1.

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»

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,

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

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


Discussion

No Comment Found

Related InterviewSolutions