The fourth year course AMATH 463 is one of the few undergraduate courses in Fluid Mechanics offered in a Mathematics program in Canada. Many students who have completed the course have commented that one component that they greatly appreciated is that it combined mathematical techniques from many preliminary courses. It requires a solid background in Vector Calculus (AMATH 231), Ordinary Differential Equations (AMATH 250), Partial Differential Equations (AMATH 353) and Introduction to Theoretical Mechanics (AMATH 271).

In the previous course on Continuum Mechanics (AMATH 361) the students have derived the famous Navier-Stokes equations that can be used to study many fluid dynamical phenomena. Proving existence and uniqueness theorems about these equations is one of the seven Clay prizes in mathematics that has a purse of one million U.S. dollars. AMATH 463 specializes the Navier-Stokes equations to many different scenarios in order to understand fluid dynamics. There is a strong emphasis on the physical phenomena as well as the mathematical techniques required.

The topics covered in AMATH 463 are as follows:

  • Review of the derivation of the equations of motion.
  • Potential or Irrotational Flow: special emphasis on flow around an aerofoil.
  • Surface Gravity Waves: study the behaviour of waves at the surface of a tank or the ocean.
  • Laminar Flow: study different exact solutions to the Navier-Stokes equations.
  • Boundary Layer Flow: viscosity is negligible within the interior of the fluid but becomes very important near solid boundaries.
  • Turbulence: study Reynolds averaging to obtain the modified equations of motion.
  • Geophysical Fluid Dynamics: introduce the effects of the Coriolis force and stratification on fluid motion.

Fluid Dynamics is a vast subject that covers a broad range of length scales. Examples include flow around phytoplankton (~10-6 m), gravity waves due to surface tension (~10-2 m), flow in pipes (~10-2 m), people swimming in a lake (~1 m), internal gravity waves shoaling in lakes (~10 m), ocean currents such as the Gulf Stream (~103 m), atmospheric weather systems (~106 m), flow in stellar interiors (~108 m) and galactic dynamics (~1022 m).

This one course cannot cover all of these topics but it aims to give the student a firm background from which they can then study other fluid dynamical systems of interest to them. Two graduate courses that our department offers that require a strong knowledge of fluid dynamics are Nonlinear Waves AMATH 867 and Hydrodynamic Stability AMATH 863.