Boundary layer

A boundary layer is a thin region next to a surface in which viscosity remains significant even when inertia dominates the outer flow.

A thermal boundary layer is a thin region in which temperature changes rapidly and thermal conduction remains important. Its thickness can be estimated by balancing thermal diffusion against advection or by requiring a local Rayleigh number to be near its critical value.

In a renewal model, a diffusive thermal boundary layer grows until a convective instability overturns it, after which growth restarts. A layer reaching thickness $\delta \sim \sqrt{\kappa \tau }$ has mean heat flux of order $k\Delta T/\delta$. A diffusive $t^{-1/2}$ flux averaged over one cycle differs only by an order-one coefficient. Distinct thermal and material layers need not share a renewal time.

Balancing $U^2/x$ with $\nu U/{\delta }^2$ gives $\delta \sim \sqrt{\nu x/U}$.

For steady incompressible flow with imposed outer pressure gradient,The normal momentum equation makes pressure approximately uniform across the layer.

For a sheet moving with $u(x,0)=\alpha x$ through otherwise stationary fluid, the layer thickness is $\delta =\sqrt{\nu /\alpha }$. Withthe boundary-layer equations reduce toThe exact solution is $f=1-e^{-\eta }$.

Boundary-layer separation occurs where near-wall flow detaches from a surface, commonly after an adverse pressure gradient drives the wall shear to zero and then reverses it.