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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) ρdVI 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»

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



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