On the Torsional Oscillations of a Solid Sphere in a Viscous Fluid

1956 ◽  
Vol 23 (4) ◽  
pp. 601-605
Author(s):  
G. F. Carrier ◽  
R. C. Di Prima
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Angular Displacement ◽  
Solid Sphere ◽  
Circumferential Velocity ◽  
Torsional Oscillations ◽  
Viscosity Measurements ◽  
First Approximation ◽  
Viscous Torque ◽  
Solid Bodies

Abstract Most treatments of the torsional oscillations of solid bodies assume that the velocity field is circumferential. In this paper the motion in planes containing the axis of oscillation is also considered. An expansion in terms of the angular displacement ϵ (assumed small) is made. The first approximation to the circumferential velocity is computed, and then used in computing the first approximation to the pumping motion. This is used to compute the correction to the circumferential velocity and, in particular, the correction to the viscous torque. For the range of parameters considered it is found that the correction to the torque is of the order of 0.04ϵ2|N0|, where N0 is the classical viscous torque. This problem is of interest in practical viscosity measurements.

On a sphere performing linear and torsional oscillations in a viscous fluid

1988 ◽  
Vol 66 (7) ◽  
pp. 576-579
Author(s):  
G. T. Karahalios ◽  
C. Sfetsos
Keyword(s):  
Boundary Layer ◽  
Viscous Fluid ◽  
Secondary Flow ◽  
Small Amplitude ◽  
Equations Of Motion ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Time Varying ◽  
Torsional Oscillations

A sphere executes small-amplitude linear and torsional oscillations in a fluid at rest. The equations of motion of the fluid are solved by the method of successive approximations. Outside the boundary layer, a steady secondary flow is induced in addition to the time-varying motion.

The elastic field outside an ellipsoidal inclusion

1959 ◽  
Vol 252 (1271) ◽  
pp. 561-569 ◽  
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Elastic Constants ◽  
Strain Energy ◽  
Elastic Field ◽  
Ellipsoidal Inclusion ◽  
Harmonic Potential ◽  
Surface Distribution ◽  
Shear Transformation ◽  
General Method

The results of an earlier paper are extended. The elastic field outside an inclusion or inhomogeneity is treated in greater detail. For a general inclusion the harmonic potential of a certain surface distribution may be used in place of the biharmonic potential used previously. The elastic field outside an ellipsoidal inclusion or inhomogeneity may be expressed entirely in terms of the harmonic potential of a solid ellipsoid. The solution gives incidentally the velocity field about an ellipsoid which is deforming homogeneously in a viscous fluid. An expression given previously for the strain energy of an ellipsoidal region which has undergone a shear transformation is generalized to the case where the region has elastic constants different from those of its surroundings. The Appendix outlines a general method of calculating biharmonic potentials.

Wave Motion of a Compressible Viscous Fluid Contained in a Cylindrical Shell

1993 ◽  
Vol 115 (3) ◽  
pp. 302-312 ◽  
Author(s):  
J. H. Terhune ◽  
K. Karim-Panahi
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Imaginary Part ◽  
Fluid Velocity ◽  
Wave Number ◽  
Mode Shapes ◽  
Infinite Length ◽  
Complex Frequency ◽  
Compressible Viscous Fluid ◽  
Fluid Velocity Field

The free vibration of cylindrical shells filled with a compressible viscous fluid has been studied by numerous workers using the linearized Navier-Stokes equations, the fluid continuity equation, and Flu¨gge ’s equations of motion for thin shells. It happens that solutions can be obtained for which the interface conditions at the shell surface are satisfied. Formally, a characteristic equation for the system eigenvalues can be written down, and solutions are usually obtained numerically providing some insight into the physical mechanisms. In this paper, we modify the usual approach to this problem, use a more rigorous mathematical solution and limit the discussion to a single thin shell of infinite length and finite radius, totally filled with a viscous, compressible fluid. It is shown that separable solutions are obtained only in a particular gage, defined by the divergence of the fluid velocity vector potential, and the solutions are unique to that gage. The complex frequency dependence for the transverse component of the fluid velocity field is shown to be a result of surface interaction between the compressional and vortex motions in the fluid and that this motion is confined to the boundary layer near the surface. Numerical results are obtained for the first few wave modes of a large shell, which illustrate the general approach to the solution. The axial wave number is complex for wave propagation, the imaginary part being the spatial attenuation coefficient. The frequency is also complex, the imaginary part of which is the temporal damping coefficient. The wave phase velocity is related to the real part of the axial wave number and turns out to be independent of frequency, with numerical value lying between the sonic velocities in the fluid and the shell. The frequency dependencies of these parameters and fluid velocity field mode shapes are computed for a typical case and displayed in non-dimensional graphs.

scholarly journals How can vorticity be produced in irrotationally forced flows?

2010 ◽  
Vol 6 (S274) ◽  
pp. 373-375
Author(s):  
Fabio Del Sordo ◽  
Axel Brandenburg
Keyword(s):  
Viscous Fluid ◽  
Interstellar Medium ◽  
Numerical Study ◽  
Supersonic Flows ◽  
First Approximation

AbstractA spherical hydrodynamical expansion flow can be described as the gradient of a potential. In that case no vorticity should be produced, but several additional mechanisms can drive its production. Here we analyze the effects of baroclinicity, rotation and shear in the case of a viscous fluid. Those flows resemble what happens in the interstellar medium. In fact in this astrophysical environment supernovae explosion are the dominant flows and, in a first approximation, they can be seen as spherical. One of the main difference is that in our numerical study we examine only weakly supersonic flows, while supernovae explosions are strongly supersonic.

scholarly journals RESEARCH ON THE INFLUENCE OF THE PROPERTIES OF INTERVERTEBRAL DISC STIFFNESS OF THE LUMBAR SPINE ON THE DISPLACEMENT OF VERETBRAE / STUBURO JUOSMENINĖS DALIES TARPSLANKSTELINIO DISKO STANDUMO CHARAKTERISTIKŲ ĮTAKOS SLANKSTELIŲ POSLINKIAMS TYRIMAS

2015 ◽  
Vol 6 (6) ◽  
pp. 595-597
Author(s):  
Artūras Linkel ◽  
Julius Griškevičius ◽  
Gintaras Jonaitis
Keyword(s):  
Lumbar Spine ◽  
Intervertebral Disc ◽  
Degrees Of Freedom ◽  
Angular Displacement ◽  
Fast Response ◽  
Intervertebral Discs ◽  
Linear Displacement ◽  
Basic Task ◽  
Non Linear ◽  
Solid Bodies

The article proposes the method for evaluating angular and linear changes in intervertebral discs of the spine depending on linear and nonlinear intervertebral disc stiffness. A dynamic made of 5 solid bodies connected by damping and stiffness components and applied for 2-D 10 degrees of freedom of the lumbar spine has been used for calculations. The system of the equation has been written in a matrix form. Lumbar intervertebral discs stiffness and damping properties have been selected from scientific articles and make from 200 N/mm to 1200 N/mm and from 229 Ns / mm to 5100 Ns/mm respectively for non-linear calculation and 800 N / mm – 2637 Ns/mm for linear displacement calculation. External loads applied to the model are 1648 N, 2957 N, 3863 N and 4542 N. The basic task of the paper is to calculate the biggest difference in linear and angular displacement considering 2 cases: linear and non-linear stiffness value. The greatest estimated difference, under the highest load, makes 0.6 mm for linear and 0.95 degrees for angular displacement. Because of the fast response of the model to the load, the damping value could not affect displacement. Tyrimo objektas yra stuburo trapslankstelinių diskų poslinkių skirtumai esant tiesiniam ir netiesiniam jų standumo koeficientui. Taikomas 10 laisvės laipsnių 2-D stuburo juosmeninės dalies dinaminis modelis, kuris susideda iš 5 juosmens slankstelių, sujungtų standumo ir slopinimo ryšiais. Modeliui nustatomos juosmens apkrovos, kurios susidaro važiuojant dviračiu. Tarpslankstelinių diskų savybės parenkamos iš mokslinės literatūros. Sudarytas matematinis modelis leido apskaičiuoti stuburo slankstelių linijinius ir kampinius poslinkius įvertinant tarpslankstelinio disko standžio netiesiškumą. Atlikti skaičiavimai parodė, kad didžiausi skirtumai susidaro esant maksimaliai apkrovai. Didžiausi linijinių poslinkių skirtumai yra 0,6 mm, o kampinių – 0,95 laipsnio. Nustatytos slopinimo koeficiento reikšmės dėl greito modelio atsako poslinkių skaičiavimams įtakos neturėjo.

On 22-Year Pole To Equator Variation of the Corona Intensity

1994 ◽  
Vol 144 ◽  
pp. 96
Author(s):  
V. I. Makarov ◽  
A. G. Tlatov
Keyword(s):  
Magnetic Field ◽  
Velocity Field ◽  
Solution Method ◽  
Least Square ◽  
Field Reversal ◽  
Torsional Oscillations ◽  
Zero Values ◽  
The Magnetic Field ◽  
Magnetic Field Reversal

AbstractKislovodsk sets of observations (1957.5-1991.0) of the corona in the line Fe XIV 5303 Å have been processed. Half annual intensities of corona have been approximated with the polynoms of the m-degree in dependence on latitude. The coefficients of these sets have been found using the least square solution method. The zone of zero values of the second derivation of the coronal intensity on latitude has been determined. It is shown that this zone emerges after the magnetic field reversal at the latitudes near 80° then migrates equatorward during 22 years with a velocity of 2 m/s. The obtained data were compared with a pattern of torsional oscillations plotted on the basis of the velocity field. A possible reason for 22 year pole to equator variations in the corona is considered.

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