Question

Consider the continuity equation \(\frac{\partial\rho}{\partial t}+\nabla.(\rho \vec{V})=0\). For an incompressible flow, this equation becomes ___________

(a) \(\nabla.(\rho \vec{V})=0\)

(b) \(\frac{\partial(\rho\vec{V})}{\partial t}=0\)

(c) \(div(\vec{V})=0\)

(d) \(div(\rho\vec{V})=0\)

The question was posed to me in quiz.

Origin of the question is Continuity Equation in portion Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

Correct answer is (c) \(div(\vec{V})=0\)

The best explanation: Taking the continuity equation,

\(\frac{\partial\rho}{\partial t}+\nabla.(\rho \vec{V})=0\)

For incompressible flow, ρ is constant

\(\frac{\partial\rho}{\partial t}+\nabla.(\rho \vec{V})=0\)

The resulting equation is

\(\nabla.(\rho \vec{V})=0\)

\(\rho\nabla.(\vec{V})=0\)

\(\nabla.(\vec{V})=0\)

Thus, for incompressible flow, divergence of \(\vec{V}=0\).