An Exact Solution for the Macroscopic Approach to Extended Thermodynamics of Dense Gases With Many Moments

Abstract

Extended Thermodynamics of Dense Gases with an arbitrary but fixed number of moments has been recently studied in literature; the arbitrariness of the number of moments is linked to a number N and the resulting model is called an (N ) − M odel. As usual in Extended Thermodynamics, in the field equations some unknown functions appear; restriction on their generalities are obtained by imposing the entropy principle, the Galilean relativity principle and some symmetry conditions. The solution of these conditions is called the "closure problem" and it has not been written explicitly because an hard notation is necessary, but it has been shown how the theory is selfgenerating in the sense that, if we know the closure of the (N ) − M odel, than we will be able to find that of the (N + 1) − M odel. Instead of this, we find here an exact solution which holds for every number N .

Extended Thermodynamics of dense gases with many moments – The macroscopic approach

“…It is not the general solution; a more general solution has been found in [24]. But this too is not the most general solution of our conditions; its determination can be found by applying the iterative procedure of the present paper starting from the known closure of the macroscopic model with N ¼1.…”

“…(14) and (15) leads to an extreme use of indexes which cause loss of elegance to the treatment. A possibility to avoid this problem can be to employ the method used in [23] which belongs to the period when only one block of the equations was used. In other word, the treatment is simplified by using a 4-dimensional notation.…”

A Numberable Set of Exact Solutions for the Macroscopic Approach to Extended Thermodynamics of Polyatomic Gases with Many Moments

“…The equivalence of (4) 5 to the Galilean relativity principle is proved in literature such as [13,20,21] and on Representation Theorems such as that of [22]. In (5), (6), and (7) we report an already known particular solution (see [23]) for the conditions (4). But it is not the general solution; in fact, in the present article we will exhibit a significative set of other solutions depending on a numberable family of arbitrary single variable functions ( ) and we report it in the next section, (8) and (9).…”

The General Exact Solution for the Many Moments Macroscopic Approach to Extended Thermodynamics of Polyatomic Gases

“…In fact, we can take for h ′ the sum of the present expression (4), of the expression (4) of [2] and of (5) in [4]. Similarly, we can take for h ′k the sum of the present expression (5), of the expression (5) of [2] and of (6) in [4]. The result is also a solution of the conditions (3) and, in the particular case N = 1, it is equal to (4) and (5) of [2].…”

“…Similarly, we can take for h ′k the sum of the present expression (5), of the expression (5) of [2] and of (6) in [4]. The result is also a solution of the conditions (3) and, in the particular case N = 1, it is equal to (4) and (5) of [2]. There is no possible further additional term for h ′ and h ′k , as outlined in the conclusions of [1].…”