Fluid Dynamics

Viscous, Unstable and Rotating Flows in 16 chapters — from “The stress tensor completed: constitutive law and fluid properties” to “Stratified flow, double diffusion and geophysical scales”.

Chapters

  1. The stress tensor completed: constitutive law and fluid properties
    • Index notation
    • Rotated axes and principal stresses
    • Mean stress and deviatoric stress
    • The isotropic constitutive law
    • Bulk viscosity and two pressures
    • The dissipation function
    • Temperature dependence of viscosity
    • Limits of the Newtonian law
    • Problems
  2. Kinematics in three dimensions and Cauchy's equation of motion
    • The Lagrangian map
    • The Reynolds transport theorem
    • Stretching of material lines
    • Vorticity as local rotation
    • Mass conservation in differential form
    • Cauchy's equation of motion
    • The kinetic-energy budget
    • The Stokes stream function
    • Problems
  3. Interfaces: capillary statics and the stress conditions at a surface
    • Kinematic condition at a moving surface
    • The general Young-Laplace law
    • The interfacial stress balance
    • Meniscus profile and puddle depth
    • Contact-angle hysteresis
    • Capillary instability of a cylinder
    • Marangoni stress and the tears of wine
    • Problems
  4. Euler's equation in Lamb's form and the four Bernoulli theorems
    • The epsilon-delta identity
    • Lamb's form of Euler's equation
    • Four Bernoulli theorems
    • Bernoulli's theorem in a rotating frame
    • The compressible Bernoulli equation
    • The momentum-flux tensor
    • The Borda-Carnot loss and the Borda mouthpiece
    • The energy flux of an ideal fluid
    • Where the Bernoulli theorems stop
    • Problems
  5. Vorticity: its sources, its field and the motion of vortices
    • The inviscid vorticity equation and frozen-in vortex lines
    • The baroclinic torque
    • The Biot-Savart law for vorticity
    • Vortex sheets
    • Point-vortex dynamics and its invariants
    • Vortex-ring speed and the core
    • Bound and trailing vortices of a wing
    • What the inviscid vorticity equation settles
    • Problems
  6. Potential flow, conformal mapping and the finite wing
    • Potential flow past a sphere
    • Added mass
    • Analytic functions and residues
    • Conformal mapping of plane flows
    • The Joukowski aerofoil and the Kutta condition
    • Blasius's theorem
    • Thin-aerofoil theory
    • The lifting line and induced drag
    • Ideal-flow forces and the Kutta condition
    • Problems
  7. The Navier-Stokes equations and unsteady exact solutions
    • The incompressible Navier-Stokes equation
    • The general Navier-Stokes equations assembled
    • The viscous vorticity equation
    • The impulsively started plate
    • The oscillating plate and the Stokes layer
    • Circular Couette flow
    • Hiemenz stagnation-point flow
    • Lamb-Oseen and Burgers vortices
    • The dissipation theorem
    • Exact solutions of the general Navier-Stokes equations
    • Problems
  8. Scaling, similarity and similarity solutions
    • The Buckingham Pi theorem and dynamical similarity
    • Non-dimensional equations and their boundary conditions
    • Dominant balance
    • Incomplete similarity in model tests
    • Similarity solutions from the Pi theorem
    • The Taylor blast-wave law
    • Data collapse
    • A catalogue of dimensionless groups
    • The reach of similarity solutions
    • Problems
  9. Stokes flow, lubrication and viscous wetting
    • Stokes drag and slow flow
    • The Stokes equations and their four properties
    • Stokes flow past a sphere
    • The Stokeslet and long-range interactions
    • Hadamard-Rybczynski drag and the Oseen correction
    • The lubrication approximation
    • Slider bearings and squeeze films
    • Viscous wetting
    • Reversibility and the scallop theorem
    • Where the Stokes equations stop
    • Problems
  10. The boundary layer: Prandtl's equations and the Blasius solution
    • A model boundary-layer problem
    • Repairing the singular inviscid limit
    • The boundary-layer equations
    • Displacement and momentum thickness
    • The Blasius solution
    • The momentum integral equation
    • Favourable and adverse pressure gradients
    • The displacement effect and the Kutta condition
    • Laminar and turbulent skin-friction drag
    • Summary
    • Problems
  11. Separation, wakes, jets and free shear flows
    • Zero wall stress and boundary-layer separation
    • Boundary-layer separation on a cylinder
    • Form drag and the drag crisis
    • Aerofoil stall
    • The wake momentum deficit
    • Vortex shedding
    • The laminar jet and the turbulent round jet
    • Mixing layers
    • Summary
    • Problems
  12. Hydrodynamic instability: the normal-mode method
    • Linear stability of a fixed point
    • The normal-mode method
    • The Rayleigh-Taylor instability
    • The Kelvin-Helmholtz instability
    • The break-up of a liquid jet
    • Rayleigh's inflection-point theorem
    • Rayleigh's circulation criterion
    • Taylor vortices
    • Summary
    • Problems
  13. Transition to turbulence
    • Pitchfork and Hopf bifurcations
    • The Landau amplitude equation
    • Onset of vortex shedding
    • Tollmien-Schlichting waves
    • Subcritical transition in a pipe
    • Routes to chaos
    • The Reynolds decomposition
    • Turbulent friction and the Kolmogorov scale
    • Problems
  14. Thermal convection and the Rayleigh-Benard problem
    • Heated layers and the thermal expansion coefficient
    • The temperature equation
    • The Boussinesq approximation
    • The conduction state and the Rayleigh number
    • Rayleigh-Benard instability with free boundaries
    • Rayleigh-Benard instability between rigid plates
    • Patterns and Benard-Marangoni convection
    • Heat transport and the Nusselt number
    • Mantle and solar convection
    • Problems
  15. Rotating fluids: geostrophy, Taylor columns and Ekman layers
    • The stirred cup and rotating frames
    • Rotating-frame equations of motion
    • Geostrophic balance
    • The Taylor-Proudman theorem and Taylor columns
    • Inertial oscillations
    • The Ekman layer
    • Ekman transport and pumping
    • Spin-down and the tea leaves
    • Potential vorticity in shallow water
    • Problems
  16. Stratified flow, double diffusion and geophysical scales
    • Stable stratification and the buoyancy frequency
    • Shear and the Richardson number
    • Stratified flow over obstacles
    • The thermal wind
    • The Rossby deformation radius
    • Double-diffusive convection
    • Thermohaline staircases
    • Geophysical scales, settled and open
    • Problems