An Invitation to Fluid Mechanics for Physicists

“Hydrodynamics and magnetohydrodynamics are fundamental, regardless of popular opinion.”

  – Eugene N. Parker (1927-2022)*

      *Quoted in the retrospective by Stuart D. Bale, Science, 376: 461 (29 Apr 2022).

“That we have written an equation does not remove from the flow of fluids its charm or mystery or its surprise.”

  – Richard P. Feynman (1918-1988)**

      ** The Feynman Lectures in Physics, vol. 2, ch. 41. (Addison-Wesley, 1964.)

“The older I get, the stranger it seems to me that my undergraduate education in physics and astronomy included virtually no instruction in fluids. I suspect I am not the only physicist who feels that way.”

  – Chanda Prescod-Weinstein***

      *** Physics Today, July 2022, p. 53.

Welcome. This site aims to outline why a physicist should learn a little bit of fluid mechanics, and how to get started doing so. It exists because the subject is usually completely neglected in the standard physics curriculum. I hope that this site will be useful to undergraduate physics majors as well as physicists at any later stage of their career.

The site is currently under construction; thank you for your patience. To further develop this site, comments and collaboration from others (physicists or otherwise) is welcome. We physicists have a checkered track record in teaching hydrodynamics, as sadly documented elsewhere on this site, so please share suggestions for improvement of this site. Like T. E. Faber, I am but an “enthusiastic amateur” in this arena, though a highly opinionated one, so constructive criticism is always and humbly welcome.   – Christopher Tong, Ph.D. (physics).

Outline:

Why should a physicist learn some fluid mechanics?

Some additional reasons physics students should consider learning some fluid mechanics:

Jerry Gollub (Physics Today, Dec. 2003, pp. 10-11) noted many of the reasons above, as well as reasons why physics faculty should consider teaching undergraduates the subject: it would “increase students’ confidence that their physics knowledge is widely applicable” and might help retain students “whom we currently lose to other fields.”

Intrigued, but not yet ready to commit?

Here are a few sampler dishes to see if you like it. Effort increases as you work your way down the list.

The items on the list are most helpful for those with some facility with vector calculus and differential equations; I recommend them for those who have already taken an intermediate electromagnetism course along the lines of Griffiths’ Introduction to Electrodynamics. (In the Feynman Lectures Richard Feynman also recommended studying hydrodynamics after studying electrodynamics. Daniel V. Schroeder’s reason #9 for studying E&M is as “practice” for topics such as fluid mechanics.) For those who haven’t yet taken intermediate E&M, there is another resource. A book by an applied mathematician, Eric Lauga, Fluid Mechanics: A Very Short Introduction (Oxford University Press, 2022) has recently been published. While longer than the others mentioned above (about 140 small pages), its technical level is lower than any of them. It can be read by anyone comfortable with the first semester of introductory physics; no math beyond high school algebra is needed. See my review for details. For a completely nonmathematical account, see Philip Ball’s Flow (Oxford University Pres, 2009).

In addition, check out the F*ck Yeah Fluid Dynamics blog, which is a completely nontechnical, often whimsical take on fluid physics, run by aerospace engineer Dr. Nicole Sharp.

Convinced, and ready to take the plunge?

Some of you will be ready to crack open Landau & Lifshitz’s Fluid Mechanics, part of which represents their (very demanding) “theoretical minimum”. For most of you, starting there would be unwise; a friendlier introduction might be in order. I discuss books on fluid and continuum mechanics written by physicists on a companion page. Please consult that list, as it is of great value to read how fellow physicists view the subject. Having said that, hydrodynamics is an interdisciplinary field, and some of the finest literature in it has been written by others, deserving of your attention. Below I will enumerate some books by non-physicists that physicists often recommend. A personal favorite is by the late oceanographer, Pijush K. Kundu (1941-1994):

This was my primary text when I began learning the subject in the summer of 1996, and subsequent experience has convined me that it remains one of the finest first books on the subject one can learn from. I am unable to vouch for later editions of the book, which were prepared by other authors after Kundu’s passing. Physicist Raymond A. Shaw has used the original edition, but would not use the later ones.

I will divide the other books into two categories: those by applied mathematicians and those by engineers.

I will also mention a few more advanced books that any serious student of the subject should consider:

Some perceptive thoughts on books for physicists to use when teaching fluid mechanics are offered by Raymond A. Shaw’s review of the second edition of Physical Hydrodynamics by Guyon et al. in the American Journal of Physics (2015). Shaw’s short review should be required reading for everyone visiting this page 🙂.

Fluid mechanics is an inherently visual discipline. Some multimedia resources include:

If you are currently based at a university, please consider taking or auditing a fluid mechanics class at your institution, even if you have to find it outside the physics department. Several online courses are also available. For instance, I once audited the MIT Introduction to Aerodynamics course on edX.org, and I notice that MIT also offers an Advanced Fluid Mechanics sequence on the same platform. The Boulder School for Condensed Matter and Materials Physics offered 2011 and 2022 summer schools on hydrodynamics; recordings of the lectures for the 2022 edition have been posted on youtube.

If you’re ready to find out what kinds of contemporary research in fluid mechanics are taking place, there are plenty of options. If you attend the American Physical Society (APS)’s March Meeting, you can easily drop into sessions co-sponsored by the APS Division of Fluid Dynamics (DFD) there. When you get more serious, the APS-DFD annual meeting is usually right before Thanksgiving. This is the most important fluid dynamics conference in the U.S., and the home of the Gallery of Fluid Motion. You will find that physicists are in the minority at this conference :smile:. The American Meteorological Society (AMS) hosts its Atmospheric and Oceanic Fluid Dynamics conference every other year, and the AIAA used to run another major annual fluid dynamics conference. ASME’s Fluids Engineering Division hosts an annual summer meeting. Indeed, most of the engineering society conferences will have talks on hydraulics/hydrodynamics. For instance, I’ve been fortunate to attend biomedical engineering conferences organized by both ASME and IEEE, and found plenty of biological fluid dynamics content at both. Internationally, keep an eye on events sponored by IUTAM (International Union of Theoretical and Applied Mechanics), EUROMECH (European Mechanics Society), and the Japan Society of Fluid Mechanics.

You can also start browing the field’s journals. I recommend starting with the venerable review journal, the Annual Review of Fluid Mechanics. Founded in 1969 under first editor William Sears, and associate editor Milton van Dyke, this is a treasure-chest of the hydrodynamics community. The leading research journal is Cambridge University Press’ Journal of Fluid Mechanics (JFM), founded by George Batchelor in 1956. It has recently been joined by a sibling journal, Flow: Applications of Fluid Mechanics. For many years, the American Institute of Physics’ journal Physics of Fluids was published in coordination with APS-DFD. Founded in 1958 under first editor Francois Frenkiel, the journal was forsaken by the DFD at the end of 2015, when DFD launched the competing Physical Review Fluids, published by the APS itself. Other notable journals include a trio of complementary Springer titles: Experiments in Fluids, Theoretical and Computational Fluid Dynamics, and Journal of Mathematical Fluid Mechanics; the Japanese journal Fluid Dynamics Research, and EUROMECH’s European Journal of Mechanics B/Fluids. There are also more specialized journals in computational fluid dynamics, heat and mass transfer, rheology, acoustics, and so on. Readers of this page may find the boutique journal Geophysical and Astrophysical Fluid Dynamics of particular interest. Of course, fluid mechanics research often appears in the journals of the engineering societies, as well as more general physics, math, and geoscience journals.

Among the engineering society journals, I would like to single out ASME’s Journal of Fluids Engineering. In the 1990s, JFE included a Technical Forum, in which a series of columns titled “Questions in Fluid Mechanics” occasionally appeared. Lloyd M. Trefethen, who initiated the column in 1992 (JFE, 114: 281-282), noted that technical journals “are long on pronouncing answers and short on posing questions.” The column was to focus on unanswered questions, for “Questions are an economical way to identify the boundaries where understanding ends and new learning begins. They shake up our ways of thinking, occasionally so severely as to force us to change a cherished paradigm, thus propelling us forward in the process of illuminating natural phemomena hitherto opaque.” Unfortunately the series did not seem to continue in the new century. I actually think a forum of this kind would be useful in any STEM field, and it is too bad that journals have no room for such thought pieces.

Some background thoughts

“Remember when discoursing on the flow of water to adduce first experience and then reason.”

  – Leonardo da Vinci (1452-1519)

Fluid mechanics consists of the statics and dynamics of fluids (hydrostatics and hydrodynamics, respectively), classically including liquids, gases, and plasmas. The same macroscopic laws of fluid mechanics are operative despite fundamentally different microscopic physics governing these states of matter, perhaps a clue that fluid mechanics is an emergent phenomenon. These macroscopic laws were developed primarily in the 18th and 19th centuries, well before the general acceptance of the atomic hypothesis, and have scarcely needed revision since then. Nonetheless, a number of fundamental problems involving fluid dynamics remains unsolved, and there is a wide range of applications throughout science, engineering, and medicine.

In classical mechanics, we typically study point particles, systems thereof, and rigid bodies. However, an oft-neglected feature of mechanics in today’s physics curriculum is continuum mechanics, the statics and dynamics of deformable solids and fluids, as well as materials with more complex rheology. The classical theory of continua makes use of both the continuum hypothesis (ignoring the discrete molecular structure of matter) and the assumption of local thermodynamic equilibrium. In fluid dynamics, the bulk transport of mass, momentum, and heat are nonequilibrium phenomena, but we posit the existence of scalar, vector, and tensor fields where mechanical/thermodynamic variables (like velocity, density, pressure, and temperature) can be locally well defined, even as they vary globally in space and time. This requires attention to focus on motions with length scales much greater than the mean free path, and time scales much greater than the relaxation time (to local thermal equilibrium). In kinetic theories of gases, however, it is possible to derive the classical equations of hydrodynamics as approximations to the Boltzmann transport equation for, e.g., dilute monatomic gases with conservative binary collisions. The approximation is often based on decomposing the number density distribution function in state space as the sum of a Maxwell-Boltzmann distribution and a small deviation, and expanding the collision integral to first order in that deviation. Thus while hydrodynamics is usually considered a branch of mechanics, it has intimate connections with statistical mechanics and nonequilibrium thermodynamics, dating back to the founding of both hydrodynamics and the kinetic theory of gases by Daniel Bernoulli in 1738.

Fluid dynamics is often divided into incompressible and compressible flows (the latter is sometimes termed “gas dynamics”). The transmission of sound waves in fluids is a link between compressible hydrodynamics and acoustics, another neglected topic of classical physics in the contemporary curriculum. Less neglected is magnetohydrodynamics, the extension of hydroynamics to electrically conducting fluids in the presence of magnetic fields, an often useful approximation in plasma physics, astrophysics, and cosmology (where relativistic fluid dynamics may also be considered). More specialized subjects include electrohydrodynamics (charged fluid in an electrostatic field, receiving recent attention in microfluidics) and ferrohydrodynamics (magnetic fluids often in the absence of either electric charge or current). Meanwhile the inherently quantum mechanical phenomenon of superfluidity (e.g., liquid helium-3’s and -4’s superfulid states, and Bose-Einstein condensates formed from dilute alkali atomic gases) have been the source of multiple Nobel Prizes in Physics. (It is even claimed that superfluids can form in the cores of neutron stars.)

Much of classical hydrodynamics focuses on Newtonian fluids, for which a linear relationship exists between the stress tensor and velocity gradients; Non-Newtonian fluids have a more complex relationship between stress and response. Rheology is the study of such materials. For Newtonian fluids, the classical equations of hydrodynamics include the continuity equation (mass balance) and the Navier-Stokes equations (momentum balance). (The special case of inviscid flow is described by the Euler equations, the Navier-Stokes equations without the viscous terms.) The no-slip condition is the most important boundary condition for viscous flow. As Oxford theoretical physicist Julia Yeomans reminds us, we celebrate the 200th anniversary of the Navier-Stokes equations in 2022 (the year this site went live). While some analytical solutions can be derived, in most realistic problems the equations must be solved numerically. Computational fluid dynamics (CFD) is one of the oldest domains of computational physics, and the arena of much applied research, and engagement with yet another discipline, computer science and engineering. For mathematicians, the Navier-Stokes equations are the focus of one of the seven Clay Mathematics Institute’s Millenium Prizes. The problem referenced there was solved in 2 dimensions by the Russian mathematical physicist, Olga Ladyzhenskaya (1922-2004), but remains unsolved in 3D. (The simplest constitutive laws for the diffusion of heat and solute concentration in fluids are those of Fourier and Fick, respectively, and can be incorporated in convection equations that supplement the Navier-Stokes.)

I have already mentioned several of the dualisms involved in fluid and flow physics: Newtonian vs. non-Newtonian, inviscid vs. viscous, and compressible vs. incompressible. Richard W. Johnson, in his Handbook of Fluid Dynamics (CRC Press, 1998) reminds us of several others: laminar vs. turbulent, single-phase vs. multi-phase, single-component vs. multicomponent, and reacting vs. non-reacting. In engineering, flow phenomena are part of the larger study of transport phenomena (e.g., the bulk transport of mass, momentum, and heat). Meanwhile hydrodynamic instabilities and the transition to turbulence provide links between fluid dynamics and other subdisciplines of physics, including nonlinear dynamics and chaos, complex systems, and pattern formation. An idiosyncratic list of other hydrodynamic topics includes combustion, flow in porous media (in which Darcy’s Law is the most famous result), micro- and nano-fluidics, bubbles and cavitation, foams, fluid-structure interaction, interfacial fluid mechanics and thin films, surface tension and capillary effects, suspensions, and sedimentation.

Fluid dynamics is of increasing interest in condensed and soft matter physics, biological and medical physics, and environmental physics. The subject has historically even deeper connections to applied mathematics, physical chemistry, all the geophysical, atmospheric, oceanic, and planetary sciences, and to civil, mechanical, aerospace, chemical, and biomedical engineering, among others. In fact it could be argued that civil engineers were the first hydrodynamicists, since all ancient civilizations were founded on river banks, and they collectively needed to solve the problems of irrigation, drainage, and flood control. By that argument, the study of fluid flow is as old as human civilization itself.

What is a fluid?

Within the context of continuum mechanics, a fluid is distinguished from other continuous materials as follows. “The characteristic property of a fluid is that it cannot support a shearing stress indefinitely, so that if a shearing stress is applied to a body of fluid and maintained, the fluid will flow and continue to do so as long as the stress remains. A solid, on the other hand, can be in equilibrium under a shear stress….Fluids have no natural configutation and, given sufficient time, will adapt to the shape of any container in which they are placed.” Quoted from A. J. M. Spencer, Continuum Mechanics (Longman, 1980). “The fundamental property which distinguishes a fluid from other continuous media is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface.” Quoted from James Serrin (1959), “Mathematical Principles of Classical Fluid Mechanics” (Handbuch der Physik, Band VIII/1, Springer, pp. 125-263).

Rheologists would point out that some solids will flow if the shear stress reaches some threshold (classical plasticity) or when observed for a sufficiently long time (such as glacial ice flows).

Solid vs. fluid

Above I mentioned Prof. Steve Simon’s excellent recorded lecture, Why Hydrodynamics? His flippant answer was “why not?” but actually his lecture is a nice advertisement of fluid mechanics for physicists. Prof. Simon is also, of course, author of The Oxford Solid State Basics (Oxford University Press, 2013; corrected reprint, 2019). In this book he includes a section titled “Why Solid State Physics?” His answers include:

To these points I cite, in counterpoint, the following passage from Emrich and Frenkiel (Physics Today, May 1968, pp. 44-46):

Since fluids, including plasmas, essentially constitute the entire content of the universe, with solids but a trace impurity, it might appear that for all but a few exceptions natural phenomena would be within the domain of the physics of fluids. Of course only a fraction of the activity in physics is organized under this topical heading, and this probably represents our inability to handle the complexities of large and complicated motions. Laws governing subnuclear, nuclear and atomic “particles” are believed to provide for the behavior of collections of “particles”. However, the ensemble of particles that constitutes a fluid produces many phenomena that are not always expected from these laws.

The end of this passage almost foreshadows P. W. Anderson’s 1972 classic essay, “More is Different” (Science, 177: 393-396). Indeed, hydrodynamics is often cited as a canonical example of an emergent phenomenon.

Some favorite fluid mechanics pieces in Physics Today

Here are some Physics Today articles about hydrology:

Speaking of Physics Today, check out this lamentation, an excerpt from a sidebar in a Physics Today article by Richard G. Fowler (June, 1966; pp. 37-42):

Alas, a similar panel could easily be held today, except (1) there would be women, not just men, on the panel; and (2) they might also have to point out the “egregious errors” in fluid physics made by physicists themselves!

Companion pages

Relevant book reviews by yours truly

Acknowledgments

I am solely responsible for the content throughout this site, including errors of fact and judgment. However I am grateful to Dr. Linda St-Cyr for providing constructive feedback.

During the construction of this site, I consulted the libraries of Iowa State University, the University of Nevada - Reno, the University of California - Davis, the University of California - Riverside, and the University of Utah.

“One thing remains about which I wish the Reader to be especially warned in advance: that I was not able to apply to this work that diligence or attention which I should have, and which I myself desired. And therefore I have no doubt but that some errors will have crept in while I was doing the calculations, which I hope no one will employ wrongly; others which met my eye while I lightly read over the treatise I myself corrected; nevertheless I am convinced that still others remain even yet.”

  – Daniel Bernoulli (1700-1782), Hydrodynamica (1738)

Disclaimers

The content on this site was developed solely on my personal time. The views expressed are solely my own, and do not necessarily represent the views, policies, or opinions of my employer or any organization with which I am affiliated.

(c) 2022-2025 by Christopher Tong