Fluid mechanics Incompressible viscous flow Theorem

Momentum integral equation

Builds on Prandtl boundary layer equations Displacement thickness

Statement

Let $(u, v)$ be a steady solution of the Prandtl boundary layer equations along the wall $y = 0$ . The density is $1$ , the kinematic viscosity is $\nu > 0$ , and the outer speed $U(x) > 0$ is smooth on an interval. Assume that $u \to U$ and $\partial_y u \to 0$ fast enough as $y \to \infty$ for all the integrals below to converge. Assume also that differentiation in $x$ may be taken under them. Let $\delta^*$ be the displacement thickness, $\theta$ the momentum thickness, $H = \delta^* / \theta$ the shape factor and $\tau_w = \nu \, \partial_y u(x, 0)$ the wall shear stress. Then $\frac{d\theta}{dx} + (2 + H) \frac{\theta}{U} \frac{dU}{dx} = \frac{\tau_w}{U^2}.$ Equivalently, $\frac{d}{dx}\big( U^2 \theta \big) + U \frac{dU}{dx} \delta^* = \tau_w$ .

Remarks

The proof integrates the momentum equation $u \, \partial_x u + v \, \partial_y u = U U' + \nu \, \partial_y^2 u$ over $0 < y < \infty$ . The term with $v$ is handled by writing $v = -\int_0^y \partial_x u$ from continuity and integrating by parts, and the viscous term gives $-\tau_w$ . Collecting terms gives the second form, and dividing by $U^2$ gives the first. Example: for a flat plate, $U$ is constant and the equation says $d\theta / dx = \tau_w / U^2$ : the growth of the momentum deficit equals the skin friction. With an assumed profile shape, such as a polynomial in $y / \delta$ , the equation becomes an ordinary differential equation for $\delta(x)$ ; this is the Kármán–Pohlhausen method.

Going deeper

For a student

The momentum integral equation is due to von Kármán in 1921. Once one guesses a profile family, it turns the boundary layer equations into a single ordinary differential equation along the wall. Such integral methods gave useful estimates of skin friction and of separation points long before numerical solution was easy. They are still used for turbulent layers, with empirical closures for $H$ and $\tau_w$ . It is exact for the Prandtl equations; the approximation enters only through the assumed profile.

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Displacement thickness concept
Prandtl boundary layer equations concept
Momentum integral equation statement

Tag	`WCGH22X05Y`
URI	`m2://WCGH22X05Y`
Kind	statement, theorem
Writing	W0 stub
MSC	76D10
Status	This copy of the site has no build data. Run `m2 status WCGH22X05Y`.
About	Prandtl boundary layer equations
Uses	Prandtl boundary layer equations, Displacement thickness
Changed	3 November 2026