Question

Consider an infinitesimally small fluid element with density ρ (of dimensions dx, dy and dz with mass δ m and volume δ V) moving along with the flow with a velocity \(\vec{V}=u\vec{i}+v \vec{j}+w\vec{k}\). The continuity equation is \(\frac{D\rho}{Dt}+\rho \nabla.\vec{V}=0\). Where does this second term come from?

(a) Integral

(b) The rate of change of element’s volume

(c) Elemental change in mass

(d) Local derivative

I have been asked this question in an online quiz.

I need to ask this question from Continuity Equation topic in section Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

Right option is (b) The rate of change of element’s volume

Easiest explanation: Applying mass conservation for this element,

time rate of change of mass = 0

\(\frac{D(\rho \delta V)}{Dt}=0 \)

\(\delta V \frac{D(\rho)}{Dt}+\rho\frac{D(\delta V)}{Dt}=0 \)

\(\frac{D(\rho)}{Dt}+\rho\frac{1}{\delta V}\frac{D(\delta V)}{Dt}=0 \)

\(\frac{D(\rho)}{Dt}+\rho\nabla.\rho=0 \)

Thus, the term arises from the rate of change of element’s volume.