Hicks equation

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.^[1][2][3] 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.^[4][5][6] The Hicks equation without swirl was first introduced by George Gabriel Stokes in 1842.^[7][8] The Grad–Shafranov equation appearing in plasma physics also takes the same form as the Hicks equation.

Representing ${\displaystyle (r,\theta ,z)}$ as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by ${\displaystyle (v_{r},v_{\theta },v_{z})}$, the stream function ${\displaystyle \psi }$ that defines the meridional motion can be defined as

${\displaystyle rv_{r}=-{\frac {\partial \psi }{\partial z}},\quad rv_{z}={\frac {\partial \psi }{\partial r}}}$

that satisfies the continuity equation for axisymmetric flows automatically. The Hicks equation is then given by ^[9]

${\displaystyle {\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}=r^{2}{\frac {\mathrm {d} H}{\mathrm {d} \psi }}-\Gamma {\frac {\mathrm {d} \Gamma }{\mathrm {d} \psi }}}$

where

${\displaystyle H(\psi )={\frac {p}{\rho }}+{\frac {1}{2}}(v_{r}^{2}+v_{\theta }^{2}+v_{z}^{2}),\quad \Gamma (\psi )=rv_{\theta }}$

where ${\displaystyle H(\psi )}$ is the total head, c.f. Bernoulli's Principle. and ${\displaystyle 2\pi \Gamma }$ is the circulation, both of them being conserved along streamlines. Here, ${\displaystyle p}$ is the pressure and ${\displaystyle \rho }$ is the fluid density. The functions ${\displaystyle H(\psi )}$ and ${\displaystyle \Gamma (\psi )}$ 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 ${\displaystyle H(\psi )}$ and ${\displaystyle \Gamma (\psi )}$ are typically unknown and therefore in those regions, Hicks equation is not useful; Prandtl–Batchelor theorem provides details about the closed streamline regions.