Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics

by Grigory Isaakovich Barenblatt

• ISBN13: 9780521435222
• ISBN10: 0521435226
• Format: Paperback
• Copyright: 1996-12-28
• Publisher: Cambridge University Press
• More Book Details

Summary

Scaling (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.

Table of Contents

Preface	xi
Foreword	xvii
Introduction	1	(27)
Dimensional analysis and physical similarity	1	(8)
Assumptions underlying dimensional analysis	9	(5)
Self-similar phenomena	14	(4)
Self-similar solutions as intermediate asymptotics. The solutions of the first and second kind. Renormalization group	18	(8)
Self-similarities and travelling waves	26	(2)
Dimensions, dimensional analysis and similarity	28	(36)
Dimensions	28	(11)
Dimensional Analysis	39	(13)
Similarity	52	(12)
The construction of intermediate-asymptotic solutions using dimensional analysis. Self-similar solutions	64	(31)
Heat propagation from a concentrated instaneous source	64	(12)
Phenomena at the initial stage of a nuclear explosion	76	(10)
Self-similarity. Intermediate asymptotics	86	(9)
Self-similarities of the second kind: first examples	95	(24)
Flow of an ideal fluid past a wedge	95	(9)
Filtration in an elasto-plastic porous medium: the modified instaneous heat source problem	104	(15)
Self-similarities of the second kind: further examples	119	(26)
Modified very intense explosion problem	119	(14)
The von Weizsacker-Zeldovich problem: an impulsive loading	133	(12)
Classification of similarity rules and self-similarity solutions. A recipe for the application of similarity analysis	145	(16)
Complete and incomplete similarity	145	(16)
Scaling and transformation groups. Renormalization group	161	(20)
Dimensional analysis and transformation groups	161	(10)
The renormalisation group and incomplete similarity	171	(10)
Self-similar solutions and travelling waves	181	(19)
Solutions of travelling-wave type	181	(2)
Burgers shock wave -- steady travelling wave of the first kind	183	(2)
Flame: steady travelling wave of the second kind	185	(7)
Nonlinear eigenvalue problem	192	(2)
Flame propagation in a reacting mixture: an intermediate asymptotics	194	(6)
Invariant solutions: asymptotic conservation laws, spectrum of eigenvalues, and stability	200	(20)
Asymptotic conservation laws	200	(3)
Spectrum of eigenvalues	203	(6)
Stability of invariant solutions	209	(11)
Scaling in the deformation and fracture of solids	220	(32)
Transition from self-similarity of the first kind to self-similarity of the second; a linear elasticity problem	220	(14)
Similarity laws for brittle and quasi-brittle fracture	234	(18)
Scaling in turbulence	252	(44)
Homogeneous and isotropic turbulence	252	(16)
Turbulent shear flows	268	(28)
Scaling in geophysical fluid dynamics	296	(38)
Scaling laws for the atmospheric surface layer	296	(3)
Flows with strongly stable stratification	299	(2)
The regime of limiting saturation of a turbulent shear flow laden with sediment	301	(5)
Upper thermocline in the ocean - the travelling thermal wave model	306	(5)
Strong interaction of turbulence with internal waves. Deepending of the turbulent region	311	(5)
The breaking of internal waves and extension of mixed-fluid patches in a stably stratified fluid	316	(13)
Several phenomena related to turbulence in a stably stratified fluid	329	(5)
Scaling: miscellaneous special problems	334	(26)
Mandelbrot fractals and incomplete similarity	334	(8)
Example: scaling relationship between the breathing rate of animals and their mass. Fractality of respiratory organs	342	(3)
The spreading of a ground-water mound	345	(15)
Afterword	360	(6)
References	366	(17)
Index	383