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.[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 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.

Derivation

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]

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

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.

References

1. Hicks, W. M. (1898). Researches in vortex motion. Part III. On spiral or gyrostatic vortex aggregates. Proceedings of the Royal Society of London, 62(379–387), 332–338. https://royalsocietypublishing.org/doi/pdf/10.1098/rspl.1897.0119
2. Hicks, W. M. (1899). II. Researches in vortex motion.—Part III. On spiral or gyrostatic vortex aggregates. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, (192), 33–99. https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.1899.0002
3. Smith, S. G. L., & Hattori, Y. (2012). Axisymmetric magnetic vortices with swirl. Communications in Nonlinear Science and Numerical Simulation, 17(5), 2101–2107.
4. Bragg, S. L. & Hawthorne, W. R. (1950). Some exact solutions of the flow through annular cascade actuator discs. Journal of the Aeronautical Sciences, 17(4), 243–249
5. Long, R. R. (1953). Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. Journal of Meteorology, 10(3), 197–203.
6. Squire, H. B. (1956). Rotating fluids. Surveys in Mechanics. A collection of Surveys of the present position of Research in some branches of Mechanics, written in Commemoration of the 70th Birthday of Geoffrey Ingram Taylor, Eds. G. K. Batchelor and R. M. Davies. 139–169
7. Stokes, G. (1842). On the steady motion of incompressible fluids Trans. Camb. Phil. Soc. VII, 349.
8. Lamb, H. (1993). Hydrodynamics. Cambridge university press.
9. Batchelor, G. K. (1967). An introduction to fluid dynamics. Section 7.5. Cambridge university press. section 7.5, p. 543-545