Extended thermodynamics of charged gases with many moments: An alternative closure

Abstract

In 2011, a macroscopic extended model with many moments for the description of relativistic gases under the influence of an external electromagnetic field has been proposed. An exact closure of this model, up to whatever order with respect to thermodynamical equilibrium, has been found. Such closure is given in terms of the entropy density at equilibrium and the expression for the higher order terms involves a family of arbitrary constants. It allows to know the value of the constitutive functions up to every desired degree of approximation, but paying the cost of a large amount of arbitrary terms. Moreover, if one wants to use the closure for applications it is necessary to cut it at a chosen order with respect to thermodynamical equilibrium. In this paper, we propose a different method to find the closure that, starting from a fixed order with respect to thermodynamical equilibrium, allows to obtain univocally all the lower order terms in terms of two arbitrary functions depending on a scalar variable and of one polynomial which degree depends on the order of approximation.

An 18 Moments Model for Dense Gases: Entropy and Galilean Relativity Principles without Expansions

“…However, for the Galilean relativity principle, they must be coincident for whatever value of v This constraint can be written explicitly more easily if we take into account that µ r C X C B (− v τ ) = µ a B , which can be written explicitly by use of (17) and reads: Consequently, the derivatives of (48,49) with respect to v i τ become (13,14), where we have omitted index a denoting variables in the absolute reference frame, because they remain unchanged if we change v i τ with −v i τ , that is if we exchange the absolute and the relative reference frames. It is not necessary to impose the derivatives of (46,47) with respect to v i τ , because they are consequences of (13,14) and (9,10). Consequently, the Galilean relativity principle amounts simply to the two equations, (13,14).…”

“…Some of the original papers on this subject are [1,2] (see, also, the book [3] for a complete description), while more recent papers are [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], and the theory has the advantage of furnishing hyperbolic field equations, with finite speeds of propagation of shock waves and very interesting analytical properties.…”

Extended Thermodynamics for Dense Gases up to Whatever Order and with Only Some Symmetries

“…In fact, by substituting Equation (10) , thanks to Equation (11). By substituting Equation (10) into (8) 2 , we obtain:…”

“…Let us prove that H = H 1 , with H 1 given by Equation (10) and ψ n constrained by Equation (11), is a particular solution of Equations (8) and (9).…”

“…Following Ruggeri's criticism [2], a new version was proposed by Liu and Müller [3] and, subsequently, for the relativistic case, by Liu, Müller and Ruggeri [4]; more recent papers in this framework are [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”