Fluid mechanics

The continuum and the material derivative, continuity, Euler’s equation, Bernoulli, Navier–Stokes and the Reynolds number, Stokes flow, and boundary layers.

A fluid is a continuum that cannot resist a sustained shear elastically: any applied shear produces flow. Its equations of motion follow from Newton’s second law applied to a deformable fluid element, together with a constitutive law for how the element responds to stress. From that single starting point come the material derivative, the continuity equation, Euler’s inviscid dynamics, Bernoulli’s integral, the viscous Navier–Stokes equation, and the dimensionless Reynolds number that organises every flow regime.

• 5.1 The continuum and the material derivative — the Eulerian and Lagrangian views, and the convective acceleration $D/Dt = \partial_t + \mathbf{u}\cdot\nabla$.
• 5.2 Conservation of mass: the continuity equation — the mass-flux balance, $\partial_t\rho + \nabla\cdot(\rho\mathbf{u}) = 0$, and the incompressible limit $\nabla\cdot\mathbf{u}=0$.
• 5.3 Euler’s equation — the pressure-gradient force and Newton’s second law for an inviscid fluid element.
• 5.4 Bernoulli’s principle — integrating Euler along a streamline, and the trade between speed, pressure, and height.
• 5.5 Viscosity, Navier–Stokes, and the Reynolds number — the viscous stress, the no-slip condition, and the single parameter $\mathrm{Re}$ that controls the flow.
• 5.6 Stokes flow and boundary layers — the reversible low-Reynolds limit, the scallop theorem, and the thin viscous layers at high Reynolds number.