Mech 301 Fluid Mechanics HW9 1. (M-P6.93) An incompressible, Newtonian fluid flows steadily between two infinitely long, concentric cylinders as shown in the sketch. The outer cylinder is fixed, but the inner cylinder moves with a longitudinal velocity V0 as shown. For what value of V0 will the drag on the inner cylinder be zero? Assume that the flow is laminar, axisymmetric and fully developed. 2. (P4.92 in the book) Modify the analysis of Fig. 4.13 to find the velocity vθ when the inner cylinder is fixed and the outer cylinder rotates at angular velocity Ω0. May this solution be added to Eq. (4.140) to represent the flow caused when both inner and outer cylinders rotate? Explain your conclusion. 3. (P4.97 in the book) SAE 10 oil at 20°C flows between parallel plates 8 mm apart, as in Fig. 4.97. A mercury manometer, with wall pressure taps 1 m apart, registers a 6-cm height, as shown. Estimate the flow rate of oil for this condition. 4. (Problem P5.72 in the Book) The power P generated by a certain windmill design depends upon its diameter D, the air density , the wind velocity V, the rotation rate , and the number of blades n. (a) Write this relationship in dimensionless form. A model windmill, of diameter 50 cm, develops 2.7 kW at sea level when V 40 m/s and when rotating at 4800 rev/min. (b) What power will be developed by a geometrically and dynamically similar prototype, of diameter 5 m, in winds of 12 m/s at 2000 m standard altitude? (c) What is the appropriate rotation rate of the prototype? 5. (M2-Q3-2004) Consider an air bubble with radius R, density a , and viscosity a rising in water of density w , and viscosity w solely due to buoyancy as sketched. Assuming that the velocity of the bubble is given by U f ( R, , a , w , g , ) where w a , g is the gravitational acceleration, and is the surface tension. a) Using the Buckingham Pi-theorem, write this equation in dimensionless form. b) It is determined experimentally that the velocity of an air bubble of radius R = 0.1 cm is 3.27 m/s. Is it possible to predict the velocity of a same-size air bubble rising in glycerin? ( w 10 3 kg / m 3 , w 10 3 kg / m s, a 1.26kg / m 3 , a 1.8 x10 5 kg / m s, glycerin 1260kg / m 3 , glycerin 1.49kg / m s, air water 0.073N / m, air glycerin 0.0633N / m)
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