Homework 1
1. Evaluate the following partial derivative
5
x sin 3xy 2 e y
x
2. For the function depicted by the bold lines below, evaluate the following closed path integral
(2 points)
B
C
D
A
A
B
C
D
f ( x)dx f ( x)dx f ( x)dx f ( x)dx f ( x)dx ?
ABCD
f
Lecture10
21612
Differentialequationofmotionforanyfluid,underthecontinuumassumption
(thesearenottheNavierStokesequationsyet)
u
u
u
u xx yx zx
gx
u
v
w
y
z
x
y
z x
t
v
v
v
v xy yy zy
t u x v y w z x y z gy
w
w xz yz zz
w
w
w
v
u
x y z gz
z
y
x
t
Lecture9
2912
u
u
u
u xx yx zx
gx
u
v
w
y
z
x
y
z x
t
v
v
v
v xy yy zy
gy
u
v
w
x
y
z
x
y
z
t
w
w xz yz zz
w
w
t u x v y w z x y z gz
Differentialequationofmotionforanyfluid,underthecontinuumassumption
(thesearenottheNavierStokesequationsyet)
u
Lecture16
3812
Re = 0.16
Re = 26
Re = 1.54
Re = 140
Re = 9.6
Re = 2000
Re = 10,000
Re x
V x
Re x , critical 5 105 , depending on roughness
Sphere
Re=15000
Re=30000
Significant drop in the
pressure drag due to the
delayed separation!
V
http:/www.youtube.c
Polymerization of Methyl Methacrylate
Experiment has many facets: a. introduction to polymers and polymerization b. use of inert atmosphere methods c. characterization of polymers Polymers and Polymerization monomers - dimers - trimers - oligomers (n = 5-
Problem 2.40
[Difficulty: 2]
Given:
Velocity distribution between flat plates
Find:
Shear stress on upper plate; Sketch stress distribution
Solution:
Basic equation
du
yx =
dy
yx =
At the upper surface
Hence
y=
du
=
dy
d
dy
u max 1
2
2 y = u 4 2 y = 8
Problem 4.38
[Difficulty: 2]
Given:
Data on flow at inlet and outlet of a reducing elbow
Find:
Find the maximum velcoity at section 1
Solution:
r
r
V dA = 0
Basic equation
CS
Assumptions: 1) Steady flow 2) Incompressible flow
Evaluating at 1, 2 and 3
h
1
Problem 3.52
Given:
Geometry of plane gate
Find:
[Difficulty: 3]
Minimum weight to keep it closed
L=3m
h
y
L/2
dF
W
w=2m
Solution:
FR = p dA
Basic equation
MO = 0
FR = pc A
or, use computing equations
dp
= g
dh
Ixx
y' = yc +
A yc
Assumptions: static fluid
Problem 5.1
[Difficulty: 1]
The list of velocity fields provided above
Given:
Which of these fields possibly represent two-dimensional, incompressible flow
Find:
We will check these flow fields against the continuity equation
Solution:
Governing
u v w 0 (
Problem 6.44
[Difficulty: 2]
Given:
Wind tunnel with inlet section
Find:
Dynamic and static pressures on centerline; compare with Speed of air at two locations
Solution:
Basic equations
p dyn =
1
2
2
air U
p
air =
Rair T
p 0 = p s + p dyn
p = w g h
p atm
Problem 8.72
Given:
Data on flow through Alaskan pipeline
Find:
[2]
Head loss
Solution:
Basic equation
2
2
p
h
p
V1
V2
lT
1
2
+
+ z1
+
+ z2 =
= HlT
g
g
2 g
2 g
g
oil
oil
Assumptions: 1) Steady flow 2) Incompressible flow 3) at 1 and 2 is approxima
Problem 9.91
Given:
Fishing net
Find:
[Difficulty: 3]
Drag; Power to maintain motion
3
8
in 9.525 mm
Solution:
Basic equations:
CD
FD
1
2
V A
2
We convert the net into an equivalent cylinder (we assume each segment does not interfere with its neighbors
Problem 9.118
[Difficulty: 4]
Problem 3.78
Given:
Gate geometry
Find:
Force on stop B
[Difficulty: 4]
x
y
Solution:
Basic equations
4R/3
R/2
D
FV
dp
= g
dh
W1
A
R
FB
MA = 0
WGate
FH
y
W2
x
W eights for computing FV
F1
Assumptions: static fluid; = constant
ME 170B: Probability and Statistics Review
V Sundararajan
October 21, 2013
1
Probability Review
A random variable is a real number associated with an outcome of an experiment. e.g. If we associate the
numbers 0 and 1 to the outcomes tails and heads of a c