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ps5_solutions

ps5_solutions - Harvard-MIT Division of Health Sciences and...

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Harvard-MIT Division of Health Sciences and Technology HST.542J: Quantitative Physiology: Organ Transport Systems Instructors: Roger Mark and Jose Venegas MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departments of Electrical Engineering, Mechanical Engineering, and the Harvard-MIT Division of Health Sciences and Technology 6.022J/2.792J/BEH.371J/HST542J: Quantitative Physiology: Organ Transport Systems PROBLEM SET 5 SOLUTIONS March 16, 2004

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Problem 1 A. The hypodermic needle in the figure below contains a saline solution. If a plunger of area A is pushed in at a steady rate ( V ), what is the mean exit velocity ( V e ) of solution leaving the needle of area A e ? Assume no leakage past the plunger. Using conservation of mass: AV = A e V e V e = V A A e B. If there is leakage back past the plunger equal to one-third the volume ﬂow rate from the needle, find an expression for V e . If one-third of the needle ﬂow rate leaks back past the plunger we would have: AV = A e V e + 1 3 A e V e = 4 3 A e V e V e = 3 4 V A A e C. Neglecting leakage past the plunger, find an expression for the pressure at the face of the plunger if the ﬂuid exits the needle at atmospheric pressure and the ﬂuid can be treated as though it were inviscid. The ﬂow can be treated as steady. V Area = A A e V e Using Bernoulli’s equation between a point on the plunger and the end of the needle: P 1 + 1 2 ρ V 2 1 = P atm + 1 2 ρ V 2 e P 1 P atm = 1 2 ρ V 2 e V 2 1 but V 1 , V e were given above in A. P 1 = 1 2 ρ V 2 A A e 2 1 6.022j—2004: Solutions to Problem Set 5 2
assuming P atm 0 1 2 2004/241 3 6.022j—2004: Solutions to Problem Set 5

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Problem 2 A common type of viscometer consists of a cone rotating against a fixed plate, as shown in Figure 1. Show from physical arguments (or otherwise) that the shear rate is independent of
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