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
(r,\theta,z)
(vr,v\theta,vz)
\psi
rvr=-
\partial\psi | |
\partialz |
, rvz=
\partial\psi | |
\partialr |
that satisfies the continuity equation for axisymmetric flows automatically. The Hicks equation is then given by [9]
\partial2\psi | |
\partialr2 |
-
1 | |
r |
\partial\psi | |
\partialr |
+
\partial2\psi | |
\partialz2 |
=r2
dH | |
d\psi |
-\Gamma
d\Gamma | |
d\psi |
where
H(\psi)=
p | |
\rho |
+
1 | |
2 |
2), | |
(v | |
z |
\Gamma(\psi)=rv\theta
where
H(\psi)
2\pi\Gamma
p
\rho
H(\psi)
\Gamma(\psi)
H(\psi)
\Gamma(\psi)
Consider the axisymmetric flow in cylindrical coordinate system
(r,\theta,z)
(vr,v\theta,vz)
(\omegar,\omega\theta,\omegaz)
\partial/\partial\theta=0
\omegar=-
\partialv\theta | |
\partialz |
, \omega\theta=
\partialvr | |
\partialz |
-
\partialvz | |
\partialr |
, \omegaz=
1 | |
r |
\partial(rv\theta) | |
\partialr |
Continuity equation allows to define a stream function
\psi(r,z)
v | ||||
|
\partial\psi | |
\partialz |
, vz=
1 | |
r |
\partial\psi | |
\partialr |
(Note that the vorticity components
\omegar
\omegaz
rv\theta
vr
vz
\psi
\omega\theta=-
1 | \left( | |
r |
\partial2\psi | |
\partialr2 |
-
1 | |
r |
\partial\psi | |
\partialr |
+
\partial2\psi | |
\partialz2 |
\right).
The inviscid momentum equations
\partial\boldsymbol{v}/\partialt-\boldsymbol{v} x \boldsymbol{\omega}=-\nablaH
H=
1 | |
2 |
2) | |
(v | |
z |
+
p | |
\rho |
p
\rho
\begin{align} v\theta\omegaz-vz\omega\theta-
\partialvr | |
\partialt |
&=
\partialH | |
\partialr |
,\\ vz\omegar-vr\omegaz-
\partialv\theta | |
\partialt |
&=0,\\ vr\omega\theta-v\theta\omegar-
\partialvz | |
\partialt |
&=
\partialH | |
\partialz |
\end{align}
in which the second equation may also be written as
D(rv\theta)/Dt=0
D/Dt
2\pirv\theta
z
If the fluid motion is steady, the fluid particle moves along a streamline, in other words, it moves on the surface given by
\psi=
H=H(\psi)
\Gamma=\Gamma(\psi)
\Gamma=rv\theta
\omegar=
v | ||||
|
, \omegaz=
v | ||||
|
The components of
\boldsymbol{v}
\boldsymbol{\omega}
\omega\theta
\omegar
\begin{align} | \omega\theta |
r |
&=
v\theta\omegar | |
rvr |
+
1 | |
rvr |
dH | |
d\psi |
\partial\psi | |
\partialz |
\\ &=
\Gamma | |
r2 |
d\Gamma | - | |
d\psi |
dH | |
d\psi |
. \end{align}
But
\omega\theta
\psi
\omega\theta
\psi
\partial2\psi | |
\partialr2 |
-
1 | |
r |
\partial\psi | |
\partialr |
+
\partial2\psi | |
\partialz2 |
=r2
dH | |
d\psi |
-\Gamma
d\Gamma | |
d\psi |
.
This completes the required derivation.
Consider the problem where the fluid in the far stream exhibit uniform axial velocity
U
\Omega
\psi=
1 | |
2 |
Ur2, \Gamma=\Omegar2, H=
1 | |
2 |
U2+\Omega2r2.
From these, we obtain
H(\psi)=
1 | |
2 |
U2+
2\Omega2 | |
U |
\psi, \Gamma(\psi)=
2\Omega | |
U |
\psi
indicating that in this case,
H
\Gamma
\psi
\partial2\psi | |
\partialr2 |
-
1 | |
r |
\partial\psi | |
\partialr |
+
\partial2\psi | |
\partialz2 |
=
2\Omega2 | |
U |
r2-
4\Omega2 | |
U2 |
\psi
which upon introducing
\psi(r,z)=Ur2/2+rf(r,z)
\partial2f | |
\partialr2 |
+
1 | |
r |
\partialf | |
\partialr |
+
\partial2f | |
\partialz2 |
+
| ||||
\left(k |
\right)f=0
where
k=2\Omega/U
For an incompressible flow
D\rho/Dt=0
(vr',v\theta',vz')=\sqrt{
\rho | |
\rho0 |
where
\rho0
\psi'
rvr'=-
\partial\psi' | |
\partialz |
, rvz'=
\partial\psi' | |
\partialr |
.
Let us include the gravitational force acting in the negative
z
\partial2\psi' | |
\partialr2 |
-
1 | |
r |
\partial\psi' | |
\partialr |
+
\partial2\psi' | |
\partialz2 |
=r2
dH | |
d\psi' |
-r2
d\rho | |
d\psi' |
g | |
\rho0 |
z-\Gamma
d\Gamma | |
d\psi' |
where
H(\psi')=
p | |
\rho0 |
+
\rho | |
2\rho0 |
2) | |
(v | |
z' |
+
\rho | |
\rho0 |
gz, \Gamma(\psi')=rv\theta'