G. I. Barenblatt

THE MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS IN BRITTLE FRACTURE

Publisher Summary In recent years, the interest in the problem of brittle fracture and, in particular, in the theory of cracks has grown appreciably in connection with various technical applications. Numerous investigations have been carried out, enlarging in essential points the classical concepts of cracks and methods of analysis. The qualitative features of the problems of cracks, associated with their peculiar nonlinearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range of problems in terms of the theory of elasticity. The chapter presents a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discusses the results obtained thereby. The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. However, there exists a fundamental distinction between these two problems, The form of a cavity undergoes only slight changes even under a considerable variation in the load acting on a body, while the cracks whose surface also constitutes a part of the body boundary can expand even with small increase of the load to which the body is subjected.

Scaling, self-similarity, and intermediate asymptotics

Preface Introduction 1. Dimensions, dimensional analysis and similarity 2. The application of dimensional analysis to the construction of intermediate asymptotic solutions to problems of mathematical physics. Self-similar solutions 3. Self-similarities of the second kind: first examples 4. Self-similarities of the second kind: further examples 5. Classification of similarity rules and self-similarity solutions. Recipe for application of similarity analysis 6. Scaling and transformation groups. Renormalization groups. 7. Self-similar solutions and travelling waves 8. Invariant solutions: special problems of the theory 9. Scaling in deformation and fracture in solids 10. Scaling in turbulence 11. Scaling in geophysical fluid dynamics 12. Scaling: miscellaneous special problems.

The Mathematical theory of combustion and explosions

Various topics in the area of combustion and explosives are discussed. The general subjects considered include: basic physical concepts of the science of combustion, the time-independent theory of thermal explosions, time-dependent statement of the problem of the initiation of chemical reaction waves in fuel mixtures, laminar flames, complex and chain reactions in flames, the gas dynamics of combustion, and diffusional combustion of gases. 561 references.

Similarity, Self-Similarity and Intermediate Asymptotics

Make more knowledge even in less time every day. You may not always spend your time and money to go abroad and get the experience and knowledge by yourself. Reading is a good alternative to do in getting this desirable knowledge and experience. You may gain many things from experiencing directly, but of course it will spend much money. So here, by reading similarity self similarity and intermediate asymptotics, you can take more advantages with limited budget.

Theory of fluid flows through natural rocks

This book demonstrates the natural connection between classical and modern hydrodynamics. The increasing specialization of the application of hydrodynamic theory to the problems of gas and oil extraction from reservoirs with complex physical and geological properties is addressed. The goals of the model builder are discussed.

Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis

The present work consists of two parts. Here in Part 1, a scaling law (incomplete similarity with respect to local Reynolds number based on distance from the wall) is proposed for the mean velocity distribution in developed turbulent shear flow. The proposed scaling law involves a special dependence of the power exponent and multiplicative factor on the flow Reynolds number. It emerges that the universal logarithmic law is closely related to the envelope of a family of power-type curves, each corresponding to a fixed Reynolds number. A skin-friction law, corresponding to the proposed scaling law for the mean velocity distribution, is derived. In Part 2 (Barenblatt & Prostokishin 1993), both the scaling law for the velocity distribution and the corresponding friction law are compared with experimental data.

The Mathematical Model of Non-Equilibrium Effects in Water-Oil Displacement

Forced oil-water displacement and spontaneous countercurrent imbibition are crucial mechanisms of secondary oil recovery. The classical mathematical models of these phenomena are based on the fundamental assumption that in both these unsteady flows a local phase equilibrium is reached in the vicinity of every point. Thus, the water and oil flows are locally redistributed over their flow paths similarly to steady flows. This assumption allowed the investigators to further assume that the relative phase permeabilities and the capillary pressure are universal functions of the local water saturation, which can be obtained from steady-state flow experiments. The last assumption leads to a mathematical model consisting of a closed system of equations for fluid flow properties (velocity, pressure) and water saturation. This model is currently used as a basis for predictions of water-oil displacement with numerical simulations. However, at the water front in the water-oil displacement, as well as in capillary imbibition, the characteristic times of both processes are comparable with the times of redistribution of flow paths between oil and water. Therefore, the nonequilibrium effects should be taken into account. We present here a refined and extended mathematical model for the nonequilibrium two-phase (e.g., water-oil) flows. The basic problem formulation as well as the more specific equations are given, and the results of comparison with experiments are presented and discussed.

Scaling laws for fully developed turbulent shear flows. Part 2. Processing of experimental data

In Part 1 of this work (Barenblatt 1993) a non-universal scaling law (depending on the Reynolds number) for the mean velocity distribution in fully developed turbulent shear flow was proposed, together with the corresponding skin friction law. The universal logarithmic law was also discussed and it was shown that it can be understood, in fact, as an asymptotic branch of the envelope of the curves corresponding to the scaling law. Here in Part 2 the comparisons with experimental data are presented in detail. The whole set of classic Nikuradze (1932) data, concerning both velocity distribution and skin friction, was chosen for comparison. The instructive coincidence of predictions with experimental data suggests the conclusion that the influence of molecular viscosity within the main body of fully developed turbulent shear flows remains essential, even at very large Reynolds numbers. Meanwhile, some incompleteness of the experimental data presented in the work of Nikuradze (1932) is noticed, namely the lack of data in the range of parameters where the difference between scaling law and universal logarithmic law predictions should be the largest.

Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers

Processing the data from a large variety of zero-pressure-gradient boundary layer flows shows that the Reynolds-number-dependent scaling law, which the present authors obtained earlier for pipes, gives an accurate description of the velocity distribution in a self-similar intermediate region adjacent to the viscous sublayer next to the wall. The appropriate length scale that enters the definition of the boundary layer Reynolds number is found for all the flows under investigation. Another intermediate self-similar region between the free stream and the first intermediate region is found under conditions of weak free-stream turbulence. The effects of turbulence in the free stream and of wall roughness are assessed, and conclusions are drawn.

The Mathematical Model of Nonequilibrium Effects in Water-Oil Displacement

Forced oil-water displacement and spontaneous countercurrent imbibition are the crucial mechanisms of secondary oil recovery. Classical mathematical models of both these unsteady flows are based on the fundamental assumption of local phase equilibrium. Thus, the water and oil flows are assumed to be locally distributed over their flow paths similarly to steady flows. This assumption allows one to further assume that the relative phase permeabilities and the capillary pressure are universal functions of the local water saturation, which can be obtained from steady-state flow experiments. The last assumption leads to a mathematical model consisting of a closed system of equations for fluid flow properties (velocity, pressure) and water saturation. This model is currently used as a basis for numerical predictions of wateroil displacement. However, at the water front in the water-oil displacement, as well as in capillary imbibition, the characteristic times of both processes are, in general, comparable with the times of redistribution of flow paths between oil and water. Therefore, the nonequilibrium effects should be taken into account. We present here a refined and extended mathematical model for the nonequilibrium two-phase (e.g., water-oil) flows. The basic problem formulation, as well as the more specific equations, are given, and the results of comparison with an experiment are presented and discussed.