Air at T 25 C and p 1 atm flows normal to a circular cylinder with D 1cm at velocity U = 10m/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_{\theta}(\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 u,(r.) is the same velocity as va(@) but expressed in curvilinear coordinates originating at the stagnation point (r, = 0 or d = #) as shown in the figure below. In the vicinity of the stagnation point the above external velocity can be approximated u_{s}\left(x_{n}\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 0 in the vicinity of the stagnation point. Hint: use Karman Eq. 7.51 in the textbook to obtain 6*.BONUS. Repeat the calculation in (b) but using the exact expression of u, in Eq. 1 (you will need to use numerical ine

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Air at T 25 C and p 1 atm flows normal to a circular cylinder with D 1cm at velocity U = 10m/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_{\theta}(\vartheta)=-2 U \sin \vartheta \quad \rightarrow \quad u_{s}\left(x_