Vorticity.

The vorticity of the flow in Cartesian coordinates in term of dimensional variables is given by

$$\omega = \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}.$$ (61)

If we non-dimensionalize the vorticity, according to

$$\omega = \frac{U}{L}\omega',$$ (62)

the non-dimensional form of the equation (3.61) is

$$\frac{U}{L}\omega' = \frac{U}{\delta}\frac{\partial u'}{\partial y'}-\frac{U\delta}{L^{2}}\frac{\partial v'}{\partial x'},$$ (63)

and since $\delta \sim O(R^{-1/2}L)$, we have that

$$\omega' = R^{1/2}\frac{\partial u'}{\partial y'}-R^{-1/2}\frac{\partial v'}{\partial x'},$$ (64)

and in the limit $R \rightarrow \infty$ we have

$$\omega' \sim R^{1/2}\frac{\partial u'}{\partial y'}.$$ (65)

In the context of the boundary-layer approximation, the vorticity in terms of dimensional variables is given by

$$\omega = \mu\frac{\partial u}{\partial y}.$$ (66)

We can write the vorticity for the Blasius boundary layer similarity solution by placing equation (3.47) for $\psi$ into the equation (3.66), which gives

$$\omega = \mu\frac{\partial^{2}\psi}{\partial y^{2}} = \mu\left(\frac{U^{3}}{2\nu x}\right)^{1/2}f''(\eta)$$ (67)