Suppose that a piece of tubing of inner radius a, half filled with a Newtonian liquid and half filled with air, has its ends joined to form a hoop of radius R, as shown in Fig. P7-7(a). When the hoop is rotated in a vertical plane at angular velocity w, a height difference h develops between the two air-liquid interfaces. Because a < < < R, locally the tube appears to be straight. The z coordinate shown in Fig. P7-7(b) corresponds to the contour length along the tube axis. (a) Assuming that the flow is steady and fully developed, determine v(r) in the liquid in terms of h, the liquid properties, and other quantities in the figure. (b) Evaluate h, assuming that surface tension effects are negligible. (c) Determine the torque that must be applied to the hoop to maintain its rotation. g h R @ + 2a Air - Liquid a @R Z @R (a) (b) Figure P7-7. Rotating hoop: (a) overall view of a vertical hoop formed from a half-filled length of tubing: and (b) enlargement of the region indicated by the dashed rectangle in (a).