On the generation of viscous toroidal eddies in a cylinder

1979 ◽  
Vol 95 (2) ◽  
pp. 209-222 ◽  
Author(s):  
J. R. Blake
Keyword(s):  
Finite Number ◽  
Stream Function ◽  
Radial Direction ◽  
Finite Cylinder ◽  
Infinite Cylinder ◽  
Cylinder Length ◽  
Infinite Set

The streamlines due to a stokeslet on the axis in a finite, semi-infinite and infinite cylinder are obtained together with the case of a Stokes-doublet and source-doublet in an infinite cylinder. In the infinite and semi-infinite cylinder examples an infinite set of toroidal eddies are obtained. The eddies alternate in sign and the magnitude of the stream function decays exponentially with distance from the driving singularity. In the finite cylinder a primary interior eddy adjacent to the singularity is always obtained and, depending on location of the singularity within the cylinder and the ratio of cylinder length to radius, a finite number of secondary interior eddies. In the case of long cylinders, the eddies are generated along the axis, whereas, for squat cylinders, secondary eddies occur in the radial direction. The interior eddies emerge from the corner as the length of the cylinder is increased. Moffatt corner eddies exist but they are very much smaller than the interior eddies.

How to Program an Infinite Abacus

1961 ◽  
Vol 4 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Finite Number ◽  
Recursive Function ◽  
Computable Function ◽  
Unlimited Supply ◽  
Countably Infinite ◽  
Infinite Set

This is an expository note to show how an “infinite abacus” (to be defined presently) can be programmed to compute any computable (recursive) function. Our method is probably not new, at any rate, it was suggested by the ingenious technique of Melzak [2] and may be regarded as a modification of the latter.By an infinite abacus we shall understand a countably infinite set of locations (holes, wires etc.) together with an unlimited supply of counters (pebbles, beads etc.). The locations are distinguishable, the counters are not. The confirmed finitist need not worry about these two infinitudes: To compute any given computable function only a finite number of locations will be used, and this number does not depend on the argument (or arguments) of the function.

Frequency Clusters in the Spectrum of Annular Cylinders

1998 ◽  
Vol 65 (4) ◽  
pp. 797-803 ◽  
Author(s):  
K. I. Tzou ◽  
J. A. Wickert ◽  
A. Akay
Keyword(s):  
Natural Frequencies ◽  
Spectral Characteristics ◽  
Three Dimensional ◽  
Longitudinal Shear ◽  
Forced Response ◽  
Infinite Cylinder ◽  
Propagation Theory ◽  
Cylinder Length ◽  
Vibration Model ◽  
Shear Modes

As the length of a traction-free annular cylinder is increased, distinct members within any family of radial or longitudinal shear modes have natural frequencies that asymptotically approach a common nonzero value. Such modes, potentially having significantly different numbers of nodes along the cylinder’s generator, can have natural frequencies that are indistinguishable from one another within the resolution of test equipment or numerical simulation. The three-dimensional vibration model discussed here predicts the formation of narrow “frequency clusters” with the cylinder’s increasing length, the converged value of which bounds from below the frequencies of all modes within a particular family. In addition to these spectral characteristics, frequency clusters have implications for the forced response of annular cylinders. For the particular families of modes that are of interest here, the steady-state harmonic response at frequencies near a cluster can be spatially confined with displacements that decay rapidly away from the point of maximum response. At other driving frequencies, the response is distributed more uniformly along the length of the cylinder. The derived analytical model is compared with results from laboratory measurements, and from the predictions of wave propagation theory in the limit of infinite cylinder length.

Correction to letter “On Electromagnetic Induction in Elongated Ore Bodies,” GEOPHYSICS, v. 38, no. 5, p. 984–985

Geophysics ◽  
1974 ◽  
Vol 39 (2) ◽  
pp. 235-235 ◽  
Author(s):  
James R. Wait
Keyword(s):  
Electromagnetic Induction ◽  
Finite Cylinder ◽  
Infinite Cylinder ◽  
Ore Bodies ◽  
Quantitative Results

If galley proofs of this letter had been sent to author, I would have indicated that the effect of the truncation of the infinite cylinder would be to “de‐emphasize” rather than to “wipe out” the monopole or m=0 term. Quantitative results to illustrate this effect for a finite cylinder are included in an article accepted for Geofisica Internacional (Mexico). Also it should be noted that the erratum to the 1952 reference was incorrectly printed in the cited letter.

Frequency Clusters in the Spectrum of Annular Cylinders

1999 ◽  
Author(s):  
K. I. Tzou ◽  
J. A. Wickert ◽  
A. Akay
Keyword(s):  
Natural Frequencies ◽  
Spectral Characteristics ◽  
Three Dimensional ◽  
Longitudinal Shear ◽  
Forced Response ◽  
Infinite Cylinder ◽  
Propagation Theory ◽  
Cylinder Length ◽  
Vibration Model ◽  
Shear Modes

Abstract As the length of a traction-free annudar cylinder is increased, distinct members within any family of radial or longitudinal shear modes have natural frequencies that asymptotically approach a common non-zero value. Such modes, potentially having significantly different numbers of nodes along the cylinder’s generator, can have natural frequencies that are indistinguishable from one another within the resolution of test equipment or numerical simulation. The three-dimensional vibration model discussed here predicts the formation of narrow “frequency clusters” with the cylinder’s increasing length, the converged value of which bounds from below the frequencies of all modes within a particular family. In addition to these spectral characteristics, frequency clusters have implications for the forced response of annular cylinders. For the particular families of modes that are of interest here, the steady state harmonic response at frequencies near a cluster can be spatially confined with displacements that decay rapidly away from the point of maximum response. At other driving frequencies, the response is distributed more uniformly along the length of the cylinder. The derived analytical model is compared with results from laboratory measurements, and from the predictions of wave propagation theory in the limit of infinite cylinder length.

Fluid-Structure Coupling Between a Finite Cylinder and a Confined Fluid

1984 ◽  
Vol 51 (4) ◽  
pp. 857-862 ◽  
Author(s):  
G. Garner ◽  
S. Chandra
Keyword(s):  
Dynamic Behavior ◽  
Finite Length ◽  
Natural Frequencies ◽  
Finite Cylinder ◽  
Mode Shapes ◽  
Infinite Cylinder ◽  
Fluid Structure ◽  
Coupled Mode ◽  
Confined Fluid ◽  
Structure Equations

The dynamic behavior of a finite length cylindrical rod in a fluid filled annulus is considered. The fluid and structure equations are solved simultaneously, with fluid-structure coupling accounted for. Coupled mode shapes and natural frequencies are obtained for various cases. It is found that for short lengths and/or higher modes, the effect of the fluid on the cylinder motion diminishes compared to the infinite cylinder case. In addition, coupled and in-vacuum mode shapes can differ in certain cases.

scholarly journals Some improvements to Turner's algorithm for bracket abstraction

1990 ◽  
Vol 55 (2) ◽  
pp. 656-669 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Finite Number ◽  
Translation Process ◽  
Special Cases ◽  
Infinite Set

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.There are several ways in which these problems can be alleviated:(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.(2) The use of an extra clause (c), appropriate to βη reduction.(3) The introduction of a finite number of extra combinators.The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

scholarly journals Cracked semi-infinite cylinder and finite cylinder problems

2006 ◽  
Vol 44 (20) ◽  
pp. 1534-1555 ◽  
Author(s):  
Mete Onur Kaman ◽  
Mehmet Rusen Gecit
Keyword(s):  
Finite Cylinder ◽  
Infinite Cylinder

Dimension theory and homogeneity for elementary extensions of a model

1982 ◽  
Vol 47 (1) ◽  
pp. 147-160 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Number ◽  
Large Class ◽  
Countable Model ◽  
Dimension Theory ◽  
Elementary Embedding ◽  
Minimal Extension ◽  
Infinite Set ◽  
Countable Models

We take a fixed countable model M0, and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M0). Assuming that S(M0) is countable, we prove that if N is a weakly minimal extension of , and if then there is an elementary embedding of N into M over M0), then N is homogeneous over M0. Moreover the condition that ∣S(M0)∣ = ℵ0 cannot be removed. Under the hypothesis that M0 contains no infinite set of tuples ordered by a formula, we prove that M0 has infinitely many countable elementary extensions up to isomorphism over M0. A preliminary result is that all types over M0 are definable, and moreover is definable over M0 if and only if is definable over M0 (forking symmetry). We also introduce a notion of relative homogeneity, and show that a large class of elementary extensions of M0 are relatively homogeneous over M0 (under the assumptions that M0 has no order and S(M0) is countable).I will now discuss the background to and motivation behind the results in this paper, and also the place of this paper relative to other conjectures and investigations. To simplify notation let T denote the complete diagram of M0. First, our result that if M0 has no order then T has infinitely many countable models is related to the following conjecture: any theory with a finite number (more than one) of countable models is unstable.

Bifurcations of meromorphic vector fields on the Riemann sphere

1995 ◽  
Vol 15 (6) ◽  
pp. 1211-1222 ◽  
Author(s):  
Jesús Muciño-Raymundo ◽  
Carlos Valero-Valdés
Keyword(s):  
Vector Field ◽  
Finite Number ◽  
Vector Fields ◽  
Riemann Sphere ◽  
Fixed Degree ◽  
Multiplicity One ◽  
Accumulation Points ◽  
Dense Sets ◽  
Infinite Set ◽  
Zeros And Poles

AbstractLet {Xθ} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {θ}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.

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