P h y s i c s o f S t r i n g s
VIGRE Seminar, CAAM 499.005
Asymptotic Methods in Analysis
Spring 2010, Wednesday 4pm, HZ 120
Participants: Noel Cavazos, Sharmaine Jennings, Aaron Levine, Shuyi Li, Devin Taylor, Brian Wigianto
Resources:
Inverse Problems for Boundary Damping
Fall 2009, Wednesday 4pm, HZ 120
Participants: Noel Cavazos, Sharmaine Jennings, Aaron Levine, Shuyi Li, Devin Taylor, Brian Wigianto
Resources:
Uncertainty Quantification for Spring Experiments
Summer 2009, MWF 12pm, DH 2014
Participants: Jordon Cavazos, Aneesh Mehta, Matthew Broussard, Jeff Bridge, Heather Williamson
Resources:

Calvetti and Somersalo Text
 Pfeiffer Applied Probablity Text
 paper of Bal and Ren
The Mathematics of Drag
Spring 2009, Tuesday 2:30pm, HZ 212
Participants: Nikolay Kostov, Aneesh Mehta, Avery Cate, Jordon Cavazos,
Carter Wang, William Li, David Garland,
Anthony Austin, Aaron Tallman, Jared Ye, Kyung Oh,
Russell Carden, Jeffrey Hokanson, James Doty, Brian Leake, Bonnie Brinkerhoff
Syllabus
I. Introduction to Low R (Reynold's number) Flow
A. Overview: The NavierStokes Equations
B. Mathematical formulation: Flow past a cylinder and a sphere,
Calculating the drag coefficient.
II. History of Low R Flow Studies
A. Experiments and numerical calculations: Measuring the drag
on a sphere and a cylinder
B. Theory: Stokes and paradoxes, Oseen's equation, Matched asymptotics,
Uniformly valid approximations.
III. Renormalization Group Applied to Low R Flow
A. Introduction to the renormalization group
B. Flow past a cylinder and a sphere: Rescaling, Naive perturbation analysis,
Secular behavior, Renormalization, Truncation, Meeting boundary conditions,
Calculating the drag coefficient.
Sources
J Veysey and N Goldenfeld,
Simple viscous flows: From boundary layers to the renormalization group,
Rev. of Modern Physics 79, 2007, pp. 883927.
L D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., 1987.
Schedule
Mar 24: Applying the method of matched asymptotic expansions to the (incompressible) NavierStokes problem of steady flow past a solid body.
Mar 17: An example of the method of matched asymptotic expansions, continued.
Maple worksheet: matching_asymp_example.mw
As html: matching_asymp_example.html
Mar 10: The method of matched asymptotic expansions; a motivating example (2nd order, 1D boundary value problem).
February 24: More on Stokes's paradox; axisymmetric flow and the stream function; Whitehead's paradox.
February 17: Flow at small Reynolds number past a cylinder. Stokes' paradox.
stokesSolnAssymp.m (asymptotic claims from last week)
A more convincing presentation of Stokes' paradox:
stokes_paradox.pdf
Accompanying Maple code:
stokes_paradox.html
The actual Maple worksheet:
stokes_paradox.mws
February 10: Flow at small Reynolds number past a sphere.
lecture 5 notes, part 1 (drag force computation)
lecture 5 notes, part 2 (Stokes solutions far from the sphere)
symdrag.m
February 3: Flow at small Reynolds number past a sphere.
lecture 4 notes
vfield.m: produces a quiver plot of the velocity field
January 27: Energy dissipation and the Reynolds number.
lecture 3 notes
January 20: Derivation of the NavierStokes equations.
lecture 2 notes
January 13: Conservation of mass, Euler's equations and Momentum flux.
Fall 2008
How do strings vibrate and what forces slow them?
How can one still a vibrating body as quickly as possible?
Can one deduce material properties of a body from measurements
of its vibrations?
Participants: Anthony Austin, Jared Ye, Hunter Gilbert,
Aaron Tallman, Toni Tullius, Russell Carden,
Jeffrey Hokanson, Derek Hansen, Mark Embree and Steve Cox.
 Seminar 14: 12/2 
Simple Viscous Flows (Cox)


Seminar 13: 11/25 
Strings in Fluid, Part V (Hansen)


Seminar 12: 11/18 
Strings in Fluid,
Part IV (Hansen)


Seminar 11: 11/11 
Strings in Fluid, Part III (Hansen)


Seminar 10: 11/4 
Strings in Fluid, Part II (Hansen)


Seminar 9: 10/28 
Strings in Fluid, Part I (Hansen)


Seminar 8: 10/21 
Eigenvalues for the wave equation on a network, Part III (
Carden)


Seminar 7: 10/13 
Fall Break


Seminar 6: 10/7 
Eigenvalues for the wave equation on a network, Part II (Carden)
von Below
(Linear Algebra and Its Applications, 1985)


Seminar 5: 9/30 
Eigenvalues for the wave equation on a network, Part I (Carden)
Tritare information and
sound samples:
1,
2,
3,
4,
5


Seminar 4: 9/23 
Recovering a damped two beaded string: A concrete
example of BP Theorem 5.1 (Cox)
Notes: bp2damp.pdf


Seminar 3: 9/16 
Eigenvalues and energy for damped systems (Embree)
For extended content, see
paper 1 and
paper 2 by Krein, and Langer
(IEOT, 1978)



Seminar 2: 9/9 
Energy decay for damped beaded strings (Cox)
See Hunter's MATLAB movie: bp4eig.m,
bp4eigmovie.m



Seminar 1: 9/2 
Introduction to damped beaded strings (Cox)
Boyko and Pivovarchik (Inverse Problems, 2008)
Report on (undamped) beaded strings, theory and experiment



Preamble: 
Several areas are ripe for investigation this semester:
 Explication and experimentation of the discrete threespectra
inverse problem of Boyko and Pivovarchik (Inverse Problems, 2008).
Excellent progress on this problem was made by the Summer 2008
students; see their VIGRE poster.
 Eigenvalues for wave equations on networks.
Following the work of last year's seminar, we are ready to
consider thermoelastic strings, viscous damping, etc.
 Physical experiments with simple string networks: tritars and spider webs;
 Mathematical modeling of strings moving in viscous fluids.
 Inverse eigenvalue problems for damped strings.

Resources:

