Consider a uniform stream of magnitude $V$ inclined at angle $\alpha$ . Assuming incompressible planar irrotational flow, find the velocity potential function and the stream function. Show all your work.

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We are going to find the velocity potential function and the stream function of magnitude V inclined at angle $\alpha$ . The flow is incompressible, planar and irrotational.

Starting with determining the stream function by using components of the speed, $\textbf{u}$ :

\begin{align*} u &= \frac{\partial \psi}{\partial y} \\\\ \frac{\partial \psi}{\partial y} &= V \cos\alpha \\\\ \xrightarrow[]{} \text{Integrating with respect to y}\\\\ \psi &= V y \cos\alpha + f(x) \end{align*}

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Consider a uniform stream of magnitude $V$ inclined at angle $\alpha$ . Assuming incompressible planar irrotational flow, find the velocity potential function and the stream function. Show all your work.