Forces on bodies moving transversely through a rotating fluid

1975 ◽  
Vol 71 (3) ◽  
pp. 577-599 ◽  
Author(s):  
P. J. Mason
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Vertical Axis ◽  
The Body ◽  
Rotating Fluid ◽  
Size And Shape ◽  
Relative Direction ◽  
Axis Of Rotation

Measurements have been made of the net force F acting on a bluff rigid body moving with velocity U (relative to a fluid rotating about a vertical axis with uniform angular velocity Ω) in a plane perpendicular to the axis of rotation. The force F is of magnitude 2ΩρVU, where ρ is the density of the fluid and V is a volume which depends on the size and shape of the body. The relative direction of F and U is found to depend on the quantity \[ {\cal S}\equiv \frac{2\Omega L}{U}\bigg(\frac{h}{D}\bigg), \] where L and h are horizontal and vertical lengths characterizing the object and D is the depth of the fluid in which the object is placed.

A study of motions in a rotating liquid

1951 ◽  
Vol 206 (1084) ◽  
pp. 108-130 ◽  
Keyword(s):  
Angular Velocity ◽  
Equations Of Motion ◽  
Vertical Axis ◽  
The Body ◽  
Forced Oscillations ◽  
Infinite Cylinder ◽  
Two Dimensional ◽  
Small Motion ◽  
Axis Of Rotation ◽  
Rotating Liquid

Starting with the equations of motion for a perfect, incompressible fluid referred to a coordinate system which rotates about a vertical axis with uniform angular velocity R , the physical condition of ‘small motion’ is determined which permits the equations to be linearized. The small motions resulting from forced oscillations of a rotating liquid are investigated. It is shown that there are three types of flow depending on the relative magnitudes of the impressed frequency β and the angular velocity R of the fluid. Two of the regimes are studied in detail. A similarity law is developed which gives the solution of a class of problems of oscillations for β > 2 R in terms of the solutions to similar irrotational problems. An attempt is made to explain how slow, two-dimensional motion can be produced by introducing a boundary condition which is three-dimensional (as observed in experiments performed by G. I. Taylor), by considering problems from the moment at which the disturbance is created from rest relative to the rotating system, with the only initial assumption that the fluid is rotating uniformly like a solid body. For the particular cases studied the results are in agreement with Taylor’s experiments, in that the flow is found to become steady and two-dimensional if the disturbance which causes it approaches a steady state. If the disturbance is due to a body which moves along the axis of rotation of the fluid, the steady two-dimensional behaviour may be expected everywhere except in the neighbourhood of the surface of an infinite cylinder which encloses the body and whose generators are parallel to the axis of rotation. To resolve an apparent disagreement between certain theoretical results by Grace on the one hand, and experimental evidence by Taylor and the author’s conclusions, on the other, arguments are advanced that the various results may be in agreement, provided Grace’s are given a new interpretation.

scholarly journals III. Researches in physical astronomy

1831 ◽  
Vol 121 ◽  
pp. 17-66
Keyword(s):  
Angular Velocity ◽  
Major Axis ◽  
The Body ◽  
Fixed Plane ◽  
Resisting Medium ◽  
Axis Of Rotation ◽  
Geographical Latitude ◽  
The Mean ◽  
The Stability ◽  
Revolving Body

In last April I had the honour of presenting to the Society a paper containing expressions for the variations of the elliptic constants in the theory of the motions of the planets. The stability of the solar system is established by means of these expressions, if the planets move in a space absolutely devoid of any resistance*, for it results from their form that however far the ap­proximation be carried, the eccentricity, the major axis, and the tangent of the inclination of the orbit to a fixed plane, contain only periodic inequalities, each of the three other constants, namely, the longitude of the node, the longitude of the perihelion, and the longitude of the epoch, contains a term which varies with the time, and hence the line of apsides and the line of nodes revolve continually in space. The stability of the system may therefore be inferred, which would not be the case if the eccentricity, the major axis, or the tangent of the inclination of the orbit to a fixed plane contained a term varying with the time, however slowly. The problem of the precession of the equinoxes admits of a similar solution; of the six constants which determine the position of the revolving body, and the axis of instantaneous rotation at any moment, three have only periodic inequalities, while each of the other three has a term which varies with the time. From the manner in which these constants enter into the results, the equilibrium of the system may be inferred to be stable, as in the former case. Of the constants in the latter problem, the mean angular velocity of rotation may be considered analogous to the mean motion of a planet, or its major axis ; the geographical longitude, and the cosine of the geographical latitude of the pole of the axis of instantaneous rotation, to the longitude of the perihelion and the eccentricity; the longitude of the first point of Aries and the obliquity of the ecliptic, to the longitude of the node and the inclination of the orbit to a fixed plane; and the longitude of a given line in the body revolving, passing through its centre of gravity, to the longitude of the epoch. By the stability of the system I mean that the pole of the axis of rotation has always nearly the same geographical latitude, and that the angular velocity of rotation, and the obliquity of the ecliptic vary within small limits, and periodically. These questions are considered in the paper I now have the honour of submitting to the Society. It remains to investigate the effect which is produced by the action of a resisting medium; in this case the latitude of the pole of the axis of rotation, the obliquity of the ecliptic, and the angular velocity of rotation might vary considerably, although slowly, and the climates undergo a con­siderable change.

Slow motion of a paraboloid of revolution in a rotating fluid

1958 ◽  
Vol 3 (4) ◽  
pp. 404-410 ◽  
Author(s):  
L. V. K. Viswanadha Sarma
Keyword(s):  
Rigid Body ◽  
Small Amplitude ◽  
Radial Displacement ◽  
Direct Consequence ◽  
Slow Motion ◽  
Uniform Motion ◽  
Rotating Fluid ◽  
Body Rotation ◽  
Axis Of Rotation ◽  
Paraboloid Of Revolution

The slow uniform motion, after an impulsive start from relative rest, of a paraboloid of revolution along the axis of a rotating fluid is investigated by using a perturbation method. The principal purpose of the note is to illustrate the mechanism by which the fluid is not subjected to any substantial radial displacement, which is a direct consequence of the requirement that the circulation round material circuits should be constant when the perturbation velocities due to the motion of the paraboloid remain small. It appears that the mechanism is an oscillatory one in which the distance between any fluid particle and the axis of rotation oscillates sinusoidally in time with small amplitude. As time progresses, the amplitude of the oscillation decays to zero everywhere except on the paraboloid. The ultimate motion is then a rigid body rotation everywhere except on the paraboloid and the axis of rotation, where the perturbation velocities continue to oscillate indefinitely with small amplitude.

scholarly journals Angular velocity of rotation of extended bodies in general relativity

1996 ◽  
Vol 172 ◽  
pp. 309-320
Author(s):  
S.A. Klioner
Keyword(s):  
General Relativity ◽  
Angular Velocity ◽  
Rigid Body ◽  
Rotational Motion ◽  
The Body ◽  
Astronomical Observations ◽  
Newtonian Approximation ◽  
Accurate Definition ◽  
Principal Axes ◽  
Definition Of

We consider rotational motion of an arbitrarily composed and shaped, deformable weakly self-gravitating body being a member of a system of N arbitrarily composed and shaped, deformable weakly self-gravitating bodies in the post-Newtonian approximation of general relativity. Considering importance of the notion of angular velocity of the body (Earth, pulsar) for adequate modelling of modern astronomical observations, we are aimed at introducing a post-Newtonian-accurate definition of angular velocity. Not attempting to introduce a relativistic notion of rigid body (which is well known to be ill-defined even at the first post-Newtonian approximation) we consider bodies to be deformable and introduce the post-Newtonian generalizations of the Tisserand axes and the principal axes of inertia.

scholarly journals Nonlinear Magneto Convection in a Rotating Fluid due to Vertical Magnetic Field and Vertical Axis of Rotation

2021 ◽  
Vol 39 (3) ◽  
pp. 775-786
Author(s):  
Avula Benerji Babu ◽  
Gundlapally Shiva Kumar Reddy ◽  
Nilam Venkata Koteswararao
Keyword(s):  
Magnetic Field ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Normal Mode Analysis ◽  
Vertical Axis ◽  
Wave Number ◽  
Mode Analysis ◽  
Vertical Magnetic Field ◽  
Rotating Fluid ◽  
Axis Of Rotation

In the present paper, linear and weakly nonlinear analysis of magnetoconvection in a rotating fluid due to the vertical magnetic field and the vertical axis of rotation are presented. For linear stability analysis, the normal mode analysis is utilized to find the Rayleigh number which is the function of Taylor number, Magnetic Prandtl number, Thermal Prandtl number and Chandrasekhar number. Also, the correlation between the Rayleigh number and wave number is graphically analyzed. The parameter regimes for the existence of pitchfork, Takens-Bogdanov and Hopf bifurcations are reported. Small-amplitude modulation is considered to derive the Newell-Whitehead-Segel equation and using its phase-winding solution, the conditions for the occurrence of Eckhaus and zigzag secondary instabilities are obtained. The system of coupled Landau-Ginzburg equations is derived. The travelling wave and standing wave solutions for the Newell-Whitehead-Segel equation are also presented. For, standing waves and travelling waves, the stability regions are identified.

Influence of rotational inertia on turning performance of theropod dinosaurs: clues from humans with increased rotational inertia

2001 ◽  
Vol 204 (22) ◽  
pp. 3917-3926
Author(s):  
David R. Carrier ◽  
Rebecca M. Walter ◽  
David V. Lee
Keyword(s):  
Average Velocity ◽  
Human Subjects ◽  
Tail Length ◽  
Vertical Axis ◽  
The Body ◽  
Rotational Inertia ◽  
Control Value ◽  
Theropod Dinosaurs ◽  
Axis Of Rotation ◽  
Average Angle

SUMMARY The turning agility of theropod dinosaurs may have been severely limited by the large rotational inertia of their horizontal trunks and tails. Bodies with mass distributed far from the axis of rotation have much greater rotational inertia than bodies with the same mass distributed close to the axis of rotation. In this study, we increased the rotational inertia about the vertical axis of human subjects 9.2-fold, to match our estimate for theropods the size of humans, and measured the ability of the subjects to turn. To determine the effect of the increased rotational inertia on maximum turning capability, five subjects jumped vertically while attempting to rotate as far as possible about their vertical axis. This test resulted in a decrease in the average angle turned to 20 % of the control value. We also tested the ability of nine subjects to run as rapidly as possible through a tight slalom course of six 90° turns. When the subjects ran with the 9.2-fold greater rotational inertia, the average velocity through the course decreased to 77 % of the control velocity. When the subjects ran the same course but were constrained as to where they placed their feet, the average velocity through the course decreased to 65 % of the control velocity. These results are consistent with the hypothesis that rotational inertia may have limited the turning performance of theropods. They also indicate that the effect of rotational inertia on turning performance is dependent on the type of turning behavior. Characters such as retroverted pubes, reduced tail length, decreased body size, pneumatic vertebrae and the absence of teeth reduced rotational inertia in derived theropods and probably, therefore, improved their turning agility. To reduce rotational inertia, theropods may have run with an arched back and tail, an S-curved neck and forelimbs held backwards against the body.

Motion of an axisymmetric body in a rotating stratified fluid confined between two parallel planes

1969 ◽  
Vol 38 (4) ◽  
pp. 833-842 ◽  
Author(s):  
D. V. Krishna ◽  
L. V. Sarma
Keyword(s):  
Steady State ◽  
Oblate Spheroid ◽  
Vertical Axis ◽  
Vertical Gradient ◽  
Steady State Solution ◽  
Stratified Fluid ◽  
Axisymmetric Body ◽  
The Body ◽  
Homogeneous Fluid ◽  
Axis Of Rotation

We consider here the flow due to the oscillation of a slender oblate spheroid in a non-homogeneous, rotating fluid confined between two parallel planes which are perpendicular to the (vertical) axis of rotation. The direction of oscillation of the spheroid is perpendicular to the axis of rotation. By solving a set of dual integrals the steady-state solution is obtained in the two cases when the plates are at an infinite distance from the body and when they are at a large but finite distance. The singular or discontinuous surfaces observed in the case of homogeneous fluid are absent here. Also, the steady-state velocity is no longer independent of the distance along the axis of rotation. The velocity has now a vertical gradient, an important feature in the case of stratified fluid. It is also found that the presence of the plane boundaries increases the force on the body.

On the motion of a liquid in a spheroidal cavity of a precessing rigid body. II

1965 ◽  
Vol 61 (1) ◽  
pp. 279-288 ◽  
Author(s):  
P. H. Roberts ◽  
K. Stewartson
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Kinematic Viscosity ◽  
Viscous Incompressible Fluid ◽  
Minor Axis ◽  
Ultimate State ◽  
Oblate Spheroidal ◽  
Axis Of Rotation ◽  
Spheroidal Cavity ◽  
Image Position

AbstractIt is supposed that a viscous incompressible fluid is contained in an oblate spheroidal cavity (major axes a, eccentricity e) in a rigid body, and that, up to time t = 0, both fluid and container rotate together with angular velocity ω about the minor axis LB of the spheroid, which is fixed in space. At t = 0, the axis of rotation is moved impulsively, and is given a motion of precession (angular velocity Ω) about an axis LS fixed in space, which makes an angle α with LB. It is required to find the ultimate state of motion of the fluid relative to the container. In a previous paper (Stewartson and Roberts (4)), this problem was solved for arbitrary α under the assumptions thatwhere v is the kinematic viscosity. In the present paper, it is shown how the problem may be solved for arbitrary e (including zero) under the assumptions that

Detached shear layers in a rotating fluid

1967 ◽  
Vol 29 (1) ◽  
pp. 39-60 ◽  
Author(s):  
R. Hide ◽  
C. W. Titman
Keyword(s):  
Empirical Relationship ◽  
Axisymmetric Flow ◽  
Axial Flow ◽  
Vertical Axis ◽  
Wave Pattern ◽  
Shear Layers ◽  
Standard Errors ◽  
Regular Pattern ◽  
Rotating Fluid ◽  
Axis Of Rotation

The occurrence of detached shear layers should, according to straightforward theoretical arguments, often characterize hydrodynamical motions in a rapidly rotating fluid. Such layers have been produced and studied in a very simple system, namely a homogeneous liquid of kinematical viscosity v filling an upright, rigid, cylindrical container mounted coaxially on a turn-table rotating at Ω0 rad/s about a vertical axis, and stirred by rotating about the same axis at Ω1 rad/s a disk of radius a cm and thickness b’ cm immersed in the liquid with its plane faces parallel to the top and bottom end walls of the container. By varying Ω0, Ω1 and a, ranges of Rossby number, the modulus of ε ≡ (Ω1 + Ω0)/½ (Ω1 + Ω0), from 0·01 to 0·3, and Ekman number, E ≡ 2v/a2(Ω1 + Ω0), from 10−5 to 5 × 10−4 were attained. Although the apparatus was axisymmetric, only when |ε| did not exceed a certain critical value, |εT|, was the flow characterized by the same property of symmetry about the axis of rotation. Otherwise, when |ε| > |εT|, non-axisymmetric flow occurred, having the form in planes perpendicular to the axis of rotation of a regular pattern of waves, M in number, when ε was positive, and of a blunt ellipse when ε was negative.The axial flow in the axisymmetric detached shear layer, and the uniform rate of drift of the wave pattern characterizing the non-axisymmetric flow when ε is positive, depend in relatively simple ways on ε and E. The dependence of|εT| on E can be expressed by the empirical relationship |εT| = AEn, where A = 16·8 ± 2·2 and n = 0·568 ± 0·013 (= (4/7) × (1·000 − (0·005 ± 0·023))!), standard errors, 25 determinations. M does not depend strongly on E but generally decreases with increasing ε.

scholarly journals On New Modifications of Some Perturbation Procedures

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
A. I. Ismail
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Velocity Vector ◽  
The Body ◽  
Large Parameter ◽  
Angular Velocity Vector ◽  
Rigid Body Dynamic ◽  
Motion Problem ◽  
Slow Motions ◽  
Body Dynamic

In this paper, we present new modifications for some perturbation procedures used in mathematics, physics, astronomy, and engineering. These modifications will help us to solve the previous problems in different sciences under new conditions. As problems, we have, for example, the rotary rigid body problem, the gyroscopic problem, the pendulum motion problem, and other ones. These problems will be solved in a new manner different from the previous treatments. We solve some of the previous problems in the presence of new conditions, new analysis, and new domains. We let complementary conditions of such studied previously. We solve these problems by applying the large parameter technique used by assuming a large parameter which inversely proportional to a small quantity. For example, in rigid body dynamic problems, we take such quantity to be one of the components of the angular velocity vector in the initial instant of the rotary body about a fixed point. The domain of our solutions will be depending on the choice of a large parameter. The problem of slow (weak) oscillations is considered. So, we obtain slow motions of the bodies instead of fast motions and find the solutions of the problem in present new conditions on both of center of gravity, moments of inertia, and the angular velocity vector or one of these parameters of the body. This study is important for aerospace engineering, gyroscopic motions, satellite motion which has the correspondence of inertia moments, antennas, and navigations.

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