Finite Element Methods in Incompressible, Adiabatic, and Compressible Flows by Mutsuto Kawahara

Author:Mutsuto Kawahara
Language: eng
Format: epub
Publisher: Springer Japan, Tokyo

Keywords

DeformationVelocityAccelerationConservation lawConservation of massConservation of momentumConstitutive equationConservation of energyClausius–Duhem inequalityHelmholtz free energyIdeal gasAdiabatic stateInterface condition

7.1 Introduction

Although numerous theories in continuum mechanics in fluid flows have been proposed in the past decades, only theories related to the later chapters will be presented in this chapter. First, definition of description and concepts of deformation, displacement, velocity, and acceleration will be discussed. Then, the main fundamental concepts of the continuum mechanics, i.e., the conservation laws of mass, momentum, and energy, are defined. We will provide the global and local forms of the conservation laws. The global forms are given by the integral equations, whereas the local forms are expressed in terms of the differential equations. Mutual relations among conservation laws are discussed. We then introduce stress as flux of momentum, and followed by the conservation low of momentum leads to the equilibrium equation, i.e., mass times acceleration is equal to force. As the constitutive equation, linear relation between stress and deformation rate is employed. We will witness that the conservation of energy is equivalent to the thermal transport equation. As consequences of the Clausius–Duhem inequality, restrictions on the thermal conduction and viscosity coefficient will be imposed. Introducing Helmholtz free energy, the conservation of energy can be transformed into the simplified form whose physical meaning is more clarified. The concept of ideal gas in the adiabatic state is described. There are many good books in the field of continuum mechanics. Among them, Eringen (1967), Leigh (1968), Sedov (1971), Mase and Mase (1999), Bonet and Wood (2008), etc. are helpful for the readers for further reading.

Except for the explanation of material description, the equations in this chapter are expressed using the spatial description. In this chapter, we denote as superscript dot .

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