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| Web Page: | http://www.math.duke.edu/~dgs |
Teaching (Fall 2012):
Granular flow is the major focus of my research, and this is what I describe in the first part of this page. In the second part, I discuss a course I began teaching in 1996 that has expanded my research horizons.
Part I: Granular flow
Besides finding granular-flow problems fascinating, I am interested in them because of their practical importance. Manufacturing industries must handle heaps (literally) of raw materials in granular form, and difficulties in handling them are very expensive. There are also important geophysical applications involving granular flow: e.g., avalanches, earthquakes, beach erosion, etc. (Incidentally, [B97] and [JNB96] in the references contain surveys of experiments in granular flow.)
Ill-posed partial differential equations are one of the mathematical challenges which arise in granular flow. (Although it is more than a decade old, I still have occasion to refer to my original demonstration of this behavior, item [S87] in the references.) Physically, ill-posedness appears at the point when sheared material cracks and then breaks into separate pieces which move independently: in technical terms, when a shear band forms. A model problem exhibiting this behavior was developed in [S92] in the references.
Currently I am struggling to understand what mathematical sense can be made of these ill-posed problems. The equations are ill-posed in a manner analogous to
Below are annotated references to work in various other directions in granular flow. The first five items, like the work described above, relate to a continuum description of granular material; the latter two relate to discrete models for granular materials. Industry uses continuum models because they are simpler. However, a granular medium consists of many individual particles, and the separation between microscales and macroscales is incomparably smaller than in a fluid. Thus, a fundamental investigation of granular flow must include its discrete nature.
(a) In [GM99a,b] of the references, an initial-value problem is solved for the (hyperbolic) steady-state equations for granular flow in a conical hopper, with Cauchy data specified at the top of the hopper. (As hinted above, it is not known whether this is the most physical boundary problem.)
(b) In [GSS99] of the references we developed an approximation for the analogue of Jenike's radial solution in a conical hopper with an inverted conical insert. In the near future we plan to study numerically solutions of the steady-state equations in this geometry.
(c) Describing the unloading that occurs near a shear band leads to an interesting free-boundary problem for the wave equation. This problem is analyzed in [SS93,94] in the references.
(d) The dramatic role of imperfections near the onset of shear banding is demostrated in [SS97] in the references.
(e) Models for liquefication of soils are analyzed in [HS98,99] in the references. In one of these problems the solution of an initial/boundary value problem depends on the boundary data in a discontinuous, fractal, manner.
(f) Although continuum models for granular materials deal exclusively with the average behavior of materials, recent experiments by Behringer and collaborators have emphasized that fluctuations from the average may be substantial. Two simple probabilistic models are studied in [SS98,SSS99] in the references; these are preliminary attempts to understand fluctuations theoretically.
(g) Several people (e.g., [A99] in the
My collaborators in studying granular flow include Tom
Witelski,
Bill
Allard, and Bob
Behringer at Duke; Michael
Shearer,
and
Pierre
Gremaud
at NC State; Brian Hayes at Stevens; and Tony Royal at
Jenike & Johanson, Inc
Part II: Research growing out of teaching
Every spring semester since 1996 I have taught a course
that has led to an expansion of my research.
The process starts by my sending a memo to the science
and engineering faculty at Duke, asking if they would
like the assistance of a group of math graduate
students working on mathematical problems arising in
their (the faculty member's) research.
I choose one area from the responses, and I teach a
case-study course for math grad students focused on
problems in that area.
In broad terms, during the first half of the course I
lecture on scientific and mathematical background for
the area; and during the second half student teams do
independent research, with my collaboration, on the
problems isolated earlier in the semester.
I also give supplementary lectures during the second
half, and at the end of the semester each team lectures
to the rest of the class on what it has discovered.
The topics and their proposers for each year have been:
Lithotripsy is an alternative to surgery for treating
kidney stones--focused ultrasound pulses are used to
break the stones into smaller pieces that can be passed
naturally.
This course was written up in [HSSZ98] in the
references;
the research of one team was published in [MTHZ97] in the
references.
Population models in ecology became the PhD-thesis
topic for A. Ashih, working jointly with myself and
Will Wilson.
Some of his results will appear in [AW99] in the
references,
and two more publications are in preparation.
The 1999 heart course focused on the experiments of
[HBG99] in the
references,
in which a small piece of cardiac tissue was subjected
to periodic stimuli at various frequencies much above
the normal pacing rate.
Martin Hall included some of the ideas developed in the
course in his Physics PhD thesis, [H99] in the
references.
Based on what I learned teaching this course, I have
undertaken a joint research project with Dan Gauthier
(Physics) and Wanda Krassowska (BME) studying
mathematical models for the electrical response of the
heart, in particular comparing their predictions with
experiment.
Although there are many models for the electrical
response of the heart, the simplest model involves a
system of only two ODE, analogous to the ODE resulting
from suppression of spatial extent in the
FitzHugh-Nagumo equations.
Such a model is well within the reach of
undergraduates, and this fall I taught an undergraduate
seminar on it.
The course is culminating in research projects by three
separate student teams.
The problems are elementary by usual research
standards, but their successful completion will further
the research begun in [HBG99].
1996 Lithotripsy
L. Howle, P. Zhong (ME)
1997 Population models in ecology
W. Wilson (Zoology)
1998 Electrophysiology of the heart I
C. Henriquez (BME)
1999 Electrophysiology of the heart II
D. Gauthier (Physics).