Boundary layers and no-slip

No-slip creates boundary layers

A viscous fluid in contact with a solid shares the wall’s tangential velocity in the no-slip model. The velocity must change from wall value to outer-flow value, concentrating gradients and viscous stress near the boundary.

Estimate the layer scale

For a boundary layer of thickness δ, viscous acceleration scales as νU/δ² while advection scales as U²/L. Balancing gives νU/δ² ~ U²/L, so δ/L ~ Re^{-1/2}, where Re = UL/ν. The estimate explains why high-Re flow can be nearly inviscid outside a thin wall region.

For U = 2 m/s, L = 0.5 m, ν = 1×10⁻⁶ m²/s, Re = 10⁶ and δ/L ~ 10⁻³, giving δ ~ 0.5 mm as an order-of-magnitude estimate.

Separation is a pressure-gradient problem

An adverse pressure gradient can slow near-wall fluid until its momentum cannot carry it downstream; the boundary layer thickens or separates. The simple scaling does not predict separation location by itself.

Work it: Calculate Re and δ estimate for water at U = 1 m/s over L = 0.2 m. Explain why a rough nozzle wall can matter even when the bulk flow looks fast.