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A vortical thread having the shape of a torus of small cross-section. The general formulas which determine the velocity of liquid particles from vortices make it possible to represent the velocity potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969401.png" /> and the Stokes function of the current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969402.png" /> of a flow generated in an unbounded liquid by a vortical ring as integrals containing the Bessel functions of order zero and one:
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A vortical thread having the shape of a torus of small cross-section. The general formulas which determine the velocity of liquid particles from vortices make it possible to represent the velocity potential $\phi(z,r)$ and the Stokes function of the current $\psi(z,r)$ of a flow generated in an unbounded liquid by a vortical ring as integrals containing the Bessel functions of order zero and one:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969403.png" /></td> </tr></table>
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$$\phi(z,r)=\frac12a\kappa\int\limits_0^\infty e^{-kz}J_0(kr)J_1(ka)dk,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969404.png" /></td> </tr></table>
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$$\phi(z,r)=-\frac12a\kappa r\int\limits_0^\infty e^{-kz}J_1(kr)J_1(ka)dk.$$
  
In these formulas, which apply if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969406.png" /> is the radius of the ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969407.png" /> is the stress of the vortical ring. The coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969408.png" /> is taken from the plane of the moving ring. The ring, when acted upon by the velocities it itself produces in the ring, moves in the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v0969409.png" />-axis at a constant velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v09694010.png" />, which is given by the following approximate formula:
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In these formulas, which apply if $z>0$, $a$ is the radius of the ring and $\kappa$ is the stress of the vortical ring. The coordinate $z$ is taken from the plane of the moving ring. The ring, when acted upon by the velocities it itself produces in the ring, moves in the direction of the $z$-axis at a constant velocity $c$, which is given by the following approximate formula:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v09694011.png" /></td> </tr></table>
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$$c=\frac{\kappa}{4\pi\epsilon}\left[\ln\frac{8\epsilon}{a}-\frac14\right],$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v09694012.png" /> is the radius of a cross-section of the vortical ring. For several vortical rings the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v09694013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096940/v09694014.png" /> are represented as sums of the respective functions of each ring.
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where $\epsilon$ is the radius of a cross-section of the vortical ring. For several vortical rings the functions $\phi$ and $\psi$ are represented as sums of the respective functions of each ring.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.M. Milne-Thomson,  "Theoretical hydrodynamics" , Macmillan  (1950)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.M. Milne-Thomson,  "Theoretical hydrodynamics" , Macmillan  (1950)</TD></TR></table>

Latest revision as of 06:39, 15 August 2014

A vortical thread having the shape of a torus of small cross-section. The general formulas which determine the velocity of liquid particles from vortices make it possible to represent the velocity potential $\phi(z,r)$ and the Stokes function of the current $\psi(z,r)$ of a flow generated in an unbounded liquid by a vortical ring as integrals containing the Bessel functions of order zero and one:

$$\phi(z,r)=\frac12a\kappa\int\limits_0^\infty e^{-kz}J_0(kr)J_1(ka)dk,$$

$$\phi(z,r)=-\frac12a\kappa r\int\limits_0^\infty e^{-kz}J_1(kr)J_1(ka)dk.$$

In these formulas, which apply if $z>0$, $a$ is the radius of the ring and $\kappa$ is the stress of the vortical ring. The coordinate $z$ is taken from the plane of the moving ring. The ring, when acted upon by the velocities it itself produces in the ring, moves in the direction of the $z$-axis at a constant velocity $c$, which is given by the following approximate formula:

$$c=\frac{\kappa}{4\pi\epsilon}\left[\ln\frac{8\epsilon}{a}-\frac14\right],$$

where $\epsilon$ is the radius of a cross-section of the vortical ring. For several vortical rings the functions $\phi$ and $\psi$ are represented as sums of the respective functions of each ring.

References

[1] L.M. Milne-Thomson, "Theoretical hydrodynamics" , Macmillan (1950)
How to Cite This Entry:
Vortical ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vortical_ring&oldid=18391
This article was adapted from an original article by L.N. Sretenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article