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28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

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

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 $\Psi$ by:^[1]

{\begin{aligned}u_{\rho }&=-{\frac {1}{\rho }}\,{\frac {\partial \Psi }{\partial z}},\\u_{z}&=+{\frac {1}{\rho }}\,{\frac {\partial \Psi }{\partial \rho }}.\end{aligned}}

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

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 $\Psi$ through:^[2]

{\begin{aligned}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{aligned}}

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:

{\boldsymbol {\omega }}=\nabla \times {\boldsymbol {v}}=\nabla \times \nabla \times {\boldsymbol {\psi }}

, where

\psi =-{\frac {\Psi }{r\sin \theta }}{\boldsymbol {\hat {\phi }}},

with ${\boldsymbol {\hat {\phi }}}$ the unit vector in the $\phi \,$ –direction.

Derivation of vorticity

{\boldsymbol {\omega }}

using a Stokes stream function

Consider the vorticity as defined by

{\boldsymbol {\omega }}=\nabla \times {\boldsymbol {v}}.

From the definition of the curl in spherical coordinates:

{\begin{aligned}\omega _{r}&={1 \over r\sin \theta }\left({\partial  \over \partial \theta }\left(v_{\phi }\sin \theta \right)-{\partial v_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {r}}},\\\omega _{\theta }&={1 \over r}\left({1 \over \sin \theta }{\partial v_{r} \over \partial \phi }-{\partial  \over \partial r}\left(rv_{\phi }\right)\right){\boldsymbol {\hat {\theta }}},\\\omega _{\phi }&={1 \over r}\left({\partial  \over \partial r}\left(rv_{\theta }\right)-{\partial v_{r} \over \partial \theta }\right){\boldsymbol {\hat {\phi }}}.\end{aligned}}

First notice that the $r$ and $\theta$ components are equal to 0. Secondly substitute $v_{r}$ and $v_{\theta }$ into $\omega _{\phi }.$ The result is:

{\begin{aligned}\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{aligned}}

Next the following algebra is performed:

{\begin{aligned}\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{aligned}}

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

{\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}}.

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.^[3]

Zero divergence

In cylindrical coordinates, the divergence of the velocity field u becomes:^[4]

{\begin{aligned}\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{aligned}}

as expected for an incompressible flow.

And in spherical coordinates:^[5]

{\begin{aligned}\nabla \cdot {\boldsymbol {u}}&={\frac {1}{r\,\sin(\theta )}}{\frac {\partial }{\partial \theta }}{\Bigl (}u_{\theta }\,\sin(\theta ){\Bigr )}+{\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{aligned}}

Notes

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References

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20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Originally published in 1879, the 6th extended edition appeared first in 1932.
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↑ Batchelor (1967), p. 78.
↑ Batchelor (1967), p. 79.
↑ E.g. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Batchelor (1967), p. 602.
↑ Batchelor (1967), p. 601.

[1] Batchelor (1967), p. 78.

[2] Batchelor (1967), p. 79.

[3] E.g. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[4] Batchelor (1967), p. 602.

[5] Batchelor (1967), p. 601.

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
+{{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'''<sub>d</sub> balances the force '''F'''<sub>g</sub> 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]] (&nbsp;''ρ''&nbsp;,&nbsp;''φ''&nbsp;,&nbsp;''z''&nbsp;), 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<sub>ρ</sub>'' and ''u<sub>z</sub>'' can be expressed in terms of the Stokes stream function <math>\Psi</math> by:<ref>Batchelor (1967), p. 78.</ref>
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+:<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 ''u<sub>φ</sub>'' does not depend on the stream function. Due to the axisymmetry, all three velocity components (&nbsp;''u<sub>ρ</sub>''&nbsp;,&nbsp;''u<sub>φ</sub>''&nbsp;,&nbsp;''u<sub>z</sub>''&nbsp;) 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&nbsp;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]] (&nbsp;''r''&nbsp;,&nbsp;''θ''&nbsp;,&nbsp;''φ''&nbsp;), ''r'' is the [[radial distance]] from the [[origin (mathematics)|origin]], ''θ'' is the [[zenith angle]] and ''φ'' is the [[azimuthal angle]]. In axisymmetric flow, with ''θ''&nbsp;=&nbsp;0 the rotational symmetry axis, the quantities describing the flow are again independent of the azimuth ''φ''. The flow velocity components ''u<sub>r</sub>'' and ''u<sub>θ</sub>'' 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 ''u<sub>φ</sub>'' 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&nbsp;ψ'', as before.
+===Vorticity===
+{{see also|Stream_function#Vorticity}}
+The [[vorticity]] is defined as:
+:<math>\boldsymbol{\omega} = \nabla \times \boldsymbol{v} = \nabla \times \nabla \times \boldsymbol{\psi}</math>, where <math>\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{v}.</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( v_\phi\sin\theta \right) - {\partial v_\theta \over \partial \phi}\right) \boldsymbol{\hat r}, \\
+\omega_\theta &= {1 \over r}\left({1 \over \sin\theta}{\partial v_r \over \partial \phi} - {\partial \over \partial r} \left( r v_\phi \right) \right) \boldsymbol{\hat \theta}, \\
+\omega_\phi   &= {1 \over r}\left({\partial \over \partial r} \left( r v_\theta \right) - {\partial v_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>v_r</math> and <math>v_\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}
+\\[1ex]
+\\[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 |last=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 }}</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}\Bigl( u_\theta\, \sin(\theta) \Bigr)
+  + \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>
+==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 }}<br>Reprinted in: {{cite book | first= G.G. | last=Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 1–16 | url= http://www.archive.org/details/mathphyspapers01stokrich }}
+{{DEFAULTSORT:Stokes stream function}}
+[[Category:Fluid dynamics]]

Wagner graph: Difference between revisions

Latest revision as of 04:43, 19 March 2013

Contents

Cylindrical coordinates

Spherical coordinates

Vorticity

Comparison with cylindrical

Alternative definition with opposite sign

Zero divergence

Notes

References

Navigation menu

Wagner graph: Difference between revisions

Latest revision as of 04:43, 19 March 2013

Cylindrical coordinates

Spherical coordinates

Vorticity

Comparison with cylindrical

Alternative definition with opposite sign

Zero divergence

Notes

References

Navigation menu

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