### EPS 236: Geological Fluid Mechanics

Syllabus, Fall 2007

This course presents an introduction to the (broad) discipline of fluid mechanics and describes the relationship between fluid mechanics and geological processes. A lecture outline is given below.

The principles of conservation of mass, momentum and energy that are the basis of fluid mechanics are relatively straightforward, and can be easily derived. Finding solutions to these equations is in general not straightforward, and only in very special cases can exact solutions be obtained. Most often (justifiable) approximations must be made to the equations and boundary conditions in order to obtain solutions. In this class we will derive or present the very simple governing equations of fluid mechanics (they only appear simple -- these equations describe the rich complexity of flows we see in everyday life: flows in rivers, the atmosphere, waves in the ocean, flows in the bathtub and kitchen sink, flying airplanes, etc.); we will then look at the different simplifications that can be made for various classes of problems that allow us to understand the main features of common problems in geological and environmental fluid mechanics.

Class meeting times:

Formal lectures are held Monday and Wednesday from 1:30-3:00 pm. There can be an optional discussion section (time to be arranged) to review basic math, and discuss progress with term projects.

Prerequisites:

We will be solving ordinary AND partial differential equations in this class. We will also be doing lots of vector calculus (sometimes involving second, third and even fourth rank tensors). The first problem set will cover some of the basic mathematical topics that will be commonly used (and are also commonly useful). Integral relations and equations are also very useful, but are unfortunately not usually covered in undergraduate classes.

Text and notes:

The most suitable book for this class is probably Fluid Physics in Geology by D.J. Furbish (Oxford University Press, 1997). You can compare its table of contents with the topics we cover in class in order to determine what pages you should read. In the outline below I provide references to books other that Furbish. I also include at the end of the outline a list of recommended references.

Instructor:

• Michael Manga (3-8532), McCone 177
• manga@seismo
• There is no GSI for this class.
Course evaluation:
• Homework 25 %
• Midterm 15 %
• Final exam 15 %
• Term paper/project 35 %
• Term paper/project presentation 10 %
Term projects:

The term project clearly accounts for a substantial part of the evaluation. The topic of the project is chosen by each student. For undergraduate students, a critical literature review is sufficient. Graduate students, however, must also describe a research project aimed at understanding some process or addressing an unsolved problem. All students are encouraged to attempt to actually solve a problem, wither numerically, or experimentally; equipment, facilities and/or computers may be available. If appropriate students may also work in groups in order to work on more involved projects.

Students who register in the class can receive a more detailed list of suggested projects as a pdf file by email -- contact manga@seismo.berkeley.edu once you register

Here are four examples of projects completed in previous years (of course only a small number of term projects lead to refereed research papers, but published papers are easier to find than unpublished term papers):

Hammer, J.E., M. Manga & K.V. Cashman (1998) Non-equilibrium and unsteady fluid degassing during slow decompression, Geophysical Research Letters, vol. 25: 4565-4568; Download pdf reprint

Weeraratne, D. & M. Manga (1998) Transitions in the style of mantle convection at high Rayleigh numbers, Earth and Planetary Science Letters, vol. 160: 563-568; Download pdf reprint

Dorsey, C. & M. Manga (1998) The spreading of drops and axisymmetric gravity currents along a free surface, Physics of Fluids, vol. 10: 3011-3013; Download pdf reprint

Gonnermann, H., M. Manga & A.M. Jellinek (2002) Dynamics and longevity of an initially stratified mantle, Geophysical Research Letters, vol. 29, paper number 10.1029/2002GL01485; Download pdf reprint

The purpose of the presentation is to provide students the experience of presenting scientific results in a format similar to those used a professional meetings.

Please point out any typographical errors and mistakes. Other comments are welcome.

Outline

Introduction and governing equations (August 27, 29)

• What are fluids? The continuum hypothesis (Batchelor, pp 1-6; Faber 2-4; Heuberger et al., Science, vol. 292, 905-908, 2001; Arfken, pp 10-37; handout -- all these will be made available)
• Vectors, tensors, stress
• Conservation equations for mass, momentum, energy (Leal, chapter 2)
• Boundary conditions
• Homework number 1 due Wednesday September 5; you can download Wood's editorial here
• Homework number 2 due Monday September 10
• Homework number 3 due Monday September 17
• Supplemental notes on the conservation equations and some useful vector identities
Scaling and unidirectional flows (September 5, 10)
• Scaling and dimensional analysis (Middleton and Wilcock, chapter 3; Fowler, chapter 1)
• Homework number 4 due Wednesday September 19
• Unidirectional flows (Leal, chapter 3)
• Homework number 5 due Monday September 24
Special limits, features of flows (September 17, 19, 24 and 26)
• Different limits, e.g., Stokes flows, incompressible flow, irrotational flows, etc. (Hinch, Chapter 3 in Disorder and Mixing; Tritton, chapters 8, 10)
• Check out NASA's airfoil page
• Vorticity (Tritton, sections 6.4-6.6)
• Boundary layers
• Lubrication theory
• Homework number 6 due Monday October 1
• Homework number 7 due Monday October 8
Waves (October 1, 3)
• Waves (Lighthill, chapter 3; Cushman-Roisin, appendix A)
• Lab, time TBD (perhaps October 29 or 31?) -- Kelvin-Helmholtz instabilities
• Homework number 8 due Monday October 17
Turbulence (October 8)
• Turbulence (Middleton and Wilcock, chapter 11; Tritton, section 20.4)
• MIDTERM ON WEDNESDAY OCTOBER 10
Gravity currents (October 15, 17)
• Gravity currents (book by Simpson)
• Rheology of geological materials and fluids
• Homework number 9 due Wednesday November 7
Convection (October 22, 24)
• Convection, including double-diffusive convection (paper by Kadanoff, Physics Today, 2001; Tritton, chapters 14, 22, 23)
• Homework number 10 due Wednesday November 14
Porous materials (November 5, 7)
Microhydrodynamics (bubbles and crystals in liquids) (November 14, 19, 26)

A few other topics (November 26, 28)

• Effects of rotation (Cushman-Roisin, chapter 1)
• Life in moving fluids (Vogel, Life in Moving Fluids)

Second midterm Dec 5, in class

Term project presentations (time and date TBD)

This is obviously a lot of material, and as a result the course will be more descriptive than most fluid mechanics classes with an emphasis on scaling analysis.

Recommended references

My favorite general fluid mechanics books:

• Batchelor, G.K., An introduction to fluid dynamics, Cambridge University Press, 1967.
• Faber, T.E., Fluid dynamics for physicists, Cambridge University Press, 1995.
• Van Dyke, M., An album of fluid motion, Parabolic, 1982.
• Tritton, D.J., Physical fluid dynamics, Oxford University Press, 1988.
• Landau, L.D. and E.M. Lifshitz, Fluid mechanics, Pergamon Press, 1987.
Some other books that are very useful references:
• Middleton, G.V. and P.R. Wilcock, Mechanics in the earth and environmental sciences, Cambridge University Press, 1994.
• Furbish, D., Geological fluid mechanics, Oxford University Press.
• Pedlosky, J., Geophysical fluid dynamics, Springer-Verlag, 1987.
• Phillips, O.M., Flow and reactions in permeable rocks, Cambridge University Press, 1991.
• Cushman-Roisin, B., Introduction to geophysical fluid dynamics, Prentice Hall, 1994.
• Lighthill, J., Waves in fluid, Cambridge University Press, 1978.
• Turcotte, D.L. and G. Schubert, Geodynamics, John Wiley and Sons, 1982.
• Leal, L.G., Laminar flow and convection transport processes, Butterworth-Heineman, 1992.
• Simpson, J.E., Gravity currents, Cambridge University Press, 1997.
• Turner, J.S., Buoyancy effects in fluids, Cambridge University Press, 1973.
• Guyon, E., et al., Physical hydrodynamics, Oxford Univ Press, 2001.
• Whitaker, S., Introduction of Fluid Mechanics, Prentice-Hall, 1968 (nice, simple explanations without sacrificing rigour).

• Interesting and sometimes useful links