A material volume carries a scalar quantity per unit mass

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

A material volume V(t)V(t) carries a scalar quantity ϕ\phi per unit mass, so the total amount of the quantity held in the volume is ∫V(t)ρϕ dV\int_{V(t)}\rho\phi\,\mathrm{d}V. Starting from the Reynolds transport theorem applied to F=ρϕF=\rho\phi, and using the differential form of mass conservation Dρ/Dt+ρ∇⋅u=0D\rho/Dt+\rho\nabla\cdot\mathbf{u}=0, show that ddt∫V(t)ρϕ dV=∫V(t)ρ DϕDt dV.\frac{\mathrm{d}}{\mathrm{d}t}\int_{V(t)}\rho\phi\,\mathrm{d}V=\int_{V(t)}\rho\,\frac{D\phi}{Dt}\,\mathrm{d}V. State which idealisations, if any, are used.

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