In fluid dynamics, Hicks equation, sometimes also referred as Bragg–Hawthorne equation or Squire–Long equation, is a partial differential equation that describes the distribution of stream function for axisymmetric inviscid fluid, named after William Mitchinson Hicks, who derived it first in 1898. The equation was also re-derived by Stephen Bragg and William Hawthorne in 1950 and by Robert R. Long in 1953 and by Herbert Squire in 1956. The Hicks equation without swirl was first introduced by George Gabriel Stokes in 1842. The Grad–Shafranov equation appearing in plasma physics also takes the same form as the Hicks equation. Representing ( r , θ , z ) {\displaystyle (r,\theta ,z)} as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by ( v r , v θ , v z ) {\displaystyle (v_{r},v_{\theta },v_{z})} , the stream function ψ {\displaystyle \psi } that defines the meridional motion can be defined as
r v r = − ∂ ψ ∂ z , r v z = ∂ ψ ∂ r {\displaystyle rv_{r}=-{\frac {\partial \psi }{\partial z}},\quad rv_{z}={\frac {\partial \psi }{\partial r}}}
that satisfies the continuity equation for axisymmetric flows automatically. The Hicks equation is then given by
∂ 2 ψ ∂ r 2 − 1 r ∂ ψ ∂ r + ∂ 2 ψ ∂ z 2 = r 2 d H d ψ − Γ d Γ d ψ {\displaystyle {\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=r^{2}{\frac {\mathrm {d} H}{\mathrm {d} \psi }}-\Gamma {\frac {\mathrm {d} \Gamma }{\mathrm {d} \psi }}}
where
H ( ψ ) = p ρ + 1 2 ( v r 2 + v θ 2 + v z 2 ) , Γ ( ψ ) = r v θ {\displaystyle H(\psi )={\frac {p}{\rho }}+{\frac {1}{2}}(v_{r}^{2}+v_{\theta }^{2}+v_{z}^{2}),\quad \Gamma (\psi )=rv_{\theta }}
where H ( ψ ) {\displaystyle H(\psi )} is the total head, cf. Bernoulli's Principle. and 2 π Γ {\displaystyle 2\pi \Gamma } is the circulation, both of them being conserved along streamlines. Here, p {\displaystyle p} is the pressure and ρ {\displaystyle \rho } is the fluid density. The functions H ( ψ ) {\displaystyle H(\psi )} and Γ ( ψ ) {\displaystyle \Gamma (\psi )} are known functions, usually prescribed at one of the boundary; see the example below. If there are closed streamlines in the interior of the fluid domain, say, a recirculation region, then the functions H ( ψ ) {\displaystyle H(\psi )} and Γ ( ψ ) {\displaystyle \Gamma (\psi )} are typically unknown and therefore in those regions, Hicks equation is not useful; Prandtl–Batchelor theorem provides details about the closed streamline regions.
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