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Stokes stream function
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<h2 id="cylindrical-coordinates">Cylindrical coordinates</h2> <p>Consider a <a href="/facts/Cylindrical_coordinate_system/1Vc6lhxD">cylindrical coordinate system</a> ( <i>ρ</i> , <i>φ</i> , <i>z</i> ), with the <i>z</i>–axis the line around which the incompressible flow is axisymmetrical, <i>φ</i> the <a href="/facts/Azimuthal_angle/kEyt9nJ3">azimuthal angle</a> and <i>ρ</i> the distance to the <i>z</i>–axis. Then the flow velocity components <i>uρ</i> and <i>uz</i> can be expressed in terms of the Stokes stream function Ψ {\displaystyle \Psi } by:<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> u ρ = − 1 ρ ∂ Ψ ∂ z , u z = + 1 ρ ∂ Ψ ∂ ρ . {\displaystyle {\begin{aligned}u_{\rho }&=-{\frac {1}{\rho }}\,{\frac {\partial \Psi }{\partial z}},\\u_{z}&=+{\frac {1}{\rho }}\,{\frac {\partial \Psi }{\partial \rho }}.\end{aligned}}} <p>The azimuthal velocity component <i>uφ</i> does not depend on the stream function. Due to the axisymmetry, all three velocity components ( <i>uρ</i> , <i>uφ</i> , <i>uz</i> ) only depend on <i>ρ</i> and <i>z</i> and not on the azimuth <i>φ</i>. </p><p>The volume flux, through the surface bounded by a constant value <i>ψ</i> of the Stokes stream function, is equal to <i>2π ψ</i>. </p> <h2 id="spherical-coordinates">Spherical coordinates</h2> <p>In <a href="/facts/Spherical_coordinate_system/5cVnNof3">spherical coordinates</a> ( <i>r</i> , <i>θ</i> , <i>φ</i> ), <i>r</i> is the <a href="/facts/Polar_coordinate_system/HBko4YoN">radial distance</a> from the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>, <i>θ</i> is the <a href="/facts/Zenith_angle/tcm21e6t">zenith angle</a> and <i>φ</i> is the <a href="/facts/Azimuthal_angle/kEyt9nJ3">azimuthal angle</a>. In axisymmetric flow, with <i>θ</i> = 0 the rotational symmetry axis, the quantities describing the flow are again independent of the azimuth <i>φ</i>. The flow velocity components <i>ur</i> and <i>uθ</i> are related to the Stokes stream function Ψ {\displaystyle \Psi } through:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> u r = + 1 r 2 sin θ ∂ Ψ ∂ θ , u θ = − 1 r sin θ ∂ Ψ ∂ r . {\displaystyle {\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}}} <p>Again, the azimuthal velocity component <i>uφ</i> is not a function of the Stokes stream function <i>ψ</i>. The volume flux through a stream tube, bounded by a surface of constant <i>ψ</i>, equals <i>2π ψ</i>, as before. </p> <h3>Vorticity</h3> <p class="note">See also: <a href="/facts/Stream_function/nEoYTGAl">Stream_function § Vorticity</a></p> <p>The <a href="/facts/Vorticity/2oQ6eT4d">vorticity</a> is defined as: </p> ω = ∇ × u = ∇ × ∇ × ψ {\displaystyle {\boldsymbol {\omega }}=\nabla \times {\boldsymbol {u}}=\nabla \times \nabla \times {\boldsymbol {\psi }}} , where ψ = Ψ r sin θ ϕ ^ , {\displaystyle {\boldsymbol {\psi }}={\frac {\Psi }{r\sin \theta }}{\boldsymbol {\hat {\phi }}},} <p>with ϕ ^ {\displaystyle {\boldsymbol {\hat {\phi }}}} the <a href="/facts/Unit_vector/GyaXneDn">unit vector</a> in the ϕ {\displaystyle \phi \,} –direction. </p> <p>As a result, from the calculation the vorticity vector is found to be equal to: </p> ω = ( 0 0 − 1 r sin θ ( ∂ 2 Ψ ∂ r 2 + sin θ r 2 ∂ ∂ θ ( 1 sin θ ∂ Ψ ∂ θ ) ) ) . {\displaystyle {\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}}.} <h3>Comparison with cylindrical</h3> <p>The cylindrical and spherical coordinate systems are related through </p> z = r cos θ {\displaystyle z=r\,\cos \theta \,} and ρ = r sin θ . {\displaystyle \rho =r\,\sin \theta .\,} <h2 id="alternative-definition-with-opposite-sign">Alternative definition with opposite sign</h2> <p>As explained in the general <a href="/facts/Stream_function/nEoYTGAl">stream function</a> article, definitions using an opposite sign convention – for the relationship between the Stokes stream function and flow velocity – are also in use.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="zero-divergence">Zero divergence</h2> <p>In cylindrical coordinates, the <a href="/facts/Divergence/zpwwkFoU">divergence</a> of the velocity field u becomes:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> ∇ ⋅ u = 1 ρ ∂ ∂ ρ ( ρ u ρ ) + ∂ u z ∂ z = 1 ρ ∂ ∂ ρ ( − ∂ Ψ ∂ z ) + ∂ ∂ z ( 1 ρ ∂ Ψ ∂ ρ ) = 0 , {\displaystyle {\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}}} <p>as expected for an incompressible flow. </p><p>And in spherical coordinates:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> ∇ ⋅ u = 1 r sin θ ∂ ∂ θ ( u θ sin θ ) + 1 r 2 ∂ ∂ r ( r 2 u r ) = 1 r sin θ ∂ ∂ θ ( − 1 r ∂ Ψ ∂ r ) + 1 r 2 ∂ ∂ r ( 1 sin θ ∂ Ψ ∂ θ ) = 0. {\displaystyle {\begin{aligned}\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{aligned}}} <h2 id="streamlines-as-curves-of-constant-stream-function">Streamlines as curves of constant stream function</h2> <p>From calculus it is known that the <a href="/facts/Gradient/TsR7uukj">gradient</a> vector ∇ Ψ {\displaystyle \nabla \Psi } is normal to the curve Ψ = C {\displaystyle \Psi =C} (see e.g. <a href="/facts/Level_set/JUmNbdNh">Level set#Level sets versus the gradient</a>). If it is shown that everywhere u ⋅ ∇ Ψ = 0 , {\displaystyle {\boldsymbol {u}}\cdot \nabla \Psi =0,} using the formula for u {\displaystyle {\boldsymbol {u}}} in terms of Ψ , {\displaystyle \Psi ,} then this proves that level curves of Ψ {\displaystyle \Psi } are streamlines. </p> Cylindrical coordinates <p>In cylindrical coordinates, </p> ∇ Ψ = ∂ Ψ ∂ ρ e ρ + ∂ Ψ ∂ z e z {\displaystyle \nabla \Psi ={\partial \Psi \over \partial \rho }{\boldsymbol {e}}_{\rho }+{\partial \Psi \over \partial z}{\boldsymbol {e}}_{z}} . <p>and </p> u = u ρ e ρ + u z e z = − 1 ρ ∂ Ψ ∂ z e ρ + 1 ρ ∂ Ψ ∂ ρ e z . {\displaystyle {\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}.} <p>So that </p> ∇ Ψ ⋅ u = ∂ Ψ ∂ ρ ( − 1 ρ ∂ Ψ ∂ z ) + ∂ Ψ ∂ z 1 ρ ∂ Ψ ∂ ρ = 0. {\displaystyle \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.} Spherical coordinates <p>And in spherical coordinates </p> ∇ Ψ = ∂ Ψ ∂ r e r + 1 r ∂ Ψ ∂ θ e θ {\displaystyle \nabla \Psi ={\partial \Psi \over \partial r}{\boldsymbol {e}}_{r}+{1 \over r}{\partial \Psi \over \partial \theta }{\boldsymbol {e}}_{\theta }} <p>and </p> u = u r e r + u θ e θ = 1 r 2 sin θ ∂ Ψ ∂ θ e r − 1 r sin θ ∂ Ψ ∂ r e θ . {\displaystyle {\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 }.} <p>So that </p> ∇ Ψ ⋅ u = ∂ Ψ ∂ r ⋅ 1 r 2 sin θ ∂ Ψ ∂ θ + 1 r ∂ Ψ ∂ θ ⋅ ( − 1 r sin θ ∂ Ψ ∂ r ) = 0. {\displaystyle \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.} <h2 id="notes">Notes</h2> <ul><li><a href="/facts/George_Batchelor/peMzQBl8">Batchelor, G.K.</a> (1967). <i>An Introduction to Fluid Dynamics</i>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-66396-2.</li> <li><a href="/facts/Horace_Lamb/sY6ok9yM">Lamb, H.</a> (1994). <i>Hydrodynamics</i> (6th ed.). Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-45868-9. Originally published in 1879, the 6th extended edition appeared first in 1932.</li> <li><a href="/facts/George_Gabriel_Stokes/7RHJJ5Lj">Stokes, G.G.</a> (1842). "On the steady motion of incompressible fluids". <i>Transactions of the Cambridge Philosophical Society</i>. 7: 439–453. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1848TCaPS...7..439S">1848TCaPS...7..439S</a>.Reprinted in: Stokes, G.G. (1880). <a href="https://archive.org/details/mathphyspapers01stokrich"><i>Mathematical and Physical Papers, Volume I</i></a>. Cambridge University Press. pp. <a href="https://archive.org/details/mathphyspapers01stokrich/page/n18">1</a>–16.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Batchelor (1967), p. 78. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Batchelor (1967), p. 79. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>E.g. Brenner, Howard (1961). "The slow motion of a sphere through a viscous fluid towards a plane surface". Chemical Engineering Science. 16 (3–4): 242–251. Bibcode:1961ChEnS..16..242B. doi:10.1016/0009-2509(61)80035-3. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Batchelor (1967), p. 602. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Batchelor (1967), p. 601. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>