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1 Contents Preface xiii I Introduction 1 1 Continuous matter Molecules The continuum approximation Newtonian mechanics Continuum physics Problems Space and time Reference frames Time Space Vector algebra Basis vectors Index notation Cartesian coordinate transformations Scalars, vectors, and tensors Scalar, vector, and tensor fields Pseudo and improper quantities Problems Gravity Mass density Gravitational acceleration

2 iv CONTENTS 3.3 Sources of gravity Gravitational potential Potential energy Problems II Fluids at rest 55 4 Pressure The concept of pressure Formal definition of pressure Hydrostatic equilibrium Equation of state Barotropic fluid states The homentropic atmosphere Problems Buoyancy Archimedes principle The gentle art of ballooning Stability of floating bodies Ship stability Problems Planets and stars Gravitational flux Spherical bodies The homentropic star Gravitational energy Problems Hydrostatic shapes Fluid interfaces in hydrostatic equilibrium Shape of rotating fluids The Earth, the Moon and the tides Shape of a rotating fluid planet Problems Surface tension Definition of surface tension Contact angle Capillary effect at a vertical wall Axially invariant shapes Problems

3 CONTENTS v III Deformable solids Stress Friction The concept of stress Nine components of stress Mechanical equilibrium Proof of symmetry of the stress tensor Problems Strain Displacement Local deformation Geometrical meaning of the strain tensor Work and energy Finite deformations Problems Linear elasticity Hooke s law Hooke s law in isotropic materials Static uniform deformation Energy of deformation Problems Elastostatics Equations of elastostatics Standing up to gravity Bending a beam Twisting a shaft Tube under pressure Problems Computational elastostatics Relaxing towards equilibrium Discretization of space Gravitational settling in two dimensions Problems Elastic vibrations Elastodynamics Refraction and reflection Surface waves Problems

4 vi CONTENTS IV Basic hydrodynamics Fluids in motion The velocity field Incompressible flow Mass conservation Moving along with the flow Continuum dynamics Big Bang Newtonian cosmology Problems Nonviscous flow The Euler equation Small-amplitude sound waves Steady incompressible flow Steady compressible flow Vorticity Circulation Potential flow Potential flow for cylinder in cross-wind Potential flow around a sphere in a stream D Alembert s paradox Problems Viscosity Shear viscosity Velocity-driven planar flow Incompressible Newtonian fluids Classification of flows Compressible Newtonian fluids Viscous attenuation of sound Problems Plates and pipes Steady, incompressible, viscous flow Pressure-driven planar flow Gravity-driven planar flow Pipe flow Phenomenology of turbulent pipe flow Circulating cylindrical flow Secondary flow and Taylor vortices Problems

5 CONTENTS vii 19 Creeping flow Steady incompressible creeping flow Creeping flow around a solid ball Beyond Stokes law Beyond spherical shape Problems Rotating fluids Fictitious forces Flow in a rotating system Geostrophic flow The Ekman layer Steady bathtub vortex in rotating container Debunking an urban legend Problems Computational fluid dynamics Unsteady, incompressible flow Temporal discretization Spatial discretization Channel entrance flow Problems V Special topics Global laws of balance Connected tubes Overview of the global laws The control volume Mass balance Momentum balance Angular momentum balance Kinetic energy balance Mechanical energy balance Energy balance in elastic fluids Problems Reactions Reaction forces Formal definition of reaction force Reaction moments Problems

6 viii CONTENTS 24 Jumps and shocks Hydraulic jumps Normal shocks in ideal gases Atmospheric blast wave Problems Small-amplitude surface waves Basic physics of surface waves Harmonic line waves Gravity waves Capillary surface waves Internal waves Energy and attenuation Statistics of wind-generated ocean waves Global wave properties Problems Whirls and vortices Free cylindrical vortices Ideal vortex dynamics Parallel line vortices Steady vortex sustained by secondary flow Advective spin-up of a vortex Bathtub-like vortices Problems Lubrication Physics of lubrication Creeping flow in a long narrow gap Flat wing Loaded journal bearing Problems Boundary layers Physics of boundary layers The Stokes layer Boundary layer theory The Blasius layer Turbulent boundary layer in uniform flow Boundary layers with varying slip-flow Laminar boundary layer separation Problems

7 CONTENTS ix 29 Subsonic flight Aircraft controls Aerodynamic forces and moments Steady flight Estimating lift Estimating drag Lift, drag, and the trailing wake Two-dimensional airfoil theory The distant laminar wake Problems Heat Energy balance Heat equation for isotropic matter at rest Heat equation for fluids in motion Advective cooling or heating Problems Convection Convection Convective instability Linear stability analysis of convection Rayleigh-Bénard convection Problems Turbulence 639 A Units and constants 641 B Newtonian particle mechanics 643 B.1 Dynamic equations B.2 Force and momentum B.3 Moment of force and angular momentum B.4 Power and kinetic energy B.5 Internal and external forces B.6 Hierarchies of particle interactions Problems C Curvilinear coordinates 651 C.1 Cylindrical coordinates C.2 Spherical coordinates

8 x CONTENTS D Thermodynamics of ideal gases 657 D.1 Internal energy D.2 Heat capacity D.3 Entropy D.4 Specific quantities Problems Answers to problems Continuous matter Space and time Gravity Fluids at rest Buoyancy Planets and stars Hydrostatic shapes Surface tension Stress Strain Elasticity Solids at rest Numeric elastostatics Vibrations Matter in motion Nearly ideal flow Viscosity Plates and tubes Creeping flow Rotating fluids Computational fluid dynamics Laws of balance Reaction forces and moments Jumps and shocks Surface waves Whirls and vortices Lubrication Boundary layers Subsonic flight Heat Convection Turbulence A Units and constants B Newtonian particle mechanics C Curvilinear coordinates D Thermodynamics of ideal gases

9 xi Bibliography 723 Index 729

10 xii

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Fluid Mechanics Theory I Last Class: 1. Introduction 2. MicroTAS or Lab on a Chip 3. Microfluidics Length Scale 4. Fundamentals 5. Different Aspects of Microfluidcs Today s Contents: 1. Introduction to