O ne of the oddities of contemporary physics education is the nearly complete absence of continuum me- chanics in the typical undergraduate or graduate curriculum. Continuum mechanics refers to field descriptions of mechanical phenomena, which are usually modeled by partial differen- tial equations. The Navier–Stokes equations for the velocity and pres- sure fields of Newtonian fluids pro- vide an important example, but con- tinuum modeling is of course also wel
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  O ne of the oddities of contemporaryphysics education is the nearlycomplete absence of continuum me-chanics in the typical undergraduateor graduate curriculum. Continuummechanics refers to field descriptionsof mechanical phenomena, which areusually modeled by partial differen-tial equations. The Navier–Stokesequations for the velocity and pres-sure fields of Newtonian fluids pro-vide an important example, but con-tinuum modeling is of course also welldeveloped for elastic and plasticsolids, plasmas, complex fluids, andother systems.Students’main experience withcontinua, at both the undergraduateand graduate level, occurs in the stan-dard courses in electromagnetism.Fields associated with simple chargedistributions are encountered, as isthe propagation of electromagneticwaves in various media. On the otherhand, parallel experience does not typ-ically occur in students’studies of me-chanics, in which continuum phenom-ena are arguably just as important. A clear understanding of stresses (not just forces) is essential for under-standing how materials stretch, bend,and break, and how fluids flow.It is easy to see why engineers areinterested in fluid and solid mechan-ics, but the subject is also increasinglyvaluable for physicists and students.Continuum modeling is widely usedin astrophysics at many scales, in-cluding both stellar interiors andlarger-scale phenomena. The subjectplays a significant role in the growing field of biological physics, in whichstructures such as membranes andthe cytoskeleton are of great interest.With exposure to the continuum wayof thinking about mechanical phe-nomena, students would have accessto many physical aspects of nature:laboratory and geophysical fluid dy-namics, the dynamics of deformablematerials, part of the growing fields of soft condensed matter physics andcomplex fluids, and much more.Given these facts, how can we ac-count for the nearly total absence of continuum mechanics in typicalphysics curricula at both the under-graduate and graduate levels? Theomission may be in part a consequenceof the historical relegation of some of these topics—for example, fluid me-chanics—to engineering. However,other factors contribute as well.  Our curricula at both the under-graduate and graduate levels are al-ready crowded with other topics. Weknow from educational research thatcrowded curricula interfere with at-taining deep understanding, so it isdifficult to make additions withoutdeleting other topics.  The most popular textbooks in-tended for courses in classical me-chanics do not generally treat fluid orsolid mechanics. It is hard to teachcourses that are not adequately sup-ported by textbooks, especially at theundergraduate level. (Although thereare a few fluid dynamics textbookssuitable for physics students, they aremainly intended for courses that areentirely devoted to that topic.)  We physicists often do not feel pre-pared to teach relatively unfamiliartopics. (Co-teaching with an engineer-ing colleague might be an intriguing solution.)  We worry about appearing to in-fringe on subjects that are taughtelsewhere in our universities.  We are not yet collectively convincedthat the need is compelling, despite thewide applicability of fluid and solid me-chanics. Perhaps this column will help.The integration of continuum me-chanics into the physics curriculumcould yield many benefits. I do not pre-sume that this integration would haveto occur as a single separate course.Greater attention to this field could bedistributed in various parts of the un-dergraduate and graduate curricula, in-cluding courses in classical mechanics,condensed matter physics, and so on. Some suggestions  At Haverford College, I generally in-clude an introduction to fluid dynam-ics in our undergraduate mechanicscourse. Time for this can be created inseveral ways, for example by treating oscillations in an earlier “waves andoptics” course or by abbreviating thetreatment of rigid bodies. I also planto teach a general interest coursecalled Fluids in Nature for non-ma- jors in 2004.I like to get students interested inhydrodynamic phenomena by show-ing them some of the images in theGallery of Fluid Motion, a competitivefeature of the annual American Phys-ical Society’s division of fluid dynam-ics meetings. The entries are avail-able online at units/dfd and will appear soon in bookform. 1 Several recent examples aregiven in figures 1 and 2, but manymore are available online. (Click onthe American Institute of Physics“AIPGallery of Fluid Motion” andthen either “2003 Gallery” or“Archives.”) Readers may also be in-terested in Steven Vogel’s  Life in Mov-ing Fluids , 2 a fascinating expositionon biological fluid mechanics.For graduate and upper-level un-dergraduate students, a modern coursein fluid mechanics could easily coversome of the applications to fields I havementioned in this column and thus pro-vide an opportunity to showcase the di-versity of physics and its connections toneighboring disciplines. It is also desir-able to explain the limits of traditionalhydrodynamics, to show how it is con-nected to atomic-scale thinking, and toindicate that it can be extended to non-Newtonian fluids. Potential benefits What are some of the potential bene-fits of including fluid and solid me-chanics in courses? First, many stu-dents have difficulty developing competence in using partial differen-tial equations in physical theories. Byapplying them to a wide range of me-chanical phenomena that can be di-rectly visualized, students might sig-nificantly improve their knowledge inthis area of applied mathematics thatis central to physical modeling. (Itwould not be a bad experience for mostinstructors, either!) Concepts such asscaling, dimensional analysis, linear Continuum Mechanics in Physics Education Jerry Gollub Jerry Gollub is a professor of physics at Haverford College and is also affili- ated with the University of Pennsylva- nia.His experiments focus on the non- linear dynamics of fluids and granular materials. 10 December 2003 Physics Today  ©  2003 American Institute of Physics, S-0031-9228-0312-210-7  http://www.physicstoday.orgDecember 2003 Physics Today 11 and nonlinear stability theory, asymptotic analysis, Fourier meth-ods, and so forth can be effectivelytaught in the context of continuummechanics.Second, continuum mechanics isone way to introduce physics studentsto nonlinear dynamics, a subject thathas wide applications. Perhaps be-cause linear phenomena appear to beso straightforward to model, physicscourses still suffer from a preoccupa-tion with them. Among the nonlinearphenomena that can be introducedwould be instabilities, chaotic dynam-ics, and complexity. Although some of these topics can also be treated at thelevel of individual particles as dis-cussed in a previous column (P HYSICS T ODAY  , January 2003, page 10), a con-tinuum treatment offers special op-portunities because fluid dynamics isinherently nonlinear. An example of how fluid mechanicscan contribute to an understanding of nonlinear dynamics occurs throughconsideration of mixing in fluids. Al-though most students learn aboutHamiltonian mechanics as under-graduates, a smaller number en-counter the fundamental concepts of hyperbolic and elliptic fixed pointsthat characterize the phase space of almost any nonlinear conservativesystem such asthe pendulum.Even fewer stu-dents (and fac-ulty) have anyrecall of these topics, as I know fromhaving asked audiences at variouscolloquia. Yet these important mathe-matical structures can be visualizedconcretely and in real space (ratherthan phase space) by considering howan impurity is mixed into a fluidwhose velocity field is time periodic. 3  Athird advantage is that acquiring the tools of continuum mechanicsgives students the potential to under-stand phenomena that are amazinglydiverse and also important—the frac-ture and failure of solids, instabilityand pattern formation in flowing flu-ids, the dynamics of the atmosphericcirculation, and the behavior of softmaterials such as membranes, emul-sions, and biological materials. It isplausible to imagine that the inclu-sion of topics from these fields in oureducational programs would highlymotivate some students whom wecurrently lose to other fields. It wouldcertainly increase students’confi-dence that their physics knowledge iswidely applicable and would con-tribute to their preparation for a vari-ety of research and employment op-portunities. Perhaps over time, ourthinking about what students reallyneed to know will evolve to includethis multidisciplinary field.  I appreciate helpful comments by Leo Kadanoff, Mike Marder, Lyle Roelofs,and Howard Stone, and research sup- port from the NSF division of materialsresearch. References 1.M. Samimy, K.S. Breuer, L.G. Leal, P.H.Steen, eds.  AGallery 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 Dynam-ics: An Introduction for Scientists and Engineers , 2nd ed. Oxford U. Press,New York (2000), p. 436. At a more ad-vanced level, see J.M. Ottino, The Kinematics of Mixing: Stretching Chaosand Transport, Cambridge U. Press,New York (1989).  Figure 1. Unstable spreading of a surfactant solution on a thin viscous film. The solution is driven by surfacetension gradients. (A.A. Darhuber, S.M. Troian, from the AIPGallery of Fluid Motion Web site,, item S9, 2003.) Figure 2. “Fluid fishbones” created by the collision of two jets at flow rates increasing from left to right. (A.E.Hasha, J.W.M. Bush, from the AIPGallery of Fluid Motion Web site,, item S8, 2002.)
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