Homework 7
(Due on March 12, 2015)
Note:
All numbered problems are from Panton.
All symbols are the same as those dened/used in class unless stated otherwise.
Copying other peoples work is considered plagiarizing, and it will be treated according to
th
MAE 250A, Winter 2013
J. D. Eldredge
Homework 3, Due Friday, February 1
0. Read chapter 4 of K&C if you have not already done so. (Please note: chapter 4 has a lot of
material on the use of the integral forms of conservation laws. Though we are not coveri
MAE 250A Foundations of Fluid Dynamics
Prof. J. D. Eldredge
Winter 2013
M.S. Comprehensive Exam Question
This exam is due at 5:00 PM on Thursday, March 21, by email to me
Be clean and concise with your answers, and please demonstrate every step clearly.
A Brief Primer on Vector and Tensor Calculus
MAE 250A, J. D. Eldredge
These notes discuss vector and tensor calculus, and the expression of this calculus in Cartesian
coordinate systems. The restriction to Cartesian coordinate expressions is for simplicit
2
3
Problem 4
See les streamline.m and veleld.m for the scripts used for this problem.
Resulting plot:
Streamlines
1
0.8
y
0.6
0.4
0.2
0
0
0.5
1
1.5
2
x
1
2.5
3
3.5
4
5 (OPTIONAL)
Streamlines at t = 0
5
4
3
2
x2
1
0
-1
2
-3
4
-5
0
0.5
1
1.5
2
2.5
x1
3
3.5
MAE 250A
FINAL EXAM
(Open textbook and lecture notes)
Clear writing helps me to understand your work better.
I cannot give you any credit if you show me nothing. Show your work and your thought
process even if you cannot complete a problem so that I can
MAE 250A, Winter 2013
J. D. Eldredge
Homework 2, Due Friday, January 25
0. Finish reading chapters 1 3 of Kundu & Cohen.
1. In cylindrical coordinates, position is described by (r, , z ), where z is the axial position, r
represents the distance from the z
Homework 3
(Due on January 29, 2015)
Note:
All numbered problems are from Panton.
All symbols are the same as those dened/used in class unless stated otherwise.
Kundu, Cohen, and
Copying other peoples work is considered plagiarizing, and it will be tre
MAE 250A
MIDTERM EXAM
(Open book and lecture notes)
Clear writing helps me to understand your work better.
I cannot give you any credit if you show me nothing. Show your work and your thought
process even if you cannot complete a problem so that I can g
MAE 250A, Winter 2013
J. D. Eldredge
Homework 4, Due Friday, February 8
0. Read chapter 9 of K&C.
1. Solve for the velocity prole u(y ) in a uid of viscosity in an innitely long two-dimensional
channel of width h, driven by a constant pressure gradient dp
Solutions to Homework 1
1.
ak = bi cki ,
allowed
aij = bi cj + ejk ,
not allowed
ak = bk c + di eik ,
allowed
ak = bi cki d + eki ,
not allowed
2.
(v) = i ( vi ) = i vi + vi i =
v+v
.
(u v) = i (ijk uj vk ) = ijk vk i uj + ijk uj i vk
= vk kij i uj + u
Homework 4
(Due on February 5, 2015)
Note:
All numbered problems are from Panton.
All symbols are the same as those dened/used in class unless stated otherwise.
Copying other peoples work is considered plagiarizing, and it will be treated according to
Kinematics of Fluids
MAE 250A, J. D. Eldredge
Once we accept the premise that a uid is a continuum, then we use the language of continuum
mechanics to describe the uids motion. In particular, we regard a uid as an innite collection
of innitesimal parcels
MAE 2.50:
Gwétmmé EQUATIONS lM FLU! 593+ Nl '
(25w HA-L. deuswm-héx. aquah'ms gvlau cm. W {pains
04: a PMch 535m [1Cyan VhJQU 01f: W4+aa PAGQIHGW
a 191ka amml volume in 41%.:
.Mwhwah, -\ +ma¢ {1% 41-1-54:
._ .. ,x 4 . A I a/g
I _\ K 75,, \
I
Reynolds Transport Theorem
MAE 250A, J. D. Eldredge
In uid dynamics we are often interested in the rate of change of the total amount of a quantity,
f , in an arbitrary control volume, V (t). The goal of this note is to derive an expression for this
rate
1
NOTE: The solution presented here nondimensionalizes the problem before
solving it. Its not necessary to do so,
but useful.
3
2
w
.2
VkJ-k
tttA4
J7~c(
avf
I)i
~
f
~J
b~
4) I-
t!fJe
~
=- 0
J.0
V-
~
o(,y-
( r~
,72-r1fJ
t!-.~
0t1\r-. )
~
(t'o
J
=:;
0
-:c l
MAE 250A, Winter 2013
J. D. Eldredge
Homework 1, Due Friday, January 18
0. Read chapters 1-3 of Kundu & Cohen.
1. Consider a transformation from an original coordinate system with basis vectors (e1 , e2 , e3 ) to
a new coordinate system with basis vectors
Homework 5
(Due on February 19, 2015)
Note:
All numbered problems are from Panton.
All symbols are the same as those dened/used in class unless stated otherwise.
Copying other peoples work is considered plagiarizing, and it will be treated according to
Homework 6
(Due on February 26, 2015)
Note:
All numbered problems are from Panton.
All symbols are the same as those dened/used in class unless stated otherwise.
Copying other peoples work is considered plagiarizing, and it will be treated according to
The Energy Equation in Fluid Dynamics
MAE 250A, J. D. Eldredge
There are many dierent expressions for the conservation of energy in a uid, so it is instructive to look
at each one and demonstrate how they are all related. The rst law of thermodynamics sta