# How to interpret nonzero pressure gradient in momentum integral equation?

The momentum integral equation for boundary layers is:

$$\frac{d}{dx}(\rho U^2 \Theta)=\tau_w +\delta^*\frac{dP}{dx}$$

which is commonly simplified to:

$$\frac{d}{dx}(U^2 \Theta)=\frac{\tau_w}{\rho} -\delta^*U\frac{dU}{dx}$$

I know how to derive this, but how can you interpret the pressure gradient term?

This equation almost makes sense, because the loss in boundary layer momentum is contributed by a shear stress (i.e., viscous drag force) term and a pressure drag force. However, why do we multiply the pressure gradient by the displacement thickness $$\delta^*$$? If it was a true force on the boundary layer, shouldn't it be multiplied by the full height of the boundary layer?

All derivations I've seen of this equation start from Navier-Stokes, so it's hard to interpret how the pressure drag acts on a boundary layer control volume.

• "the full height of the boundary layer" is arbitrarily defined, unlike displacement thickness & momentum thickness. – D. Halsey Jan 28 at 20:49