Extended Hydrodynamics from Enskog’s Equation for a Two-Dimensional System General Formalism

Abstract

Balance equations are derived from Enskog's kinetic equation for a two-dimensional system of hard disks using Grad's moment expansion method. This set of equations constitute an extended hydrodynamics for moderately dense bi-dimensional fluids. The set of independent hydrodynamic fields in the present formulations are: density, velocity, temperature and also-following Grad's original idea-the symmetric and traceless pressure tensor p i j and the heat flux vector q k . An approximation scheme similar in spirit to one made by Grad in his original work is made. Once the hydrodynamics is derived it is used to discuss the nature of a simple one-dimensional heat conduction problem. It is shown that, not too far from equilibrium, the nonequilibrium pressure in this case only depends on the density, temperature and heat flux vector.

Entropy Principle and Galilean Relativity for Dense Gases, the General Solution without Approximations

“…In [30], Ugawa and Cordero obtained extended hydrodynamic equations derived from Enskog's equation by using Grad's moment expansion method in the bi-dimensional case; among other results, they discussed the nature of a simple one-dimensional heat conduction problem and were able to show that, not too far from equilibrium, the non-equilibrium pressure in this case depends on the density, temperature and heat flux vector.…”

On some remarkable properties of an extended thermodynamic model for dense gases and macromolecular fluids

“…Based on this last paper, Marques & Kremer (1991) obtained linearized hydrodynamic equations involving the second-order terms of the collision integral; in this way, they improved the results previously known in literature and, furthermore, they obtained linearized Burnett equations for monoatomic gases. Ugawa & Cordero (2007) obtained extended hydrodynamic equations derived from Enskog's equation by using Grad's moment expansion method in the bidimensional case; among other results, they discussed the nature of a simple one-dimensional heat conduction problem and were able to show that, not too far from equilibrium, the non-equilibrium pressure in this case depends on the density, temperature and heat flux vector.…”

Pressure and entropy of hard spheres in the weakly nonequilibrium heat-conduction steady state

“…Marques and Kremer [9] solved the Enskog kinetic equation in the higher approximations using both Grad's and Chapman-Enskog's methods. The calculated pressure tensor contains contributions from the temperature gradients of the second order, which is referred to the Burnett level [10].The Grad method was also used in solving the Enskog equation for two-dimensional hard disks [11]. The set of transport equations of the derived extended hydrodynamics was applied to two problems: 43002-1 Y.A.…”

“…The set of transport equations of the derived extended hydrodynamics was applied to two problems: 43002-1 Y.A. Humenyuk 1) the first one [12] was on the pressure difference between the equilibrium and nonequilibrium stationary heat-conduction hard-disk gases separated by a porous wall; the phenomenological conclusion on the pressure difference claimed in [13] had not been confirmed; 2) the second problem concerned the description of hard disks between two parallel walls with different temperatures [11]; for the weakly nonequilibrium case, the pressure correction was estimated to be quadratic in the heat flux.These results follow from the nonstationary Enskog equation. In application to the steady case, the pressure calculations were determined by both stationary and time-dependent parts of the nonequilibrium distribution function.…”

“…The Grad method was also used in solving the Enskog equation for two-dimensional hard disks [11]. The set of transport equations of the derived extended hydrodynamics was applied to two problems: 43002-1 Y.A.…”