Continuum Mechanics in Physics Education

Gollub 
One
of the oddities of contemporary physics education is the nearly
complete absence of continuum mechanics in the typical undergraduate or
graduate curriculum. Continuum mechanics refers to field descriptions
of mechanical phenomena, which are usually modeled by partial
differential equations. The NavierStokes equations for the velocity
and pressure fields of Newtonian fluids provide an important example,
but continuum modeling is of course also well developed for elastic and
plastic solids, plasmas, complex fluids, and other systems.
Students' main experience with continua, at both the
undergraduate and graduate level, occurs in the standard courses in
electromagnetism. Fields associated with simple charge distributions
are encountered, as is the propagation of electromagnetic waves in
various media. On the other hand, parallel experience does not
typically occur in students' studies of mechanics, in which continuum
phenomena are arguably just as important. A clear understanding of
stresses (not just forces) is essential for understanding how materials
stretch, bend, and break, and how fluids flow.
It is easy to see why engineers are interested in fluid and
solid mechanics, but the subject is also increasingly valuable for
physicists and students. Continuum modeling is widely used in
astrophysics at many scales, including both stellar interiors and
largerscale phenomena. The subject plays a significant role in the
growing field of biological physics, in which structures such as
membranes and the cytoskeleton are of great interest. With exposure to
the continuum way of thinking about mechanical phenomena, students
would have access to many physical aspects of nature: laboratory and
geophysical fluid dynamics, the dynamics of deformable materials, part
of the growing fields of soft condensed matter physics and complex
fluids, and much more.
Given these facts, how can we account for the nearly total
absence of continuum mechanics in typical physics curricula at both the
undergraduate and graduate levels? The omission may be in part a
consequence of the historical relegation of some of these topicsfor
example, fluid mechanicsto engineering. However, other factors
contribute as well.
 Our curricula at both the undergraduate and graduate levels are already
crowded with other topics. We know from educational research that crowded
curricula interfere with attaining deep understanding, so it is difficult
to make additions without deleting other topics.
 The most popular textbooks intended for courses in classical mechanics
do not generally treat fluid or solid mechanics. It is hard to teach courses
that are not adequately supported by textbooks, especially at the undergraduate
level. (Although there are a few fluid dynamics textbooks suitable for physics
students, they are mainly intended for courses that are entirely devoted
to that topic.)
 We physicists often do not feel prepared to teach relatively unfamiliar
topics. (Coteaching with an engineering colleague might be an intriguing
solution.)
 We worry about appearing to infringe on subjects that are taught elsewhere
in our universities.
 We are not yet collectively convinced that the need is compelling, despite
the wide applicability of fluid and solid mechanics. Perhaps this column
will help.
The integration of continuum mechanics into the physics
curriculum could yield many benefits. I do not presume that this
integration would have to occur as a single separate course. Greater
attention to this field could be distributed in various parts of the
undergraduate and graduate curricula, including courses in classical
mechanics, condensed matter physics, and so on.
Some suggestions
At Haverford College, I generally include an introduction to fluid
dynamics in our undergraduate mechanics course. Time for this can be
created in several ways, for example by treating oscillations in an
earlier "waves and optics" course or by abbreviating the treatment of
rigid bodies. I also plan to teach a general interest course called
Fluids in Nature for nonmajors in 2004.
I
like to get students interested in hydrodynamic phenomena by showing
them some of the images in the Gallery of Fluid Motion, a competitive
feature of the annual American Physical Society's division of fluid
dynamics meetings. The entries are available online at http://www.aps.org/units/dfd and will appear soon in book form.^{1} Several recent examples are given in figures 1 and 2,
but many more are available online. (Click on the American Institute of
Physics "AIP Gallery of Fluid Motion" and then either "2003 Gallery" or
"Archives.") Readers may also be interested in Steven Vogel's Life in Moving Fluids,^{2} a fascinating exposition on biological fluid mechanics.
For
graduate and upperlevel undergraduate students, a modern course in
fluid mechanics could easily cover some of the applications to fields I
have mentioned in this column and thus provide an opportunity to
showcase the diversity of physics and its connections to neighboring
disciplines. It is also desirable to explain the limits of traditional
hydrodynamics, to show how it is connected to atomicscale thinking,
and to indicate that it can be extended to nonNewtonian fluids.
Potential benefits
What are some of the potential benefits of including fluid and solid
mechanics in courses? First, many students have difficulty developing
competence in using partial differential equations in physical
theories. By applying them to a wide range of mechanical phenomena that
can be directly visualized, students might significantly improve their
knowledge in this area of applied mathematics that is central to
physical modeling. (It would not be a bad experience for most
instructors, either!) Concepts such as scaling, dimensional analysis,
linear and nonlinear stability theory, asymptotic analysis, Fourier
methods, and so forth can be effectively taught in the context of
continuum mechanics.
Second, continuum mechanics is one way to introduce physics
students to nonlinear dynamics, a subject that has wide applications.
Perhaps because linear phenomena appear to be so straightforward to
model, physics courses still suffer from a preoccupation with them.
Among the nonlinear phenomena that can be introduced would be
instabilities, chaotic dynamics, and complexity. Although some of these
topics can also be treated at the level of individual particles as
discussed in a previous column (Physics
Today, January 2003, page 10), a continuum treatment offers special opportunities because fluid dynamics is inherently nonlinear.
An example of how fluid mechanics can contribute to an
understanding of nonlinear dynamics occurs through consideration of
mixing in fluids. Although most students learn about Hamiltonian
mechanics as undergraduates, a smaller number encounter the fundamental
concepts of hyperbolic and elliptic fixed points that characterize the
phase space of almost any nonlinear conservative system such as the
pendulum. Even fewer students (and faculty) have any recall of these
topics, as I know from having asked audiences at various colloquia. Yet
these important mathematical structures can be visualized concretely
and in real space (rather than phase space) by considering how an
impurity is mixed into a fluid whose velocity field is time periodic.^{3}
A third advantage is that acquiring the tools of continuum
mechanics gives students the potential to understand phenomena that are
amazingly diverse and also importantthe fracture and failure of
solids, instability and pattern formation in flowing fluids, the
dynamics of the atmospheric circulation, and the behavior of soft
materials such as membranes, emulsions, and biological materials. It is
plausible to imagine that the inclusion of topics from these fields in
our educational programs would highly motivate some students whom we
currently lose to other fields. It would certainly increase students'
confidence that their physics knowledge is widely applicable and would
contribute to their preparation for a variety of research and
employment opportunities. Perhaps over time, our thinking about what
students really need to know will evolve to include this
multidisciplinary field.
I appreciate helpful comments by Leo Kadanoff, Mike Marder,
Lyle Roelofs, and Howard Stone, and research support from the NSF
division of materials research.
Jerry Gollub is a
professor of physics at Haverford College and is also affiliated with
the University of Pennsylvania. His experiments focus on the nonlinear
dynamics of fluids and granular materials.
1. M. Samimy, K. S. Breuer, L. G. Leal, P. H. Steen, eds. A Gallery of Fluid Motion, Cambridge U. Press, New York (2003).
2. S. Vogel, Life in Moving Fluids: The Physical Biology of Flow, Princeton U. Press, Princeton, N.J. (1994).
3. For an elementary discussion, see R. C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, 2nd ed. Oxford U. Press, New York (2000), p. 436. At a more advanced level, see J. M. Ottino, The Kinematics of Mixing: Stretching Chaos and Transport, Cambridge U. Press, New York (1989).
2003 American Institute of Physics


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