2011 Fluid Mechanics Midterm Exam  
Fluid Mechanics
Midterm Quiz
November 4th 2011
20:00 — 22:00

NO NOTES OR BOOKS; USE FLUID MECHANICS TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; TOTAL POINTS: 100 PTS.
05.02.14
Question #1
Consider the following embankment made of concrete, with the shape of a right-angled triangle with the base $L$ and the height $h$, and fixed to the ground at point A:
figure1.png  ./download/file.php?id=1384&sid=de9353516d2fb75010c8462bf5f33d8a  ./download/file.php?id=1384&t=1&sid=de9353516d2fb75010c8462bf5f33d8a
Determine the minimum length $L$ the concrete embankment must have to avoid tipping over point A. Take $h=1$ m, $\rho_{\rm w}=1000$ kg/m$^3$ and $\rho_{\rm c}=2000$ kg/m$^3$.
Question #2
Water with a density of 1000 kg/m$^3$ flows steadily through the following elbow:
figure2.png  ./download/file.php?id=1385&sid=de9353516d2fb75010c8462bf5f33d8a  ./download/file.php?id=1385&t=1&sid=de9353516d2fb75010c8462bf5f33d8a
At the inlet to the elbow (station 1), the pressure is $5\times 10^5$ Pa and the inner diameter is 80 mm. At the outlet to the elbow (station 2), the inner diameter is 50 mm and the pressure is $4.75 \times 10^5$ Pa. The mass flow rate is 15 kg/s. Around the elbow, the atmospheric pressure is 101300 Pa. Knowing that $\theta=30^\circ$, do the following tasks:
(a)  Determine the $x$ and $y$ components of the force vector $\vec{F}$ that is required to hold the elbow in place.
(b)  How much of this force is due to the atmospheric pressure?
Question #3
A flat plate is placed in front of a horizontal jet of water. The plate is inclined with an angle $\theta=135^\circ$ as shown below:
figure3.png  ./download/file.php?id=1386&sid=de9353516d2fb75010c8462bf5f33d8a  ./download/file.php?id=1386&t=1&sid=de9353516d2fb75010c8462bf5f33d8a
Knowing that $q_1=10$ m/s and that $P_1=P_{\rm atm}=101300$ Pa, determine the ratio of mass fluxes going in the two directions (i.e., find $\dot{m}_3/\dot{m}_1$ and $\dot{m}_2/\dot{m}_1$). The flow is assumed to be inviscid and two-dimensional, and the gravity force can be neglected.
Question #4
Small droplets of liquid are formed when a liquid jet breaks up in spray and fuel injection processes. The resulting droplet diameter, $d$, is thought to depend on liquid density, liquid viscosity, and liquid surface tension, as well as jet speed $V$, and jet diameter $D$. How many dimensionless ratios are required to characterize this process? Determine these ratios. Hint: the surface tension has dimensions of force per unit length or of energy per unit area.

Answers
1.  $1~{\rm m}$.
2.  $F_x=-1314~{\rm N}$, $F_y=424~{\rm N}$, $F_x^{\rm atm}=337~{\rm N}$, $F_y^{\rm atm}=-99.5~{\rm N}$
3.  $0.854$, $0.146$
4.  $\Pi_1=\mu/\rho V D$, $\Pi_2=\sigma/V^2 D \rho$, $\Pi_3=d/D$
PDF 1✕1 2✕1 2✕2
$\pi$