Question

Consider a model of finite control volume (volume V and surface area) moving along the flow with elemental volume dV, vector elemental surface area d\(\vec{S}\), density ρ and flow velocity \(\vec{V}\). What is the time rate of change of mass inside the control volume?

(a) \(\iiint_V\rho dV\)

(b) \(\frac{\partial}{\partial t} \iiint_V\rho dV\)

(c) \(\frac{D}{Dt} \iiint_V\rho dV\)

(d) ρdV

I have been asked this question in quiz.

My doubt is from Continuity Equation in chapter Governing Equations of Fluid Dynamics of Computational Fluid Dynamics

Answer 1

Right option is (c) \(\frac{D}{Dt} \iiint_V\rho dV\)

Explanation: Substantial derivative is used as the model is moving.

mass=density × volume

mass inside dV=ρdV

mass inside \( V=\iiint_V\rho dV\)

time rate of change of mass inside \(\frac{D}{Dt} \iiint_V\rho dV\).