According to equation (2.22), the pressure across the boundary layer is constant in the boundary-layer approximation, and its value at any point is therefore determined by the corresponding main-stream conditions. If *U*(*x*,*t*) now denotes the main stream velocity, so that

Elimination of the pressure from equation (2.21) gives in terms of dimensional variables the boundary layer momentum equation

and from equation (2.23), we have the mass conservation equation

In most physical problems the solutions of the boundary layer equations (4.77) and (4.78) are such that the velocity component

The scale of the boundary layer thickness can, however, be specified adequately by certain lengths capable of precise definition, both for experimental measurement and for theoretical study. These measures of boundary layer thickness are defined as follows:

- Displacement thickness
:

- Momentum thickness
:

- Energy thickness
:

The upper limit of integration is taken as infinity owing to the asymptotic approach of *u*/*U* to 1, but in practice the upper limit is the point beyond which the integrand is negligible.

is the diminution, due to the boundary layer, of the volume flux across a normal to the surface; the streamlines of the outer flow are thus displaced away from the surface through a distance . Similarly, is the flux of the defect of momentum, and is the flux of defect of kinetic energy.

Two other quantities related to these boundary layer thickness are the skin friction
and the dissipation integral *D*. The skin friction is defined as the shearing stress exerted by the fluid on the surface over which it flows, and is therefore the value of
at *y* = 0, which by (3.74) is

in terms of dimensional variables. The rate at which energy is dissipated by the action of viscosity has been shown to be per unit time per unit volume, and

Consequently,