Methods studio: derive a conservation law

A conservation law shrinks a control volume to a point

Mass balance is first written for a finite region: accumulation equals inflow minus outflow plus sources. The local differential equation appears when the boundary flux is expanded and the volume is allowed to shrink.

Derive continuity

For density ρ and velocity u in a fixed volume V, d/dt∫_V ρ dV = −∮_S ρu·n dA. Apply the divergence theorem: ∫_V [∂ρ/∂t + ∇·(ρu)]dV = 0. Because this holds for every volume, the integrand vanishes: ∂ρ/∂t + ∇·(ρu) = 0.

For constant density, continuity reduces to ∇·u = 0. In a pipe with area changing from 4 cm² to 1 cm², Q = Av gives outlet speed four times inlet speed in steady incompressible flow.

Sources alter the balance

A chemical reaction, evaporation, or injection adds a source term. A leaking basin is not described by ∇·u = 0 unless the leak is included as a boundary flux or volumetric sink.

Work it: Start from a one-dimensional tank balance d(ρA dx)/dt = ρAu|_in − ρAu|_out. Expand the outlet term to first order and recover the 1D continuity equation.