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Difference between revisions of "Stream function"

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The [[Continuity equation|continuity equation]] for an incompressible fluid with velocity vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103001.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103002.png" />, or
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The [[Continuity equation|continuity equation]] for an incompressible fluid with velocity vector $v=(v_x,v_y,v_z)$ is $\operatorname{div}(v)=0$, or
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103003.png" /></td> </tr></table>
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$$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}=0.$$
  
For two-dimensional motion in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103004.png" />-plane, this gives
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For two-dimensional motion in the $(x,y)$-plane, this gives
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103005.png" /></td> </tr></table>
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$$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}=0,$$
  
and there is thus a stream function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103006.png" /> such that
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and there is thus a stream function $\psi$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110300/s1103007.png" /></td> </tr></table>
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$$v_x=\frac{\partial\psi}{\partial y},\quad v_y=-\frac{\partial\psi}{\partial x}.$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  "Modern developments in fluid dynamics"  S. Goldstein (ed.) , '''1''' , Dover, reprint  (1965)  pp. Chapt. III</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1967)  pp. Chapt. 2.2</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  "Modern developments in fluid dynamics"  S. Goldstein (ed.) , '''1''' , Dover, reprint  (1965)  pp. Chapt. III</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.K. Batchelor,  "An introduction to fluid dynamics" , Cambridge Univ. Press  (1967)  pp. Chapt. 2.2</TD></TR></table>

Latest revision as of 15:08, 30 July 2014

The continuity equation for an incompressible fluid with velocity vector $v=(v_x,v_y,v_z)$ is $\operatorname{div}(v)=0$, or

$$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}=0.$$

For two-dimensional motion in the $(x,y)$-plane, this gives

$$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}=0,$$

and there is thus a stream function $\psi$ such that

$$v_x=\frac{\partial\psi}{\partial y},\quad v_y=-\frac{\partial\psi}{\partial x}.$$

References

[a1] "Modern developments in fluid dynamics" S. Goldstein (ed.) , 1 , Dover, reprint (1965) pp. Chapt. III
[a2] G.K. Batchelor, "An introduction to fluid dynamics" , Cambridge Univ. Press (1967) pp. Chapt. 2.2
How to Cite This Entry:
Stream function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stream_function&oldid=32571
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article