Discussion
For a two-dimensional incompressible flow field given by \(\vec{u} = A(x\hat{i} - y\hat{j})\), where \(A \gt 0\), which one of the following statements is FALSE?
Answer: C
The false statement is Its streamlines are given by x = y.
For \(u = Ax\), \(v = -Ay\), the streamlines satisfy
\[ \begin{aligned} \frac{dx}{Ax} &= \frac{dy}{-Ay} \\ \ln x &= -\ln y + c \\ xy &= \text{constant} \end{aligned} \]
so they are rectangular hyperbolas, not the straight line x = y. This is stagnation-point flow.
The other statements are true. \(\partial u/\partial x + \partial v/\partial y = A - A = 0\), so continuity holds. The vorticity \(\partial v/\partial x - \partial u/\partial y = 0\), so the flow is irrotational. And as x tends to 0 the x-velocity vanishes, leaving flow along y only, which is unidirectional.
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