AME 538: Summary

AME 538 List of Topics

Formulation of the governing equations for a compressible viscous fluid

Philosophy of continuum mechanics
Some necessary mathematics

Vectors and Cartesian tensors
Gibbs and Einstein notation
Eigenvalues and eigenvectors
Div, grad, and curl
Gauss, Stokes, Leibnitz, and Reynolds

Kinematics

Lagrangian description
Eulerian description
Material derivatives
Kinematic decomposition of motion: translation, rotation, deformation

Conservation axioms

Mass
Linear Momenta (Newton's second law of motion)
Energy (first law of thermodynamics)
Entropy inequality (second law of thermodynamics)

Associated topics

Stress tensor
Conservative and non-conservative formulations
Mechanical and thermal energy equations
First integrals of linear momenta equations (Bernoulli's equation)

Integral forms
Constitutive relations

Frame and material indifference
Second law restrictions
Stress-strain rate relationship for isotropic, compressible, Newtonian fluids
Stokes' assumption

Boundary and interface conditions
Complete set of well-posed dimensionnless model equations

Vorticity dynamics

Helmholtz vorticity transport equation
Kelvin's circulation theorem
Ideal rotational and irrotational vortices
Bending and stretching of vortex tubes
Vortex pairs and vortex sheets
Wall generation of vorticity

Compressible flow

Formulation of generalized one-dimensional flow equations
Nozzles, diffusers, and choking conditions
Expansion and compression waves: acoustic limit
Normal shocks: ideal and non-ideal gas
Rarefactions and the method of characteristics

Potential flow

Stream functions and velocity potentials
Mathematics of complex variables

Euler's formula
Polar and Cartesian representations
Cauchy-Riemann equations
Contour integration and the Cauchy integral theorem

Elementary complex potentials

Uniform flow
Sources and sinks
Point vortices
Doublets

Method of superposition

Rankine half body
Flow over a cylinder

Blasius force theorem
Kutta-Zhukhovski lift theorem

Viscous, laminar flow

Fully developed, one dimensional solutions

Hagen-Poiseuille flow (velocity and temperature fields)
Couette flow with pressure gradient (velocity and temperature fields)

Similarity solutions

Stokes' first problem (velocity and temperature fields)

Blasius boundary layer problem (velocity and temperature fields)