In the boundary layer, the momentum equation in the $y$ direction reduces to $\partial p/\partial y = 0$. The pressure in the boundary layer remains constant along the normal direction. So the pressure depends only on the $x$ coordinate, and the pressure-gradient term in the momentum equation in the $x$ direction can be written in ordinary differential form \[ u\,\frac{\partial u}{\partial x} + v\,\frac{\partial u}{\partial y} = -\,\frac{1}{\rho}\,\frac{dP}{dx} + \frac{\mu}{\rho}\left(\frac{\partial^{2}u}{\partial y^{2}}\right) \] where $P$ is the fluid pressure outside the boundary layer that might change in the streamwise direction, it is also called Prandtl’s boundary-layer equation

Applying Newton’s law of viscosity: $\tau=\mu\,(\partial u/\partial y)$, the boundary-layer equation for steady, incompressible, laminar flow \(\boxed{u\,\frac{\partial u}{\partial x} + v\,\frac{\partial u}{\partial y} = U\,\frac{\partial U}{\partial x} + \frac{\mu}{\rho}\,\frac{\partial^{2}u}{\partial y^{2}}}\) can be rewritten as \[ u\,\frac{\partial u}{\partial x}+v\,\frac{\partial u}{\partial y} =U\,\frac{\partial U}{\partial x}+\frac{1}{\rho}\,\frac{\partial \tau}{\partial y} \]

This equation is applicable to both laminar and averaged turbulent boundary layers. For the turbulent boundary-layer equation, the velocity and shear-force terms are replaced by time-averaged values \[ \overline{u}\,\frac{\partial \overline{u}}{\partial x} +\overline{v}\,\frac{\partial \overline{u}}{\partial y} =U\,\frac{\partial U}{\partial x} +\frac{1}{\rho}\,\frac{\partial \overline{\tau}}{\partial y} \] The averaged shear stress is not equal to the gradient of the averaged velocity \(\overline{\tau}\not\propto \frac{\partial \overline{u}}{\partial y} \)

It can be proved that the average shear stress for turbulent flows is \[ \underbrace{\overline{\tau}}_{\text{average shear stress}} = \underbrace{\mu\,\dfrac{\partial \overline{u}}{\partial y}}_{\text{gradient of the averaged velocity}} \;+\; \underbrace{\left(-\,\rho\,\overline{u'v'}\right)}_{\substack{\text{streamwise and wall-normal} \\ \text{components of the fluctuating velocity}\\ \text{(difference between instantaneous and averaged)}}} \]

In many of the simpler turbulence models, turbulent shear stress is expressed in an analogous manner as suggested by the French mathematician Joseph Boussinesq as \[ -\rho\,\overline{\underbrace{u'}_{\text{streamwise fluctuation}}\; \underbrace{v'}_{\text{wall-normal fluctuation}} } \]