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.
as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by
that defines the meridional motion can be defined as that satisfies the continuity equation for axisymmetric flows automatically.
is the circulation, both of them being conserved along streamlines.
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
are typically unknown and therefore in those regions, Hicks equation is not useful; Prandtl–Batchelor theorem provides details about the closed streamline regions.
Consider the axisymmetric flow in cylindrical coordinate system
in axisymmetric flows, the vorticity components are Continuity equation allows to define a stream function
such that (Note that the vorticity components
Therefore the azimuthal component of vorticity becomes
The inviscid momentum equations
is the fluid density, when written for the axisymmetric flow field, becomes in which the second equation may also be written as
round a material curve in the form of a circle centered on
If the fluid motion is steady, the fluid particle moves along a streamline, in other words, it moves on the surface given by
The above expressions can be substituted into either the radial or axial momentum equations (after removing the time derivative term) to solve for
For instance, substituting the above expression for
into the axial momentum equation leads to[9] But
as shown at the beginning of this derivation.
, we get This completes the required derivation.
Consider the problem where the fluid in the far stream exhibit uniform axial velocity
and rotates with angular velocity
This upstream motion corresponds to From these, we obtain indicating that in this case,
are simple linear functions of
, but with variable density, Chia-Shun Yih derived the necessary equation.
is some reference density, with corresponding Stokes streamfunction
defined such that Let us include the gravitational force acting in the negative