Question

For infinitesimally small element (with volume δV) moving along with the flow with velocity \(\vec{V}\), which of these equations represent the divergence of velocity?

(a) \(\nabla.\vec{V} = \frac{1}{\delta V}\frac{d(\delta V)}{dt} \)

(b) \(\nabla.\vec{V} = \frac{D(\delta V)}{dt} \)

(c) \(\nabla.\vec{V} = \frac{1}{\delta V}\frac{D(\delta V)}{dt} \)

(d) \(\nabla.\vec{V} = \frac{d(\delta V)}{dt} \)

This question was posed to me in my homework.

Origin of the question is Governing Equations topic in section Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

The correct choice is (c) \(\nabla.\vec{V} = \frac{1}{\delta V}\frac{D(\delta V)}{dt} \)

To explain: For control volumes,

\(\frac{DV}{Dt}=\iiint_{V}(\nabla.\vec{V})dV\)

For infinitesimal elements, V can be represented as δ V. Therefore,

\(\frac{D(\delta V)}{Dt}=\iiint_{\delta V}(\nabla.\vec{V})dV\)

As the elements are infinitesimally small, ∇.\(\vec{V}\) is the same for all elements. Hence,

\(\frac{D(\delta V)}{Dt}=(\nabla.\vec{V})\delta V\)

\(\nabla \vec{V}=\frac{1}{\delta V} \frac{D(\delta V)}{Dt}\).