Hicks equation


In fluid dynamics, Hicks equation or 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 as coordinates in the sense of cylindrical coordinate system with corresponding flow velocity components denoted by, the stream function that defines the meridional motion can be defined as
that satisfies the continuity equation for axisymmetric flows automatically. The Hicks equation is then given by
where
where is the total head and is the circulation, both of them being conserved along streamlines. Here, is the pressure and is the fluid density. The functions and are known functions, usually prescribed at one of the boundary.

Derivation

Consider the axisymmetric flow in cylindrical coordinate system with velocity components and vorticity components. Since in axisymmetric flows, the vorticity components are
Continuity equation allows to define a stream function such that
. Therefore the azimuthal component of vorticity becomes
The inviscid momentum equations, where is the Bernoulli constant, is the fluid pressure and is the fluid density, when written for the axisymmetric flow field, becomes
in which the second equation may also be written as, where is the material derivative. This implies that the circulation round a material curve in the form of a circle centered on -axis is constant.
If the fluid motion is steady, the fluid particle moves along a streamline, in other words, it moves on the surface given by constant. It follows then that and, where. Therefore the radial and the azimuthal component of vorticity are
The components of and are locally parallel. The above expressions can be substituted into either the radial or axial momentum equations to solve for. For instance, substituting the above expression for into the axial momentum equation leads to
But can be expressed in terms of as shown at the beginning of this derivation. When is expressed in terms of, we get
This completes the required derivation.