scholarly journals XX. The potential of an anchor ring.-Part II

1893 ◽  
Vol 184 ◽  
pp. 1041-1106 ◽  
Keyword(s):  
Vortex Ring ◽  
Cross Section ◽  
Steady Motion ◽  
Vortex Rings ◽  
Laplace’S Equation ◽  
Single Vortex ◽  
Laplace's Equation ◽  
The Stability ◽  
Work Done

This paper is a continuation of that at pp. 43-95 suprd , on “The Potential of an Anchor Bing.” In that paper the potential of an anchor ring was found at all external points; in this/its value is determined at internal points. The annular form of rotating gravitating fluid was also discussed in that paper; here the stability of such a ring is considered. In addition, the potential of a ring whose cross-section is elliptic, being of interest in connection with Saturn, is obtained. The similarity of the methods employed, as well as of the analysis, has led me to give in this paper also a determination of the steady motion of a single vortex-ring in an infinite fluid, and of several fine vortex rings on the same axis. In Section I. solutions of Laplace’s equation applicable to space inside an anchor ring are obtained. These results are applied to obtain the potential of a solid ring at internal points, and also of a distribution of matter on the surface of the ring. The work done in collecting the ring from infinity is obtained.

scholarly journals XVII. The complete system of the periods of a hollow vortex ring

1895 ◽  
Vol 186 ◽  
pp. 603-619 ◽  
Keyword(s):  
Kinetic Theory ◽  
Vortex Ring ◽  
Cross Section ◽  
Steady Motion ◽  
Dynamical Theory ◽  
Vortex Theory ◽  
Wide Range ◽  
The Cross ◽  
Hollow Vortex ◽  
The Stability

According to the vortex theory of matter, atoms consist of vortex rings in an infinite perfect liquid, the æther. These rings may be either hollow or filled with otating liquid. The cross section of the hollow or rotating core is in the simplest ase small and the ring is circular. Such vortices have been investigated. It has been hown that they can exist, and that they are stable for certain types of deformation, in this paper the stability of the hollow vortex ring is investigated further, with a view to proving that it is stable for all small deformations of its surface. An attempt also made to make the vortex theory of matter agree with the kinetic theory of ases as regards the relation between the velocity and the energy of an atom. On he latter theory the energy of an atom varies as the square of its velocity, while on he former theory the energy decreases as the velocity increases. As the two theories liffer on a fundamental point, while the consequences of the kinetic theory agree over wide range with experiment, those of the vortex theory are likely to be in discrepancy therewith. However, no account has been taken of the electric change which an atom must hold if electrolysis is to be explained. This electrification will evidently alter the relation between the energy and the velocity. The nature of the change thus produced is here discussed for the case of a hollow vortex, the surface of which behaves as a conductor of electricity, a representation which is dynamically realised by the theory of a rotationally-elastic fluid æther developed in Mr. Larmor’s paper, “A Dynamical Theory of the Electric and Luminiferous Medium.” The small oscillations also are worked out with a view to the discussion of the stability of an electrified vortex. 2. The velocity of translation of the vortex in its steady motion is constant and perpendicular to its plane. By impressing on the whole liquid a velocity equal and opposite to this, the hollow is reduced to rest. Since the cross section of the hollow is small, any small length of it may be regarded as cylindrical. A cylindrical vortex must, by reason of symmetry, have its cross section a circle, so that the cross section of the hollow of the annular vortex is approximately circular, and the hollow itself approximately a tore.

scholarly journals III. The complete system of the periods of a hollow vortex ring

1895 ◽  
Vol 58 (347-352) ◽  
pp. 155-156 ◽  
Keyword(s):  
Vortex Ring ◽  
Cross Section ◽  
Electric Charge ◽  
Complete System ◽  
Steady Motion ◽  
Azimuth Angle ◽  
Axis Of Symmetry ◽  
Hollow Vortex ◽  
The Stability ◽  
Small Deformations

The author discusses the stability of a hollow annular vortex in an infinite perfect liquid, and also the effect of an electric charge on the steady motion and the stability of such a vortex. It is known that a hollow vortex ring (without electric charge) is stable for such small deformations as are symmetrical about the axis of symmetry of the ring, and for such as consist in displacement of the axis of the hollow without alteration of the size or shape of its cross section. This investigation shows that, in addition to the fluted and sinuous vibrations above referred to, the vortex is capable of beaded vibrations, in which the hollow is enlarged and contracted at regular intervals along its length, and also of vibration of a more general type, in which the displacement at any moment consists of waves on the surface of the hollow, of which the crests are circles parallel to the axis of the hollow, and the amplitude a sine or cosine of a multiple of the azimuth angle.

scholarly journals The steady motion and stability of a helical vortex

1928 ◽  
Vol 120 (786) ◽  
pp. 670-690 ◽  
Keyword(s):  
Reynolds Number ◽  
Vortex Motion ◽  
Steady Motion ◽  
Vortex Rings ◽  
Cross Sectional ◽  
Flow Past A Cylinder ◽  
Two Dimensional Flow ◽  
The Stability ◽  
Sectional Shape ◽  
Stable Formation

1. Introductory .—This is the third of a series of papers dealing with the stability or instability of certain forms of vortex motion associated with the wake of a body moving in a fluid. In the earlier papers we examined the case of a system of equal vortex rings in parallel planes, as they might form in the rear of a sphere in steady motion. Nisi and Porter have shown that the lowest speed at which the vortex ring forms is 8·14 v / d where v is the kinematic viscosity of the fluid and d is the diameter of the sphere. Such a system of vortices has been proved to be only partially stable, and it is therefore to be inferred that their production occurs at a transition stage to a more stable type of flow. Now it is well known that in the case of two dimensional flow past a cylinder of any cross-sectional shape, eddies are formed in symmetrical pairs at low values of Reynolds' number, whereas at higher values asymmetry sets in and the eddying is formed alternately at one side of the cylinder and then at the other with regular periodicity. This latter stage occurs over a range of values of Reynolds’ number extending from about 70 to 10 5 . Detailed explorations of the field for some distance behind the cylinder have established that the centres of eddying approximately assume the stable formation which has come to be known as the “Kármán vortex street."

scholarly journals On the stability of Laplace’s equation

2013 ◽  
Vol 26 (5) ◽  
pp. 549-552 ◽  
Author(s):  
Balázs Hegyi ◽  
Soon-Mo Jung
Keyword(s):  
Laplace’S Equation ◽  
Laplace's Equation ◽  
The Stability

scholarly journals The vibrations of an infinite system of vortex rings

1927 ◽  
Vol 116 (774) ◽  
pp. 352-379 ◽  
Keyword(s):  
Infinite System ◽  
Vortex Rings ◽  
Critical Ratio ◽  
Single Vortex ◽  
Natural Modes ◽  
Finite Section ◽  
Modes Of Vibration ◽  
Central Filament ◽  
Set Up ◽  
The Stability

The vibrations that may be set up and maintained in the central filament of a single vortex ring of small but finite section have been investi­gated by Thomson and others, but no corresponding investigation appears to have been undertaken for a system of parallel rings, although the matter is of some importance in connection with the state of motion behind a moving body. In a previous paper the authors have examined the stability of an infinite system of equal vortex rings situated in parallel planes with their centres evenly spaced along an infinite line and with their planes at right angles to that line. Instability was there found to occur for disturbances confined to displacements of the centre of each ring along the central axis, the filament of each ring still remaining circular. In the present paper the investigation is extended to deformation of the vortex filaments, and some interesting conclusions are drawn regarding natural modes of vibration of the infinite system of vortex rings, such as may occur without the longitudinal instability referred to in the previous paper becoming apparent. It is found, for example, that for any given ratio of radius of ring section to radius of ring there exists a critical ratio of ring spacing to radius, separating the region of stable oscillation from that of instability, a result in some respects closely analogous to that found by Kármán for the stability of two infinite parallel rows of rectilinear vortices.

A universal time scale for vortex ring formation

1998 ◽  
Vol 360 ◽  
pp. 121-140 ◽  
Author(s):  
MORTEZA GHARIB ◽  
EDMOND RAMBOD ◽  
KARIM SHARIFF
Keyword(s):  
Vortex Ring ◽  
Vortex Rings ◽  
Water Tank ◽  
Single Vortex ◽  
Piston Cylinder ◽  
Vortex Ring Formation ◽  
Image Velocimetry ◽  
Wide Range ◽  
Formation Number ◽  
Piston Stroke

The formation of vortex rings generated through impulsively started jets is studied experimentally. Utilizing a piston/cylinder arrangement in a water tank, the velocity and vorticity fields of vortex rings are obtained using digital particle image velocimetry (DPIV) for a wide range of piston stroke to diameter (L/D) ratios. The results indicate that the flow field generated by large L/D consists of a leading vortex ring followed by a trailing jet. The vorticity field of the leading vortex ring formed is disconnected from that of the trailing jet. On the other hand, flow fields generated by small stroke ratios show only a single vortex ring. The transition between these two distinct states is observed to occur at a stroke ratio of approximately 4, which, in this paper, is referred to as the ‘formation number’. In all cases, the maximum circulation that a vortex ring can attain during its formation is reached at this non-dimensional time or formation number. The universality of this number was tested by generating vortex rings with different jet exit diameters and boundaries, as well as with various non-impulsive piston velocities. It is shown that the ‘formation number’ lies in the range of 3.6–4.5 for a broad range of flow conditions. An explanation is provided for the existence of the formation number based on the Kelvin–Benjamin variational principle for steady axis-touching vortex rings. It is shown that based on the measured impulse, circulation and energy of the observed vortex rings, the Kelvin–Benjamin principle correctly predicts the range of observed formation numbers.

X. On the vibrations of a vortex ring, and the action of two vortex rings upon each other

1882 ◽  
Vol 33 (216-219) ◽  
pp. 145-147 ◽  
Keyword(s):  
Angular Velocity ◽  
Vortex Ring ◽  
Cross Section ◽  
Vortex Core ◽  
Molecular Rotation ◽  
Vortex Rings ◽  
The Cross ◽  
Circular Axis

In the first part of the paper it is shown that if the circular axis of a vortex ring be displaced so as to be represented by the equations— ρ = a + α n cos nd , z = β u cos nd . when ρ is the distance of a point on the circular axis from the straight axis, and z the distance of a point on the circular axis from its mean plane, then— α n = A cos (ω e 2 /2 a 2 log 2 a / e n √ n 2 —1 . t + B), β n = A √ n 2 —1/ n sin ((ω e 2 /2 a 2 log 2 a / e n √ n 2 —1 . t + B), when ω is the angular velocity of molecular rotation, e the radius of the cross section of the vortex core, and a the radius of the aperture. The cross section is supposed small compared with the aperture so that e is small compared with a .

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