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Stream function

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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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