Fluid Mechanics: Navier–Stokes
Continuity, Navier–Stokes, Bernoulli — and the Reynolds-number transition to turbulence.
Treat a fluid as a continuum with density $\rho$ and velocity field $\mathbf u(\mathbf x, t)$. Mass conservation gives the continuity equation:
$$\partial_t \rho + \nabla \cdot (\rho \mathbf u) = 0.$$Momentum balance (Newton's second law per unit volume) is the Navier–Stokes equation:
$$\rho \left(\partial_t \mathbf u + (\mathbf u \cdot \nabla)\mathbf u\right) = -\nabla p + \mu \nabla^2 \mathbf u + \mathbf f.$$For incompressible flow ($\nabla \cdot \mathbf u = 0$) and along a streamline of an inviscid steady flow, Bernoulli's principle holds:
$$\tfrac{1}{2}\rho u^2 + p + \rho g z = \text{const}.$$The dimensionless Reynolds number $\mathrm{Re} = \rho U L / \mu$ measures the ratio of inertial to viscous forces; transition from laminar to turbulent flow happens around $\mathrm{Re} \sim 10^3$ depending on geometry.