Discussion
If \(\phi = (x - t)(y - t)\) represents the velocity potential of an incompressible two-dimensional fluid then the stream lines at time \(t\) are the curves :
Answer: A
The stream lines are \((x - t)^{2} - (y - t)^{2} = \) constant.
With \(\mathbf{q} = -\nabla\phi\), \(u = -(y - t)\) and \(v = -(x - t)\). Stream lines at a fixed time satisfy \(dx/u = dy/v\):
\[ \begin{aligned} \frac{dx}{y - t} &= \frac{dy}{x - t} \\ (x - t)\,dx &= (y - t)\,dy \\ (x - t)^{2} - (y - t)^{2} &= c \end{aligned} \]
(The sign convention for \(\phi\) does not change the curves.) These are rectangular hyperbolas centred at \((t, t)\). The sum of squares would describe circles, which is what a vortex, not this potential, gives, and the options in \(x^{2}\) and \(y^{2}\) have lost the shift by \(t\).
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