**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.
**

**
Main Texts:
**

**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).
**

**
Additional Texts:
**

**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,
**

**Web: http://www.math.psu.edu/belmonte/
**

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