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 ${\displaystyle (r,\theta ,\phi )}$ with corresponding velocity components ${\displaystyle (v_{r},v_{\theta },0)}$. The velocity components are identified from Stokes stream function ${\displaystyle \psi (r,\theta )}$ as follows

${\displaystyle v_{r}={\frac {1}{r^{2}\sin \theta }}{\frac {\partial \psi }{\partial \theta }},\quad v_{\theta }=-{\frac {1}{r\sin \theta }}{\frac {\partial \psi }{\partial r}}.}$

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

${\displaystyle \psi ={\begin{cases}-{\frac {3U}{4}}\left(1-{\frac {r^{2}}{a^{2}}}\right)r^{2}\sin ^{2}\theta \quad {\text{in}}\quad r\leq a\\{\frac {U}{2}}\left(1-{\frac {a^{3}}{r^{3}}}\right)r^{2}\sin ^{2}\theta \quad {\text{in}}\quad r\geq a\end{cases}}}$

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

${\displaystyle \omega _{\phi }=-{\frac {15U}{2a^{2}}}r\sin \theta .}$

Note that here the parameters ${\displaystyle U}$ and ${\displaystyle a}$ 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 ${\displaystyle (v_{r},v_{\theta })}$ in the axial (or, meridional) plane is described by the Stokes stream function ${\displaystyle \psi }$ as before. The azimuthal motion ${\displaystyle v_{\phi }}$ is given by

${\displaystyle v_{\phi }={\frac {\pm k\psi }{r\sin \theta }}}$

where

${\displaystyle \psi ={\begin{cases}-{\frac {3U}{2}}{\frac {J_{3/2}(ka)}{J_{5/2}(ka)}}\left[\left({\frac {a}{r}}\right)^{3/2}{\frac {J_{3/2}(kr)}{J_{3/2}(ka)}}-1\right]r^{2}\sin ^{2}\theta \quad {\text{in}}\quad r\leq a\\{\frac {U}{2}}\left(1-{\frac {a^{3}}{r^{3}}}\right)r^{2}\sin ^{2}\theta \quad {\text{in}}\quad r\geq a\end{cases}}}$

where ${\displaystyle J_{3/2}}$ and ${\displaystyle J_{5/2}}$ 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 ${\displaystyle ka}$. 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