Viscosity & Turbulence

A real fluid does not slide freely past a solid surface: the layer touching a wall moves with the wall — the no-slip condition — so the speed must change between wall and free stream, and a Newtonian fluid resists that shearing with a stress τ = η (dv/dy) proportional to the velocity gradient. The dynamic viscosity η is measured in Pa s, or kg m⁻¹ s⁻¹: water at 20 °C is 1.0 × 10⁻³ Pa s, air is 1.8 × 10⁻⁵ Pa s. Viscosity is not wall friction: the stress acts between every pair of layers — momentum carried sideways from faster fluid to slower, which is why heating a liquid lowers its viscosity (molecules escape each other's attraction more easily) while heating a gas raises it (molecules cross between layers faster).

In laminar flow the fluid moves in ordered layers that do not mix, and while it is steady each element traces the path of the one ahead. Drive a viscous fluid along a circular pipe and balance the pressure force on a cylindrical element against the shear on its surface: out comes a parabolic speed profile — zero at the wall, maximum on the axis — and a volume flow rate Q = πΔP r⁴/(8ηL). Q is only proportional to ΔP, so doubling the flow rate merely doubles the pressure difference; the striking part is the fourth power, which cuts the flow to one sixteenth when the radius is halved. That strict proportionality is also a test: a straight line through the origin on a ΔP against Q graph says the flow is still laminar.

Solving the viscous equations for a sphere moving slowly through an unbounded fluid gives Fd = 6πηrv — linear in speed, and with no fluid density in it at all. Release the sphere and it accelerates until drag plus buoyancy balance weight, so vt = 2r²(ρs − ρf)g/(9η). A 1.0 mm-radius steel ball (ρs = 7800 kg m⁻³) in an oil with η = 1.0 Pa s and ρf = 900 kg m⁻³ settles at vt = 2(1.0 × 10⁻³)²(6900)(9.81)/(9 × 1.0) = 1.5 × 10⁻² m s⁻¹, about 1.5 cm s⁻¹. Falling-ball viscometers measure η exactly this way. Every step assumed the flow around the sphere stays viscous-dominated, so the result is only trustworthy once you check the Reynolds number.

Viscosity does not act alone; what matters is its size relative to inertia. For a flow of characteristic speed v across a characteristic length L, the inertial term in the equation of motion scales as ρv²/L and the viscous term as ηv/L², so their ratio is Re = ρvL/η = vL/ν. It is dimensionless, so it can be compared between systems of wildly different scale. A swimming bacterium (L ≈ 2 μm, v ≈ 30 μm s⁻¹ in water) has Re ≈ 6 × 10⁻⁵ and lives in pure viscosity — stop pushing and it stops instantly. Water at 0.10 m s⁻¹ in a 20 mm bore pipe has Re = (0.10)(0.020)/(1.0 × 10⁻⁶) = 2.0 × 10³. Two geometrically similar flows at the same Re behave identically, which is why scale models in wind tunnels tell you anything at all.

Raise Re and small disturbances that viscosity used to damp out begin to grow instead. In a circular pipe, with L taken as the diameter, flow is reliably laminar below Re ≈ 2300, unstable through roughly 2300 to 4000, and fully turbulent above that. The same water in the same 20 mm pipe is laminar at 0.10 m s⁻¹ and turbulent at 0.50 m s⁻¹ (Re ≈ 1.0 × 10⁴). Turbulent flow is unsteady, three-dimensional and strongly mixing, with eddies of every size passing energy down to the smallest scales, where viscosity finally turns it into heat. The velocity profile flattens, wall shear rises, and the pressure drop stops being proportional to Q, climbing roughly as Q to the power 1.75 in smooth pipes and as Q² in rough ones.

Bernoulli's equation describes a fluid with no viscosity: no shear stress, no dissipation, so p + ½ρv² + ρgh is conserved along a streamline. Real flow always breaks that near a solid boundary, where a thin boundary layer carries the whole change from zero at the wall to the free-stream speed. At low Re that layer fills the pipe and Bernoulli is useless. At high Re it stays thin and Bernoulli works well across most of the cross-section — but the layer can separate, shed a turbulent wake, and dissipate energy anyway. Engineers restore the missing physics by adding a measured head loss to Bernoulli's equation. State the assumption, estimate Re, and you know in advance which model you are allowed to use.