5.5 Viscosity, Navier–Stokes, and the Reynolds number

Euler’s equation describes an idealised fluid with no internal friction. Real fluids resist shear: adjacent layers moving at different speeds exert tangential forces on one another, transporting momentum across the flow. Adding this viscous stress to Euler’s equation produces the Navier–Stokes equation, and rescaling that equation reveals that a single dimensionless number controls the entire character of incompressible flow.

The viscous stress

In a Newtonian fluid — water, air, and most simple liquids and gases — the viscous shear stress is proportional to the rate of strain, the gradient of velocity. A layer of fluid sliding over a slower neighbour drags it forward and is itself dragged back; the constant of proportionality is the dynamic viscosity $\mu$ . Summing the net viscous force on a fluid element from the imbalance of these stresses across it gives, for an incompressible flow, a force per unit volume of $\mu\,\nabla^2\mathbf{u}$ — the viscous redistribution of momentum is diffusive, governed by a Laplacian, exactly as heat conduction and molecular diffusion are. The full treatment of the shear stress and its molecular origin belongs to the transport chapter; here we take the force per unit volume as given.

The Navier–Stokes equation

Adding the viscous term to Euler’s equation gives the Navier–Stokes equation for an incompressible Newtonian fluid:

\rho\,\frac{D\mathbf{u}}{Dt} \;=\; -\nabla p \;+\; \mu\,\nabla^2\mathbf{u} \;+\; \rho\mathbf{g}, \qquad \nabla\cdot\mathbf{u} = 0.

The combination $\nu = \mu/\rho$ — the kinematic viscosity — has units of $\text{m}^2/\text{s}$ , the units of a diffusivity: $\nu$ is the diffusion coefficient for momentum. It is what governs how fast a velocity disturbance spreads sideways through the fluid.

The viscous term also supplies the boundary condition that Euler’s equation could not enforce: at a solid surface a viscous fluid sticks, matching the wall velocity exactly. This no-slip condition is the physical origin of drag and of the boundary layers of the next lesson.

Nondimensionalising: the Reynolds number

How much the viscous term matters relative to inertia is not absolute — it depends on the scale of the flow. Make the equation dimensionless to see this directly.

▶ Rescaling Navier–Stokes Derivation

Choose a characteristic length $L$ , speed $U$ , time $L/U$ , and pressure scale $\rho U^2$ . Write $\tilde{\mathbf{u}} = \mathbf{u}/U$ , $\tilde{\nabla} = L\nabla$ , $\tilde t = tU/L$ , $\tilde p = p/\rho U^2$ . Substituting into Navier–Stokes (dropping gravity) and dividing through by $\rho U^2/L$ ,

\frac{D\tilde{\mathbf{u}}}{D\tilde t} \;=\; -\tilde\nabla\tilde p \;+\; \frac{\mu}{\rho U L}\,\tilde\nabla^2\tilde{\mathbf{u}}.

Every term is now order unity except the dimensionless coefficient on the viscous term. Its reciprocal is the Reynolds number.

\mathrm{Re} \;\equiv\; \frac{\rho U L}{\mu} \;=\; \frac{U L}{\nu}.

The Reynolds number is the only dimensionless parameter of incompressible Newtonian flow (refresher: dimensional analysis →). It measures the ratio of inertial forces to viscous forces. Two flows with the same $\mathrm{Re}$ , however different in size or speed or fluid, are governed by the identical dimensionless equation and are therefore geometrically similar — the principle of dynamic similarity that lets a scale model in a wind tunnel stand in for a full-size aircraft.

Alternating vortices shed periodically; pressure on the cylinder oscillates.

Re100

C_d1.40

Strouhal St0.165

log₁₀ Re = 2.00 (Re = 100)

The diagram shows steady flow past a smooth cylinder across nine decades of Reynolds number. Distinct regimes appear in turn: creeping flow with no separation, a pair of attached recirculating eddies, the periodic shedding of a von Kármán vortex street, a fully turbulent wake, and — past the drag crisis — a sharp narrowing of the wake when the boundary layer itself turns turbulent. One equation governs all of them; the only thing that changes is the single number weighing inertia against viscosity.

⏳ The history — Euler, Navier, Stokes, and the slow domestication of viscosity

Leonhard Euler in 1755 wrote down the inviscid equation of fluid motion in a memoir to the Berlin Academy, deriving the entire formalism deductively from Newton’s laws applied to fluid elements. For nearly a century Euler’s equation was the fluid equation, and its persistent disagreements with experiment — most famously d’Alembert’s paradox, that a body in steady inviscid flow feels no drag at all — were treated as embarrassments rather than as evidence of a missing term.

The missing term is viscosity. Claude-Louis Navier in 1822 and George Gabriel Stokes in 1845 independently added the viscous-stress term, producing the equation now bearing both their names. The molecular justification — that microscopic momentum transport across velocity gradients yields a stress proportional to the rate of strain — was supplied later by Maxwell and Boltzmann through the kinetic theory of gases.

The mathematics of Navier–Stokes remains uneven. Whether smooth three-dimensional solutions always exist is still an open question, one of the Clay Millennium Prize problems. Yet for prediction the equations are unambiguously right: they reproduce every flow regime, every transition, and every measured drag curve.