summation has enlarged amplitude while other spectrum components in its vicinity have been damped. It underlines the usefulness of the synchronous analysis in the vibration diagnostics of the rotating machinery.

System for measuring small angular velocities

System for measuring small angular velocities

Second-class motions of a shallow liquid

When a basin containing a shallow liquid is rotating with angular velocity Ω and the dimensionless number is small (L and M are typical horizontal and vertical dimensions), then, to a first approximation, the second-class motions behave as if the free surface of the liquid were fixed in its equilibrium position. The lower second-class modes of such a liquid, contained in a paraboloid, are relatively easy to describe on the basis of this approximation. When the liquid is rotating within an elliptical paraboloid and the sense of rotation is opposite to that of the container itself, the motion is unstable for a range of small angular velocities. Such unstable motions always exert a couple tending to oppose the rotation of the container.

The effect of viscosity near the cylindrical boundaries of a rotating fluid with a horizontal temperature gradient

Under certain conditions, the motion caused in an annulus of fluid by rotating it about its (vertical) axis of symmetry and at the same time subjecting it to a radial temperature gradient has been shown by Hide (1958) to be mostly concentrated in a narrow jet stream which meanders between the inner and outer cylindrical boundaries of the fluid in a regular wave pattern: this wave pattern has a small angular velocity relative to the cylindrical walls containing the fluid. A theoretical solution has been found by Davies (1959) which is valid in the main body of the fluid: this solution neglects viscosity (which is permissible except near the boundaries of the fluid), and is related to the absolute angular velocity of the wave pattern. The present paper introduces viscous boundary layers between the main body of the fluid and the cylindrical walls, in an attempt to find a relation between the angular velocity of the wave pattern and that of the walls. That this is only partially successful is due to the presence of the boundary layer at the rigid surface at the bottom of the fluid (which is rotating with the same angular velocity as the cylindrical walls): this layer is ignored in the present theory. In addition to this contribution towards a complete explanation of the steady motion, the theory describes qualitatively certain periodic oscillations (vacillation) which were observed by Hide in his experiments.

Time-dependent Hartree-Fock equations and rotational states of nuclei

The coneection between translational and rotational degeneracies in the solutions of the time independent Hartree-Fock equations and the existence of solutions of the time-dependent equations which represent uniform translational or rotational motions is investigated. With a non-spherical equilibrium shape, the equations give rotating solutions in which the self-consistent rotating potentials are determined by the motion of the particles. With no axial symmetry, and neglecting higher order effects in the angular velocities, there are time-dependent solutions corresponding to the general motion of a classical top. For an equilibrium shape with axial symmetry, the only rotational motions are uniform rotations about an axis perpendicular to the symmetry axis. The equations are studied in the density matrix formulation in which the results can be immediately extended to include the effect of pairing correlations. Solutions representing a uniform translational motion can be explicitly constructed. For the rotational motion, the changes in the density matrix are determined for small angular velocities by a linear equation. This contains a correction to the cranking model equation which expresses the effect of a transformation to a moving frame of reference on the velocity-dependent self-consistent potentials. The connection between vibrational and rotational solutions of the time-dependent Hartree-Fock equations is discussed. The rotational models are related to zero-frequency solutions of the equations for small oscillations, and in the linear space of small variations the infinitesimal rotational and translational changes define an independent subspace. This leads to a separation of the rotational and translational energies in the Hamiltonian of the random phase approximation.

Climatic change through geological time in the light of the palaeomagnetic evidence for polar wandering and continental drift

AbstractThe key to the understanding of the gross changes of climate throughout geological time indicated by the geological record in the various continents depends on finding a physical method of determining latitudes in the geological past. The palaeomagnetic method seems to be the most hopeful, but for this one must assume that the average geomagnetic field is a dipole along the axis of rotation. The palaeomagnetic evidence shows this to be true for late Tertiary times – the poles derived from the mean palaeomagnetic directions of Tertiary rocks cluster about the present geographical pole and are distinctly different from the present position of the magnetic pole. There are strong theoretical grounds for believing that it has been true throughout the earth's history. The theory of the irregular fluctuations in the length of the day indicates that the rotation of the core and mantle are only weakly coupled. The electro‐magnetic torque to which this coupling is ascribed is variable because of the short term changes in the geomagnetic field. These fluctuations appear to accumulate with time and so there is always a small differential angular velocity between core and mantle, probably of either sign. Therefore, to an observer fixed on the earth's crust, the earth's magnetic field is axisymmetrical when averaged over periods of the order of the secular variation, i.e., extremely small times geologically speaking.The palaeomagnetic surveys of the various continents show systematic differences which, it appears, cannot be explained even by assuming that the earth's magnetic field was non‐dipolar, though axial, in times earlier than the Tertiary. Continental drift, as well as polar wandering, must therefore be inferred.There is a prima facie case that the latitudes thus inferred for the various continents from palaeomagnetic observations do agree with the commonly held inferences concerning the past climates of the earth made by geologists from the study of evaporite deposits, red beds, zonation of fossils, and glacial deposits. A new important check has been provided by palaeowind observations. The inference from the palaeomagnetic studies of Europe and North America that these continents were in low latitude between the Devonian and Triassic is consistent with the wind directions inferred from dune sandstones in Britain and the western United States. The consistency of the directions in the western United States over large distances seems to show that the winds were not affected by local geography.It appears that an explanation of Continental drift involves the hypothesis of convection currents in the earth's mantle which requires that the mantle is fluid when subjected to small stresses existing for many millions of years.

On the rotation and the figure of celestial bodies

AbstractDetermining the form of celestial bodies one usually assumes according to celestial mechanics, that they rotate as solid bodies, i. e. with a constant angular velocity over the whole body. The equations of rotation of a free fluid subjected to the influence of the forces of its own gravitation and of pressure considered as a solid body are not the general solution of EULER's hydrodynamical equations for the case of rotation. As POINCARÉ and JEANS have shown there exists an infinite number of other solutions describing the movement of a rotating fluid with angular velocity depending on the distance from the axis of rotation. We examine in the present paper these more general solutions generalizing the well known POINCARÉ theorem of the maximum possible angular velocity. Within the framework of these generalizations we examine ROCHE's model which treats the whole mass of the fluid concentrated in the center of gravity. Some of our results are shown to be applicable for the determination of the form and the gravitational field of non‐uniformly rotating celestial bodies, for instance the Sun, Jupiter and Saturn.On the basis of LIAPOUNOFF's classical works (1903 and 1904) we prove the existence of unique solutions of the generalized problem of a non‐uniformly rotating fluid at small angular velocities. We arrive at a system of integrodifferential equations which generalize the CLAIRAUT‐LIAPOUNOFF equations for the case of a non‐uniformly rotating fluid. By means of these exact solutions we examine a model which enables us, assuming some restrictive conditions, to refute the objection put forward against KANT‐LAPLACE's theory concerning the distribution of moment of momentum in the Solar system.

Pinch with Rotating Plasma

The magnetohydrodynamic stability of an infinitely conducting incompressible plasma, in the shape of an infinitely long circular cylinder within an external conductor, compressed by external Bθ and Bz against internal Bz, is investigated in the circumstance that the plasma has an angular velocity about the z axis of the cylinder. Two different cases are examined in detail: (i) arbitrarily large constant angular velocity, in which case the differential equation for the perturbed total (magnetic plus hydrodynamic) pressure inside the plasma is solved exactly in terms of Bessel functions; (ii) nonconstant but small angular velocity, in which event perturbation theory can be employed. For both these cases the rotating plasma is less stable than the nonrotating plasma.

Thermal Conduction in Rotating Liquid Helium II

When a heat current flows in a wide channel filled with liquid helium II, the resulting temperature gradient is approximately proportional to the cube of the heat current density. The establishment of this gradient requires a time $\ensuremath{\tau}$ which is a function of heat current, temperature, and past history of the helium. The present experiments concern the effect of uniform slow rotation, about an axis normal to the direction of heat flow, upon $\mathrm{grad}T$ and $\ensuremath{\tau}$. $\mathrm{Grad}T$ was measured at 0 and 1.3 rad/sec, the highest angular velocity at which this measurement could conveniently be carried out. No effect of rotation could be observed; however, an approximate calculation suggests that $\mathrm{grad}T$ might increase detectably at somewhat higher rates of rotation. $\ensuremath{\tau}$ was measured at a number of angular velocities between 0 and 4 rad/sec; it was found that $\ensuremath{\tau}$ was appreciably reduced by rotation, the effect being greatest at small heat currents and high angular velocities. These results can be explained on the assumption that mutual friction results from turbulence in the superfluid component, taking the form of a tangled mass of vortex line. The delay time $\ensuremath{\tau}$ characterizes the rate of growth of this turbulence when a heat current is switched on; rotation reduces $\ensuremath{\tau}$ by introducing an initial length of vortex line which accelerates this growth.

On almost rigid rotations

In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

A stator-fed half-speed synchronous motor

It is well known that an induction motor with an asymmetrically connected secondary circuit will run at a fractional slip rather greater than 0.5 but that the asynchronous performance is not wholly satisfactory. The defects arise chiefly, though not entirely, from a rotor field component which rotates with a small angular velocity (1–2s)ω, and is responsible for reflecting undesirable low-frequency currents into the primary supply system. These disadvantages are avoided by superimposing a stationary magnetic field which fixes the (1–2s)ω field in space and thus compels the rotor to run at s=0.5, i.e. at exactly half speed. Further, the power factor can be controlled by altering the d.c. excitation in the usual way, and good efficiencies can be achieved. Test results are quoted for three different types of rotor and compared with a first-order theory which provides suggestions for improved rotor design and further developments.