Fluid mechanics: a geometrical point of view
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford University Press
2018
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| Edition: | First edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 250 Seiten Illustrationen, Diagramme |
| ISBN: | 9780198805038 9780198805021 |
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MARC
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Record in the Search Index
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Contents List of figures 1 ХѴІІ Vector Fields 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 The velocity field Space-time approach Eulerian vs Lagrangian picture Integral curves The method of characteristics Conservation law Densities vs scalars Steady flows Incompressible flow Irrotational flow Irrotational and incompressible flow Jacobi matrix Flow near a fixed point 1 1 2 3 3 4 5 6 8 8 8 9 10 11 Euler’s Equations 14 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 14 15 16 17 18 18 19 21 24 25 26 28 Conservation of momentum The stress tensor Incompressible flow Conservation of energy Helmholtz equation Steady flows Equations of state Transport form of Euler’s equation Linearization of Euler’s equations: sound Inviscid Burgers equation Scale invariance d’Alembert’s paradox: limitations the ideal fluid model The Navier-Stokes Equations 3.1 3.2 3.3 3.4 Viscosity Viscous incompressible flow Dissipation of energy at constant density Dissipation of energy for compressible flows 30 30 33 33 34
xii Contents 3.5 3.6 3.7 3.8 3.9 3.10 4 Ideal Fluid Flows 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 5 Pipe Poiseuille flow Circular Couette flow Stokes flow Stokes flow past a sphere Vortex with dissipating core Shocks 6.1 6.2 6.3 6.4 6.5 7 Statics Solutions of Laplace’s equation Complex analytic methods Fluid with a stirrer Flow past a cylinder The d’Alembert paradox Joukowski airfoil Surface waves Viscous Flows 5.1 5.2 5.3 5.4 5.5 6 Scale invariance: Reynolds number Navier-Stokes in curvilinear coordinates Diffusion and advection The diffusion kernel Growth of entropy in diffusion The advection-diffusion kernel The Burgers equations The Cole-Hopf transformation The limit of small viscosity Maxwell-Lax-Oleneik minimum principle The Riemann problem Boundary Layers 7.1 7.2 7.3 7.4 7.5 Prandtl’s theory The Blasius reduction Weyl’s method Drag on a flat plate Limitations of boundary layer theory 8 Instabilities 8.1 8.2 8.3 8.4 The Rayleigh-Taylor instability Linearization of Navier-Stokes equations Orr-Sommerfeld equation Transient solutions of linear equations 35 36 37 38 41 42 45 45 49 51 52 56 58 59 60 64 64 65 66 66 71 72 72 74 78 78 80 83 83 84 88 89 90 91 91 94 95 99
Contents 8.5 8.6 8.7 8.8 8.9 8.10 Normal operators A non-normal operator A nonlinear model with transients Stability regained Rapidly changing external force The Kapitza pendulum Integrable Models 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 9.11 9.12 KdV The soliton solution Multi-soliton solutions Lax pair Hamiltonian formalism of Fadeev and Zakharov The hamiltonian formalism of Magri The vortex filament Geometry of curves Velocity from vorticity: the Biot-Savart law The velocity field of a vortex filament Regularization and renormalization Relation to the Heisenberg model xiii 100 101 102 104 105 107 109 109 110 111 111 113 114 114 115 116 117 118 121 Hamiltonian Systems Based on a Lie Algebra 123 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 123 124 126 133 135 138 139 140 142 143 Rigid body mechanics Vector spaces Lie algebra The Virasoro algebra Hamiltonian for the two-dimensional Euler equations Spectral discretization of two-dimensional Euler Clebsch variables Hamiltonian form of 3D Euler equations Poisson brackets of velocity Analogy with angular momentum Curvature and Instability 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 Riemannian geometry Covariant derivative Geodesic deviation and curvature Curvature as a bi-quadratic Adding dissipation Lie groups Infinite dimensional Lie groups Geometry of left-invariant metrics Geodesics on a group manifold 144 144 145 146 148 149 151 152 152 153
xiv Contents 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 Covariant derivative on a group Curvature of a left-invariant metric Geodesics on 50(3) The diffeomorphism group Lie algebra of vector fields Diffeomorphisms of the circle The L2 -metric for vector fields on R3 Incompressible diffeomorphisms Curvature of the diffeomorphism group 12 Singularities 12.1 12.2 12.3 12.4 12.5 Norms: L2, Lf, Sobolev The dissipation of energy Solution of Navier-Stokes by perturbation theory Leray: finite time regularity Scale invariant solutions 13 Spectral Methods 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 The Chebychev basis Spectral discretization Sampling Interpolation Differentiation Integration The basic ODE Downsampling Spectral solution of the Orr-Sommerfeld equation Higher dimensions 14 Finite Difference Methods 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 Differential and difference operators in one dimension Pádé approximant Boundary value problems Explicit scheme for the diffusion equation Numerical stability Implicit schemes Physical explanation The Poisson equation Discrete version of the Clebsch formulation Radial basis functions A Lagrangian discretization 15 Geometric Integrators 15.1 Lie group methods 15.2 Exponential coordinates 15 5 156 157 158 160 160 161 162 164 166 167 169 170 171 173 175 175 176 177 178 178 179 180 181 183 185 186 186 188 189 190 192 192 194 195 197 199 201 203 203 204
Contents 15.3 Differentiating the matrix exponential 15.4 Magnus expansion Appendix A Dynamical Systems A. 1 A.2 A.3 A.4 A.5 A.6 A. 7 A.8 Jacobi matrix at a fixed point Stable and unstable manifolds Arnold’s cat map The homoclinic tangle The Smale horse shoe Binary code Iterated function systems on the interval Normal form of the horse shoe Appendix В B. 1 B.2 B.3 Stirred fluid inside a cylinder (again) Moving stirrers Aref’s protocol Appendix C C.l C.2 C.3 C.4 C.5 C.6 C.7 C.8 C.9 Chaotic Advection Renormalization The Ising model Transfer matrix Renormalization dynamics Cayley tree Spontaneous magnetization of the Ising model on the Cayley tree The Ising model on square and cubic lattices Iterations of a function Feigenbaum’s renormalization dynamics The Feigenbaum-Cvitanonic equation xv 205 206 209 209 210 211 213 214 215 217 219 224 224 225 226 228 228 230 231 234 236 239 239 241 241 Bibliography 245 Index 249 |
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| institution | BVB |
| isbn | 9780198805038 9780198805021 |
| language | English |
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| physical | xv, 250 Seiten Illustrationen, Diagramme |
| publishDate | 2018 |
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| publisher | Oxford University Press |
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| spelling | Rajeev, Sarada G. 1959- Verfasser (DE-588)1038558972 aut Fluid mechanics a geometrical point of view S. G. Rajeev, Department of Physics and Astronomy, Department of Mathematics, University of Rochester, Rochester, NY First edition Oxford Oxford University Press 2018 xv, 250 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Geometrische Methode (DE-588)4156715-8 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 s Geometrische Methode (DE-588)4156715-8 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030465289&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rajeev, Sarada G. 1959- Fluid mechanics a geometrical point of view Geometrische Methode (DE-588)4156715-8 gnd Strömungsmechanik (DE-588)4077970-1 gnd |
| subject_GND | (DE-588)4156715-8 (DE-588)4077970-1 |
| title | Fluid mechanics a geometrical point of view |
| title_auth | Fluid mechanics a geometrical point of view |
| title_exact_search | Fluid mechanics a geometrical point of view |
| title_full | Fluid mechanics a geometrical point of view S. G. Rajeev, Department of Physics and Astronomy, Department of Mathematics, University of Rochester, Rochester, NY |
| title_fullStr | Fluid mechanics a geometrical point of view S. G. Rajeev, Department of Physics and Astronomy, Department of Mathematics, University of Rochester, Rochester, NY |
| title_full_unstemmed | Fluid mechanics a geometrical point of view S. G. Rajeev, Department of Physics and Astronomy, Department of Mathematics, University of Rochester, Rochester, NY |
| title_short | Fluid mechanics |
| title_sort | fluid mechanics a geometrical point of view |
| title_sub | a geometrical point of view |
| topic | Geometrische Methode (DE-588)4156715-8 gnd Strömungsmechanik (DE-588)4077970-1 gnd |
| topic_facet | Geometrische Methode Strömungsmechanik |
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