Wagner graph

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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 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.

Cylindrical coordinates

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 u_ρ and u_z can be expressed in terms of the Stokes stream function $\mathrm{\Psi }$ by:

$\begin{array}{rl}{u}_{\rho }& =-\frac{1}{\rho }\frac{\partial \mathrm{\Psi }}{\partial z},\\ u_z& =+\frac{1}{\rho }\frac{\partial \mathrm{\Psi }}{\partial \rho }.\end{array}$

The azimuthal velocity component u_φ does not depend on the stream function. Due to the axisymmetry, all three velocity components ( u_ρ , u_φ , u_z ) 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

A point plotted using the spherical coordinate system

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 u_r and u_θ are related to the Stokes stream function $\mathrm{\Psi }$ through:

$\begin{array}{rl}{u}_r& =+\frac{1}{r^2\sin (\theta )}\frac{\partial \mathrm{\Psi }}{\partial \theta },\\ u_{\theta }& =-\frac{1}{r\sin (\theta )}\frac{\partial \mathrm{\Psi }}{\partial r}.\end{array}$

Again, the azimuthal velocity component u_φ 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

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The vorticity is defined as:

$\mathit{\omega }=\mathrm{\nabla }\times \mathit{v}=\mathrm{\nabla }\times \mathrm{\nabla }\times \mathit{\psi }$, where $\psi =-\frac{\mathrm{\Psi }}{r\sin \theta }\hat{\mathit{\varphi }},$

with $\hat{\mathit{\varphi }}$ the unit vector in the $\varphi$–direction.

Derivation of vorticity $\mathit{\omega }$ using a Stokes stream function

Consider the vorticity as defined by $\mathit{\omega }=\mathrm{\nabla }\times \mathit{v}.$ From the definition of the curl in spherical coordinates: $\begin{array}{rl}{\omega }_r& =\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(v_{\varphi }\sin \theta )-\frac{\partial v_{\theta }}{\partial \varphi })\hat{\mathit{r}},\\ {\omega }_{\theta }& =\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial v_r}{\partial \varphi }-\frac{\partial }{\partial r}\left(r{v}_{\varphi }\right))\hat{\mathit{\theta }},\\ {\omega }_{\varphi }& =\frac{1}{r}(\frac{\partial }{\partial r}\left(r{v}_{\theta }\right)-\frac{\partial v_r}{\partial \theta })\hat{\mathit{\varphi }}.\end{array}$ First notice that the $r$ and $\theta$ components are equal to 0. Secondly substitute $v_r$ and $v_{\theta }$ into ${\omega }_{\varphi }.$ The result is: $\begin{array}{rl}{\omega }_r& =0,\\ {\omega }_{\theta }& =0,\\ {\omega }_{\varphi }& =\frac{1}{r}(\frac{\partial }{\partial r}\left(r(-\frac{1}{r\sin \theta }\frac{\partial \mathrm{\Psi }}{\partial r})\right)-\frac{\partial }{\partial \theta }\left(\frac{1}{r^2\sin \theta }\frac{\partial \mathrm{\Psi }}{\partial \theta }\right)).\end{array}$ Next the following algebra is performed: $\begin{array}{rl}{\omega }_{\varphi }& =\frac{1}{r}(-\frac{1}{\sin \theta }\left(\frac{\partial }{\partial r}\left(\frac{\partial \mathrm{\Psi }}{\partial r}\right)\right)-\frac{1}{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{\sin \theta }\frac{\partial \mathrm{\Psi }}{\partial \theta }\right))\\ & =\frac{1}{r}(-\frac{1}{\sin \theta }\left(\frac{{\partial }^2\mathrm{\Psi }}{\partial r^2}\right)-\frac{\sin \theta }{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\frac{1}{\sin \theta }\frac{\partial \mathrm{\Psi }}{\partial \theta }\right))\\ & =-\frac{1}{r\sin \theta }(\frac{{\partial }^2\mathrm{\Psi }}{\partial r^2}+\frac{\sin \theta }{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{\sin \theta }\frac{\partial \mathrm{\Psi }}{\partial \theta }\right)).\end{array}$

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

$\mathit{\omega }=\left(\begin{array}{c}0\\ 0\\ -\frac{1}{r\sin \theta }(\frac{{\partial }^2\mathrm{\Psi }}{\partial r^2}+\frac{\sin \theta }{r^2}\frac{\partial }{\partial \theta }\left(\frac{1}{\sin \theta }\frac{\partial \mathrm{\Psi }}{\partial \theta }\right))\end{array}\right).$

Comparison with cylindrical

The cylindrical and spherical coordinate systems are related through

$z=r\cos (\theta )$ Template:Pad and Template:Pad $\rho =r\sin (\theta ).$

Alternative definition with opposite sign

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.

Zero divergence

In cylindrical coordinates, the divergence of the velocity field u becomes:

$\begin{array}{rl}\mathrm{\nabla }\cdot \mathit{u}& =\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho u_{\rho }\right)+\frac{\partial u_z}{\partial z}\\ & =\frac{1}{\rho }\frac{\partial }{\partial \rho }(-\frac{\partial \mathrm{\Psi }}{\partial z})+\frac{\partial }{\partial z}\left(\frac{1}{\rho }\frac{\partial \mathrm{\Psi }}{\partial \rho }\right)=0,\end{array}$

as expected for an incompressible flow.

And in spherical coordinates:

$\begin{array}{rl}\mathrm{\nabla }\cdot \mathit{u}& =\frac{1}{r\sin (\theta )}\frac{\partial }{\partial \theta }(u_{\theta }\sin (\theta ))+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{u}_r\right)\\ & =\frac{1}{r\sin (\theta )}\frac{\partial }{\partial \theta }(-\frac{1}{r}\frac{\partial \mathrm{\Psi }}{\partial r})+\frac{1}{r^2}\frac{\partial }{\partial r}\left(\frac{1}{\sin (\theta )}\frac{\partial \mathrm{\Psi }}{\partial \theta }\right)=0.\end{array}$

Notes

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References

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