ELEMENTARY FLUID MECHANICS
CONTENTS 1 FLOWS 1 1 WHAT ARE ?OWS? 1 2 FLUID PARTICLE AND ?ELDS 1 3 STREAM-LINE, PARTICLE-PATH AND STREAK-LINE 1 3 1 STREAM-LINE 1 3 2 PARTICLE-PATH (PATH-LINE) 1 3 3 STREAK-LINE 1 3 4 LAGRANGE DERIVATIVE 1 4 RELATIVE MOTION 1 4 1 DECOMPOSITION 1 4 2 SYMMETRIC PART (PURE STRAINING MOTION) 1 4 3 ANTI-SYMMETRIC PART (LOCAL ROTATION) 1 5 PROBLEMS 2 FLUIDS 2 1 CONTINUUM AND TRANSPORT PHENOMENA 2 2 MASS DI?USION IN A ?UID MIXTURE 2 3 THERMAL DI?USION 2 4 MOMENTUM TRANSFER 2 5 AN IDEAL ?UID AND NEWTONIAN VISCOUS ?UID 2 6 VISCOUS STRESS 2 7 PROBLEMS CONTENTS 3 FUNDAMENTAL EQUATIONS OF IDEAL ?UIDS 3 1 MASS CONSERVATION 3 2 CONSERVATION FORM 3 3 MOMENTUM CONSERVATION 3 3 1 EQUATION OF MOTION 3 3 2 MOMENTUM ?UX 3 4 ENERGY CONSERVATION 3 4 1 ADIABATIC MOTION 3 4 2 ENERGY ?UX 3 5 PROBLEMS 4 VISCOUS ?UIDS 4 1 EQUATION OF MOTION OF A VISCOUS ?UID 4 2 ENERGY EQUATION AND ENTROPY EQUATION 4 3 ENERGY DISSIPATION IN AN INCOMPRESSIBLE ?UID 4 4 REYNOLDS SIMILARITY LAW 4 5 BOUNDARY LAYER 4 6 PARALLEL SHEAR ?OWS 4 6 1 STEADY ?OWS 4 6 2 UNSTEADY ?OW 4 7 ROTATING ?OWS 4 8 LOW REYNOLDS NUMBER ?OWS 4 8 1 STOKES EQUATION 4 8 2 STOKESLET 4 8 3 SLOW MOTION OF A SPHERE 4 9 FLOWS AROUND A CIRCULAR CYLINDER 4 10 DRAG COE?CIENT AND LIFT COE?CIENT 4 11 PROBLEMS 5 FLOWS OF IDEAL ?UIDS 5 1 BERNOULLI’S EQUATION 5 2 KELVIN’S CIRCULATION THEOREM 5 3 FLUX OF VORTEX LINES 5 4 POTENTIAL ?OWS 5 5 IRROTATIONAL INCOMPRESSIBLE ?OWS (3D) CONTENTS 5 6 EXAMPLES OF IRROTATIONAL INCOMPRESSIBLE ?OWS (3D) 5 6 1 SOURCE (OR SINK) 5 6 2 A SOURCE IN A UNIFORM ?OW 5 6 3 DIPOLE 5 6 4 A SPHERE IN A UNIFORM ?OW 5 6 5 A VORTEX LINE 5 7 IRROTATIONAL INCOMPRESSIBLE ?OWS (2D) 5 8 EXAMPLES OF 2D ?OWS REPRESENTED BY COMPLEX POTENTIALS 5 8 1 SOURCE (OR SINK) 5 8 2 A SOURCE IN A UNIFORM ?OW 5 8 3 DIPOLE 5 8 4 A CIRCULAR CYLINDER IN A UNIFORM ?OW 5 8 5 POINT VORTEX (A LINE VORTEX) 5 9 INDUCED MASS 5 9 1 KINETIC ENERGY INDUCED BY A MOVING BODY 5 9 2 INDUCED MASS 5 9 3 D’ALEMBERT’S PARADOX AND VIRTUAL MASS 5 10 PROBLEMS 6 WATER WAVES AND SOUND WAVES 6 1 HYDROSTATIC PRESSURE 6 2 SURFACE WAVES ON DEEP WATER 6 2 1 PRESSURE CONDITION AT THE FREE SURFACE 6 2 2 CONDITION OF SURFACE MOTION 6 3 SMALL AMPLITUDE WAVES OF DEEP WATER 6 3 1 BOUNDARY CONDITIONS 6 3 2 TRAVELING WAVES 6 3 3 MEANING OF SMALL AMPLITUDE 6 3 4 PARTICLE TRAJECTORY 6 3 5 PHASE VELOCITY AND GROUP VELOCITY 6 4 SURFACE WAVES ON WATER OF A ?NITE DEPTH 6 5 KDV EQUATION FOR LONG WAVES ON SHALLOW WATER CONTENTS 6 6 SOUND WAVES 6 6 1 ONE-DIMENSIONAL ?OWS 6 6 2 EQUATION OF SOUND WAVE 6 6 3 PLANE WAVES 6 7 SHOCK WAVES 6 8 PROBLEMS 7 VORTEX MOTIONS 7 1 EQUATIONS FOR VORTICITY 7 1 1 VORTICITY EQUATION 7 1 2 BIOT–SAVART’S LAW FOR VELOCITY 7 1 3 INVARIANTS OF MOTION 7 2 HELMHOLTZ’S THEOREM 7 2 1 MATERIAL LINE ELEMENT AND VORTEX-LINE 7 2 2 HELMHOLTZ’S VORTEX THEOREM 7 3 TWO-DIMENSIONAL VORTEX MOTIONS 7 3 1 VORTICITY EQUATION 7 3 2 INTEGRAL INVARIANTS 7 3 3 VELOCITY ?ELD AT DISTANT POINTS 7 3 4 POINT VORTEX 7 3 5 VORTEX SHEET 7 4 MOTION OF TWO POINT VORTICES 7 5 SYSTEM OF N POINT VORTICES (A HAMILTONIAN SYSTEM) 7 6 AXISYMMETRIC VORTICES WITH CIRCULAR VORTEX-LINES 7 6 1 HILL’S SPHERICAL VORTEX 7 6 2 CIRCULAR VORTEX RING 7 7 CURVED VORTEX ?LAMENT 7 8 FILAMENT EQUATION (AN INTEGRABLE EQUATION) 7 9 BURGERS VORTEX (A VISCOUS VORTEX WITH SWIRL) 7 10 PROBLEMS 8 GEOPHYSICAL ?OWS 8 1 FLOWS IN A ROTATING FRAME 8 2 GEOSTROPHIC ?OWS CONTENTS 8 3 TAYLOR–PROUDMAN THEOREM 8 4 A MODEL OF DRY CYCLONE (OR ANTICYCLONE) 8 5 ROSSBY WAVES 8 6 STRATI?ED ?OWS 8 7 GLOBAL MOTIONS BY THE EARTH SIMULATOR 8 7 1 SIMULATION OF GLOBAL ATMOSPHERIC MOTION BY AFES CODE 8 7 2 SIMULATION OF GLOBAL OCEAN CIRCULATION BY OFES CODE 8 8 PROBLEMS 9 INSTABILITY AND CHAOS 9 1 LINEAR STABILITY THEORY 9 2 KELVIN–HELMHOLTZ INSTABILITY 9 2 1 LINEARIZATION 9 2 2 NORMAL-MODE ANALYSIS 9 3 STABILITY OF PARALLEL SHEAR ?OWS 9 3 1 INVISCID ?OWS (? = 0) 9 3 2 VISCOUS ?OWS 9 4 THERMAL CONVECTION 9 4 1 DESCRIPTION OF THE PROBLEM 9 4 2 LINEAR STABILITY ANALYSIS 9 4 3 CONVECTION CELL 9 5 LORENZ SYSTEM 9 5 1 DERIVATION OF THE LORENZ SYSTEM 9 5 2 DISCOVERY STORIES OF DETERMINISTIC CHAOS 9 5 3 STABILITY OF ?XED POINTS 9 6 LORENZ ATTRACTOR AND DETERMINISTIC CHAOS 9 6 1 LORENZ ATTRACTOR 9 6 2 LORENZ MAP AND DETERMINISTIC CHAOS 9 7 PROBLEMS 10 TURBULENCE 10 1 REYNOLDS EXPERIMENT 10 2 TURBULENCE SIGNALS CONTENTS 10 3 ENERGY SPECTRUM AND ENERGY DISSIPATION 10 3 1 ENERGY SPECTRUM 10 3 2 ENERGY DISSIPATION 10 3 3 INERTIAL RANGE AND ?VE-THIRDS LAW 10 3 4 SCALE OF VISCOUS DISSIPATION 10 3 5 SIMILARITY LAW DUE TO KOLMOGOROV AND OBOUKOV 10 4 VORTEX STRUCTURES IN TURBULENCE 10 4 1 STRETCHING OF LINE-ELEMENTS 10 4 2 NEGATIVE SKEWNESS AND ENSTROPHY ENHANCEMENT 10 4 3 IDENTI?CATION OF VORTICES IN TURBULENCE 10 4 4 STRUCTURE FUNCTIONS 10 4 5 STRUCTURE FUNCTIONS AT SMALL S 10 5 PROBLEMS 11 SUPER?UID AND QUANTIZED CIRCULATION 11 1 TWO-?UID MODEL 11 2 QUANTUM MECHANICAL DESCRIPTION OF SUPER?UID ?OWS 11 2 1 BOSE GAS 11 2 2 MADELUNG TRANSFORMATION AND HYDRODYNAMIC REPRESENTATION 11 2 3 GROSS–PITAEVSKII EQUATION 11 3 QUANTIZED VORTICES 11 3 1 QUANTIZED CIRCULATION 11 3 2 A SOLUTION OF A HOLLOW VORTEX-LINE IN A BEC 11 4 BOSE–EINSTEIN CONDENSATION (BEC) 11 4 1 BEC IN DILUTE ALKALI-ATOMIC GASES 11 4 2 VORTEX DYNAMICS IN ROTATING BEC CONDENSATES 11 5 PROBLEMS CONTENTS 12 GAUGE THEORY OF IDEAL ?UID ?OWS 12 1 BACKGROUNDS OF THE THEORY 12 1 1 GAUGE INVARIANCES 12 1 2 REVIEW OF THE INVARIANCE IN QUANTUM MECHANICS 12 1 3 BRIEF SCENARIO OF GAUGE PRINCIPLE 12 2 MECHANICAL SYSTEM 12 2 1 SYSTEM OF N POINT MASSES 12 2 2 GLOBAL INVARIANCE AND CONSERVATION LAWS 12 3 FLUID AS A CONTINUOUS ?ELD OF MASS 12 3 1 GLOBAL INVARIANCE EXTENDED TO A ?UID 12 3 2 COVARIANT DERIVATIVE 12 4 SYMMETRY OF ?OW ?ELDS I: TRANSLATION SYMMETRY 12 4 1 TRANSLATIONAL TRANSFORMATIONS 12 4 2 GALILEAN TRANSFORMATION (GLOBAL) 12 4 3 LOCAL GALILEAN TRANSFORMATION 12 4 4 GAUGE TRANSFORMATION (TRANSLATION SYMMETRY) 12 4 5 GALILEAN INVARIANT LAGRANGIAN 12 5 SYMMETRY OF ?OW ?ELDS II: ROTATION SYMMETRY 12 5 1 ROTATIONAL TRANSFORMATIONS 12 5 2 IN?NITESIMAL ROTATIONAL TRANSFORMATION 12 5 3 GAUGE TRANSFORMATION (ROTATION SYMMETRY) 12 5 4 SIGNI?CANCE OF LOCAL ROTATION AND THE GAUGE ?ELD 12 5 5 LAGRANGIAN ASSOCIATED WITH THE ROTATION SYMMETRY 12 6 VARIATIONAL FORMULATION FOR ?OWS OF AN IDEAL ?UID 12 6 1 COVARIANT DERIVATIVE (IN SUMMARY) 12 6 2 PARTICLE VELOCITY 12 6 3 ACTION PRINCIPLE 12 6 4 OUTCOMES OF VARIATIONS 12 6 5 IRROTATIONAL ?OW 12 6 6 CLEBSCH SOLUTION CONTENTS 12 7 VARIATIONS AND NOETHER’S THEOREM 12 7 1 LOCAL VARIATIONS 12 7 2 INVARIANT VARIATION 12 7 3 NOETHER’S THEOREM 12 8 ADDITIONAL NOTES 12 8 1 POTENTIAL PARTS 12 8 2 ADDITIONAL NOTE ON THE ROTATIONAL SYMMETRY 12 9 PROBLEM CONTENTS REFERENCES