MA 591 Introduction to Fluid Dynamics

Mon-Wed 4.30-5.45

    Class-room: 1108 SAS Hall

    Instructor: Mette S Olufsen
    Office: SAS 3216
    Office Hours: By appointment via email.
    Phone Number: 515-2678
    Email address: msolufse@ncsu.edu

Course Information

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, Oxford, UK.
Format: 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 20%).

Prerequisites:  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.


NOTES

    1.    About the course
    2.    Syllabus
    3.    Course policies
    4.    Errors (old edition of book)
    5.    Isentropic flow
    6.    Navier Stokes equations
    7.    Fluid flow images
    8.    Hyperbolic systems
    9.    Chapter 4
  10.    Swimming and flying (pdf)

PROJECT

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

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 15)

Homework 3 (Due Wednesday 2/6)
1.    Couette-flow (problem 1.2-2 page 31).
2.    Cylindrical 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 example.
       Discuss the problem and note weather it should be described using Stokes or Navier Stokes equations.
       Also note what coordinate system would be more appropriate for describing this flow.
2.    Redo the example we started in class (pages 42-43)
3.    Problem 1.3-4 page page 46.

Homework 5 (Due Wednesday 2/27)
1.    Solve the two problems related to blood flow

Homework 6 (Due Wednesday 3/13)
1.    In class we studied the example for the piston pulled with constant velocity U, for which p = A rho^g (g = gamma).
       For the right state I u_r = 0, rho_r = rho_0, p_r = A rho_0^g
       For the left state III u_l = U

       Solve 1D gas dynamics flow equations predicting c_r, c_l, u, and c (within state II). We did this example in part in class.

2.    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 break.

3.    For the gas dynamics analogy to the blood flow model, show that for gamma =/= 1 that
      
       A(p) = ( A(p*)^{g-1} + (g-a)/g (p - p*)/(K rho^g )^(1/g - 1)

       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, and p0.

Homework 7 (Due Monsday 3/25)
1.    Show that the Hugoniot function (page 125) for a gamma-gas law for which eps = p tau / (gamma - 1) is given by

        2 mu H(tau, p) = (tau - mu^2 tau_0) p - (tau_0 - mu^2 tau) p_0

2.    Show that p (tau) for H = 0 forms the hyperbolic curve given in figure 3.2.3 page 126.

3.    Show that for the gamma-gas law p = A rho^gamma, for which p rho^(-gamma) = p_0 rho_0^(-gamma) or equivalently p tau^gamma = p_0 tau_0^gamma
       H(tau,p) = 0 gives that p rho^(-gamma) cannot be constant. (Check argument page 126).

Homework 8 (Due Monday 4/15)
1.    Exercise 2.1.-4 page 66 in Chorin and Marsden
2.    Exercise 4.7 (Acheson) - except final question related to finding the torque on a flat plate.
 
NOTE: IF YOU DO NOT HAVE MATLAB ON YOUR PERSONAL COMPUTERS, THE COMPUTERS IN SAS HALL LABS ALL HAVE MATLAB!

GRADES

    Homework 60%
    Project and presentation 40%

This page was last modified February 19'th 2013.