2A07:7E81:7A08:0:183D:3E14:45A6:675C: /* Vorticity */

2025-06-02T10:01:01Z

Vorticity

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{{for|two-dimensional flows|Stream function}}
[[File:Stokes sphere.svg|right|thumb|150px|Streamlines around a [[sphere]] in [[axisymmetric]] [[Stokes flow]]. At [[terminal velocity]] the [[drag force]] '''F'''d balances the force '''F'''g propelling the object.]]
In [[fluid dynamics]], the '''Stokes stream function''' is used to describe the [[Streamlines, streaklines, and pathlines|streamlines]] and [[flow velocity]] in a three-dimensional [[incompressible flow]] with [[axisymmetry]]. A surface with a constant value of the Stokes stream function encloses a [[streamtube]], everywhere [[tangential]] to the flow velocity vectors. Further, the [[volume]] [[flux]] within this streamtube is constant, and all the streamlines of the flow are located on this surface. The [[velocity field]] associated with the Stokes stream function is [[solenoidal vector field|solenoidal]]—it has zero [[divergence]]. This stream function is named in honor of [[George Gabriel Stokes]].

==Cylindrical coordinates==
[[File:Cylindrical with grid.svg|thumb|300px|A point plotted with cylindrical coordinates.]]
Consider a [[cylindrical coordinate system]] ( ''ρ'' , ''φ'' , ''z'' ), with the ''z''–axis the line around which the incompressible flow is axisymmetrical, ''φ'' the [[azimuthal angle]] and ''ρ'' the distance to the ''z''–axis. Then the flow velocity components ''uρ'' and ''uz'' can be expressed in terms of the Stokes stream function <math>\Psi</math> by:<ref>Batchelor (1967), p. 78.</ref>

:<math>
\begin{align}
u_\rho &= - \frac{1}{\rho}\, \frac{\partial \Psi}{\partial z},
\\
u_z &= + \frac{1}{\rho}\, \frac{\partial \Psi}{\partial \rho}.
\end{align}
</math>

The azimuthal velocity component ''uφ'' does not depend on the stream function. Due to the axisymmetry, all three velocity components ( ''uρ'' , ''uφ'' , ''uz'' ) only depend on ''ρ'' and ''z'' and not on the azimuth ''φ''.

The volume flux, through the surface bounded by a constant value ''ψ'' of the Stokes stream function, is equal to ''2π ψ''.

==Spherical coordinates==
[[File:spherical with grid.svg|thumb|right|300px|A point plotted using the spherical coordinate system]]
In [[spherical coordinate system|spherical coordinates]] ( ''r'' , ''θ'' , ''φ'' ), ''r'' is the [[Polar coordinate system|radial distance]] from the [[origin (mathematics)|origin]], ''θ'' is the [[zenith angle]] and ''φ'' is the [[azimuthal angle]]. In axisymmetric flow, with ''θ'' = 0 the rotational symmetry axis, the quantities describing the flow are again independent of the azimuth ''φ''. The flow velocity components ''ur'' and ''uθ'' are related to the Stokes stream function <math>\Psi</math> through:<ref>Batchelor (1967), p. 79.</ref>

:<math>
\begin{align}
u_r &= + \frac{1}{r^2\, \sin \theta}\, \frac{\partial \Psi}{\partial \theta},
\\
u_\theta &= - \frac{1}{r\, \sin \theta}\, \frac{\partial \Psi}{\partial r}.
\end{align}
</math>

Again, the azimuthal velocity component ''uφ'' is not a function of the Stokes stream function ''ψ''. The volume flux through a stream tube, bounded by a surface of constant ''ψ'', equals ''2π ψ'', as before.

===Vorticity===
{{see also|Stream_function#Vorticity}}
The [[vorticity]] is defined as:

:<math>\boldsymbol{\omega} = \nabla \times \boldsymbol{u} = \nabla \times \nabla \times \boldsymbol{\psi}</math>, where <math>\boldsymbol{\psi}=\frac{\Psi}{r\sin\theta}\boldsymbol{\hat \phi},</math>

with <math>\boldsymbol{\hat \phi}</math> the [[unit vector]] in the <math>\phi\,</math>–direction.
:{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
!Derivation of vorticity <math>\boldsymbol{\omega}</math> using a Stokes stream function
|-
|Consider the vorticity as defined by

:<math>\boldsymbol{\omega} = \nabla \times \boldsymbol{u}.</math>

From the definition of the [[del in cylindrical and spherical coordinates|curl in spherical coordinates]]:

:<math>
\begin{align}
\omega_r &= {1 \over r\sin\theta}\left({\partial \over \partial \theta} \left( u_\phi\sin\theta \right) - {\partial u_\theta \over \partial \phi}\right) \boldsymbol{\hat r}, \\
\omega_\theta &= {1 \over r}\left({1 \over \sin\theta}{\partial u_r \over \partial \phi} - {\partial \over \partial r} \left( r u_\phi \right) \right) \boldsymbol{\hat \theta}, \\
\omega_\phi &= {1 \over r}\left({\partial \over \partial r} \left( r u_\theta \right) - {\partial u_r \over \partial \theta}\right) \boldsymbol{\hat \phi}.
\end{align}
</math>

First notice that the <math>r</math> and <math>\theta</math> components are equal to 0. Secondly substitute <math>u_r</math> and <math>u_\theta</math> into <math>\omega_\phi.</math> The result is:

:<math>
\begin{align}
\omega_r &= 0, \\
\omega_\theta &= 0, \\
\omega_\phi &= {1 \over r}\left({\partial \over \partial r} \left( r \left(-\frac{1}{r \sin\theta}\frac{\partial\Psi}{\partial r}\right) \right) -
{\partial \over \partial \theta}\left(\frac{1}{r^2 \sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right).
\end{align}
</math>

Next the following algebra is performed:

:<math>
\begin{align}
\omega_\phi &= {1 \over r}\left(-\frac{1}{\sin\theta}\left({\partial \over \partial r} \left(\frac{\partial\Psi}{\partial r}\right)\right) -
\frac{1}{r^2 }{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right) \\
&= {1 \over r}\left(-\frac{1}{\sin\theta}\left(\frac{\partial^2\Psi}{\partial r^2}\right) -
\frac{\sin\theta}{r^2 \sin\theta}{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right) \\
&= -\frac{1}{r\sin\theta} \left(\frac{\partial^2\Psi}{\partial r^2} + \frac{\sin\theta}{r^2}{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right).
\end{align}
</math>
|}

As a result, from the calculation the vorticity vector is found to be equal to:

:<math>\boldsymbol{\omega} =
\begin{pmatrix}
0 \\[1ex]
0 \\[1ex]
\displaystyle -\frac{1}{r\sin\theta} \left(\frac{\partial^2\Psi}{\partial r^2} + \frac{\sin\theta}{r^2}{\partial \over \partial \theta}\left(\frac{1}{\sin\theta}\frac{\partial\Psi}{\partial \theta}\right)\right)
\end{pmatrix}.
</math>

===Comparison with cylindrical===
The cylindrical and spherical coordinate systems are related through

:<math>z = r\, \cos\theta\,</math> {{pad|3em}} and {{pad|3em}} <math>\rho = r\, \sin\theta.\,</math>

== Alternative definition with opposite sign ==
As explained in the general [[Stream_function#Alternative_definition_.28opposite_sign.29|stream function]] article, definitions using an opposite sign convention – for the relationship between the Stokes stream function and flow velocity – are also in use.<ref>E.g. {{cite journal |last1=Brenner |year=1961 |journal=Chemical Engineering Science |volume=16 |pages=242–251 |doi=10.1016/0009-2509(61)80035-3 |title=The slow motion of a sphere through a viscous fluid towards a plane surface |first1=Howard |issue=3–4 |bibcode=1961ChEnS..16..242B }}</ref>

==Zero divergence==
In cylindrical coordinates, the [[divergence]] of the velocity field '''u''' becomes:<ref>Batchelor (1967), p. 602.</ref>

:<math>
\begin{align}
\nabla \cdot \boldsymbol{u} &=
\frac{1}{\rho} \frac{\partial}{\partial \rho}\Bigl( \rho\, u_\rho \Bigr)
+ \frac{\partial u_z}{\partial z}
\\
&=
\frac{1}{\rho} \frac{\partial}{\partial \rho} \left( - \frac{\partial \Psi}{\partial z} \right)
+ \frac{\partial}{\partial z} \left( \frac{1}{\rho} \frac{\partial \Psi}{\partial \rho} \right)
= 0,
\end{align}
</math>
as expected for an incompressible flow.

And in spherical coordinates:<ref>Batchelor (1967), p. 601.</ref>

:<math>
\begin{align}
\nabla \cdot \boldsymbol{u} &=
\frac{1}{r\, \sin\theta} \frac{\partial}{\partial \theta}( u_\theta\, \sin\theta)
+ \frac{1}{r^2} \frac{\partial}{\partial r}\Bigl( r^2\, u_r \Bigr)
\\
&=
\frac{1}{r\, \sin\theta} \frac{\partial}{\partial \theta} \left( - \frac{1}{r} \frac{\partial \Psi}{\partial r} \right)
+ \frac{1}{r^2} \frac{\partial}{\partial r} \left( \frac{1}{\sin\theta} \frac{\partial \Psi}{\partial \theta} \right)
= 0.
\end{align}
</math>

==Streamlines as curves of constant stream function==
From calculus it is known that the [[gradient]] vector <math>\nabla \Psi</math> is normal to the curve <math>\Psi = C</math> (see e.g. [[Level set#Level sets versus the gradient]]). If it is shown that everywhere <math>\boldsymbol{u} \cdot \nabla \Psi = 0,</math> using the formula for <math>\boldsymbol{u}</math> in terms of <math>\Psi,</math> then this proves that level curves of <math>\Psi</math> are streamlines.

;Cylindrical coordinates:
In cylindrical coordinates,

:<math>\nabla \Psi = {\partial \Psi \over \partial \rho} \boldsymbol{e}_\rho + {\partial \Psi \over \partial z} \boldsymbol{e}_z</math>.

and

:<math>
\boldsymbol{u} = u_\rho \boldsymbol{e}_\rho + u_z \boldsymbol{e}_z = - {1 \over \rho} {\partial \Psi \over \partial z} \boldsymbol{e}_\rho + {1 \over \rho} {\partial \Psi \over \partial \rho} \boldsymbol{e}_z.
</math>

So that

:<math>\nabla \Psi \cdot \boldsymbol{u} = {\partial \Psi \over \partial \rho} (- {1 \over \rho} {\partial \Psi \over \partial z}) + {\partial \Psi \over \partial z} {1 \over \rho} {\partial \Psi \over \partial \rho} = 0. </math>

;Spherical coordinates:
And in spherical coordinates

:<math>\nabla \Psi = {\partial \Psi \over \partial r} \boldsymbol{e}_r + {1 \over r} {\partial \Psi \over \partial \theta} \boldsymbol{e}_\theta</math>

and

:<math>
\boldsymbol{u} = u_r \boldsymbol{e}_r + u_\theta \boldsymbol{e}_\theta = {1 \over r^2 \sin \theta} {\partial \Psi \over \partial \theta} \boldsymbol{e}_r - {1 \over r \sin \theta} {\partial \Psi \over \partial r} \boldsymbol{e}_\theta .
</math>

So that

:<math>\nabla \Psi \cdot \boldsymbol{u} = {\partial \Psi \over \partial r} \cdot {1 \over r^2 \sin \theta} {\partial \Psi \over \partial \theta} + {1 \over r} {\partial \Psi \over \partial \theta} \cdot \Big( - {1 \over r \sin \theta} {\partial \Psi \over \partial r} \Big) = 0 .</math>

==Notes==
{{reflist}}

==References==
*{{cite book | first=G.K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 }}
*{{cite book | first=H. | last=Lamb | authorlink=Horace Lamb | year=1994 | title=Hydrodynamics | publisher=Cambridge University Press | edition=6th| isbn=978-0-521-45868-9 }} Originally published in 1879, the 6th extended edition appeared first in 1932.
*{{cite journal | first=G.G. | last=Stokes | authorlink=George Gabriel Stokes | year= 1842 | title= On the steady motion of incompressible fluids | journal= Transactions of the Cambridge Philosophical Society | volume= 7 | pages= 439–453 | bibcode=1848TCaPS...7..439S }} Reprinted in: {{cite book | first= G.G. | last=Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= [https://archive.org/details/mathphyspapers01stokrich/page/n18 1]–16 | url= https://archive.org/details/mathphyspapers01stokrich }}

{{DEFAULTSORT:Stokes stream function}}
[[Category:Fluid dynamics]]

Stokes stream function - Revision history

2A07:7E81:7A08:0:183D:3E14:45A6:675C: /* Vorticity */