Discussion
A two-dimensional source flow (with stream function, \(\psi_{1} = m\tan^{-1}\frac{y}{x}\)) is placed at the origin in a uniform flow (with stream function, \(\psi_{2} = Uy\)). Here, the strength of the source is m and the freestream velocity is U. The velocity components u and v of the combined flow in x- and y-directions, respectively, are
Answer: A
The velocity components are \(u = U+\frac{mx}{x^{2}+y^{2}}\), \(v = \frac{my}{x^{2}+y^{2}}\).
Stream functions add, so \(\psi = m\tan^{-1}(y/x) + Uy\). With \(u = \partial\psi/\partial y\) and \(v = -\partial\psi/\partial x\):
\[ \begin{aligned} u &= m\frac{x}{x^{2}+y^{2}} + U \\ v &= -m\frac{-y}{x^{2}+y^{2}} = \frac{my}{x^{2}+y^{2}} \end{aligned} \]
A check: the source part points radially outward, \((x, y)/(x^{2}+y^{2})\) times m, and the uniform stream adds U in x only.
Putting U in the y-component treats the stream as vertical, and the minus sign in option C comes from forgetting the minus in \(v = -\partial\psi/\partial x\).
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