1.

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 derivativeI 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»

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.



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