Difference between revisions of "Stream function"
From Encyclopedia of Mathematics
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| − | The [[Continuity equation|continuity equation]] for an incompressible fluid with velocity vector | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}=0.$$ | |
| − | For two-dimensional motion in the | + | 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 | + | 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==== | ====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=18599
Stream function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stream_function&oldid=18599
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article