Discussion
The velocity components for two-dimensional flow are given by \(u = y^{2} - x^{2}\), \(v = 2xy\). The stream function \(\psi\) and the velocity potential \(\phi\) of the flow are :
Answer: C
The answer is option C: \(\phi = \tfrac{x^{3}}{3} - xy^{2}\), \(\psi = x^{2}y - \tfrac{y^{3}}{3}\).
Use the convention of most Indian fluid dynamics texts (the one the paper's own Q76 key follows): \(u = -\partial\phi/\partial x\), \(v = -\partial\phi/\partial y\), and \(u = -\partial\psi/\partial y\), \(v = \partial\psi/\partial x\). For option C:
\[ \begin{aligned} -\phi_{x} &= -(x^{2} - y^{2}) = y^{2} - x^{2} = u \\ -\phi_{y} &= 2xy = v \end{aligned} \]
\[ \begin{aligned} -\psi_{y} &= -(x^{2} - y^{2}) = u \\ \psi_{x} &= 2xy = v \end{aligned} \]
Both pairs reproduce \(u\) and \(v\), and the flow is irrotational and incompressible, so both functions exist. With the opposite sign convention (\(u = \phi_{x}\), \(u = \psi_{y}\)) the functions are \(\phi = xy^{2} - \tfrac{x^{3}}{3}\) and \(\psi = \tfrac{y^{3}}{3} - x^{2}y\), which is option B with its two labels interchanged, so B is wrong under either convention. Options A and D have the wrong signs on the cubic terms.
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