Fluid mechanics reaches from practical engineering problems to difficult mathematical questions. Mathematics is involved in understanding and predicting observed phenomena and in designing reliable numerical methods. This course will include a basic treatment of the theoretical foundations of fluid mechanics, and some additional topics, with emphasis on the incompressible case (liquids, or gases at low speeds). Basic topics will include (at least) the formulation and significance of the Euler and Navier-Stokes equations for flow without or with viscosity; stress; vorticity; conservation of circulation etc. in inviscid flow; potential flow and its relation to Laplace's equation; boundary layers and the Prandtl equations; and an introduction to turbulence models.
We expect to make a connection with work in the Nicholas School of the Environment on turbulent air flow through a forest and its effect on deforestation, and also work on the dynamics of aerosols in the atmosphere. Other topics may be chosen as time permits.
Students should have some familiarity with partial differential equations at either undergraduate or graduate level, and the mathematical maturity of a graduating math major. Students who are unsure whether their background is adequate should consult the instructor.
There will be two textbooks: