Fluid Mechanics

How fluids move, from Bernoulli to turbulence

About this course

Fluid mechanics from the continuum hypothesis through ideal and viscous flow, boundary layers, vorticity and lift, to waves, compressible flow, instabilities, turbulence and rotating and stratified fluids.

What you will be able to do

  • Describe flow fields with the material derivative and apply Euler's equation and Bernoulli's principle
  • Solve simple potential flows and explain where they fail
  • Use the Navier–Stokes equations and the Reynolds number to characterise viscous flow
  • Explain boundary layers, and relate circulation and vorticity to lift
  • Analyse surface waves, sound and flow instabilities

What this course covers

  • Euler's Equation: Newton's Law for an Ideal Fluid
  • Bernoulli's Equation: Streamline Integration of Euler
  • The Venturi: Pressure Drop Across a Throat
  • Static vs. Stagnation Pressure: Clarifying 'Fast Means Low Pressure'
  • The Pitot Tube and Indicated Airspeed
  • The Equal-Transit Lift Fallacy
  • Bernoulli's Failure in a Long Pipe: The Kitchen Tap
  • Dispersion relation derivation and capillary-gravity waves
  • Particle motion and Stokes drift under a wave
  • Group velocity and wave-packet energy transport
  • Wave energy density and energy flux
  • Shallow-water waves, tsunamis, and Green's Law
  • The Kelvin wake and the universal wedge angle
  • Wave power worked example and master summary
  • Newton's Law of Viscosity and the Couette Film
  • The Navier-Stokes Equation
  • Molecular Origin and Temperature Dependence of Viscosity
  • Poiseuille's Parabola and the Fourth-Power Law
  • Diffusion of Momentum: Stokes' First Problem
  • Fermi Estimate: Circulation and Heart Power
  • Coronary Artery Stenosis: Fourth-Power Worked Example
  • Linear stability analysis method
  • Rayleigh-Taylor instability
  • Kelvin-Helmholtz instability and least wind
  • Nonlinear evolution and limits of linear theory
  • Rayleigh-Plateau instability and jet breakup
  • Rayleigh-Bénard convection and the Rayleigh number
  • Worked example: straw, glass, and card
  • The Continuum Hypothesis and Knudsen Number
  • Pressure at Rest and Hydrostatic Equation
  • Atmospheric Structure: Scale Height and Lapse Rate
  • Archimedes' Principle and Buoyancy
  • Surface Tension and the Laplace Pressure
  • Weighing the Atmosphere with a Barometer
  • Fluid Statics as the First Clause of the Navier–Stokes Spine
  • Speed of Sound: Stiffness, Inertia, and Laplace's Correction
  • Mach Number and the Compressibility Threshold
  • Isentropic Flow, Choking, and the Area–Mach Relation
  • Normal Shock: Conservation Laws and Entropy Constraint
  • Shock Tube: Three Waves from One Diaphragm Burst
  • Mach Cone and the Sonic Boom
  • Rocket Nozzle Worked Example and Exhaust Speed Dependence
  • Reynolds Averaging and the Closure Problem
  • Richardson's Energy Cascade
  • Kolmogorov −5/3 Spectrum from Dimensional Analysis
  • Kolmogorov Scale and Degrees of Freedom
  • Pipe Friction Factor and the Moody Chart
  • Law of the Wall and the Logarithmic Velocity Profile
  • Turbulent Mixing: the Tea-Cup Example
  • Velocity potential and Laplace's equation
  • Potential flow past a cylinder and d'Alembert's paradox
  • Circulation, lift, and the Kutta-Joukowski theorem
  • Why potential flow is useful despite d'Alembert's paradox
  • Wing circulation, speed excess, and the Kutta condition
  • Flettner rotor ship as a Kutta-Joukowski application
  • Lecture synthesis: lift exactly, drag not at all
  • Reynolds Number from Scaling Navier-Stokes
  • Reynolds Pipe Experiment and Laminar-Turbulent Transition
  • Life at Low Reynolds Number (Purcell's Scallop Theorem)
  • Stokes Flow and Terminal Velocity
  • Drag Coefficient Curve and the Drag Crisis
  • Dynamic Similarity and Model Testing
  • Worked Capstone: Ball Bearing in Honey and Air
  • Eulerian vs. Lagrangian description of flow
  • The material derivative
  • Streamlines, path lines, and streak lines
  • Velocity gradient: strain rate and vorticity
  • The Rankine vortex
  • Continuity and incompressibility
  • Material derivative applied: the boat thermometer
  • The Rotating Frame and Coriolis Force
  • The Draining Bath and Hemisphere Myth
  • Geostrophic Balance and Buys Ballot's Rule
  • Ekman Spiral and Perpendicular Transport
  • Coriolis Does No Work — Yet Holds Up the Gulf Stream
  • Inertial Circles and Brunt–Väisälä Frequency
  • Gulf Stream as Capstone: Geostrophic Balance in Full
  • Prandtl's boundary layer idea
  • Blasius profile and layer thickness
  • Skin friction and transition to turbulence
  • Role of the boundary layer beyond friction
  • Boundary layer separation and the drag crisis
  • Streamlining: shaping the tail, not the nose
  • Drag ledger Fermi: car, cyclist, airliner, golf ball
  • Vorticity Equation: Transport, Stretching, Diffusion
  • Helmholtz–Kelvin Theorems and the Vortex Ring
  • Vorticity Origin at Walls and the Bathtub Interlude
  • Kármán Vortex Street and Strouhal's Law
  • Magnus Effect and the Kutta Condition
  • Airliner Circulation, Induced Drag, and Wake (Fermi Estimate)
  • Circulation as the Unifying Quantity
  • Blood circulation: Poiseuille, heart power, and Murray's law
  • Lungs as a Murray's-law tree
  • Reynolds ladder and the scallop theorem
  • Strouhal number and the universal beat band
  • Péclet number and why bacteria swim
  • Betz limit and pipeline power (machines)
  • Fourteen numbers on one axis: the course spine

Syllabus

  1. Foundations and Kinematics

    1. L1 · The Continuum and Pressure
    2. L2 · Kinematics: Fields and the Material Derivative

  2. Ideal Flow

    1. L3 · Euler and Bernoulli
    2. L4 · Potential Flow: Dry Water

  3. Viscous Flow

    1. L5 · Viscosity and Navier-Stokes
    2. L6 · The Reynolds Number
    3. L7 · Boundary Layers
    4. L8 · Vorticity, Circulation and Lift

  4. Waves and Compressibility

    1. L9 · Surface Waves
    2. L10 · Compressible Flow and Sound

  5. Instability, Turbulence and Geophysical Flows

    1. L11 · Instabilities
    2. L12 · Turbulence
    3. L13 · Rotating and Stratified Fluids
    4. L14 · Fluids of Life and Machines, and the Close