Discussion
Consider the two-dimensional velocity field given by \(\vec{V} = (5 + a_{1}x + b_{1}y)\hat{i} + (4 + a_{2}x + b_{2}y)\hat{j}\), where \(a_{1}, b_{1}, a_{2}\) and \(b_{2}\) are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?
Answer: B
The condition is \(a_{1} + b_{2} = 0\).
For two-dimensional incompressible flow the divergence of the velocity must vanish:
\[ \begin{aligned} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &= a_{1} + b_{2} \\ &= 0 \end{aligned} \]
since u = 5 + a1x + b1y gives \(\partial u/\partial x = a_{1}\) and v = 4 + a2x + b2y gives \(\partial v/\partial y = b_{2}\).
The other combinations pair the wrong derivatives. \(a_{2} - b_{1}\) (not offered) would be the vorticity; \(a_{2} + b_{1} = 0\) is not a condition of continuity at all.
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