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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.
 +
 +
If the flow is incompressible, then the continuity equation is identically satisfied
 +
 +
:<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


d\psi = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy = 0

The equation of a streamline in two-dimensions is


v dx - u dy = 0

Comparing the two equations, we have


u = - \frac{\partial \psi}{\partial y}

v = \frac{\partial \psi}{\partial x}

Conversely, the stream function at any point P 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 ]

where P_o is some reference point and one can assume \psi(P_o) = 0 since the stream function is determined only upto a constant.

If the flow is incompressible, then the continuity equation is identically satisfied


\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
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