2018 Viscous Flow Midterm Exam  
Friday 2nd November 2018
16:00 — 18:00


NO NOTES OR BOOKS; USE VISCOUS FLOW TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
10.18.18
Question #1
Show that, for a 1D problem along $y$ in which the forces are limited to viscous and pressure forces, the following is correct: $$ − \frac{\rho\delta W}{m \Delta t} = \tau_{yy} \frac{\partial v }{\partial y} $$ with $\delta W/\Delta t$ the rate of work done by the fluid element on the environment by forces other than pressure and $m$ the mass of the fluid element.
Question #2
Two equally big circular plates rotate very close to each other in a viscous fluid as follows:
figure3.png  ./download/file.php?id=4536&sid=33b9a2bc3a5eea3351866ff1a0de6b05  ./download/file.php?id=4536&t=1&sid=33b9a2bc3a5eea3351866ff1a0de6b05
One of the plates is driven by a constant power ${\cal P}_{\rm L}$ and at a constant angular speed $\omega_{\rm L}$. The other plate is braked with a power ${\cal P}_{\rm R}$. For a plate radius $R$ much larger than the distance between the plates $H$, do the following:
(a)  Determine the angular speed $\omega_{\rm R}$ as a function of $\omega_{\rm L}$ and if the breaking power ${\cal P}_{\rm R}=\frac{1}{2}{\cal P}_{\rm L}$
Question #3
Consider a steel flat plate sinking in a pool of water as follows:
Q3.png  ./download/file.php?id=4511&sid=33b9a2bc3a5eea3351866ff1a0de6b05  ./download/file.php?id=4511&t=1&sid=33b9a2bc3a5eea3351866ff1a0de6b05
Knowing that the plate is made of steel with a density of $\rho_{\rm s}=8000$ kg/m$^3$, a thickness of $D=1$ cm, a length $L=2.12$ m, and a height $H=14.25$ cm, that the gravitational acceleration $g=9.81$ m/s$^2$, and that the water has a viscosity $\mu_{\rm w}=0.001$ kg/ms and a density of $\rho_{\rm w}=1000$ kg/m$^3$, do the following:
(a)  Derive an expression for the force felt by the plate due to viscous effects.
(b)  Determine the plate velocity in m/s.
Hint: It is known that the velocity of the falling plate is not more than 30 m/s and at least 5 m/s.
Question #4
Consider a piston of length $L$ and radius $R_1$ moving within a casing of radius $R_2$ as follows:
Q4.png  ./download/file.php?id=4521&sid=33b9a2bc3a5eea3351866ff1a0de6b05  ./download/file.php?id=4521&t=1&sid=33b9a2bc3a5eea3351866ff1a0de6b05
Knowing that $R_2-R_1$ can not be assumed much less than $R_2$, that a force $F$ is applied on the piston making it move at the speed $q$ with respect to the casing, and that the viscosity of the oil located between the piston and the casing is $\mu$, do the following:
(a)  Starting from the momentum and mass conservation equations, derive an expression for the velocity within the oil as a function of the piston speed $q$, and the radii $r$, $R_1$, and $R_2$.
(b)  Find an expression for the oil viscosity $\mu$ as a function of the force $F$, the radii $R_2$ and $R_1$, the piston length $L$, and the piston speed $q$.
Note: $L$ can be assumed much greater than $R_2$.
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