Stream function
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On a streamline in two-dimensional flow | On a streamline in two-dimensional flow | ||
| - | <math> | + | :<math> |
d\psi = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy = 0 | d\psi = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy = 0 | ||
</math> | </math> | ||
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The equation of a streamline in two-dimensions is | The equation of a streamline in two-dimensions is | ||
| - | <math> | + | :<math> |
v dx - u dy = 0 | v dx - u dy = 0 | ||
</math> | </math> | ||
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Comparing the two equations, we have | Comparing the two equations, we have | ||
| - | <math> | + | :<math> |
u = - \frac{\partial \psi}{\partial y} | u = - \frac{\partial \psi}{\partial y} | ||
</math> | </math> | ||
| - | <math> | + | :<math> |
v = \frac{\partial \psi}{\partial x} | v = \frac{\partial \psi}{\partial x} | ||
</math> | </math> | ||
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Conversely, the stream function at any point <math>P</math> can be obtained from the velocity field by a line integral | Conversely, the stream function at any point <math>P</math> can be obtained from the velocity field by a line integral | ||
| - | <math> | + | :<math> |
\psi(P) = \psi(P_o) + \int_{P_o}^P [ v(x,y,t) dx - u(x,y,t) dy ] | \psi(P) = \psi(P_o) + \int_{P_o}^P [ v(x,y,t) dx - u(x,y,t) dy ] | ||
</math> | </math> | ||
where <math>P_o</math> is some reference point and one can assume <math>\psi(P_o) = 0</math> since the stream function is determined only upto a constant. | where <math>P_o</math> is some reference point and one can assume <math>\psi(P_o) = 0</math> since the stream function is determined only upto a constant. | ||
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| + | If the flow is incompressible, then the continuity equation is identically satisfied | ||
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| + | :<math> | ||
| + | \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = -\frac{\partial^2 \psi}{\partial x \partial y} + \frac{\partial^2 \psi}{\partial y \partial x} = 0</math> | ||
Latest revision as of 11:39, 12 September 2005
The stream function is a scalar field variable which is constant on each streamline. It exists only in two-dimensional and axisymmetric flows.
On a streamline in two-dimensional flow
The equation of a streamline in two-dimensions is
Comparing the two equations, we have
Conversely, the stream function at any point 
can be obtained from the velocity field by a line integral
![\psi(P) = \psi(P_o) + \int_{P_o}^P [ v(x,y,t) dx - u(x,y,t) dy ]](/W/images/math/0/6/1/061c8e98a4c2cb20abf21f9183e0332e.png)
where 
is some reference point and one can assume 
since the stream function is determined only upto a constant.
If the flow is incompressible, then the continuity equation is identically satisfied
