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A Course in Fluid Dynamics

Prof. Epifanio G. Virga
Dr. André M. Sonnet
Dr. Riccardo Rosso
Dipartimento di Matematica, Università di Pavia,
via Ferrata 1, 27100 Pavia, Italy

Contents


Preliminaries


  • Tensors and Vectors.
  • Kinematics.

Balance Equations


  • Conservation of mass.
  • Transport Theorems.
  • Stress Tensor. Contact and body forces: CAUCHY hypothesis; CAUCHY theorem. Consequences: theorem of expended power.
  • Constitutive Equations. General requirements: Frame indifference, causality, locality. Constitutive assumptions for Eulerian and Newtonian fluids. EULER and NAVIER-STOKES equations. The vorticity equation. Variational characterization of irrotational motions: KELVIN theorem. Bernoullian theorems. Simple solutions of NAVIER-STOKES equations. Similarity: REYNOLD's number. STOKES paradox; OSEEN solution.
Two-dimensional motions
of Eulerian fluids

  • Steady Motions. General concepts: stream function, stagnation points. Velocity potential; complex potential and complex velocity: examples. Rigid boundaries: the circle theorem; the rôle of conformal transformations: ZUKOVSKY's transform. Sources and sinks: the image method.
  • Forces on obstacles. D'ALEMBERT paradox. The BLASIUS-KUTTA-ZUKOVSKY theorem; aerofoils.
  • Motion of bodies in a fluid. Virtual mass.


Vortex Motions


  • Different types of vortices. Circular vortex; vortex filaments: centroid of a vortex filament; vortex pairs; vortex sheets; vortex patches; VON KÁRMÁN street. Line and ring vortices. Conserved quantity in vortex motions; helicity. HILL and KIRCHHOFF vortices.
  • Stability of vortices. KELVIN-HELMHOLTZ instability.
  • Vorticity and viscosity. Diffusion of vorticity in a Newtonian fluid. Effects of viscosity on vortex motions.


Advanced Topics


  • Stability of fluid motions.
  • Dynamics of liquid Crystals.
  • Dynamics of deformable bodies in a fluid.


Bibliography
K. R. Rajagopal, C. A. Truesdell An Introduction to the Mechanics of Fluids, Birkäuser (1999).

C. A. Truesdell A First Course in Rational Continuum Mechanics, Vol.1, second edition, Academic Press (1991).

P. G. Saffmann Vortex Dynamics, Cambridge University Press (1992).


Calendar


Classes meet on Tuesday (4-6 p.m.) and (only in October) Thursday (3-5 p.m.) in Sala Conferenze at floor C in the Department of Mathematics.