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. The vortex is named after Micaiah John Muller Hill who discovered the exact solution in 1894. The two-dimensional analogue of this vortex is the Lamb–Chaplygin dipole. The solution is described in the spherical polar coordinates system ( r , θ , ϕ ) {\displaystyle (r,\theta ,\phi )} with corresponding velocity components ( v r , v θ , 0 ) {\displaystyle (v_{r},v_{\theta },0)} . The velocity components are identified from Stokes stream function ψ ( r , θ ) {\displaystyle \psi (r,\theta )} as follows
v r = 1 r 2 sin θ ∂ ψ ∂ θ , v θ = − 1 r sin θ ∂ ψ ∂ r . {\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
ψ = { − 3 U 4 ( 1 − r 2 a 2 ) r 2 sin 2 θ in r ≤ a U 2 ( 1 − a 3 r 3 ) r 2 sin 2 θ in r ≥ a {\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 U {\displaystyle U} is a constant freestream velocity far away from the origin and a {\displaystyle a} is the radius of the sphere within which the vorticity is non-zero. For r ≥ a {\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 a {\displaystyle a} . The only non-zero vorticity component for r ≤ a {\displaystyle r\leq a} is the azimuthal component that is given by
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