Bringing the concepts of dimensional analysis, self–similarity, andfractal dimensions together in a logical and self–contained manner,this book reveals the close links between modern theoreticalphysics and applied mathematics.The author focuses on the classic applications of self–similarsolutions within astrophysical systems, with some general theory ofself–similar solutions, so as to provide a framework forresearchers to apply the principles across all scientificdisciplines. He discusses recent advances in theoretical techniquesof scaling while presenting a uniform technique that encompassesthese developments, as well as applications to almost any branch ofquantitative science.The result is an invaluable reference for active scientists,featuring examples of dimensions and scaling in condensed matterphysics, astrophysics, fluid mechanics, and general relativity, aswell as in mathematics and engineering. INDICE: Preface XI.Acknowledgments XIII.Introduction XV.1 Arbitrary Measures of the PhysicalWorld 1.1.1 Similarity 1.1.2 Dimensional Similarity 3.1.3 Physical Equations and the Pi Theorem 6.1.4 Applications of the PiTheorem 10.1.4.1 Plane Pendulum 11.1.4.2 Pipe Flow of a Fluid 16.1.4.3 Steady Motion of a Rigid Object in Viscous Fluid 18.1.4.4 Diffusion and Self–Similarity 20.1.4.5 ShipWave Drag 26.1.4.6 Adiabatic Gas Flow 28.1.4.7 Time–Dependent Adiabatic Flow 30.1.4.8 Point Explosion in a Gaseous Medium 33.1.4.9 Applications in Fundamental Physics 35.1.4.10 Drag on a Flexible Object in Steady Motion 41.1.4.11 Dimensional Analysis of Mammals 42.1.4.12 Trees 47.References 51.2 Lie Groups and Scaling Symmetry 53.2.1 The Rescaling Group 53.2.1.1 Rescaling Physical Objects 55.2.1.2 Reconciliation with the Buckingham PiTheorem 59.2.1.3 Rescaling and Self–Similarity as a Lie Algebra 60.2.1.4 Practical Lie Self–Similarity 63.2.2 Familiar Physical Examples 68.2.2.1 Line Vortex Diffusion: Reprise 69.2.2.2 Burgers Equation 71.2.3 Less Familiar Examples 77.2.3.1 Self–Gravitating Collisionless Particles: The Boltzmann–Poisson Problem 77.References 84.3 Poincaré Group Plus Rescaling Group 87.3.1 Galilean Space–Time 87.3.2 Minkowski Space–Time 96.3.2.1 Self–Similar Lorentz Boost 96.3.2.2 Self–Similar Boost/Rotation 102.3.3 Kinematic General Relativity 108.References 119.4 Instructive Classic Problems 121.4.1 Introduction 121.4.2 Ideal Fluid Flow Past aWedge: Self–Similarity of the Second Kind 121.4.3 Boundary Layer on a Flat Plate: the Blasius Problem 126.4.4 Adiabatic Self–Similarity in the Diffusion Equation 133.4.5 Waves in a Uniformly Rotating Fluid 140.References 146.5 Variations on Lie Self–Similarity 147.5.1 Variations on the Boltzmann Poisson System 147.5.1.1 Infinite Self–Gravitating Collisionless Spheres 147.5.1.2 Finite Self–Gravitating Collisionless Spheres 155.5.1.3 Other Approaches to Finite Spheres 159.5.2 Hydrodynamic Examples 164.5.2.1 General Navier StokesTheory 164.5.2.2 Modified Couette Flow 166.5.2.3 Flow at Large Scale inside a LaminarWake 170.5.3 Axi–Symmetric Ideal Magnetohydrodynamics 178.5.3.1 Incomplete Self–Similarity as Separable Multi–variable Self–Similarity 182.5.3.2 Isothermal Collapse 185.References 187.6 Explorations 189.6.1 Anisotropic Self–Similarity 189.6.1.1 Anisotropic Similarity 192.6.2 Mathematical Variations 193.6.3 Periodicity and Similarity 198.6.3.1 Log Periodicity and Self–Similarity: Diffusion Equation 203.References 207.7 Renormalization Group and Noether Invariants 209.7.1 Hybrid Lie Self–Similarity/Renormalization Group 209.7.1.1 Renormalizing More Complicated Equations 216.7.1.2 Schrödinger: Adiabatic and Fractal 219.7.1.3 Noether Invariants and Self–Similarity 223.References 229.8 Scaling in Hydrodynamical Turbulence 231.8.1 General Introduction 231.8.2 Homogeneous, Isotropic, Decaying Turbulence 232.8.2.1 Third–Order Correlation Negligible 236.8.2.2 Renormalization and Homogeneous, Isotropic, Turbulence 240.8.3 Dimensional Phenomenology of Stationary Turbulence 242.8.4 Structure in 2D Turbulence 246.8.4.1 Similarity of Time–Dependent 2D Vortical Fluid Flow 248.8.4.2 Similarity in Physically Steady, Inviscid Vortical Fluid Flow 258.References 264.Epilogue 267.Appendix: Examples from the literature 269.Index 273
- ISBN: 978-3-527-41335-5
- Editorial: Wiley VCH
- Encuadernacion: Cartoné
- Páginas: 304
- Fecha Publicación: 22/04/2015
- Nº Volúmenes: 1
- Idioma: Inglés