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by Joseph M. Powers
Department of Aerospace and Mechanical Engineering
University of Notre Dame

TOC

1 Governing equations
1.1 Philosophy of rational continuum mechanics
1.1.1 Approaches to fluid mechanics
1.1.2 Mechanics
1.1.3 Continuum mechanics
1.1.4 Rational continuum mechanics
1.1.5 Notions from Newtonian continuum mechanics
1.2 Some necessary mathematics
1.2.1 Vectors and Cartesian tensors
1.2.2 Eigenvalues and eigenvectors
1.2.3 Grad, div, curl, etc
1.3 Kinematics
1.3.1 Lagrangian description
1.3.2 Eulerian description
1.3.3 Material derivatives
1.3.4 Streamlines
1.3.5 Pathlines
1.3.6 Streaklines
1.3.7 Kinematic decomposition of motion
1.3.8 Expansion rate
1.3.9 Invariants of the strain rate tensor
1.3.10 Invariants of the velocity gradient tensor
1.3.11 Two-dimensional kinematics
1.4 Conservation axioms
1.4.1 Mass
1.4.2 Linear momenta
1.4.3 Angular momenta
1.4.4 Energy
1.4.4.1 Total energy term
1.4.4.2 Work term
1.4.4.3 Heat transfer term
1.4.4.4 Conservative form of the energy equation
1.4.4.5 Secondary forms of the energy equation
1.4.5 Entropy inequality
1.4.6 Summary of axioms in differential form
1.4.7 Complete system of equations?
1.4.8 Integral forms
1.5 Constitutive equations
1.5.1 Frame and material indifference
1.5.2 Second law restrictions and Onsager relations
1.5.3 Fourier’s law
1.5.4 Stress-strain rate relation for a Newtonian fluid
1.5.4.2 Analysis for isotropic Newtonian fluid
1.5.4.3 Stokes’ assumption
1.5.4.4 Second law restrictions
1.5.5 Equations of state
1.6 Boundary and interface conditions
1.7 Complete set of compressible Navier-Stokes equations
1.8 Incompressible Navier-Stokes equations with constant properties
1.8.1 Mass
1.8.2 Linear momenta
1.8.3 Energy
1.8.4 Summary of incompressible constant property equations
1.8.5 Limits for one-dimensional diffusion
1.9 Dimensionless compressible Navier-Stokes equations
1.10 First integrals of linear momentum
1.10.1 Bernoulli’s equation
1.10.2 Crocco’s theorem

2 Vortex dynamics
2.1 Transformations to cylindrical coordinates
2.1.1 Centripetal and Coriolis acceleration
2.1.2 Grad and div for cylindrical systems
2.1.3 Incompressible Navier-Stokes equations in cylindrical coordinates
2.2 Ideal rotational vortex
2.3 Ideal irrotational vortex
2.4 Helmholtz vorticity transport equation
2.4.1 General development
2.4.2 Limiting cases
2.4.3 Physical interpretations
2.4.3.1 Baroclinic (non-barotropic) effects
2.4.3.2 Bending and stretching of vortex tubes: three-dimensional effects
2.5 Kelvin’s circulation theorem
2.6 Potential flow of ideal point vortices
2.6.1 Two interacting ideal vortices
2.6.2 Image vortex
2.6.3 Vortex sheets
2.6.4 Potential of an ideal vortex
2.6.5 Interaction of multiple vortices
2.6.6 Pressure field
2.6.6.1 Single stationary vortex
2.6.6.2 Group of N vortices
2.7 Influence of walls
2.7.1 Streamlines and vortex lines at walls
2.7.2 Generation of vorticity at walls

3 One-dimensional compressible flow
3.1 Generalized one-dimensional equations
3.1.1 Mass
3.1.2 Momentum
3.1.3 Energy
3.1.4 Summary of equations
3.1.5 Influence coefficients
3.2 Flow with area change
3.2.1 Isentropic Mach number relations
3.2.2 Sonic properties
3.2.3 Effect of area change
3.2.4 Choking
3.3 Normal shock waves
3.3.1 Rankine-Hugoniot equations
3.3.2 Rayleigh line
3.3.3 Hugoniot curve
3.3.4 Solution procedure for general equations of state
3.3.5 Calorically perfect ideal gas solutions
3.3.6 Acoustic limit
3.4 Flow with area change and normal shocks
3.4.1 Converging nozzle
3.4.2 Converging-diverging nozzle
3.5 Rarefactions and the method of characteristics
3.5.1 Inviscid one-dimensional equations
3.5.2 Homeoentropic flow of an ideal gas
3.5.3 Simple waves
3.5.4 Prandtl-Meyer rarefaction
3.5.5 Simple compression
3.5.6 Two interacting expansions
3.5.7 Wall interactions
3.5.8 Shock tube
3.5.9 Final note on method of characteristics

4 Potential flow
4.1 Stream functions and velocity potentials
4.2 Mathematics of complex variables
4.2.1 Euler’s formula
4.2.2 Polar and Cartesian representations
4.2.3 Cauchy-Riemann equations
4.3 Elementary complex potentials
4.3.1 Uniform flow
4.3.2 Sources and sinks
4.3.3 Point vortices
4.3.4 Superposition of sources
4.3.5 Flow in corners
4.3.6 Doublets
4.3.7 Rankine half body
4.3.8 Flow over a cylinder
4.4 More complex variable theory
4.4.1 Contour integrals
4.4.1.1 Simple pole
4.4.1.2 Constant potential
4.4.1.3 Uniform flow
4.4.1.4 Quadrapole
4.4.2 Laurent series
4.5 Pressure distribution for steady flow
4.6 Blasius force theorem
4.7 Kutta-Zhukovsky lift theorem
4.8 Conformal mapping

5 Viscous incompressible laminar flow
5.1 Fully developed, one dimensional solutions
5.1.1 Pressure gradient driven flow in a slot
5.1.2 Couette flow with pressure gradient
5.2 Similarity solutions
5.2.1 Stokes’ first problem
5.2.2 Blasius boundary layer

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