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 \(\vec{V}=u\vec{i}+v\vec{j}+w\vec{k}\). What is the time rate of change of mass of this element?

(a) \(\frac{D(\rho \delta V)}{Dt}\)

(b) \(\frac{\partial(\rho \delta m)}{\partial t}\)

(c) \(\frac{\partial(\rho \delta V)}{\partial t}\)

(d) \(\frac{D(\rho \delta m)}{Dt}\)

I got this question in examination.

This is a very interesting question from Continuity Equation in division Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

Right answer is (a) \(\frac{D(\rho \delta V)}{Dt}\)

Easiest explanation: Substantial derivative is used as the model is moving.

mass = ρ δ V

time rate of change of mass=\(\frac{D(\rho \delta V)}{Dt}\)