Elements of Continuum Mechanics and Conservation Laws

Abstract

This English edition draws on the original work of S. Godunov [5]. This book was published in Russia in 1978 and received the M. A. Lavrent'ev prize of the Russian Academy of Science in 1993. Later, the material of [5] was included in the book by both authors [7, Chapters I-IV] published in 1998. This edition also presents a new concept developed by S. Godunov on the basis of examples due to E. Romenskii. This concept was essentially developed while preparing the English translation of [7] and was intensively discussed with specialists in France, Russia, and USA last year. Four years ago we could only conjecture as to the existence of relationships between the structure of thermodynamical conservation laws and representations of the rotation group. Now, we can say more owing to the very recent results of S. Godunov and V. Gordienko (cf. Appendix and references there). Chapters I-IV were based on lectures given by S. Godunovat the Physical Department of the Novosibirsk State University. This course was based on the original results of both authors and concerned formulations of differential equations describing high-rate deformations of metals. To study such processes, equations of elasticity should be stated in one zone, whereas equations of hydrodynamics should be used in other zones where, for example, cumulative jets appear. Plastic deformations can occur in transition zones, which causes the residual stresses. The suggested model differs from the equations of elasticity by relaxation terms simulating plastic deformations and transistion of an v

Mathematical and numerical model for nonlinear viscoplasticity

“…This hypothesis was proposed by Godunov [3], Godunov & Romenskii [9] and Godunov & Peshkov [12]. It is related to the fact that the description of the Maxwell-like models in terms of deformation is not direct.…”

Numerical simulation of the free convection in a viscous compressible fluid

“…The bulk viscosity effects are not considered. The law of conservation of energy follows from equations (4), (6)- (8).…”

“…Equations of state of a viscous compressible fluid, closing the dynamic equations (6)-(8) are adopted in the linear approximation [8] The density of the medium and the entropy density are determined by the temperature and pressure:…”

Modeling the multiphase flows in deformable porous media

“…The description of such hydrodynamic systems requires a consistent model of the flow of compressible media and deformation of the porous medium, and reliability of results of mathematical and numerical modeling should be provided both by the methods of deriving the coordinated equations of the model and the methods for their solution. In this study, the model of the flow of a liquid mixture of compressible phases through a porous deformable medium is constructed in the framework of the method of thermodynamically consistent system of conservation laws [1]. The equations of motion satisfy the fundamental laws: the laws of conservation of mass, momentum, and energy, invariance relative to Galileo transformations, and fundamentals of thermodynamics.…”