2019 Fundamentals of Fluid Mechanics A Midterm Exam  
Thursday November 7th 2019
11:00 — 12:15
INSTRUCTIONS
•  NO NOTES OR BOOKS; USE FUNDAMENTALS OF FLUID MECHANICS TABLES THAT WERE DISTRIBUTED.
•  ALL QUESTIONS HAVE EQUAL VALUE; ANSWER ALL 2 QUESTIONS.
•  YOU MUST USE THE UNIVERSITY OF ARIZONA BLUE EXAMINATION BOOK TO WRITE YOUR SOLUTIONS.
•  USE THE LEFT PAGE OF THE BLUE BOOK FOR THE DRAFT, AND THE RIGHT PAGE FOR THE SOLUTION. I WILL NOT CORRECT ANYTHING WRITTEN ON THE LEFT PAGE.
•  WRITE YOUR SOLUTIONS IN SINGLE COLUMN FORMAT, WITH ONE STATEMENT FOLLOWING ANOTHER VERTICALLY.
•  WRITE YOUR SOLUTIONS NEATLY SO THAT THEY ARE EASY TO READ AND VERIFY.
•  DON'T WRITE ONE LINE WITH TWO EQUAL SIGNS.
•  HIGHLIGHT YOUR ANSWERS USING A BOX.
11.04.19
Question #1
Consider a rigid pressurized vessel travelling in outer space. The vessel contains the air required to support the life of its passengers. A small hole forms on the surface of vessel and air escapes to the vacuum of space. It is known that the mass flow rate of air coming out the hole can be well approximated as $$ \dot{m}_{\rm out}=\frac{1}{4} \rho \overline{q} A $$ with $\overline{q}$ the average molecular speed of the air in the vessel, $\rho$ the density of the air in the vessel, and $A$ the hole area. Derive an equation giving the average temperature $T$ in the vessel as a function of the time $t$ since the hole was created, the initial temperature $T_{\rm i}$, the area $A$ of the hole, and the volume $V$ of the vessel. Consider the air within the vessel to be a perfect molecular gas. Your expression for $T$ must be as accurate as possible. Outline all assumptions.
Question #2
Consider a water reservoir with the upper wall freely rotating around a pivot as follows:
Q2.png
The dimensions $a$, $b$, and $f$ are given. In order to prevent the upper wall from rotating (and hence, the water from escaping the reservoir), you decide to attach a deflector on it at a distance $c$ from the pivot. Then, you drop some water on the deflector such that the change of momentum of the water when deflected will create a force on the upper wall, hence preventing it from rotating. Because the amount of water being dropped on the deflector is limited, you need to optimize your design so that the amount of water that can be contained within the reservoir is as large as possible for a given mass flow rate $\dot{m}$. That is, you need to find the optimal value for the length $c$ and for the angle $\theta$ that maximize the height $h$ for a given $\dot{m}$ and a given $f$. Specifically, do the following:
(a)  Find the value for $c$ that results in an optimal design (largest possible $h$ for given $\dot{m}$, $a$, $b$, and $f$).
(b)  Find the value for $\theta$ that results in an optimal design (largest possible $h$ for given $\dot{m}$, $a$, $b$, and $f$). You must give a proof that your chosen $\theta$ is optimal.
(c)  Find an expression for the height $f$ that will result in the deflected water creating just enough force on the upper wall that the water in the reservoir does not escape from the reservoir. Your expression should be a function of the water density $\rho_{\rm w}$, the air density $\rho_{\rm a}$, the dimensions $a$, $b$, $c$, $h$, the mass flow rate per unit depth $\dot{m}$, the angle $\theta$, and the gravitational acceleration $g$.
Note that because this is a 2D problem, $\dot{m}$ is the mass flow rate per unit depth and has units of kg/ms. Also note that (a), (b), and (c) can be answered independently of each other.
Answers
2.  $\frac{\pi}{2}-{\rm atan}\left(\frac{a}{b}\right)$, $\frac{1}{2g}\left( \frac{e}{\dot{m} \left( \sin(\phi+\theta) + \cos(\phi) \right)}\right)^2$ with $e=-\frac{\rho_{\rm w} g (a-h)(a^2+b^2)}{2c}+\frac{\rho_{\rm w} g a (a^2+b^2)}{3c}-\frac{\rho_{\rm w}g(a-h)(\sqrt{a^2+b^2}-h \cos(\phi))^2}{2c}+\frac{\rho_{\rm w} g a(\sqrt{a^2+b^2}-h \cos(\phi))^3}{3c\sqrt{a^2+b^2}}$ and $\phi={\rm atan}(a/b)$
10.13.20
Due on Wednesday October 21st at 23:00. Do problem #2 only.
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$\pi$