Question

the equation of continuity

Answer 1

The continuity equation expresses local conservation of some scalar quantity (e.g., mass, charge, probability). In words: the rate of change of the quantity inside a control volume equals minus the net flux out of the volume (plus any sources/sinks).

Integral form (no sources):
d/dt ∫_V ρ dV + ∮_∂V ρ v · n dS = 0

Using the divergence theorem and assuming V is arbitrary gives the differential form:
∂ρ/∂t + ∇·(ρ v) = 0

Alternate (material) form:
Dρ/Dt + ρ ∇·v = 0
where D/Dt = ∂/∂t + v·∇ is the material derivative.

Special cases and variants
- Incompressible flow (ρ = constant): ∇·v = 0.
- Steady flow (∂/∂t = 0): ∇·(ρ v) = 0.
- One-dimensional: ∂ρ/∂t + ∂(ρ u)/∂x = 0.
- With sources/sinks S(x,t): ∂ρ/∂t + ∇·(ρ v) = S.
- For electric charge (current density J): ∂ρ/∂t + ∇·J = 0.

Physical meaning: local production/destruction of the quantity is zero (or given by S), so any local change must be balanced by a net flux across the boundary.