In fluid dynamics, the Stokes stream function is used to describe the 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—it has zero divergence. This stream function is named in honor of George Gabriel Stokes.
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
\Psi
\begin{align} u\rho&=-
| 1 | |
| \rho |
| \partial\Psi | |
| \partialz |
, \\ uz&=+
| 1 | |
| \rho |
| \partial\Psi | |
| \partial\rho |
. \end{align}
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.
In spherical coordinates ( r , θ , φ ), r is the radial distance from the 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
\Psi
\begin{align} ur&=+
| 1 | |
| r2\sin\theta |
| \partial\Psi | |
| \partial\theta |
, \\ u\theta&=-
| 1 | |
| r\sin\theta |
| \partial\Psi | |
| \partialr |
. \end{align}
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.
The vorticity is defined as:
\boldsymbol{\omega}=\nabla x \boldsymbol{u}=\nabla x \nabla x \boldsymbol{\psi}
| \boldsymbol{\psi}=- | \Psi |
| r\sin\theta |
\boldsymbol{\hat\phi},
with
\boldsymbol{\hat\phi}
\phi
As a result, from the calculation the vorticity vector is found to be equal to:
\boldsymbol{\omega}=\begin{pmatrix} 0\\[1ex] 0\\[1ex] \displaystyle-
| 1 | \left( | |
| r\sin\theta |
| \partial2\Psi | |
| \partialr2 |
+
| \sin\theta | |
| r2 |
{\partial\over\partial\theta}\left(
| 1 | |
| \sin\theta |
| \partial\Psi | |
| \partial\theta |
\right)\right) \end{pmatrix}.
The cylindrical and spherical coordinate systems are related through
z=r\cos\theta
\rho=r\sin\theta.
As explained in the general stream function article, definitions using an opposite sign convention – for the relationship between the Stokes stream function and flow velocity – are also in use.[3]
In cylindrical coordinates, the divergence of the velocity field u becomes:[4]
\begin{align} \nabla ⋅ \boldsymbol{u}&=
| 1 | |
| \rho |
| \partial | |
| \partial\rho |
l(\rhou\rhor)+
| \partialuz | |
| \partialz |
\\ &=
| 1 | |
| \rho |
| \partial | |
| \partial\rho |
\left(-
| \partial\Psi | |
| \partialz |
\right) +
| \partial | |
| \partialz |
\left(
| 1 | |
| \rho |
| \partial\Psi | |
| \partial\rho |
\right) =0, \end{align}
And in spherical coordinates:[5]
\begin{align} \nabla ⋅ \boldsymbol{u}&=
| 1 | |
| r\sin\theta |
| \partial | |
| \partial\theta |
(u\theta\sin\theta) +
| 1 | |
| r2 |
| \partial | |
| \partialr |
l(r2urr)\\ &=
| 1 | |
| r\sin\theta |
| \partial | |
| \partial\theta |
\left(-
| 1 | |
| r |
| \partial\Psi | |
| \partialr |
\right) +
| 1 | |
| r2 |
| \partial | |
| \partialr |
\left(
| 1 | |
| \sin\theta |
| \partial\Psi | |
| \partial\theta |
\right) =0. \end{align}
From calculus it is known that the gradient vector
\nabla\Psi
\Psi=C
\boldsymbol{u} ⋅ \nabla\Psi=0,
\boldsymbol{u}
\Psi,
\Psi
\nabla\Psi={\partial\Psi\over\partial\rho}\boldsymbol{e}\rho+{\partial\Psi\over\partialz}\boldsymbol{e}z
and
\boldsymbol{u}=u\rho\boldsymbol{e}\rho+uz\boldsymbol{e}z=-{1\over\rho}{\partial\Psi\over\partialz}\boldsymbol{e}\rho+{1\over\rho}{\partial\Psi\over\partial\rho}\boldsymbol{e}z.
So that
\nabla\Psi ⋅ \boldsymbol{u}={\partial\Psi\over\partial\rho}(-{1\over\rho}{\partial\Psi\over\partialz})+{\partial\Psi\over\partialz}{1\over\rho}{\partial\Psi\over\partial\rho}=0.
\nabla\Psi={\partial\Psi\over\partialr}\boldsymbol{e}r+{1\overr}{\partial\Psi\over\partial\theta}\boldsymbol{e}\theta
and
\boldsymbol{u}=ur\boldsymbol{e}r+u\theta\boldsymbol{e}\theta={1\overr2\sin\theta}{\partial\Psi\over\partial\theta}\boldsymbol{e}r-{1\overr\sin\theta}{\partial\Psi\over\partialr}\boldsymbol{e}\theta.
So that
\nabla\Psi ⋅ \boldsymbol{u}={\partial\Psi\over\partialr} ⋅ {1\overr2\sin\theta}{\partial\Psi\over\partial\theta}+{1\overr}{\partial\Psi\over\partial\theta} ⋅ (-{1\overr\sin\theta}{\partial\Psi\over\partialr})=0.
. George Batchelor . An Introduction to Fluid Dynamics . 1967 . Cambridge University Press . 0-521-66396-2 .
. Horace Lamb . 1994 . Hydrodynamics . Cambridge University Press . 6th. 978-0-521-45868-9 . Originally published in 1879, the 6th extended edition appeared first in 1932.