MASC 560/PHYS 660/GEOL 560/ENVR 452 - Fluid Dynamics Spring 2015 Instructor: Prof. John Bane Tel: (919) 962-0172 Email: bane@unc.edu. Office: 3117F Venable Hall Office hours: email for an appointment Course goals and objectives This course is a quantitative introduction to the motion of fluid substances. Over the last century and a half, the study of fluid dynamics has spurred a large number of ideas that have found application in different areas of science. For example, in the late 1960 s, Ed Lorentz, while studying the properties of a simple flow confined between two plates kept at different temperatures, made the discovery that lies at the heart of chaos theory. Knowledge concerning the flows of fluids is fundamental in many areas of engineering (aerodynamics, turbo-machinery, chemical reactors, etc.), medicine and applied sciences (oceanography, geology & geophysics, meteorology,...). Finally, the equations that govern the motion of fluids pose fundamental mathematical problems that have not, so far, found answers. Hence, fluid dynamics lies at a crossroad where many different paths converge. In this course, the goal is to provide an introduction to the theoretical tools necessary to obtain a quantitative description of the motion of fluid substances under a variety of conditions. The course is divided into three parts: The starting point for a quantitative approach is to derive an appropriate set of dynamical equations valid for fluids as different as helium and mercury. The equations express the fundamental conservation laws of classical physics: conservation of mass, linear momentum, angular momentum and energy. Next, we will consider the properties of fluid flows in relatively simple situations. Finally, we will consider more complex (and realistic) flows. Below you will find a tentative sketch of the program that we will follow during the semester. Depending on many factors (interest, average range of preparation of the students in the class, etc.) some topics might be added or dropped. Almost all readings will be from the textbook Fluid Mechanics by Kundu and Cohen (2 nd Edition). Prerequisites Students are expected to know the fundamental concepts of Newtonian Mechanics (mass, momentum, energy, force,...). We will make heavy use of mathematical tools such as derivatives (ordinary and partial), integrals (uni-dimensional, multi-dimensional, surface and volume integrals), partial differential equations, vector calculus and so on. If a student feels that his/her preparation is not adequate, he/she should contact me at the beginning of the semester. As a rule of thumb, courses in general physics, calculus and algebra/ geometry should be counted among the prerequisites.
Class attendance and participation I expect you to attend each class and to be present in class before the beginning of it. Since I expect this class to have a relatively small attendance (10 or so students), I will encourage each and every one of you to actively contribute to the lesson. This means the following: 1. Ask questions. 2. Come up to the board to discuss some points and/or solve some problems. 3. Contribute ideas, suggestions, thoughts. 4. Correct my mistakes. I understand that some of you may feel uncomfortable in doing some of these things, but as scientists, you have to get used to the idea that you have to present, explain and defend your work in front of an audience, and I see this as part of your training. You are, of course, expected to adhere to the University Honor Code. Grading The final course grade will be obtained by adding the contribution of weekly homework (25%), one mid-term exam (35%) and a final exam (40%). The grading system is the UNC standard A to F for undergraduates and H, P, L, F for graduates. I remind graduate students of the dire consequences that an F (or 9 hours of L) can represent for their graduate status at UNC. Semester schedule The subjects to be covered during the semester are as follows. Readings are from Kundu and Cohen. 1. Introduction and discussion of simple flows. The point is to realize that fluid dynamics covers phenomena that are immediately accessible to our experience. Chapter 1, sec. 1 3. 2. The continuum hypothesis and fluid kinematics. This is where it all begins. We need to achieve the qualitative jump from a description based on a few pointlike objects typical of elementary Newtonian mechanics (e.g., the motion of a planet around a star) to a discussion in terms of fields. We look at Lagrangian vs. Eulerian descriptions of fluid flows, which are complementary views of the same phenomenon. Chapter 1, sec. 4-5 and Ch 3, sec. 1 5. 3. Review of vector/tensor calculus (including Gauss and Stokes theorems). The nature of the problem introduces new mathematical tools. You may be already familiar with them if you have studied electromagnetic theory. You should know though that they have been developed in the context of fluid dynamics (and you will probably understand why things are named as they are). Chapter 2. 4. Derivation of the equation of motion. Now we are ready to build the tools that will be used to achieve (at least in principle) the desired quantitative description: the Navier-Stokes equations. Chapter 3, sec. 5-10 and Chapter 4, sec. 1 10 and 19. 5. Vorticity. Together with momentum, vorticity (the spin of a fluid parcel) is a fundamental property of any flow. Chapter 5, all but sec. 7.
6. Laminar flows. When viscosity dominates the force balance on a parcel of fluid, the flow is usually smooth and amenable to analytical description. Lubrication theory is a special application. Chapter 9. 7. Dimensional analysis. We need to develop ways to understand the relative importance of different forces acting on fluids. Dimensional analysis is the tool. At its roots, it relies on the apparently trivial observations that the way nature works does not depend on how we define the units of measure. Chapter 8. See also: http://en.wikipedia.org/wiki/dimensionlessnumbers.) 8. Boundary layer theory. This is a very important application of all of the ideas investtigated during the class. What happens at the interface between a fluid and solid boundary? How does the fluid respond? Chapter 10, sec. 1 13. Multimedia fluid mechanics CD-ROM and movies. Ideally, a fluid dynamics class should have a lab where students can experience first hand what is shown in class. Alas, we live in a non-ideal world and the range of experiments that we might do is very limited. Moved in part by this consideration, in the mid-1960s, several leaders in the field (Corrsin, Taylor,...) formed a group to produce a number of movies illustrating different aspects of the fluid flows and properties. The original 16 mm movies are now available as videos and we are lucky enough to have the majority of them available at the library. In addition to that, Cambridge University Press has finally released a CD-ROM that represents the 21 st century version of the movies. It takes full advantage of multimedia capabilities, and I strongly recommend that you get one. It features clips from the old movies, too. Suggested readings Fluid dynamics is an old science. As a consequence, a large number of books exists. This is a very small list of books that I suggest, along with some personal comments. Books H. M. Schey, Div, Grad, Curl and All That, Norton & Co (1997). It is a little book about vector calculus. If you have never seen this stuff before, I strongly recommend that you get a copy. D. J. Tritton, Physical Fluid Dynamics, Oxford University Press, (1988). This book emphasizes the physical aspect of the science. A good reference. E. Guyon, J-P Hulin, L. Petit and C. D. Mitescu, Physical Hydrodynamics, Oxford University Press (2001). Similar to the book by Tritton in its emphasis on basic physics principles, but with a more in depth coverage of the standard topics. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, (1967). It is considered by many an excellent book, one that will be still around many years from now. However, it is not an easy book to read, and, despite the title, it is an advanced text. Moreover, it avoids completely the problem of turbulence.
R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice Hall (1962). It covers in great detail the (tensorially correct) derivation of the equations used in fluid mechanics. If you ever need to know how to do fluid mechanics in a non-euclidean space (e.g., on the surface of a soap bubble), this is the book you need. R. L. Panton, Incompressible Flow, Wiley Interscience, (1984). This is a good textbook with a strong bias towards engineering applications. R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Volume II, Chapters 39, 40 and 41, Addison-Wesley, (1963). Great book, lots of fun reading it! Feynman s lectures are a masterpiece of pedagogical literature. Here I just point to the chapters dealing with fluids, but you should consider buying the whole series (three books), if you haven t already. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press (1975). Legend has it that when Landau learned that Enrico Fermi had died he exclaimed: Now I am the only living being left that knows all about physics! This bold claim is supported by 10 impressive volumes that collectively are known simply as The Landau and represent a summa covering basically every aspect of classical and modern physics. The sixth volume is entirely devoted to fluid mechanics. The presentation is highly theoretical, and I am not too crazy about the notation, but it still is a good book. Besides, having a few volumes of the Landau on your shelf qualifies you immediately as a serious scientist. Beware of the chapter about transition to turbulence. Unless you have the last edition (which I believe was printed after his death), what he says is basically wrong. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, (1970). Although this books deals primarily with a rather specific subfield of fluid dynamics, one that we will not deal with, if not briefly at the end, it is nonetheless a good reference for particular forms of the equations of motion. Dover is known for bringing back, in paperback form, good books that are out of print at a bargain price. Journals As you can imagine, the number of publications dealing with fluid dynamics is staggering. Below is just a very short list. Annual Review of Fluid Mechanics, Annual Reviews Inc., published yearly. Every year, the editorial board asks a bunch of big wigs in the field to contribute an article summarizing the state of the art is his/her subfield. If you start working on new topic involving fluid mechanics, it is a good idea to browse the subject index to see if it has been covered in the Review. It will save you a lot of time. All of the readings come from this series. Journal of Fluid Mechanics, Cambridge University Press, bimonthly. JFM for the practitioners, it is widely regarded as the journal in the field. It publishes article covering all aspect of basic fluid dynamics.
Physics of Fluids, American Institute of Physics, monthly. It is not surrounded by the aura of JFM, but it is considered a very solid journal, with a slight bias towards the engineering applications of fluid dynamics. Physical Review Letters. American Physical Society, monthly. Due to the interdisciplinary nature of fluid dynamics, it is often the case that discoveries originally made in fluid dynamics apply to other branches of physics as well. In this case Phys. Rev. Let. is the venue of choice. The refereeing process is severe, which ensures the quality of the articles. Physical Review E. American Physical Society, monthly. Originally, Phys. Rev. Let. was meant to be a quick way to disseminate results (the articles have a 4 page limit). More comprehensive articles were meant to be published in Physical Review (which is divided in 5 series, E being the one devoted, among other things, to fluid dynamics). Over the years, the publication criteria became somewhat relaxed, with the results that now Phys. Rev. E is an oversized publication where the quality of the articles varies considerably (from ground breaking to mundane). Reader Caveat. Physica D. Elsevier, bimonthly. The journal deals with the general subject of non-linear physics, of which fluid dynamics is an important part. The general tone of the articles stresses the mathematical aspects. It is often the case that results of applied nature are published in specialized journals, such as Journal of Physical Oceanography, Journal of Geophysical Research in the field of geosciences. Fortunately, there exist good search engines, like the Scientific Citation Index (SCI), which can help.