Uniform flow at angle \(\alpha\)
\( \phi = U(x\cos\alpha + y\sin\alpha) \)
\( \psi = U(y\cos\alpha - x\sin\alpha) \)
\( u=\dfrac{\partial\phi}{\partial x}=\dfrac{\partial\psi}{\partial y}=U\cos\alpha \)
\( v=\dfrac{\partial\phi}{\partial y}=-\dfrac{\partial\psi}{\partial x}=U\sin\alpha \)
Differentiating the velocity potential gives the velocity components directly: \(u\) in the \(x\)-direction and \(v\) in the \(y\)-direction. The stream function gives the same components through \(u=\partial\psi/\partial y\) and \(v=-\partial\psi/\partial x\). For this uniform flow, the derivatives are constants, so \(\alpha\) sets the direction of the flow while \(U\) sets its speed.
The velocity vector has constant magnitude everywhere. Lines of constant \(\psi\) are streamlines, while lines of constant \(\phi\) are perpendicular equipotential lines.
\(\phi=\) constant
\(\psi=\) constant
The plot domain is fixed so the pattern rotates smoothly as \(\alpha\) changes.
u = U cos α
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v = U sin α
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φ at probe
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ψ at probe
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This page treats a two-dimensional, inviscid, irrotational flow whose speed and direction are spatially uniform. Rotating the flow changes the orientation of the flow net, not the speed at any point.