2017 Viscous Flow Midterm Exam  
When is the best time for you?
Thu Nov 2 18:00-20:00  36% of the voters. 10% of the votes.    8
Fri Nov 3 11:00-13:00  55% of the voters. 15% of the votes.    12
Fri Nov 3 14:00-16:00  73% of the voters. 21% of the votes.    16
Sat Nov 4 11:00-13:00  41% of the voters. 12% of the votes.    9
Mon Nov 6 12:45-14:45  55% of the voters. 15% of the votes.    12
Wed Nov 8 12:45-14:45  32% of the voters. 9% of the votes.    7
Wed Nov 8 18:00-20:00  23% of the voters. 6% of the votes.    5
Thu Nov 9 18:00-20:00  41% of the voters. 12% of the votes.    9
Poll ended at 2:44 pm on Monday October 30th 2017. Total votes: 78. Total voters: 22.
Sunday 5th November 2017
19:00 — 21:00


NO NOTES OR BOOKS; USE VISCOUS FLOW TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
10.27.17
Question #1
Starting from Newton's law $\vec{F}_y=m\frac{dv}{dt}$ and the mass conservation equation show that the $y$-component of the momentum transport equation for a viscous fluid corresponds to: $$ \frac{\partial \rho v}{\partial t} + \frac{\partial \rho u v}{\partial x} + \frac{\partial \rho v^2}{\partial y} + \frac{\partial \rho w v}{\partial z} = -\frac{\partial P}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{zy}}{\partial z} $$ with $P$ the pressure and $\tau_{ij}$ the shear stress vector along $j$ acting on the faces perpendicular to $i$.
Question #2
An air stream with a speed of $50$ m/s and density of $\rho=1.0$ kg/m$^3$ flows parallel to a flat plate with a length of 45 cm and a width of 100 cm. Determine the total drag force on the flat plate and calculate the boundary layer thickness 10 and 45 cm from the leading edge. Take the kinematic viscosity as $15\times 10^{-6}$ m$^2$/s.
Question #3
Consider a journal bearing as follows:
Q3.png  ./download/file.php?id=3890&sid=1f9a190fc5373f59f97830091b3e22dd  ./download/file.php?id=3890&t=1&sid=1f9a190fc5373f59f97830091b3e22dd
Starting from the Navier-Stokes equations in cylindrical coordinates, derive an expression for $v_\theta$ as a function of $r$. It is known that $R_2-R_1=\frac{1}{4} R_1$. Indicate clearly the assumptions (including terms dropped) and explain why this is valid in this case.
Question #4
A large piece of styrofoam floats on water and is being pulled by the force $F$ as follows:
Q4.png
Knowing that the force pulling the styrofoam is of $F=9.37$ N, and that the length, height, and depth of the styrofoam block are $L=10$ m, $H=0.3$ m, and $D=2$ m respectively, find the speed $q$ of the styrofoam with respect to the water. You can neglect air resistance.
The midterm will take place Sunday November 5th 19:00 — 21:00 in room 9301.
10.30.17
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