Fluid Dynamics Lecture notes Shreyas Mandre Department of Engineering, University of Cambridge sdm63@cam.ac.uk ©2020-2022 Shreyas Mandre CC-BY-NC-ND with contribution from Thomasina Ball (Chapter 9: Stokes Flow, Chapter 10: Lubrication theory, Chapter 11: Complex Fluids and Non-Newtonian Rheology and Chapter 12: Hydrodynamic Instability) (Last updated: 27 May 2022 ) Contents 1 Preliminaries 1.1 Newtonian mechanics 1.2 Units and dimensions 1.3 Cartesian tensors 1.4 Tensor Calculus 2 Mathematical Modelling of Fluid Flow 2.1 Definition of a fluid 2.2 Fluids in Continuum Mechanics 2.3 Material properties of fluids 2.4 Lagrangian and Eulerian Descriptions 2.4.1 Lagrangian description 2.4.2 Eulerian description 2.4.3 Eulerian versus Lagrangian descriptions 2.4.4 The Material Derivative and Acceleration 2.4.5 Reynolds transport theorem 2.5 Flow Visualisation 2.6 Deformation of infinitesimal fluid elements 2.7 Conservation Laws in Continuum Mechanics 2.7.1 The dimensional inconsistency of the surface source term 2.7.2 Cauchy’s tetrahedron argument 2.8 Conservation of Mass a.k.a. Mass Balance 2.9 Conservation of Momentum a.k.a. Momentum Balance 2.10 Conservation of Angular Momentum a.k.a. Torque Balance 2.11 Constitutive laws 2.11.1 General considerations 2.11.2 An Ideal or Inviscid fluid 2.11.3 A Newtonian Fluid 2.11.4 Navier-Stokes in cylindrical coordinates 2.12 Boundary and initial conditions 2.12.1 Initial conditions 2.12.2 Boundary conditions 3 One-dimensional flows 3.1 Steady flows 3.1.1 One-dimensionality 3.1.2 The driving force 3.2 Select solutions of Navier-Stokes 3.2.1 Plane Poiseuille flow 3.2.2 Hagen-Poiseuille 3.3 Unsteady flows 3.3.1 Stokes first problem 4 Dimensional Analysis 4.1 Reducing parameters in a set of equations 4.2 Deducing dependence on parameters 5 Hydrostatics and Bernoulli equation 5.1 Hydrostatics 5.1.1 Archimedes principle 5.2 Bernoulli equation 5.2.1 Steady incompressible inviscid flow 5.2.2 Unsteady incompressible potential flow 5.3 Applications and examples 5.3.1 Flow through an opening in a tank 5.3.2 A wind turbine 5.3.3 A water wave 6 Potential flow 6.1 Conditions for and implications of potential flow 6.2 Incompressible and irrotational flow in two-dimensions 6.2.1 Stream function and streamlines 6.2.2 Stream function and velocity potential 6.3 Elementary complex potentials 6.3.1 Uniform flow 6.3.2 Point source 6.3.3 Point vortex 6.3.4 Point dipole 6.3.5 Higher multipoles 6.4 Potential flow past immersed bodies 6.4.1 Rankine half-body 6.4.2 Rankine oval 6.4.3 Circle 6.5 Force on a immersed body 6.5.1 Force in a steady potential flow 6.5.2 Added mass from the unsteady term 6.6 Flow around an airfoil 7 Boundary layers 7.1 Flow next to a flat plate 7.1.1 Qualitative description of the flow 7.1.2 Derivation of the Prandtl boundary layer equations 7.1.3 Self-similar flow past a semi-infinite flat plate 7.1.4 Drag on the plate 8 Acoustics and Incompressibility 8.1 Acoustic limit of Navier-Stokes equations 8.1.1 The Acoustic limit 8.2 Incompressible approximation 8.2.1 Variations in density due to pressure 8.2.2 The finite propagation-speed of sound 9 Stokes Flow 9.1 Recap: Governing Equations 9.1.1 Navier-Stokes Equations 9.1.2 Reynolds number 9.1.3 Energy and Dissipation 9.1.4 Additional comments 9.2 Properties of Stokes Flows 9.3 Stokes Flow Solutions 9.3.1 Representation as Harmonic Functions 9.3.2 Papkovich-Neuber Solutions 9.3.3 Spherical Harmonic Solutions 9.3.4 Tensors and Pseudotensors 9.3.5 Point Force and Source Flow 9.3.6 Motion of a rigid particle 9.3.7 Translating Rigid Sphere 10 Lubrication Theory 10.1 Lubrication approximation for flow in a thin layer 10.2 Squeeze flow 10.2.1 Thrust bearing 10.2.2 Cylinder approaching a wall 10.3 Free surface flow 10.3.1 Surface Tension 10.3.2 Bond number 10.3.3 Capillary number 10.3.4 Capillary drop 10.3.5 Gravitational spreading 10.3.6 Gravitational spreading down an inclined plane 11 Complex Fluids and Non-Newtonian Rheology 11.1 Constitutive laws 11.1.1 Newtonian fluid 11.1.2 Generalised Newtonian fluid 11.2 Poiseuille flow 11.3 Taylor-Couette flow 11.4 Viscoelastic fluids 11.4.1 Maxwell fluid 11.4.2 Oldroyd B fluid 11.4.3 Shear flow 11.4.4 Extensional flow 11.4.5 Weakly non-linear viscoelastic fluids 11.4.6 Rod climbing for a second-order fluid 11.4.7 Words of caution 12 Hydrodynamic Instability 12.1 The Kelvin-Helmholtz Instability 12.2 The Rayleigh-Plateau Instability 12.3 The Rayleigh Instability 13 Linear waves and instabilities 13.1 Interfacial waves and instabilities 13.1.1 The governing equations 13.1.2 Steady state and linearization 13.1.3 Fourier transform 13.1.4 Solution to Laplace equation 13.2 Waves and instabilities 13.2.1 Deep water gravity waves 13.2.2 Capillary and gravity-capillary waves 13.2.3 Rayleigh-Taylor instability 13.2.4 Kelvin-Helmholtz instability