## Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate AsymptoticsScaling (power-type) laws reveal the fundamental property of the phenomena--self similarity. Self-similar (scaling) phenomena repeat themselves in time and/or space. The property of self-similarity simplifies substantially the mathematical modeling of phenomena and its analysis--experimental, analytical and computational. The book begins from a non-traditional exposition of dimensional analysis, physical similarity theory and general theory of scaling phenomena. Classical examples of scaling phenomena are presented. It is demonstrated that scaling comes on a stage when the influence of fine details of initial and/or boundary conditions disappeared but the system is still far from ultimate equilibrium state (intermediate asymptotics). It is explained why the dimensional analysis as a rule is insufficient for establishing self-similarity and constructing scaling variables. Important examples of scaling phenomena for which the dimensional analysis is insufficient (self-similarities of the second kind) are presented and discussed. A close connection of intermediate asymptotics and self-similarities of the second kind with a fundamental concept of theoretical physics, the renormalization group, is explained and discussed. Numerous examples from various fields--from theoretical biology to fracture mechanics, turbulence, flame propagation, flow in porous strata, atmospheric and oceanic phenomena are presented for which the ideas of scaling, intermediate asymptotics, self-similarity and renormalization group were of decisive value in modeling. |

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### Contents

Dimensions Dimensional Analysis and Similarity | 28 |

12 Dimensional Analysis | 39 |

13 Similarity | 52 |

The construction of intermediate asymptotic solutions using dimensional analysis Selfsimilar solutions | 64 |

22 Phenomena at the initial stage of a nuclear explosion | 76 |

23 Selfsimilarity Intermediate asymptotics | 86 |

Selfsimilarities of the second kind first examples | 95 |

the modified instantaneous heat source problem | 104 |

83 Stability of invariant solutions | 209 |

Scaling in the deformation and fracture of solids | 220 |

92 Similarity laws for brittle and quasibrittle fracture | 234 |

Scaling in turbulence | 252 |

102 Turbulent shear flows | 268 |

Scaling in geophysical fluid dynamics | 296 |

112 Flows with strongly stable stratification | 299 |

113 The regime of limiting saturation of a turbulent shear flow laden with sediment | 301 |

Selfsimilarities of the second kind further examples | 119 |

an impulsive loading | 133 |

Classification of similarity rules and selfsimilar solutions A recipe for the application of similarity analysis | 145 |

Scaling and transformation groups Renormalization group | 161 |

62 The renormalisation group and incomplete similarity | 171 |

Selfsimilar solutions and travelling waves | 181 |

72 Burgers shock wave steady travelling wave of the first kind | 183 |

steady travelling wave of the second kind | 185 |

74 Nonlinear eigenvalue problem | 192 |

an intermediate asymptotics | 194 |

Invariant solutions asymptotic conservation laws spectrum of eigenvalues and stability | 200 |

82 Spectrum of eigenvalues | 203 |

114 Upper thermocline in the ocean the travelling thermal wave model | 306 |

115 Strong interaction of turbulence with internal waves Deepening of the turbulent region | 311 |

116 The breaking of internal waves and extension of mixedfluid patches in a stably stratified fluid | 316 |

117 Several phenomena related to turbulence in a stably stratified fluid | 329 |

Scaling miscellaneous special problems | 334 |

scaling relationship between the breathing rate of animals and their mass Fractality of respiratory organs | 342 |

123 The spreading of a groundwater mound | 345 |

Afterword | 360 |

366 | |

383 | |

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### Common terms and phrases

assumed assumption Barenblatt basic boundary conditions broken line chapter coefficient conservation laws considered const constant corresponding crack curve decrease density depend derivative determined differential equations dimension function dimensional analysis equal to zero example exists exponent fact factor Figure finite flux fractal front fundamental units Goldenfeld governing parameters grid Hausdorff dimension heat conduction homogeneous incomplete similarity independent dimensions initial conditions initial-value problem integral intense explosion internal waves layer length scale limit linear loading mass mathematical motion mound non-self-similar problem nonlinear eigenvalue problem obtained ordinary differential equations particles patch perturbed physical porous medium pressure properties quantity radius region relation renormalization group Reynolds number satisfies scaling law second kind segment self-similar solutions self-similar variables shear flow shock wave similarity parameters speed of propagation stage stratification stress system of units tends to zero thermal thermocline tion transformation travelling waves travelling-wave type turbulent energy velocity viscous wedge Zeldovich