At high Reynolds number, viscosity can be weak through most of an external flow yet remain essential in a thin region next to a no-slip wall. The Prandtl boundary-layer approximation exploits this separation of scales.

Consider steady two-dimensional incompressible flow along a smooth wall. Let $x$ be tangent to the wall, $y$ normal to it, $u(x,y)$ and $v(x,y)$ the velocity components, $L$ a streamwise length scale, $\delta\ll L$ the boundary-layer thickness, and $U$ the outer-flow speed scale.

Thin-layer scaling

Continuity is

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0.$$

With

$$u\sim U,\quad x\sim L,\quad y\sim\delta,\quad v\sim V,$$

continuity gives

$$\boxed{\frac{V}{U}\sim\frac{\delta}{L}\ll1}.$$

Although $v$ is small, the term $v,\partial u/\partial y$ remains comparable with $u,\partial u/\partial x$ because transverse velocity gradients are large.

The streamwise inertial scale is

$$\frac{U^2}{L},$$

while transverse viscous diffusion scales as

$$\nu\frac{U}{\delta^2}.$$

Balancing them gives

$$\frac{U^2}{L}\sim\nu\frac{U}{\delta^2},$$

so

$$\boxed{\frac{\delta}{L}\sim Re_L^{-1/2}},$$

where

$$\boxed{Re_L=\frac{UL}{\nu}}.$$

Thus a high-Reynolds-number boundary layer is thin. The streamwise viscous term $\nu u_{xx}$ is smaller than $\nu u_{yy}$ by a factor of order $(\delta/L)^2$ and is neglected at leading order.

Pressure matching to the outer flow

The normal momentum equation implies, to leading order,

$$\boxed{\frac{\partial p}{\partial y}\approx0}.$$

Hence the boundary-layer pressure is imposed by the outer inviscid flow:

$$\boxed{p(x,y)\approx p_e(x)}.$$

If the outer tangential speed is $U_e(x)$, steady Euler flow gives

$$\boxed{U_e\frac{dU_e}{dx}=-\frac1\rho\frac{dp_e}{dx}}.$$

Prandtl boundary-layer equations

The leading equations are

$$\boxed{u_x+v_y=0},$$

and

$$\boxed{u,u_x+v,u_y=U_e,U_e'+\nu,u_{yy}},$$

where subscripts denote partial derivatives and

$$U_e'=\frac{dU_e}{dx}.$$

Equivalently,

$$u\frac{\partial u}{\partial x} +v\frac{\partial u}{\partial y} =U_e\frac{dU_e}{dx} +\nu\frac{\partial^2u}{\partial y^2}.$$

The pressure is

$$\boxed{p=p_e(x)}.$$

Boundary and matching conditions

For a stationary impermeable no-slip wall,

$$\boxed{u(x,0)=0},$$

$$\boxed{v(x,0)=0}.$$

Outside the viscous layer,

$$\boxed{u(x,y\to\infty)\to U_e(x)}.$$

The outer inviscid solution supplies $U_e(x)$ and the pressure gradient; the inner viscous solution supplies the wall shear and near-wall velocity deficit.

Favorable and adverse pressure gradients

Because

$$\frac{dp_e}{dx}=-\rho U_e\frac{dU_e}{dx},$$

an accelerating outer flow $(dU_e/dx>0)$ has a favorable pressure gradient, while a decelerating outer flow $(dU_e/dx<0)$ has an adverse pressure gradient.

An adverse pressure gradient removes momentum from the already slow near-wall fluid. The wall shear can fall to zero and the near-wall flow can reverse, leading toward boundary-layer separation.

Worked scaling estimate

Air with

$$\nu=1.5\times10^{-5},\mathrm{m^2/s}$$

flows at

$$U=20,\mathrm{m/s}$$

over a length

$$L=0.50,\mathrm m.$$

Then

$$Re_L=\frac{(20)(0.50)}{1.5\times10^{-5}} \approx6.67\times10^5.$$

Therefore

$$\frac{\delta}{L}\sim Re_L^{-1/2} \approx1.22\times10^{-3},$$

so the scale estimate is

$$\delta\sim0.61,\mathrm{mm}.$$

This is only the characteristic $\sqrt{\nu L/U}$ scale. A precise thickness definition such as $\delta_{99}$ introduces a numerical factor supplied by a particular solution.

Prandtl's approximation explains how a flow can be nearly inviscid over most of its domain while a thin viscous region still controls wall shear, separation, and drag.