FERROHYDRODYNAMICS R. E. ROSENSWEIG DOVER PUBLICATIONS, INC. Mineola, New York
CONTENTS Preface 1 Introduction 1.1 Scope of ferrohydrodynamics 1.2 Ferromagnetic solids 1.3 Magnetic fluids 1.4 Ferromagnetic concepts and units Definition of the fleld External field of a dipole source Magnetic force and torque on dipolar matter Interaction energy of two dipoles 1.5 Concepts of fluid mechanics Continuity equation Substantial derivative 1.6 Generalized Bernoulli equation 1.7 Stress tensor and its physical meaning Force resulting from a stress tensor Addendum: equivalence ofdipolar and polar representations 2 Magnetic fluids 2.1 Stability requirements Stability in a magnetic-field gradient Stability against settling in a gravitational field Stability against magnetic agglomeration Necessity to guard against the van der Waals attractive force 2.2 Preparation of magnetic colloids by size reduction 2.3 Preparation of ferrofluids by chemical precipitation Magnetite precipitation with steric stabilization Cobalt particles in an organic carrier Charge-stabilized magnetite 2.4 Other magnetic fluids Paramagnetic sah Solutions
vm Metallic-base ferrofluid 2.5 Surfaceadsorptionandstericstabilization Stenc repulsion mechanism Net interaction curve Dispersant structural guidelines 2.6 Ferrofluid modification Phenomenological basis Carrier liquid exchange Surfactant exchange 2.7 Physical properties Equilibrium magnetization: superparamagnetism «Magnetic relaxation 55 Viscosity 61 Concentrated suspensions 2.8 Correlation phenomena ^ 2.9 Tabulated physical properties 3 Electromagnetism and flelds 3.1 Magnetostaticfieldequations Scalar potential 74 3.2 Magnetic-fieldboundaryconditions 77 3.3 Maxwell stress tensor 77 Portrait ofthe Maxwell stresstensor 81 3.4 Maxwell's equations 83 Integral equations 84 Differential equations 87 3.5 Energy density of the electromagnetic field ^ 3.6 94 Transformed expression for the field energy 95 QQ 98 4 Stress tensor and the equation of motion 4.1 Thermodynamic background 100 4.2 Formulation of the magnetstress tensor 101 Stress tensor of a magnetizable fluid 103 What is the "pressure" in a magnetized fluid?,"* 4.3 Magnetic body-force density 108 Alternative general forms 110 Alternative reduced forms 111 Remarksconcerningstriction in compressible media!" 4.4 Equation of motion for magnetic fluid 7 Alternative forms ofthe equation of motion 119 121 5 5.1 The ferrohydrodynamic BernouIIi equation Derivation nn 5.2 Boundary conditions 5.3 Categories of equilibrium inviscid flows 111 122 l ** 124 126 131 45 46 46 48 48 50 50 51 54 63 67 70 72 74
ix 5.4 Applications of the FHD Bernoulli equation 132 Classical Quincke problem 132 Surface elevation in a normal field 133 Magnetic nozzle 134 Modified Gouy experiment 136 Conical meniscus 137 Origin of the radial force 141 Magnetic-fluid rotary-shaft seals 142 5.5 Eamshaw's theorem and magnetic levitation 146 Simplined treatment of the levitation of a nonmagnetic body 149 Phenomenon of self-levitation 150 Analysis of forces on an immersed body 153 5.6 Striction effect 157 158 6 Magnetocaloric energy conversion 161 6.1 Thermodynamics of magnetic materials 162 6.2 Mechanism for power generation 164 6.3 Linear equation of State 166 6.4 General cycle analysis 168 6.5 Cycle efficiency for a linear material 171 6.6 Cycle efficiency with regeneration 172 6.7 Implementing the cycle. 172 6.8 Summary 174 175 7 Ferrohydrodynamic instabilities 177 7.1 Normal-field instability 178 Equation set 178 Interfacial force balance 179 Kinematics 182 Flow-field analysis 183 Perturbed magnetic field 185 Boundary conditions on the field vectors 185 Applying the magnetic boundary conditions 187 Theoretical predictions 188 Normal-field instability experiments 193 Nonlinear analysis 196 7.2 General dispersion relation for moving media with oblique magnetic field 199 7.3 Rayleigh-Taylor problem 200 Experimental confirmation 202 7.4 Kelvin-Helmholtz instability 202 Criterion for Kelvin-Helmholtz instability in a ferrofluid 204 7.5 Gradient-field stabilization 206 7.6 Labyrinthine instability 208 Labyrinth spacing 212 7.7 Stabilityof fluid cylinders 216 Magnetic-fluid jet in a uniform field 217
x Cylindrical column in a radial gradient field 7.8 Porous-medium flow: fingering instability 7.9 Thermoconvective stability 7.10 Retrospective 222 225 232 Addendum: representation of disturbance waves 232 235 8 Magnetic fluids and asymmetric stress 237 8.1 Phenomena 8.2 Cauchy stress principle and conservation of momentum 8.3 Integral equation for conservation of angular momentum for nonpolar materials 8.4 Reynolds' transport theorem 8.5 Cauchy equation ofmotion 8.6 Symmetry of the stress tensor for nonpolar fluids 8.7 Analysis for polar fluids 8.8 Summary of the basic laws of continuum mechanics 8.9 Constitutive relations Magnetization relaxation process 8.10 Analogs of the Navier-Stokes equations for fluids with internal angular momentum 8.11 Ferrohydrodynamic-torque-drivenflow 8.12 Effective viscosity of a magnetized fluid 9 Magnetic two-phase flow 9.1 Background 9.2 Ordinary fluidization 9.3 Some fundamental problems in fluidization engineering 9.4 Basic relationships in two-phase flow Definition of averages Averaged continuity equations Averaged magnetostatic relationships Averaged momentum balances Constitutive relations 9.5 Summary of the averaged equations 9.6 Magnetized fluidized solids Solution in the steady State with a uniform magnetic field 9.7 Stability of the steady-state Solution Derivation of the linearized equations Form of the linearized equations for plane waves General Solution for the voidage perturbation Nature of the predicted behavior 9.8 Experimental behavior Transition velocity Influence of the field orientation Rheology of magnetically stabilized fluidized solids M 237 240 242 243 245 246 249 252 253 255 257 258 263 269 272 272 273 274 276 276 279 280 288 289 290 291 291 293 297 301 304 304 306 307
xi Directions for further study Appendixes 315 1 Vector and tensor notation 315 2 Application of the Maxwell stress tensor: analysis of a sheet jet 318 References Citation index 335 Subject index 339 312 323 3 * 2