Discussion
If \(\psi\) is the stream function, the Laplace's equation \(\nabla^{2}\psi = 0\) is true when the flow is
Answer: B
\(\nabla^{2}\psi = 0\) holds when the flow is incompressible and irrotational.
A stream function exists for two-dimensional incompressible flow, because defining \(u = \partial\psi/\partial y\) and \(v = -\partial\psi/\partial x\) satisfies continuity automatically. Substituting these into the vorticity gives
\[ \begin{aligned} \omega_{z} &= \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \\ &= -\nabla^{2}\psi \end{aligned} \]
so \(\nabla^{2}\psi = 0\) is exactly the condition of zero vorticity. Both conditions are needed: incompressibility for \(\psi\) to exist in this form, and irrotationality for it to satisfy Laplace's equation. Neither alone is enough, and compressible flow does not have this stream function.
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