Revision as of 10:07, 1 July 2020

Abstract

Most of the equations used to describe the behaviour of continua are of the form:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u,t +\nabla x (Fa - Fv) = S(u) , (9.1) where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \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:Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p = (\gamma - 1)\rho [e- 1 2 vjvj ] , T = cv[e - 12 vjvj ]

, (9.3 a, b) where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma , cv are the ratio of specific heats and the specific heat at constant volume respectively. Furthermore, the relationship between the stress tensor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma ij

and the deformation rate must be supplied. For water and almost all gases, Newton’s hypothesis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma ij = \mu( \partial kvi \partial xj+ \partial vj \partial xi) + \lambda \partial vk \partial x\sigma kij
(9.4) complemented with Stokes’ hypothesis Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lambda = −2\mu 3
(9.5)is an excellent approximation. The compressible Euler equations are obtained by neglecting the viscous fluxes, i.e., setting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 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).