Hicks equation

Consider the axisymmetric flow in cylindrical coordinate system ${\displaystyle (r,\theta ,z)}$ with velocity components ${\displaystyle (v_{r},v_{\theta },v_{z})}$ and vorticity components ${\displaystyle (\omega _{r},\omega _{\theta },\omega _{z})}$. Since ${\displaystyle \partial /\partial \theta =0}$ in axisymmetric flows, the vorticity components are

${\displaystyle \omega _{r}=-{\frac {\partial v_{\theta }}{\partial z}},\quad \omega _{\theta }={\frac {\partial v_{r}}{\partial z}}-{\frac {\partial v_{z}}{\partial r}},\quad \omega _{z}={\frac {1}{r}}{\frac {\partial (rv_{\theta })}{\partial r}}}$.

Continuity equation allows to define a stream function ${\displaystyle \psi (r,z)}$ such that

${\displaystyle v_{r}=-{\frac {1}{r}}{\frac {\partial \psi }{\partial z}},\quad v_{z}={\frac {1}{r}}{\frac {\partial \psi }{\partial r}}}$

(Note that the vorticity components ${\displaystyle \omega _{r}}$ and ${\displaystyle \omega _{z}}$ are related to ${\displaystyle rv_{\theta }}$ in exactly the same way that ${\displaystyle v_{r}}$ and ${\displaystyle v_{z}}$ are related to ${\displaystyle \psi }$). Therefore the azimuthal component of vorticity becomes

${\displaystyle \omega _{\theta }=-{\frac {1}{r}}\left({\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}\right).}$

The inviscid momentum equations ${\displaystyle \partial {\boldsymbol {v}}/\partial t-{\boldsymbol {v}}\times {\boldsymbol {\omega }}=-\nabla H}$, where ${\displaystyle H={\frac {1}{2}}(v_{r}^{2}+v_{\theta }^{2}+v_{z}^{2})+{\frac {p}{\rho }}}$ is the Bernoulli constant, ${\displaystyle p}$ is the fluid pressure and ${\displaystyle \rho }$ is the fluid density, when written for the axisymmetric flow field, becomes

${\displaystyle {\begin{aligned}v_{\theta }\omega _{z}-v_{z}\omega _{\theta }-{\frac {\partial v_{r}}{\partial t}}&={\frac {\partial H}{\partial r}},\\v_{z}\omega _{r}-v_{r}\omega _{z}-{\frac {\partial v_{\theta }}{\partial t}}&=0,\\v_{r}\omega _{\theta }-v_{\theta }\omega _{r}-{\frac {\partial v_{z}}{\partial t}}&={\frac {\partial H}{\partial z}}\end{aligned}}}$

in which the second equation may also be written as ${\displaystyle D(rv_{\theta })/Dt=0}$, where ${\displaystyle D/Dt}$ is the material derivative. This implies that the circulation ${\displaystyle 2\pi rv_{\theta }}$ round a material curve in the form of a circle centered on ${\displaystyle z}$-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 ${\displaystyle \psi =}$constant. It follows then that ${\displaystyle H=H(\psi )}$ and ${\displaystyle \Gamma =\Gamma (\psi )}$, where ${\displaystyle \Gamma =rv_{\theta }}$. Therefore the radial and the azimuthal component of vorticity are

${\displaystyle \omega _{r}=v_{r}{\frac {\mathrm {d} \Gamma }{\mathrm {d} \psi }},\quad \omega _{z}=v_{z}{\frac {\mathrm {d} \Gamma }{\mathrm {d} \psi }}}$.

The components of ${\displaystyle {\boldsymbol {v}}}$ and ${\displaystyle {\boldsymbol {\omega }}}$ are locally parallel. The above expressions can be substituted into either the radial or axial momentum equations (after removing the time derivative term) to solve for ${\displaystyle \omega _{\theta }}$. For instance, substituting the above expression for ${\displaystyle \omega _{r}}$ into the axial momentum equation leads to[9]

${\displaystyle {\begin{aligned}{\frac {\omega _{\theta }}{r}}&={\frac {v_{\theta }\omega _{r}}{rv_{r}}}+{\frac {1}{rv_{r}}}{\frac {\mathrm {d} H}{\mathrm {d} \psi }}{\frac {\partial \psi }{\partial z}}\\&={\frac {\Gamma }{r^{2}}}{\frac {\mathrm {d} \Gamma }{\mathrm {d} \psi }}-{\frac {\mathrm {d} H}{\mathrm {d} \psi }}.\end{aligned}}}$

But ${\displaystyle \omega _{\theta }}$ can be expressed in terms of ${\displaystyle \psi }$ as shown at the beginning of this derivation. When ${\displaystyle \omega _{\theta }}$ is expressed in terms of ${\displaystyle \psi }$, we get

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

This completes the required derivation.

Consider the problem where the fluid in the far stream exhibit uniform axial velocity ${\displaystyle U}$ and rotates with angular velocity ${\displaystyle \Omega }$. This upstream motion corresponds to

${\displaystyle \psi ={\frac {1}{2}}Ur^{2},\quad \Gamma =\Omega r^{2},\quad H={\frac {1}{2}}U^{2}+\Omega ^{2}r^{2}.}$

From these, we obtain

${\displaystyle H(\psi )={\frac {1}{2}}U^{2}+{\frac {2\Omega ^{2}}{U}}\psi ,\qquad \Gamma (\psi )={\frac {2\Omega }{U}}\psi }$

indicating that in this case, ${\displaystyle H}$ and ${\displaystyle \Gamma }$ are simple linear functions of ${\displaystyle \psi }$. The Hicks equation itself becomes

${\displaystyle {\frac {\partial ^{2}\psi }{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}={\frac {2\Omega ^{2}}{U}}r^{2}-{\frac {4\Omega ^{2}}{U^{2}}}\psi }$

which upon introducing ${\displaystyle \psi (r,z)=Ur^{2}/2+rf(r,z)}$ becomes

${\displaystyle {\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {\partial ^{2}f}{\partial z^{2}}}+\left(k^{2}-{\frac {1}{r^{2}}}\right)f=0}$

where ${\displaystyle k=2\Omega /U}$.

For an incompressible flow ${\displaystyle D\rho /Dt=0}$, but with variable density, Chia-Shun Yih derived the necessary equation. The velocity field is first transformed using Yih transformation

${\displaystyle (v_{r}',v_{\theta }',v_{z}')={\sqrt {\frac {\rho }{\rho _{0}}}}(v_{r},v_{\theta },v_{z})}$

where ${\displaystyle \rho _{0}}$ is some reference density, with corresponding Stokes streamfunction ${\displaystyle \psi '}$ defined such that

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

Let us include the gravitational force acting in the negative ${\displaystyle z}$ direction. The Yih equation is then given by[10][11]

${\displaystyle {\frac {\partial ^{2}\psi '}{\partial r^{2}}}-{\frac {1}{r}}{\frac {\partial \psi '}{\partial r}}+{\frac {\partial ^{2}\psi '}{\partial z^{2}}}=r^{2}{\frac {\mathrm {d} H}{\mathrm {d} \psi '}}-r^{2}{\frac {\mathrm {d} \rho }{\mathrm {d} \psi '}}{\frac {g}{\rho _{0}}}z-\Gamma {\frac {\mathrm {d} \Gamma }{\mathrm {d} \psi '}}}$

where

${\displaystyle H(\psi ')={\frac {p}{\rho _{0}}}+{\frac {\rho }{2\rho _{0}}}(v_{r}'^{2}+v_{\theta }'^{2}+v_{z}'^{2})+{\frac {\rho }{\rho _{0}}}gz,\quad \Gamma (\psi ')=rv_{\theta }'}$