Question

For an incompressible flow field, \(\vec V\) , which one of the following conditions must be satisfied?
1. \(\nabla \cdot \vec V = 0\)
2. \(\nabla \times \vec V = 0\)
3. \(\left( {\vec V \cdot \nabla } \right)\vec V = 0\)
4. \(\frac{{\partial \vec V}}{{\partial t}} + \left( {\vec V \cdot \nabla } \right)\vec V = 0\)

Answer 1

Correct Answer - Option 1 : \(\nabla \cdot \vec V = 0\)

Explanation:
Any flow must satisfy the continuity equation if continuity equation violates then such flow is not possible.

continuity equation for incompressible flow is

\(\frac{{\partial u}}{{\partial x}} + \;\frac{{\partial v}}{{\partial y}} + \;\frac{{\partial w}}{{\partial z}} = 0\) which is \(\nabla \cdot \vec V = 0\)

∴ For an incompressible flow field, divergence must be zero, i.e. \(\nabla \cdot \vec V = 0\)