On Gollub's “Continuum mechanics in physics education”

Continuum mechanics is the study of deformable media, including solid and fluid dynamics, using a macroscopic, continuum approximation, as opposed to a microscopic, molecular approach. Like electromagnetism, continuum mechanics can be formulated as a local field theory, using a handful of partial differential equations (PDEs) for vector and scalar fields, supplemented with boundary conditions. Thermodynamic equation(s) of state may also be needed. In fluid dynamics, for instance, the basic PDEs are statements of conservation laws, such as the balances of mass, momentum, and energy. For Newtonian fluids, these basic equations are called the Navier-Stokes equations. When dealing with electrically conducting fluids, the basic PDEs must also be coupled with Maxwell's equations.

In the early 21^st century U.S. physics curriculum, continuum mechanics is notably absent, except in certain corners of plasma physics, astrophysics, and perhaps biological physics. Most of the teaching in solid and fluid dynamics occurs in schools of engineering and the geosciences, with occasional coverage by applied mathematicians. A similar fate has befallen another great branch of classical physics, acoustics, which could be thought of a special branch of continuum mechanics.

Jerry Gollub

Long ago, James Lighthill (1962) made the case that fluid dynamics is a bona fide branch of physics, and Jerry Gollub (2003, 2008) has argued in favor of restoring some role for continuum mechanics in the physics curriculum. Gollub notes that there are a number of texts that could be used for a dedicated course in fluid dynamics for physics students. Several others, by and for physicists, either focused on fluids or tackling continuum mechanics as a whole, have been published since he first wrote. However, given the already crowded physics curriculum, Gollub has tried to incorporate continuum mechanics in an existing course, rather than creating a new one. By curtailing discussion of other topics, Gollub covers selected topics in continuum mechanics in an introductory physics course for physics majors, as well as in the more advanced undergrad mechanics course. He acknowledges the challenges for other physicists to do so, since not many have expertise of their own in continuum mechanics, and few of the popular mechanics texts include these topics. I know of two major graduate level mechanics texts that cover continua: Fetter and Walecka (1980/2003), and José and Saletan (1998). At the undergrad level, Taylor (2005) and Chaichian, et al. (2012) take a brief look at continua in their final chapters. (Readers, do you know of any others?)

The Navier-Stokes equations occasionally show up in graduate level texts in statistical mechanics, such as Huang (1987, ch. 5) and Reichl (2008, Ch. 8). They have even made appearances in advanced electromagnetism texts, such as the extremely cursory section on magnetohydrodynamic waves in Jackson (1999. Sec. 7.7), where viscosity is ignored. A far more substantive treatment integrating electromagnetism and continuum mechanics may be found in Kovetz (2000). In my view, the undergrad courses in statistical physics and electromagnetism are probably less appropriate places to introduce fluids than a classical mechanics course, although fluid dynamics should be considered a good option at the grad level for any of these courses.

Another appealing way to tackle fluid dynamics is by combining it with a course on nonlinear dynamics and chaos theory, as reflected in textbooks like Hilborn (2000). (A truly dual-topic course might have two texts, one each for fluids and nonlinear dynamics.) It is also possible, as Gollub suggests, to incorporate further coverage of fluid and solid mechanics in more advanced courses such as condensed matter theory and materials science. As mentioned, this approach is already frequently taken in plasma physics, astrophysics, and (hopefully) biological physics. Aref (2008) notes that “The somewhat applied subfield of fluid dynamics known as fluidics has taken on new life in the context of microfluidics and nanofluidics,” presenting yet another opportunity.

Gollub (2008) rightly suggests the use of the Multimedia Fluid Mechanics DVD-ROM as a supplement to the textbooks (Homsy, et al., 2008); this item is now often found packaged with conventional fluid dynamics textbooks. Fluid or solid mechanics projects could be considered in an advanced lab course as well.

Realistically, it is unlikely that continuum mechanics will find a place among the standard topics of the core physics curricula. We are lucky if it gets included as a special topic in another course, but even this will be hit-or-miss across physics departments nationwide. This may be a blessing in disguise, if physics students serious about learning continuum mechanics are sent to good engineering or geoscience courses offered on their campuses. One of the greatest rewards of working in fluid dynamics for me has been the relentless interdisciplinarity of the enterprise. Meetings of the American Physical Society's Division of Fluid Dynamics, the Acoustical Society of America, and the Society of Rheology (all member societies of the American Institute of Physics) are places where physicists are often outnumbered by engineers and applied mathematicians.

References

 

H. Aref, 2008: Something old, something new. Phil. Trans. R. Soc. A, 366: 2649-2670.

M. Chaichian, I. Merches, and A. Tureanu (2012): Mechanics: An Intensive Course. Springer.

A.L. Fetter and J.D. Walecka, 1980/2003: Theoretical Mechanics of Particles and Continua. McGraw-Hill (original) and Dover (reprint).

J. Gollub, 2003: Continuum mechanics in physics education. Physics Today, 56 (12), 10-11.

J. Gollub, 2008: Teaching about fluids. Physics Today, 61 (10), 8-9.

J.D. Jackson, 1999: Classical Electrodynamics, 3d ed. Wiley.

R.C. Hilborn, 2000: Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers, 2d ed. (Oxford University Press).

G.M. Homsy, et al, 2008: Multimedia Fluid Mechanics, 2d ed. Cambridge University Press.

K. Huang, 1987: Statistical Mechanics, 2d ed. Wiley.

J.V. José and E.J. Saletan, 1998: Classical Dynamics: A Contemporary Approach. Cambridge University Press.

A. Kovetz, 2000: Electromagnetic Theory. Oxford University Press.

M.J. Lighthill, 1962: Fluid dynamics as a branch of physics. Physics Today, 15 (2): 17-20.

L.E. Reichl, 2008: A Modern Course in Statistical Physics, 3d ed. Wiley-VCH.

J.R. Taylor, 2005: Classical Mechanics. University Science Books.