Conservation of mass, the continuity equation and incompressibility

Mass is neither created nor destroyed. Take any fixed region \(V\) with boundary \(\partial V\) and outward normal \(\mathbf{n}\). The mass inside changes only by what flows across the boundary, and the rate at which mass leaves through a small patch \(dS\) is \(\rho\,\mathbf{u}\cdot\mathbf{n}\,dS\). So \[ \frac{d}{dt}\int_V \rho\,dV = -\oint_{\partial V}\rho\,\mathbf{u}\cdot\mathbf{n}\,dS = -\int_V \nabla\cdot(\rho\mathbf{u})\,dV, \] by the divergence theorem. Since \(V\) is arbitrary and the integrand is continuous, the integrand itself vanishes: \[ \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{u}) = 0. \] This is the continuity equation. Expanding the divergence gives the equivalent Lagrangian form \(D\rho/Dt + \rho\,\nabla\cdot\mathbf{u} = 0\).

A flow is incompressible when the density of each particle does not change, \(D\rho/Dt = 0\). Because \(\rho \gt 0\), the continuity equation then says exactly \[ \nabla\cdot\mathbf{u} = 0. \] By Euler's expansion formula \(DJ/Dt = (\nabla\cdot\mathbf{u})J\), this is the same as \(J \equiv 1\): the flow map preserves volume. A blob of fluid can be stretched into a long thread, but its volume stays fixed. The density need not be constant everywhere (salt water under fresh water is incompressible but layered); when it is, \(\rho = \rho_0\), the flow is called homogeneous, and that is the setting of the Navier-Stokes equations studied in this course.

No real fluid is exactly incompressible. The approximation is good when the Mach number \(\mathrm{Ma} = U/c\) is small, where \(c\) is the speed of sound: density changes are then of relative size about \(\mathrm{Ma}^2\). Air moving at \(30\) m/s has \(\mathrm{Ma} \approx 0.09\) and density variations below one per cent, so a car in a wind tunnel is an incompressible flow problem.

Checking incompressibility is a calculation. The strain \(\mathbf{u} = (x,-y,0)\) has \(\nabla\cdot\mathbf{u} = 1 - 1 + 0 = 0\): it squeezes in one direction exactly as fast as it stretches in the other. The radial field \((x,y,z)\) has divergence \(3\) and is a source. Incompressibility also changes the role of pressure. There is no longer an equation of state relating \(p\) to \(\rho\); instead the pressure adjusts instantly, everywhere, to whatever value keeps \(\nabla\cdot\mathbf{u} = 0\). We will see this made precise when the pressure is eliminated from the Navier-Stokes equations.

Picture it: a square of dye in the strain flow becomes a long thin rectangle with the same area. Think it: divergence-free vector fields are exactly those whose flow maps preserve volume, and they form a linear subspace; projecting onto that subspace is what the pressure does.