Discussion
If \(\bar{q}\) denotes velocity field of an incompressible fluid in motion, then the mass conservation will not imply :
Answer: D
Mass conservation does not imply \(\nabla \times \bar{q} = 0\).
The equation of continuity is \(\partial\rho/\partial t + \nabla \cdot (\rho\bar{q}) = 0\). For an incompressible fluid with constant density this reduces to \(\nabla \cdot \bar{q} = 0\), and \(\partial\rho/\partial t = 0\) gives the form in option B as well. So A, B and C all follow from conservation of mass.
\(\nabla \times \bar{q} = 0\) says the flow is irrotational. That is a separate kinematic property, not a consequence of mass conservation: a rigid rotation \(\bar{q} = (-y, x, 0)\) is incompressible but has vorticity 2.
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