Spessore di quantità di moto

$$\vartheta = \int_0^\infty{\frac{u}{U}\left(1-\frac{u}{U}\right)dy}$$

$$= \int_0^\infty{\frac{x f'}{k x}\left(1-\frac{x f'}{k x}\right)\sqrt{\frac{\nu}{k}}d\eta}$$

$$= \sqrt{\frac{\nu}{k}}\int_0^\infty{\frac{k F'}{k }\left(1-\frac{k F'}{k }\right)d\eta}$$

$$= \sqrt{\frac{\nu}{k}}\int_0^\infty{F'\left(1-F'\right)d\eta}$$

$$= 0.292\sqrt{\frac{\nu}{k}}$$

$$= 0.292\sqrt{\frac{\nu x^2}{k x^2}}$$

$$= 0.292\ x\ \sqrt{\frac{\nu}{U x}}$$

$$= 0.292\ x\ \frac{1}{\sqrt{\mathcal{R}e_x}}$$

$$\frac{\vartheta}{x}\sqrt{\mathcal{R}e_x}=0.292$$ (8.10)