Final ME 106 - Saturday Professor Philip S Marcus ME 106...

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Saturday, December 12, 2009 Professor Philip S. Marcus ME 106 Final Examination 1) 20 points You have built a small Couette-Taylor apparatus. In this apparatus, an incompressible flow with constant density ρ and kinematic viscosity ν is confined between two concentric cylinders. The inner cylinder has radius R inner and outer cylinder has radius R outer . The inner cylinder rotates with an angular velocity of Ω inner and the outer with angular velocity Ω outer . The flow is not only confined by the cylinders, but also by two end plates that are perpendicular to the cylinders at z = 0 and z = H . The end plates rotate with angular velocity Ω outer . The gravitational acceleration g is perpendicular to the z -axis. In your small apparatus, ρ = 1 gram/cm 3 , ν = . 01 cm 2 /sec, R inner = 5 cm, R outer = 7 cm, H = 100 cm, Ω inner = 1 radian/second and Ω outer = 0. With these parameter values, the flow is periodic in time with period τ = 10 seconds. The flow is filled with di ff erent colored particles (which do not a ff ect the flow) and makes a very “cool” pattern that is also periodic in time with the same period as the flow. To brighten up the entrance to Etcheverry Hall, you have been requested to make a big version of the apparatus. The three lengths, R inner , R outer and H must all be 10 times bigger than their values in your small Couette apparatus. However, for the pattern to look “cool”, is is necessary that the flow and the pattern be periodic in time with period 25 seconds. You are free to change the values of Ω inner and Ω outer . In addition, you can change the (constant) values of ρ and ν by blending three fluids together. Specifically, you can choose the ratios α and β of the masses of the three fluids in the mixture that you will use in the big apparatus such that the values of ρ and ν become: ρ = (0 . 5 + 2 . 0 α + 3 . 0 β ) gram/cm 3 and ν = (0 . 003 + 0 . 001 α + 0 . 002 β ) cm 2 /sec. It must be noted that two of the fluids are very common and therefore free but the third fluid is very expensive, so the total cost of the fluid is 600 β dollars. a) Using dimensional analysis, write an expression for the period τ of the flow in terms of the given parameters. b) Can you determine the needed parameter values for ρ , ν , Ω inner and Ω outer to make the big apparatus have the period of 25 seconds, or must you first do some experiments in your small apparatus that might involve changing the values of ρ , ν , Ω inner and Ω outer in the small apparatus?
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