Discussion
In the following equations, u and v are the velocities in the x- and y- directions, respectively and t is time. The flow field that CANNOT be termed as incompressible is
Answer: A
The flow that is not incompressible is \(u = x^{3} + xy^{2},\ v = y^{3} + yx^{2}\).
A two-dimensional flow is incompressible when its divergence vanishes, \(\partial u/\partial x + \partial v/\partial y = 0\). For this flow
\[ \begin{aligned} \nabla \cdot \vec{V} &= (3x^{2} + y^{2}) + (3y^{2} + x^{2}) \\ &= 4(x^{2} + y^{2}) \end{aligned} \]
which is not zero except at the origin.
The others pass: 10t - 10t = 0 for the second; in the third u depends only on y and v = 0, so both derivatives vanish; and in the fourth u = 2y, v = 2x give 0 + 0.
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