Instructor: Mette S Olufsen
Office: SAS 3216
Office Hours: By appointment via email.
Phone Number: 515-2678
Email address: email@example.com
This course will
offer an introduction to fluid dynamics and its applications in
biology. In particular we will spend time discussing blood flow in
arterial networks. The course will study fluid mechanics from a
mathematical perspective, including examples from biological
applications. Students will work on an independent project that
can be taken from their research. The course is suitable for
advanced undergraduate and graduate students in mathematics,
bio-engineering, and mathematical biology. Topics will include
physical concepts such as viscosity, vorticity, shock waves, flow
in tubes and pipes, stokes flow, boundary layers, and potential
flow. Biological applications will include swimming, flying, and
blood flow in networks of arteries.
Text: A mathematical introduction
to fluid mechanics, third edition, corrected
fourth printing, by A.J. Chorin, and J.E. Marsden, Springer,
2000. Another excellent reference is Elementary
Fluid Dynamics by D.J. Acheson, Oxford Applied
Mathematics and Computing Science Series, Claredon Press,
During the course a number of smaller and larger homework
projects will be assigned. Students should be prepared to
discuss results of homework problems on the board or using
electronic media in class. While homework problems can be worked
out in a group, everybody should be able to discuss the solution
to a given homework. In addition to homework problems, one
in-class (mid-term) and one take-home (final) test will be
given. Results of the take-home test should be typed up and
handed in electronically.
Evaluation criteria:The course will be
graded based on the work performed in homework (40% of grade),
group project (20% of grade), and tests (midterm 20% and final
Basic understanding of mechanics, vector calculus, matrices,
ordinary differential equations. Other mathematics and physics
will be reviewed as needed.
Homework: I encourage you to work with others on the
homework, but remember - if you work with someone else, put
all names on the paper for a common grade. If computers are
used for solving aspects of an homework problem (e.g., Maple,
Matlab, XPP, or Mathematica) you must indicate so on the
homework. Nicely commented computer code should be attached as
part of the homework. Points will be deducted for homework
handed in late - graded homework can be reworked for some
extra credit. Some homework problems will be discussed in
class, and I will call on you to go through your solution on
the board. For homework, anyone in the group can be called to
discuss the solution of any problem.
Step 1 (Due Wednesday 3/20) Write
short introduction to project and include 2-3 questions that
you plan to address during the project.
Homework 1 (Due Monday 1/14) 1.
Verify the divergence theorem
2. Problem 1.1.1 (page 18)
3. Write out Euler's equation and the equation for
conservation of mass. Assume x = (x(t), y(t), z(t)) and u
= (u1, u2, u3).
Include gravity (a body force)
assuming g = (0, 0, g)
4. Prove the lemma page 8-9 (the proof is in the
book, but I want you guys to go through it and write it up with all
details). Homework 2 (Due Wednesday 1/23) 1.
Show Eulers equations for isentropic flow given BE (page
3 (Due Wednesday 2/6) 1.
Couette-flow (problem 1.2-2
coordinates (problem 1.3-2 page 45)
3. The stress tensor
(problem 6.1 page 217 - Acheson)
4. Simple shear flow
example (problem 6.3 page 217 - Acheson)
Homework 4 (Due
Wednesday 2/13) 1.
Predict the Reynolds number for
a flow present either in your
research project, or find an
Discuss the problem and note
weather it should be described
using Stokes or Navier Stokes
Also note what coordinate system
would be more appropriate for
describing this flow.
2. Redo the
example we started in class
1.3-4 page page 46.
(Due Wednesday 2/27) 1.
blood flow Homework
6 (Due Wednesday
In class we studied
the example for the
piston pulled with
constant velocity U,
for which p = A rho^g
(g = gamma).
right state I u_r = 0,
rho_r = rho_0, p_r = A
For the left state III
u_l = U
gas dynamics flow
c_r, c_l, u, and c
(within state II). We
did this example in
part in class.
Explain in words and
using an illustration,
what happens when the
piston is moved in
rather than out?
You do not have to do
this rigorously using
math, we'll discuss
this case in more
detail after the
For the gas dynamics
analogy to the blood
flow model, show that
for gamma =/= 1 that
Furthermore show that
the state equation p =
Eh/r_0 (1 -
sqrt(A0/A)) + p0 fits
into the above form.
Find values for g, K,
Homework 7 (Due Monsday 3/25) 1. Show that the
125) for a
for which eps
= p tau /
(gamma - 1) is
2 mu H(tau, p)
= (tau - mu^2
tau_0) p -
(tau_0 - mu^2
Show that p
(tau) for H =
0 forms the
curve given in
page 126. 3. Show that for
law p = A
H(tau,p) = 0
gives that p
8 (Due Monday
4/15) 1. Exercise
2.1.-4 page 66
in Chorin and
torque on a
NOTE: IF YOU DO NOT HAVE MATLAB ON YOUR PERSONAL
COMPUTERS, THE COMPUTERS IN SAS HALL LABS ALL HAVE MATLAB!