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AER210 VECTOR CALCULUS and FLUID MECHAICS
Quiz 4
25 November 2010
9:15 am – 10:15 am
Closed Book, no aid sheets
Non-programmable calculators allowed
Instructor: Alis Ekmekci
Family Name: __________________________________________
Given Name: __________________________________________
Student #:
__________________________________________
TA Name/Tutorial #: ____________________________________
FOR MARKER USE ONLY
Question
Marks
Earned
1
8
2
7
3
9
4
10
5
10
6
10
TOTAL
54
/50
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1) a) A company produces a perishable product in a factory located at x = 0 and sells the product
along a certain distribution route. The selling price (P) of the product is a function of the length of
time (t) after it was produced, and the distance (x) from the factory it was produced. That is, P =
P(x,t). At a given location the price of the product decreases in time (as it is perishable) according
to
= − 12 dollars/hr. In addition, because of shipping costs the price increases with distance
from the factory according to
= 0.2 dollars/km. If the manufacturer wishes to sell the product
at the same constant price everywhere along the distribution route, determine how fast he must
travel along the route. (4 points)
1) b) Explain the difference between surface forces and body forces. (2 points)
1) c) Consider a control volume of arbitrary shape with volume V. If the pressure distribution
over the control surface is constant, use the gradient theorem to prove that the resultant pressure
force on the control surface is zero. (2 points)
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2) A piezometer and a Pitot tube are tapped into a horizontal pipe, as shown in the Figure, to
measure static and stagnation (=static+dynamic) pressures. For the indicated water column
heights, determine the velocity of the pipe. (7 points)
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3) A U-tube of constant area acts as a water siphon as shown in the figure. Water exits from the
bottom of the siphon (2) as a free jet at atmospheric pressure. Determine, after listing all the
necessary assumptions, the speed of the free jet, and the absolute pressure at (A), the highest part
of the tube. (Density of the water ρ = 1000 kg/m3 and the gravitational acceleration g = 10 m/s2,
and the atmospheric pressure is 101.3 kPa). (9 points)
Assumptions:
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4) Water flows through a horizontal 180° pipe bend as illustrated in Figure below. The flow cross
section area is constant at a value of 9000mm2. The flow velocity everywhere in the bend is 15
m/s. The pressures at the entrance and exit of the bend are 210 and 165 kPa, respectively.
Calculate the x and y components of the anchoring force needed to hold the bend in place. (The
density of water is 1000kg/m3) (10 points)
Solution:
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5) An engineer is measuring the lift and drag on an airfoil section mounted in a two-dimensional
wind tunnel. The wind tunnel is 0.5 m high and 0.5 m deep (into the paper). The upstream wind
velocity is uniform at 10 m/s, and the downstream velocity is 12 m/s and 8 m/s as shown. The
vertical component of velocity is zero at both stations. The test section is 1m long. The engineer
measures the pressure distribution in the tunnel along the upper and lower walls and finds:
Pu = 100 - 10x - 20x(1-x)
(Pa, gage)
Pl = 100 - 10x + 20x(1-x)
(Pa, gage)
where x is the distance in meters measured from the beginning of the test section. The gas density
is homogeneous throughout and equal to 1.2 kg/m3. The lift and drag are the vectors indicated on
the figure, and are shown with symbols L and D respectively. The forces acting on the fluid are in
the opposite direction to these vectors. Find the lift and drag forces acting on the airfoil section.
(10 points)
Solution:
Apply the momentum principle to the control volume below:
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6) A steady, incompressible, two-dimensional velocity field is given by
= (u,v) = (0.5 + 0.8x) ̂ + (1.5 – 0.8y) ̂
V
where the x- and y- coordinates are in meters and the magnitude of velocity is in m/s.
(a) Determine if there are any stagnation points in this flow field, and if so, where? (2 points)
(b)For this velocity field, generate an analytical expression for the flow streamlines (6 points)
(c) and draw a couple of streamlines in the right half of the flow (x > 0) (2 points).
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