Balance of mass

Mass balance is universal before constitutive choice

Mass cannot appear or disappear inside a material volume except through an explicitly modeled source or phase change. This balance does not care whether the material is elastic, viscous, or granular.

Convert material conservation to local form

For a material volume, d/dt∫_V(t) ρ dV = 0. Applying the transport theorem yields ∂ρ/∂t + ∇·(ρu) = 0. Expanding the divergence gives Dρ/Dt + ρ∇·u = 0, showing that parcel density changes when velocity field divergence compresses or expands it.

If ρ is constant along a parcel, Dρ/Dt = 0, so ∇·u = 0. A converging incompressible pipe still has changing velocity because area and direction adjust while volume is preserved.

Constant density needs justification

Liquids are often approximately incompressible at modest pressure, but gases and strong pressure waves are not. A source, evaporation, or mixing interface requires extra mass bookkeeping.

Work it: For ρ(t) = 1000e^{-0.02t} kg/m³ in a parcel, compute Dρ/Dt at t = 5 s and infer ∇·u from continuity.