Boundary layers

At high Reynolds number the viscous term is multiplied by the small number \(1/\mathrm{Re}\), and it is tempting to drop it and use the Euler equations. But the Euler equations cannot satisfy no-slip, and real fluids do. Prandtl's resolution (1904) is that viscosity matters only in a thin boundary layer next to the wall, where the velocity changes rapidly from zero to the outer inviscid value. In that layer the second derivative across it is large enough to balance inertia.

A scaling argument gives the thickness \(\delta\). Along a wall of length \(L\) with outer speed \(U\), inertia is \(u\,\partial_x u \sim U^2/L\) and viscosity across the layer is \(\nu\,\partial_y^2 u \sim \nu U/\delta^2\). Balancing them, \[ \delta \sim \sqrt{\frac{\nu L}{U}}, \qquad \frac{\delta}{L} \sim \mathrm{Re}^{-1/2}. \] An aircraft wing at \(\mathrm{Re} = 10^7\) has a boundary layer about a three-thousandth of its chord. Rescaling \(y = \delta Y\) and \(v = (U\delta/L)V\) and keeping leading-order terms gives the Prandtl boundary layer equations: \[ u\,u_x + v\,u_y = -\frac{1}{\rho}p_x + \nu\,u_{yy}, \qquad p_y = 0, \qquad u_x + v_y = 0. \] The streamwise second derivative \(\nu u_{xx}\) has dropped out, and the pressure is constant across the layer, so it is imposed by the outer potential flow: \(-p_x/\rho = U_e U_e'\).

For a flat plate in a uniform stream, Blasius (1908) found a similarity solution \(u = U f'(\eta)\) with \(\eta = y\sqrt{U/(\nu x)}\), where \(f^{\prime\prime\prime} + \tfrac12 f f^{\prime\prime} = 0\), \(f(0) = f'(0) = 0\), \(f'(\infty) = 1\). Solving it numerically gives the working results: the layer reaches 99% of the outer speed at \[ \delta_{99} \approx \frac{5.0\,x}{\sqrt{\mathrm{Re}_x}}, \qquad \mathrm{Re}_x = \frac{Ux}{\nu}, \] and the local skin-friction coefficient is \(c_f = 0.664/\sqrt{\mathrm{Re}_x}\). Air at 10 m/s over the first 50 cm of a plate builds a layer about 4 mm thick.

When the outer flow decelerates, the pressure rises downstream (an adverse pressure gradient). The slow fluid near the wall has too little momentum to push against it, stops, and reverses: the boundary layer separates and the flow leaves the surface, forming a wake of low pressure. That wake is the drag that potential flow could not produce, and the resolution of d'Alembert's paradox. As mathematics the theory is delicate: Oleinik (1960s) proved well-posedness of the Prandtl equations for monotone profiles, while without monotonicity they can be ill-posed in Sobolev spaces (Gérard-Varet and Dormy, 2010), and whether Navier-Stokes solutions converge to Euler solutions as viscosity vanishes in a domain with walls remains largely open.

Picture it: a thin skin of slow fluid clinging to a wing, thickening with distance from the leading edge like a square root, and peeling away where the pressure rises. Think it: the boundary layer is a singular perturbation: the small parameter multiplies the highest derivative, the limit problem loses a boundary condition, and a thin layer of width \(\mathrm{Re}^{-1/2}\) restores it.