Stress.

The normal components of the stress perpendicular and parallel to the flat plate expressed non-dimensionally are

$$\frac{\tau_{xx}}{\rho U^{2}} = -\frac{p}{\rho U^{2}}+\underse... ...underbrace{2\frac{\nu}{U^{2}}\frac{\partial u}{\partial x}}} ,$$ (68)

$$\frac{\tau_{yy}}{\rho U^{2}} = -\frac{p}{\rho U^{2}}+\underse... ...underbrace{2\frac{\nu}{U^{2}}\frac{\partial v}{\partial y}}} .$$ (69)

Therefore, in the limit $R \rightarrow \infty$, we have that

$$\tau_{xx} = \tau_{yy} = -p$$ (70)

The shearing stress over surfaces parallel to the wall is

$$\tau_{xy} = \mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right),$$ (71)

which is approximated in the same way as the vorticity, as follows.

$$\begin{split} \rho U^{2}\tau'_{xy} & = \rho \nu\left(\underse... ...}+\rho U^{2}R^{-3/2}\frac{\partial v'}{\partial x'} \end{split}$$ (72)

In the limit $R \rightarrow \infty$, the shear stress is given by

$$\tau'_{xy} \sim R^{-1/2}\frac{\partial u'}{\partial y'}$$ (73)

or in terms of dimensional variables

$$\tau_{xy} = \mu\frac{\partial u}{\partial y}.$$ (74)

For the Blasius laminar boundary layer similarity solution given by equation (3.47), the shear stress $\tau_{xy}$ is given by

$$\tau_{xy} \sim \mu\frac{\partial^{2} \psi}{\partial y^{2}} = \mu\left(\frac{U^{3}}{2\nu x}\right)^{1/2}f''(\eta),$$ (75)