Air at T = 25 C° and p = 1 atm flows normal to a circular cylinder with D = 1 cm at velocity U = 10 m/s. According to

the inviscid theory, the external velocity on the surface of the cylinder profile is given by Eq. 8.39 in the textbook with K = 0, v_{\vartheta}(\vartheta)=-2 U \sin \vartheta \quad \rightarrow \quad u_{s}\left(x_{s}\right)=2 U \sin \left(2 \frac{x_{s}}{D}\right) where us(xs) is the same velocity as va(9) but expressed in curvilinear coordinates originating at the stagnation point (rs = 0 or v = T) as shown in the figure below. In the vicinity of the stagnation point the above external velocity can be approximated as u_{s}\left(x_{s}\right)=\frac{4 U x_{s}}{D} (a) Derive Eq. 2 starting from Eq. 1; (b) use Thwaite's method and Eq. 2 to calculate the wall shear stress Tw, the displacement thickness &* and momentum thickness theta in the vicinity of the stagnation point. Hint: use Karman Eq. 7.51 in the textbook to obtain 8*.