M676: Mathematical Foundations of Fluid Mechanics
Fall 2007,  Instructor: Dr. I. Oprea, http://www.math.colostate.edu/~juliana, Reference number -  CRN is 65789;
classroom: E104, class time: 4-5:15PM, MW

Without fluid motion, life in the form we know it on Earth could not exist. Blood moves through the vessels in our bodies, air flows into our lungs, and we simply cannot live without air and water. Not only are fluids and their complex physics of general interest, but it is widely recognized that fluid mechanics is an essential part of the comprehensive design and manufacture of nearly all modern machinery, structures, and devices.

This course is an entry-level graduate course on the mathematical foundations of fluid mechanics, a central theme in modern applied mathematics. The aim is to help students acquire an understanding of some of the basic concepts of fluid dynamics, and give them a good working knowledge of mathematical modeling.

Prerequisite: M340 or knowledge in ordinary differential equations.
Grading  will be based on homeworks (approx 70%) and  a final take home exam (30%), and there will be no more than 2 homeworks per month .
Textbook: lecture notes will be provided in advance,  please find here handwritten notes that cover the first week of classes:
lecture notes 0, lecture notes 1, lecture notes 2,  lecture notes 3, some thermodynamics
recommended lecture:
1) Elementary Fluid Dynamics, D. J. Acheson, Oxford Applied Mathematics and Computing Science Series, 1990
2) An Introduction to Fluid Dynamics, G. K. Batchelor, Cambridge University Press, last reprint 1979
3) Mathematical Methods in Aerodynamics, L. Dragos, Kluwer 2003
4) L D Landau and E M Lifshitz, Fluid Mechanics, Butterworth-Heinemann, 1995

Syllabus:

A. The continuous medium
-Kinematics of deformable media, Eulerian and Lagrangian description, the expression of stress tensor in different configurations.

B Field equations of continuum mechanics
-Conservation of mass, balance of momentum, Cauchy's stress principle, energy and entropy, constitutive equations.

C. The equations of ideal fluids
-The equations of motion;  the potential flow;   Helmholtz equation, Bernoulli integral, the shock wave theory;  jump equations, Hugoniot equations; ideal incompressible fluids; the equations of aerodynamics.

D. Viscous fluids
-Navier Stokes equations: exact solutions, stability of laminar flows, brief introduction to Prandtl boundary layer and   Stokes-Oseen model.

E: Applications
-This is a special chapter, it could be an introduction to one of the following: Magneto-hydrodynamics equations, Complex fluids, Rotating/convective motions, Vortex Dynamics, Microfluids, or vary with student interest.