Latest revision as of 10:10, 1 July 2020

Abstract

Most of the equations used to describe the behaviour of continua are of the form:${\displaystyle u,t+\nabla x(Fa-Fv)=S(u)}$, (9.1) where ${\displaystyle u,Fa,Fv,S(u)}$ denote the vector of unknowns, advective and diffusive flux tensors, and source-terms respectively. In the case of compressible gases, we have ${\displaystyle u=(\rho ;\rho vi;\rho e),Faj=(\rho vj;\rho vivj+p\delta ij;vj(\rho e+p)),Fvj=(0;\sigma ij;vl\sigma lj+kT,j)}$. (9.2) Here ${\displaystyle \rho ,p,e,T,k,vi}$ denote the density, pressure, specific total energy, temperature, conductivity and fluid velocity in direction xi respectively. This set of equations is closed by providing an equation of state, e.g., for a polytropic gas:${\displaystyle p=(\gamma -1)\rho [e-12vjvj],T=cv[e-12vjvj]}$, (9.3 a, b) where ${\displaystyle \gamma }$, cv are the ratio of specific heats and the specific heat at constant volume respectively. Furthermore, the relationship between the stress tensor ${\displaystyle \sigma ij}$ and the deformation rate must be supplied. For water and almost all gases, Newton’s hypothesis ${\displaystyle \sigma ij=\mu (\partial kvi\partial xj+\partial vj\partial xi)+\lambda \partial vk\partial x\sigma kij}$ (9.4) complemented with Stokes’ hypothesis ${\displaystyle \lambda =-2\mu 3}$ (9.5)is an excellent approximation. The compressible Euler equations are obtained by neglecting the viscous fluxes, i.e., setting ${\displaystyle Fv=0}$. The incompressible Euler or Navier-Stokes equations are obtained by assuming that the density is constant and that pressure does not depend on temperature. The Maxwell equations of electromagnetics, the heat conduction equations of solids, and the equations describing elastic solids undergoing small deformation can readily be written in the form given by Equation (9.1).