Hill's spherical vortex is an exact solution of the Euler equations that is commonly used to model a vortex ring. The solution is also used to model the velocity distribution inside a spherical drop of one fluid moving at a constant velocity through another fluid at small Reynolds number.[1] The vortex is named after Micaiah John Muller Hill who discovered the exact solution in 1894.[2] The two-dimensional analogue of this vortex is the Lamb–Chaplygin dipole.

The solution is described in the spherical polar coordinates system with corresponding velocity components . The velocity components are identified from Stokes stream function as follows

The Hill's spherical vortex is described by[3]

where is a constant freestream velocity far away from the origin and is the radius of the sphere within which the vorticity is non-zero. For , the vorticity is zero and the solution described above in that range is nothing but the potential flow past a sphere of radius . The only non-zero vorticity component for is the azimuthal component that is given by

Note that here the parameters and can be scaled out by non-dimensionalization.

Hill's spherical vortex with a swirling motion

The Hill's spherical vortex with a swirling motion is provided by Keith Moffatt in 1969.[4] Earlier discussion of similar problems are provided by William Mitchinson Hicks in 1899.[5] The solution was also discovered by Kelvin H. Pendergast in 1956, in the context of plasma physics,[6] as there exists a direct connection between these fluid flows and plasma physics (see the connection between Hicks equation and Grad–Shafranov equation). The motion in the axial (or, meridional) plane is described by the Stokes stream function as before. The azimuthal motion is given by

where

where and are the Bessel functions of the first kind. Unlike the Hill's spherical vortex without any swirling motion, the problem here contains an arbitrary parameter . A general class of solutions of the Euler's equation describing propagating three-dimensional vortices without change of shape is provided by Keith Moffatt in 1986.[7]

References

  1. Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge university press. page 526
  2. Hill, M. J. M. (1894). VI. On a spherical vortex. Philosophical Transactions of the Royal Society of London.(A.), (185), 213–245.
  3. Acheson, D. J. (1991). Elementary fluid dynamics. page. 175
  4. Moffatt, H. K. (1969). The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics, 35(1), 117–129.
  5. Hicks, W. M. (1899). Ii. researches in vortex motion.—part iii. on spiral or gyrostatic vortex aggregates. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, (192), 33–99.
  6. Prendergast, K. H. (1956). The Equilibrium of a Self-Gravitating Incompressible Fluid Sphere with a Magnetic Field. I. The Astrophysical Journal, 123, 498.
  7. Moffatt, H. K. (1986). On the existence of localized rotational disturbances which propagate without change of structure in an inviscid fluid. Journal of Fluid Mechanics, 173, 289–302.


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.