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Next: Mathematical Models of Fluid

Fluid Mechanics

Richard Fitzpatrick

Professor of Physics

The University of Texas at Austin

Mathematical Models of Fluid Motion
- Introduction
- What is a Fluid?
- Volume and Surface Forces
- General Properties of Stress Tensor
- Stress Tensor in a Static Fluid
- Stress Tensor in a Moving Fluid
- Viscosity
- Conservation Laws
- Mass Conservation
- Convective Time Derivative
- Momentum Conservation
- Navier-Stokes Equation
- Energy Conservation
- Equations of Incompressible Fluid Flow
- Equations of Compressible Fluid Flow
- Dimensionless Numbers in Incompressible Flow
- Dimensionless Numbers in Compressible Flow
- Fluid Equations in Cartesian Coordinates
- Fluid Equations in Cylindrical Coordinates
- Fluid Equations in Spherical Coordinates
- Exercises
Hydrostatics
- Introduction
- Hydrostatic Pressure
- Buoyancy
- Equilibrium of Floating Bodies
- Vertical Stability of Floating Bodies
- Angular Stability of Floating Bodies
- Determination of Metacentric Height
- Energy of a Floating Body
- Curve of Buoyancy
- Rotational Hydrostatics
- Equilibrium of a Rotating Liquid Body
- Maclaurin Spheroids
- Jacobi Ellipsoids
- Roche Ellipsoids
- Exercises
Surface Tension
- Introduction
- Young-Laplace Equation
- Spherical Interfaces
- Capillary Length
- Angle of Contact
- Jurin's Law
- Capillary Curves
- Axisymmetric Soap-Bubbles
- Exercises
Incompressible Inviscid Flow
- Introduction
- Streamlines, Stream Tubes, and Stream Filaments
- Bernoulli's Theorem
- Euler Momentum Theorem
- d'Alembert's Paradox
- Flow Through an Orifice
- Sub-Critical and Super-Critical Flow
- Flow over Shallow Bump
- Stationary Hydraulic Jumps
- Tidal Bores
- Flow over a Broad-Crested Weir
- Vortex Lines, Vortex Tubes, and Vortex Filaments
- Circulation and Vorticity
- Kelvin Circulation Theorem
- Irrotational Flow
- Exercises
Two-Dimensional Incompressible Inviscid Flow
- Introduction
- Two-Dimensional Flow
- Velocity Potentials and Stream Functions
- Two-Dimensional Uniform Flow
- Two-Dimensional Sources and Sinks
- Two-Dimensional Vortex Filaments
- Two-Dimensional Irrotational Flow in Cylindrical Coordinates
- Flow Past a Cylindrical Obstacle
- Motion of a Submerged Cylinder
- Inviscid Flow Past a Semi-Infinite Wedge
- Inviscid Flow Over a Semi-Infinite Wedge
- Two-Dimensional Jets
- Exercises
Two-Dimensional Potential Flow
- Introduction
- Complex Functions
- Cauchy-Riemann Relations
- Complex Velocity Potential
- Complex Velocity
- Method of Images
- Conformal Maps
- Schwarz-Christoffel Theorem
- Free Streamline Theory
- Complex Line Integrals
- Blasius Theorem
- Exercises
Axisymmetric Incompressible Inviscid Flow
- Introduction
- Axisymmetric Flow
- Stokes Stream Function
- Axisymmetric Velocity Fields
- Axisymmetric Irrotational Flow in Spherical Coordinates
- Uniform Flow
- Point Sources
- Dipole Point Sources
- Flow Past a Spherical Obstacle
- Motion of a Submerged Sphere
- Conformal Maps
- Flow Around a Submerged Oblate Spheroid
- Flow Around a Submerged Prolate Spheroid
- Exercises
Incompressible Boundary Layers
- Introduction
- No Slip Condition
- Boundary Layer Equations
- Self-Similar Boundary Layers
- Boundary Layer on a Flat Plate
- Wake Downstream of a Flat Plate
- Von Kármán Momentum Integral
- Boundary Layer Separation
- Criterion for Boundary Layer Separation
- Approximate Solutions of Boundary Layer Equations
- Exercises
Incompressible Aerodynamics
- Introduction
- Theorem of Kutta and Zhukovskii
- Cylindrical Airfoils
- Zhukovskii's Hypothesis
- Vortex Sheets
- Induced Flow
- Three-Dimensional Airfoils
- Aerodynamic Forces
- Ellipsoidal Airfoils
- Simple Flight Problems
- Exercises
Incompressible Viscous Flow
- Introduction
- Flow Between Parallel Plates
- Flow Down an Inclined Plane
- Poiseuille Flow
- Taylor-Couette Flow
- Flow in Slowly-Varying Channels
- Lubrication Theory
- Stokes Flow
- Axisymmetric Stokes Flow
- Axisymmetric Stokes Flow Around a Solid Sphere
- Axisymmetric Stokes Flow In and Around a Fluid Sphere
- Exercises
Waves in Incompressible Fluids
- Introduction
- Gravity Waves
- Gravity Waves in Deep Water
- Gravity Waves in Shallow Water
- Energy of Gravity Waves
- Wave Drag on Ships
- Ship Wakes
- Gravity Waves in a Flowing Fluid
- Gravity Waves at an Interface
- Steady Flow over a Corrugated Bottom
- Surface Tension
- Capillary Waves
- Capillary Waves at an Interface
- Wind Driven Waves in Deep Water
- Exercises
Terrestrial Ocean Tides
- Introduction
- Tide Generating Potential
- Decomposition of Tide Generating Potential
- Expansion of Tide Generating Potential
- Surface Harmonics and Solid Harmonics
- Planetary Rotation
- Total Gravitational Potential
- Planetary Response
- Laplace Tidal Equations
- Harmonics of Forcing Term in Laplace Tidal Equations
- Response to Equilibrium Harmonic
- Global Ocean Tides
- Non-Global Ocean Tides
- Useful Lemma
- Transformation of Laplace Tidal Equations
- Another Useful Lemma
- Basis Eigenfunctions
- Auxilliary Eigenfunctions
- Gyroscopic Coefficients
- Proudman Equations
- Hemispherical Ocean Tides
- Exercises
Equilibrium of Compressible Fluids
- Introduction
- Isothermal Atmosphere
- Adiabatic Atmosphere
- Atmospheric Stability
- Eddington Solar Model
- Exercises
One-Dimensional Compressible Inviscid Flow
- Introduction
- Thermodynamic Considerations
- Isentropic Flow
- Sound Waves
- Bernoulli's Theorem
- Mach Number
- Sonic Flow through a Nozzle
- Normal Shocks
- Piston-Generated Shock Wave
- Piston-Generated Expansion Wave
- Exercises
Two-Dimensional Compressible Inviscid Flow
- Introduction
- Oblique Shocks
- Supersonic Flow in Corner or over Wedge
- Weak Oblique Shocks
- Supersonic Compression by Turning
- Supersonic Expansion by Turning
- Detached Shocks
- Shock-Expansion Theory
- Thin-Airfoil Theory
- Crocco's Theorem
- Homenergic Homentropic Flow
- Small-Perturbation Theory
- Subsonic Flow Past a Wave-Shaped Wall
- Supersonic Flow Past a Wave-Shaped Wall
- Linearized Subsonic Flow
- Linearized Supersonic Flow
- Flat Lifting Wings
- Exercises
Vectors and Vector Fields
- Introduction
- Scalars and Vectors
- Vector Algebra
- Cartesian Components of a Vector
- Coordinate Transformations
- Scalar Product
- Vector Area
- Vector Product
- Rotation
- Scalar Triple Product
- Vector Triple Product
- Vector Calculus
- Line Integrals
- Vector Line Integrals
- Surface Integrals
- Vector Surface Integrals
- Volume Integrals
- Gradient
- Grad Operator
- Divergence
- Laplacian Operator
- Curl
- Useful Vector Identities
- Exercises
Cartesian Tensors
- Introduction
- Tensors and Tensor Notation
- Tensor Transformation
- Tensor Fields
- Isotropic Tensors
- Exercises
Non-Cartesian Coordinates
- Introduction
- Orthogonal Curvilinear Coordinates
- Cylindrical Coordinates
- Spherical Coordinates
- Exercises
Ellipsoidal Potential Theory
- Introduction
- Analysis
- Exercises
Calculus of Variations
- Indroduction
- Euler-Lagrange Equation
- Conditional Variation
- Multi-Function Variation
- Exercises
Bibliography
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Richard Fitzpatrick 2016-03-31