Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 221A: Mathematical Fluid Dynamics
Approved: 2009-03-01, Albert Fannjiang

Units/Lecture:

Fall, alternate years; 4 units; lecture/term paper or discussion

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
I.M. Cohen and P. K. Kundu, "Fluid Mechanics," Chapters 1, 3-7, ($90)
Search by ISBN on Amazon: 0123737354

Prerequisites:

MAT 118B or consent of instructor

Course Description:

Kinematics and dynamics of fluids. The Euler and Navier-Stokes equations. Vorticity dynamics. Irrotational flow. Low Reynolds number flows and the Stokes equations. High Reynolds number flows and boundary layers. Compressible fluids. Shock waves.

Suggested Schedule:

Lectures Sections Topics/Comments
1 Chapter 1 Introduction
1 3.1 - 3.3 Lagrangian and Eulerian descriptions
1 3.4 - 3.5 Streamline, path line, streak line
1 3.6 - 3.7 Strain rate
1 3.8 - 3.9 Vorticity, circulation, principal axes
1 Chapter 3.10 - 3.13 Shear flows, vortex flows, streamfunction
2 4.1 - 4.9 Conservation laws
1 4.10 - 4.11 Constitutive equation, Navier-Stokes equation
1 Chapter 4.13 Mechanical energy equation
1 4.14 - 4.15 Thermodynamics
1 4.16 - 4.17 Bernoulli equation
1 4.18 - 4.19 Boussinesq approximation, boundary conditions
1 5.1 - 5.3 Vortex lines, vortex tubes
2 5.4 Kelvin's Circulation Theorem, Helmholtz Vortex Theorems
1 5.5 Vorticity equation
1 5.6 Biot-Savart law
1 5.8 - 5.9 Vortex sheet
1 6.1 - 6.7 Irrotational flows
2 6.8 - 6.11 Flows and drags past 2-d body, Blasius Theorem, Kutta-Zhukhovsky Lift Theorem
1 6.20 - 6.22 Flows and drags around a 3-d body, d'Alembert paradox
1 7.1 - 7.5 Surface gravity waves
1 7.6 Deep and shallow water approximations
2 7.8 - 7.10 Group velocity, energy flux, wave dispersion
1 7.13 The Stokes wave
2 7.15 - 7.17 Internal waves

Additional Notes:

COMMENT: I taught 221AB from this book in 2007-2008 and appreciated the clarity, readability and variety of interesting topics covered in the book. Cohen & Kundu also has a lot more material than one can cover in a two-quarter course. It is, however, not written in the style of a typical "mathematical fluids" book. For the latter, one can consult, e.g., the well written Mathematical Theory of Incompressible Nonviscous Fluids by Marchioro and Pulvirenti. Such books are often more focused but limited in scope.