The full boundary-layer equations determine a velocity profile $u(x,y)$ throughout the layer. If the main goal is wall shear or overall growth, a more economical description can be obtained by integrating the streamwise momentum balance across the layer.

For a steady incompressible two-dimensional boundary layer with outer velocity $U_e(x)$, define

$$\delta^* =\int_0^\infty\left(1-\frac{u}{U_e}\right)dy,$$

$$\theta =\int_0^\infty\frac{u}{U_e} \left(1-\frac{u}{U_e}\right)dy,$$

and wall shear

$$\boxed{\tau_w=\mu\left.\frac{\partial u}{\partial y}\right|_{y=0}}.$$

Integrating the Prandtl momentum equation and using continuity gives the von Karman momentum-integral equation

$$\boxed{ \frac{\tau_w}{\rho} =U_e^2\frac{d\theta}{dx} +U_e\frac{dU_e}{dx} \left(2\theta+\delta^*\right) }.$$

Define the local skin-friction coefficient

$$\boxed{c_f(x)=\frac{\tau_w}{\tfrac12\rho U_e^2}}.$$

Using the shape factor

$$H=\frac{\delta^*}{\theta},$$

the integral equation becomes

$$\boxed{ \frac{d\theta}{dx} +(2+H)\frac{\theta}{U_e}\frac{dU_e}{dx} =\frac{c_f}{2} }.$$

It relates boundary-layer momentum growth to wall friction and to acceleration or deceleration of the outer flow.

Zero pressure gradient

For a flat plate in a uniform outer stream,

$$U_e=U_\infty=\text{constant},$$

so

$$\frac{dU_e}{dx}=0.$$

The equation reduces to

$$\boxed{\frac{d\theta}{dx}=\frac{c_f}{2}}.$$

Thus the growth of momentum thickness is directly tied to the wall shear.

This gives a useful physical interpretation: the plate removes streamwise momentum from the near-wall fluid, and the momentum deficit accumulated downstream is exactly the effect measured by $\theta$.

Pressure gradients

If the outer flow accelerates,

$$\frac{dU_e}{dx}>0,$$

the pressure gradient is favorable. The second term in the integral equation is positive and changes how much momentum-thickness growth is required for a given wall shear.

If the outer flow decelerates,

$$\frac{dU_e}{dx}<0,$$

the pressure gradient is adverse. The pressure-gradient term can oppose the wall-friction contribution strongly enough that the wall shear decreases toward zero.

The condition

$$\boxed{\tau_w=0}$$

marks incipient separation in the classical boundary-layer picture. Once substantial reverse flow and separation have developed, the simple attached boundary-layer approximation is no longer sufficient.

Approximate-profile method

The integral equation is especially useful when an exact velocity profile is unavailable. One can

1. assume a profile shape satisfying the important boundary conditions;
2. compute $\delta^*$ and $\theta$ from that profile;
3. compute $\tau_w$ from the wall gradient;
4. substitute those expressions into the momentum-integral equation;
5. solve for the unknown thickness parameter.

This is the basis of von Karman-Pohlhausen methods.

Worked example: approximate flat-plate growth

For zero pressure gradient, use the polynomial profile

$$\frac{u}{U_\infty} =\frac32\eta-\frac12\eta^3, \qquad \eta=\frac y\delta,$$

for $0\le\eta\le1$, with $u=U_\infty$ above the layer.

Its integral thicknesses are

$$\delta^*=\frac38\delta,$$

$$\theta=\frac{39}{280}\delta.$$

The wall gradient is

$$\left.\frac{\partial u}{\partial y}\right|{0} =\frac{U\infty}{\delta} \left.\frac{d}{d\eta} \left(\frac32\eta-\frac12\eta^3\right)\right|{0} =\frac{3U\infty}{2\delta}.$$

Therefore

$$\tau_w=\frac{3\mu U_\infty}{2\delta},$$

and

$$\frac{c_f}{2} =\frac{\tau_w}{\rho U_\infty^2} =\frac{3\nu}{2U_\infty\delta}.$$

The zero-pressure-gradient integral equation gives

$$\frac{39}{280}\frac{d\delta}{dx} =\frac{3\nu}{2U_\infty\delta}.$$

Hence

$$\delta\frac{d\delta}{dx} =\frac{140}{13}\frac{\nu}{U_\infty}.$$

With $\delta=0$ at the ideal leading edge,

$$\boxed{ \delta(x) =\sqrt{\frac{280}{13}} \sqrt{\frac{\nu x}{U_\infty}} \approx4.64\sqrt{\frac{\nu x}{U_\infty}} }.$$

The corresponding local friction coefficient is

$$c_f =\frac{3\nu}{U_\infty\delta} \approx\boxed{\frac{0.647}{\sqrt{Re_x}}},$$

where

$$Re_x=\frac{U_\infty x}{\nu}.$$

This approximate result is already close to the exact Blasius values. The integral method succeeds because it enforces global momentum conservation even though the assumed profile is only approximate.

The von Karman equation is therefore a bridge between detailed field equations and engineering boundary-layer estimates: it retains the momentum physics while reducing the unknown flow to a few integral measures.