For an irrotational, incompressible planar flow, the velocity components $(u, v)$ in terms of velocity potential $\Phi$ are related to stream function $\psi$ by:

(A) $u = \frac{\partial \Phi}{\partial x} = \frac{\partial \psi}{\partial y}$, $v = \frac{\partial \Phi}{\partial y} = -\frac{\partial \psi}{\partial x}$

(B) $u = \frac{\partial \Phi}{\partial y} = \frac{\partial \psi}{\partial x}$, $v = -\frac{\partial \Phi}{\partial x} = \frac{\partial \psi}{\partial y}$

(C) $u = -\frac{\partial \Phi}{\partial x} = \frac{\partial \psi}{\partial y}$, $v = -\frac{\partial \Phi}{\partial y} = -\frac{\partial \psi}{\partial x}$

(D) $u = \frac{\partial \psi}{\partial x}$, $v = \frac{\partial \psi}{\partial y}$

Answer Key:

A

### Solution: By Cauchy-Riemann conditions for potential flow: $u = \frac{\partial \Phi}{\partial x} = \frac{\partial \psi}{\partial y}$ and $v = \frac{\partial \Phi}{\partial y} = -\frac{\partial \psi}{\partial x}$. Correct: A.

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