We studied the case of the lifting flow over a circular cylinder. In real life, a rotating cylinder in a flow will produce lift; such real flow fields are shown in the photographs in Figure 3.34b and \(c\). Here, the viscous shear stress acting between the flow and the surface of the cylinder drags the flow around in the direction of rotation of the cylinder.

For a cylinder of radius \(R\) rotating with an angular velocity \(\omega\) in an otherwise stationary fluid, the viscous flow solution for the velocity field obtained from the Navier-Stokes equations is

\[ V_{\theta}=\frac{R^{2} \omega}{r} \]

where \(V_{\theta}\) is the tangential velocity along the circular streamlines and \(r\) is the radial distance from the center of the cylinder. (See Schlichting, Boundary-Layer Theory, 6th ed., McGraw Hill, 1968, page 81.) Note that \(V_{\theta}\) varies inversely with \(r\) and is of the same form as the inviscid flow velocity for a point vortex given by Equation (3.105). If the rotating cylinder has a radius of \(1 \mathrm{~m}\) and is flying at the same velocity and altitude as the airfoil in Problem 4.11, what must its angular velocity be to produce the same lift as the airfoil in Problem 4.11?

Data From Problem 4.11:
Consider again the NACA 2412 airfoil discussed in Problem 4.10. The airfoil is flying at a velocity of \(60 \mathrm{~m} / \mathrm{s}\) at a standard altitude of \(3 \mathrm{~km}\). The chord length of the airfoil is \(2 \mathrm{~m}\). 

Data From Problem  4.10:

For the NACA 2412 airfoil, the lift coefficient and moment coefficient about the quarter-chord at \(-6^{\circ}\) angle of attack are -0.39 and -0.045 , respectively. At \(4^{\circ}\) angle of attack, these coefficients are 0.65 and -0.037 , respectively.