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Register a new userOn a non approximated approach to Extended Thermodynamics for dense gases and macromolecular fluids
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Recently the 14 moments model of Extended Thermodynamics for dense gases and macromolecular fluids has been considered and an exact solution, of the restrictions imposed by the entropy principle and that of Galilean relativity, has been obtained through a non relativistic limit. Here we prove uniqueness of the above solution and exploit other pertinent conditions such us the convexity of the function $h$ related to the entropy density, the problem of subsystems and the fact that the flux in the conservation law of mass must be the moment of order 1 in the conservation law of momentum. Also the solution of this last condition is here obtained without using expansions around equilibrium. The results present interesting aspects which were not suspected when only approximated solutions of this problem were known.
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The 14 moments model for dense gases, introduced in the last years by Ruggeri, Sugiyama and collaborators, is here considered. They have found the closure of the balance equations up to second order with respect to equilibrium; subsequently, Carrisi has found the closure up to whatever order with respect to equilibrium, but for a more constrained system where more symmetry conditions are imposed. Here the closure is obtained up to whatever order and without imposing the supplementary conditions. It comes out that the first non symmetric parts appear only at third order with respect to equilibrium, even if Ruggeri and Sugiyama found a non symmetric part proportional to an arbitrary constant also at first order with respect to equilibrium. Consequently, this constant must be zero, as Ruggeri, Sugiyama assumed in the applications and on an intuitive ground.
Extended Thermodynamics is the natural framework in which to study the physics of fluids, because it leads to symmetric hyperbolic systems of field laws, thus assuming important properties such as finite propagation speeds of shock waves and well posedness of the Cauchy problem. The closure of the system of balance equations is obtained by imposing the entropy principle and that of galilean relativity. If we take the components of the mean field as independent variables, these two principles are equivalent to some conditions on the entropy density and its flux. The method until now used to exploit these conditions, with the macroscopic approach, has not been used up to whatever order with respect to thermodynamical equilibrium. This is because it leads to several difficulties in calculations. Now these can be overcome by using a new method proposed recently by Pennisi and Ruggeri. Here we apply it to the 14 moments model. We will also show that the 13 moments case can be obtained from the present one by using the method of subsystems.
We investigate the properties of two- and three-dimensional non-commutative fermion gases with fixed total z-component of angular momentum, J_z, and at high density for the simplest form of non-commutativity involving constant spatial commutators. Analytic expressions for the entropy and pressure are found. The entropy exhibits non-extensive behaviour while the pressure reveals the presence of incompressibility in two, but not in three dimensions. Remarkably, for two-dimensional systems close to the incompressible density, the entropy is proportional to the square root of the system size, i.e., for such systems the number of microscopic degrees of freedom is determined by the circumference, rather than the area (size) of the system. The absence of incompressibility in three dimensions, and subsequently also the absence of a scaling law for the entropy analogous to the one found in two dimensions, is attributed to the form of the non-commutativity used here, the breaking of the rotational symmetry it implies and the subsequent constraint on J_z, rather than the angular momentum J. Restoring the rotational symmetry while constraining the total angular momentum J seems to be crucial for incompressibility in three dimensions. We briefly discuss ways in which this may be done and point out possible obstacles.
In this paper, we present a Lagrangian formalism for nonequilibrium thermodynamics. This formalism is an extension of the Hamilton principle in classical mechanics that allows the inclusion of irreversible phenomena in both discrete and continuum systems (i.e., systems with finite and infinite degrees of freedom). The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of entropy production associated to all the irreversible processes involved. Hence from a mathematical point of view, our variational formalism may be regarded as a generalization of the Lagrange-dAlembert principle used in nonholonomic mechanics. In order to formulate the nonholonomic constraint, we associate to each irreversible process a variable called the thermodynamic displacement. This allows the definition of a corresponding variational constraint. Our theory is illustrated with various examples of discrete systems such as mechanical systems with friction, matter transfer, electric circuits, chemical reactions, and diffusion across membranes. For the continuum case, the variational formalism is naturally extended to the setting of infinite dimensional nonholonomic Lagrangian systems and is expressed in material representation, while its spatial version is obtained via a nonholonomic Lagrangian reduction by symmetry. In the continuum case, our theory is systematically illustrated by the example of a multicomponent viscous heat conducting fluid with chemical reactions and mass transfer.
We utilize group-theoretical methods to develop a matrix representation of differential operators that act on tensors of any rank. In particular, we concentrate on the matrix formulation of the curl operator. A self-adjoint matrix of the curl operator is constructed and its action is extended to a complex plane. This scheme allows us to obtain properties, similar to those of the traditional curl operator.
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