This one semester course will give an introduction and grounding in the mathematical aspects of fluid motion. The course is intended for beginning graduate students in mathematics, physics, engineering, and related fields. After introducing the governing partial differential equations for a flowing continuum (from conservation laws to the stress tensor), we will survey some of the many successes of inviscid fluid dynamics, described by Euler's equation. We will discuss various exact solutions for steady and unsteady flows, then focus on particular aspects of 2D flows such as airfoil theory, conformal mapping techniques, streamlining, Milne-Thompson's Circle Theorem, etc. The limitations of the inviscid approach will lead us to the concept of a boundary layer, which will be presented along with an overview of asymptotic techniques. We will then consider the flow of viscous fluids, examining various important results in the Stokes and Hele-Shaw limits. The course will conclude with a discussion of the dynamics of complex fluids (e.g., polymers, liquid crystals, emulsions), including fluid elasticity, covariational derivatives, and memory effects.
A common theme throughout the course will be to build and strengthen intuition for the structure of the equations through the presentation and connection to experimental observations. In many cases dimensional analysis and scaling arguments serve to encapsulate the physical problem and orient the mathematical approach.
Prerequisites: No previous knowledge of fluids will be assumed beyond everyday experience. Familiarity with ordinary and partial differential equations and complex variables (especially conformal mapping) is recommended - contact the instructor for more details.
D. J. Acheson, Elementary Fluid Dynamics, (Oxford, 1990).
L. M. Milne-Thomson, Theoretical Hydrodynamics, (Dover, 1996).
H. Ockendon and J. R. Ockendon, Viscous Flow, (Cambridge, 1995).
M. Van Dyke, An Album of Fluid Motion, (Parabolic Press, 1982).
M. Renardy, Mathematical Analysis of Viscoelastic Flow, (SIAM, 2000).
Office: 302 McAllister Bldg, Phone: 865-2491, Email: belmontemath.psu.edu,