2 Governing Equations of Incompressible Flow
2.1 Conservation of mass
For a constant-density incompressible fluid in a domain \(\Omega\), the velocity field satisfies
\[\nabla\cdot\mathbf u=0.\]
This condition restricts volumetric expansion and contraction of fluid parcels in the continuum model.
2.2 Conservation of momentum
For a Newtonian fluid with constant density \(\rho\) and dynamic viscosity \(\mu\), the incompressible Navier–Stokes equations can be written
\[\rho\left(\frac{\partial\mathbf u}{\partial t}+(\mathbf u\cdot\nabla)\mathbf u\right)-\mu\Delta\mathbf u+\nabla p=\mathbf f.\]
These equations must be accompanied by an initial velocity and suitable boundary conditions. No-slip walls, prescribed inflow velocity, outflow traction, and pressure-reference conventions need careful specification.
2.3 Nondimensionalization
Using characteristic velocity \(U\) and length \(L\), the Reynolds number is
\[\mathrm{Re}=\frac{\rho UL}{\mu}.\]
It compares inertial and viscous scales; it does not, by itself, determine whether any given flow is laminar or turbulent.
Question: Which term vanishes in a steady Stokes-flow approximation, and when might that approximation be useful?
Living chapter. This is an initial working treatment. Derivations, examples, and references will be expanded in future editions.