%'e construct a consistent solution of the Bhatnagar-Gross-Krook (BGK) model kinetic equation des... more %'e construct a consistent solution of the Bhatnagar-Gross-Krook (BGK) model kinetic equation describing a system in a steady state with constant pressure and nonuniform temperature. The thermal profile is not linear and depends on the interaction potential. All the moments of the distribution function are given as polynomials in the local thermal gradient. In particular, the heat flux always obeys the (linear} Fourier law.
A dilute binary mixture under uniform shear flow is considered in the tracer limit. The analysis ... more A dilute binary mixture under uniform shear flow is considered in the tracer limit. The analysis is made from an exact solution of a generalized Gross-Krook model for r Ϫ␥ repulsive forces. The results show that the partial contribution of the tracer species to the total properties of the mixture becomes finite if the shear rate is larger than a certain critical value a c , which is a function of the mass ratio, force constant ratios, and the interaction potential considered. This phenomenon can be interpreted as a nonequilibrium phase transition in velocity space. For non-Maxwell molecules (␥ 5), the corresponding order parameter is discontinuous at the critical point ͑first order transition͒; on the other hand, the transition becomes continuous at a c in the special case of Maxwell molecules (␥ϭ5). ͓S1063-651X͑97͒16008-1͔
... ` Departement de Physique, Universite Moulay Ismail, Meknes, Morocco ? ? Vicente Garzo and An... more ... ` Departement de Physique, Universite Moulay Ismail, Meknes, Morocco ? ? Vicente Garzo and Andres Santos ? ... The analysis is performed from an exact perturbation solution of the Boltzmann equation for Maxwell molecules through second order in the field. ...
Nonlinear transport properties of a d-dimensional dilute gas subjected to a planar Couette flow a... more Nonlinear transport properties of a d-dimensional dilute gas subjected to a planar Couette flow are determined. The results are obtained from a kinetic model that accounts for the correct value of the Prandtl number. The solution is characterized by constant pressure and linear velocity and parabolic temperature profiles with respect to a scaled variable. The main transport coefficients are explicitly obtained as nonlinear functions of the reduced shear rate. A comparison with recent molecular-dynamics simulations of a bidimensional gas of hard disks ͓D. Risso and P. Cordero, Phys. Rev. E 56, 489 ͑1997͔͒ is carried out. Such a comparison shows that our results are in better agreement with the computer simulations than those previously derived from other approximations, especially in the case of the thermal conductivity tensor. ͓S1063-651X͑98͒03308-X͔
The e1ect of gravity on the tracer particles immersed in a dilute gas of mechanically di1erent pa... more The e1ect of gravity on the tracer particles immersed in a dilute gas of mechanically di1erent particles and subjected to the steady planar Couette 3ow is analyzed. The results are obtained from the Gross-Krook (GK) kinetic model of a binary mixture and the description applies for arbitrary values of both velocity and temperature gradients. The GK equation is solved by means of a perturbation method in powers of the 7eld around a nonequilibrium state which retains all the hydrodynamic orders in the shear rate a and the thermal gradient � . To 7rst order in the gravity 7eld, we explicitly determine the hydrodynamic pro7les and the partial contributions to the momentum and heat 3uxes associated with the tracer species. All these quantities are given in terms of a; � , and the mass and size ratios. The shear-rate dependence of some of these quantities is illustrated for several values of the mass ratio showing that in general, the e1ect of gravity is more signi7cant when the particles of the gas are lighter than the tracer particles. c 2001 Elsevier Science B.V. All rights reserved.
The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dyn... more The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dynamics of an impurity (or intruder) immersed in a granular gas driven by a uniform shear flow. The analysis is based on an exact solution of the Boltzmann equation for a granular binary mixture. It applies for conditions arbitrarily far from equilibrium (arbitrary values of the shear rate a) and for arbitrary values of the parameters of the mixture (particle masses mi, mole fractions xi, and coefficients of restitution αij). In the tracer limit where the mole fraction of the intruder species vanishes, a non equilibrium phase transition takes place. We thereby identity ordered phases where the intruder bears a finite contribution to the properties of the mixture, in a region of parameter space that is worked out in detail. These findings extend previous results obtained for ordinary Maxwell gases, and further show that dissipation leads to new ordered phases.
Physica A: Statistical Mechanics and its Applications, 1990
The influence of a nonconservative force proportional to the velocity acting on a system describe... more The influence of a nonconservative force proportional to the velocity acting on a system described by the Boltzmann equation is analyzed. When this force is the only external action on the system, an H-theorem is proved. showing that the distrihution function tends towards a Maxwellian with a time-dependent temperature. Self-diffusion in such a state is analyzed in the case of Maxwell molecules. It is shown that the external force can even prevent the system to reach a hydrodynamic stage. Next. self-diffusion in a system under uniform shear How is considered. For Maxwell molecules. the conditions under which a hydrodynamic regime is reached arc discussed. In the hydrodynamic regime, a self-diffusion tensor is obtained to first order in the concentration gradient. This tensor is a highly nonlinear function of both the shear rate and the strength of the external force. Comparison with previous work is carried out.
ABSTRACT Expressions for the heat had momentum transport around the state of uniform shear flow a... more ABSTRACT Expressions for the heat had momentum transport around the state of uniform shear flow are given for a dilute gas. The results are obtained from a kinetic model recently proposed: the Liu model. This model improves some insufficiencies of the well-known Bhatnagar, Gross and Krook (BGK) equation. It is shown that the coexistence between both velocity and temperature gradients is only possible for interaction models with uniform collision frequency. In this case, the heat flux verifies a generalized Fourier's law where the thermal conductivity tensor depends on the shear rate. When considering systems where the collision frequency is not uniform, a perturbation method around the shear flow state is proposed. The irreversible fluxes are evaluated explicitly up to first order in the expansion for Maxwell molecules. The transport coefficients obtained are highly nonlinear functions of the shear rate. The results are compared with those previously obtained from the BGK and Boltzmann equations.
The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform... more The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degree k >/4 diverge for shear rates larger than a critical value a(~}), which behaves for large k as a~)~k-I. This divergence is consistent with an algebraic tail of the form f(V) ~ V 4 o~a), where a is a decreaging function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.
The hierarchy of moments of the Boltzmann equation for a binary mixture of mechanically different... more The hierarchy of moments of the Boltzmann equation for a binary mixture of mechanically different Maxwell molecules is exactly solved. The solution corresponds to a nonequilibrium homogeneous steady state generated by an external force that accelerates particles of each species (or "color") along opposite directions. As a consequence, macroscopic fluxes are induced in spite of the absence of concentration gradients. Explicit expressions for the fluxes of mass and momentum as functions of the field strength, the mass ratio, the molar fractions, and the interaction constant ratio are obtained. In particular, the color conductivity coefficient reduces to the mutual diffusion coefficient in the zero-field limit. Some physically interesting limiting cases are discussed. The maximum-entropy method is used to construct an approximate velocity distribution function from the exact knowledge of the mass and momentum fluxes. This distribution is exact up to second order in the color field and also in the limit of large color field.
A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat max... more A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat maximizes the average phase-space compression factor A in nonequilibrium steady states is analyzed for a dilute gas under uniform shear flow. Three routes have been followed: (i) an exact solution of the Bhatnagar-Gross-Krook kinetic equation for arbitrary shear rate, (ii) an exact solution of the Boltzmann equation through super-Burnett order, and (iii) a numerical solution of the Boltzmann equation for finite shear rates. The results show that the above hypothesis does not exactly hold for arbitrary shear rates, although the thermostat that maximizes A is close to the Gaussian one. In addition, the influence of the thermostat considered on the nonlinear shear viscosity is also analyzed.
It is shown that an exact solution of the non-linear Boltzmann equation does not verify the varia... more It is shown that an exact solution of the non-linear Boltzmann equation does not verify the variational principle for non-equilibrium steady states proposed by Evans and Baranyai (Phys. Rev. Lett., 67 (1991) 2597). Therefore, the principle does not provide a basis for the understanding of far-from-equilibrium steady states. A few years ago, Evans and Baranyai (EB)[l] proposed a variational principle to characterize non-equilibrium steady states. The EB principle is formulated for systems arbitrarily far from equilibrium and reduces to the principle of minimum-entropy production [2] for near-equilibrium situations. To support their hypothesis, Evans and Baranyai provided simulation data for a thermostatted dense fluid under uniform shear flow. Within statistical uncertainties, the data agreed with the predictions of the principle. Given the essential role that a variational principle would play in the development of a general theory for far-from-equilibrium systems, it is very important to investigate the validity of the EB principle. In a previous paper[3], we used the BGK model kinetic equation [41 to analyse it, and found significant, although small, deviations. Nevertheless, since the BGK equation is a simplified model of the Boltzmann equation, no definitive conclusion about the validity of the principle was obtained. The aim of this letter is to carry out a similar analysis, but now using the non-linear Boltzmann equation, and without introducing any kind of approximation. In this way, the calculations we present here can be considered as exact in the context of kinetic theory. As in ref. [l] and[3], we consider a fluid under uniform shear flow. This state is
The dynamics of an impurity (or tracer particle) immersed in a dilute granular gas under uniform ... more The dynamics of an impurity (or tracer particle) immersed in a dilute granular gas under uniform shear flow is investigated. A non-equilibrium phase transition is identified from an exact solution of the inelastic Boltzmann equation for a granular binary mixture in the tracer limit, where the impurity carries either a vanishing (disordered phase) or a finite (ordered phase) fraction of the total kinetic energy of the system. In the disordered phase, the granular temperature ratio (impurity "temperature" over that of the host fluid) is finite, while it diverges in the ordered phase. To correctly capture this extreme violation of energy equipartition, we show that the picture of an impurity enslaved to the host fluid is insufficient.
A hydrodynamic description for an s-component mixture of inelastic, smooth hard disks (two dimens... more A hydrodynamic description for an s-component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first portion of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth-and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper. I.
%'e construct a consistent solution of the Bhatnagar-Gross-Krook (BGK) model kinetic equation des... more %'e construct a consistent solution of the Bhatnagar-Gross-Krook (BGK) model kinetic equation describing a system in a steady state with constant pressure and nonuniform temperature. The thermal profile is not linear and depends on the interaction potential. All the moments of the distribution function are given as polynomials in the local thermal gradient. In particular, the heat flux always obeys the (linear} Fourier law.
A dilute binary mixture under uniform shear flow is considered in the tracer limit. The analysis ... more A dilute binary mixture under uniform shear flow is considered in the tracer limit. The analysis is made from an exact solution of a generalized Gross-Krook model for r Ϫ␥ repulsive forces. The results show that the partial contribution of the tracer species to the total properties of the mixture becomes finite if the shear rate is larger than a certain critical value a c , which is a function of the mass ratio, force constant ratios, and the interaction potential considered. This phenomenon can be interpreted as a nonequilibrium phase transition in velocity space. For non-Maxwell molecules (␥ 5), the corresponding order parameter is discontinuous at the critical point ͑first order transition͒; on the other hand, the transition becomes continuous at a c in the special case of Maxwell molecules (␥ϭ5). ͓S1063-651X͑97͒16008-1͔
... ` Departement de Physique, Universite Moulay Ismail, Meknes, Morocco ? ? Vicente Garzo and An... more ... ` Departement de Physique, Universite Moulay Ismail, Meknes, Morocco ? ? Vicente Garzo and Andres Santos ? ... The analysis is performed from an exact perturbation solution of the Boltzmann equation for Maxwell molecules through second order in the field. ...
Nonlinear transport properties of a d-dimensional dilute gas subjected to a planar Couette flow a... more Nonlinear transport properties of a d-dimensional dilute gas subjected to a planar Couette flow are determined. The results are obtained from a kinetic model that accounts for the correct value of the Prandtl number. The solution is characterized by constant pressure and linear velocity and parabolic temperature profiles with respect to a scaled variable. The main transport coefficients are explicitly obtained as nonlinear functions of the reduced shear rate. A comparison with recent molecular-dynamics simulations of a bidimensional gas of hard disks ͓D. Risso and P. Cordero, Phys. Rev. E 56, 489 ͑1997͔͒ is carried out. Such a comparison shows that our results are in better agreement with the computer simulations than those previously derived from other approximations, especially in the case of the thermal conductivity tensor. ͓S1063-651X͑98͒03308-X͔
The e1ect of gravity on the tracer particles immersed in a dilute gas of mechanically di1erent pa... more The e1ect of gravity on the tracer particles immersed in a dilute gas of mechanically di1erent particles and subjected to the steady planar Couette 3ow is analyzed. The results are obtained from the Gross-Krook (GK) kinetic model of a binary mixture and the description applies for arbitrary values of both velocity and temperature gradients. The GK equation is solved by means of a perturbation method in powers of the 7eld around a nonequilibrium state which retains all the hydrodynamic orders in the shear rate a and the thermal gradient � . To 7rst order in the gravity 7eld, we explicitly determine the hydrodynamic pro7les and the partial contributions to the momentum and heat 3uxes associated with the tracer species. All these quantities are given in terms of a; � , and the mass and size ratios. The shear-rate dependence of some of these quantities is illustrated for several values of the mass ratio showing that in general, the e1ect of gravity is more signi7cant when the particles of the gas are lighter than the tracer particles. c 2001 Elsevier Science B.V. All rights reserved.
The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dyn... more The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dynamics of an impurity (or intruder) immersed in a granular gas driven by a uniform shear flow. The analysis is based on an exact solution of the Boltzmann equation for a granular binary mixture. It applies for conditions arbitrarily far from equilibrium (arbitrary values of the shear rate a) and for arbitrary values of the parameters of the mixture (particle masses mi, mole fractions xi, and coefficients of restitution αij). In the tracer limit where the mole fraction of the intruder species vanishes, a non equilibrium phase transition takes place. We thereby identity ordered phases where the intruder bears a finite contribution to the properties of the mixture, in a region of parameter space that is worked out in detail. These findings extend previous results obtained for ordinary Maxwell gases, and further show that dissipation leads to new ordered phases.
Physica A: Statistical Mechanics and its Applications, 1990
The influence of a nonconservative force proportional to the velocity acting on a system describe... more The influence of a nonconservative force proportional to the velocity acting on a system described by the Boltzmann equation is analyzed. When this force is the only external action on the system, an H-theorem is proved. showing that the distrihution function tends towards a Maxwellian with a time-dependent temperature. Self-diffusion in such a state is analyzed in the case of Maxwell molecules. It is shown that the external force can even prevent the system to reach a hydrodynamic stage. Next. self-diffusion in a system under uniform shear How is considered. For Maxwell molecules. the conditions under which a hydrodynamic regime is reached arc discussed. In the hydrodynamic regime, a self-diffusion tensor is obtained to first order in the concentration gradient. This tensor is a highly nonlinear function of both the shear rate and the strength of the external force. Comparison with previous work is carried out.
ABSTRACT Expressions for the heat had momentum transport around the state of uniform shear flow a... more ABSTRACT Expressions for the heat had momentum transport around the state of uniform shear flow are given for a dilute gas. The results are obtained from a kinetic model recently proposed: the Liu model. This model improves some insufficiencies of the well-known Bhatnagar, Gross and Krook (BGK) equation. It is shown that the coexistence between both velocity and temperature gradients is only possible for interaction models with uniform collision frequency. In this case, the heat flux verifies a generalized Fourier's law where the thermal conductivity tensor depends on the shear rate. When considering systems where the collision frequency is not uniform, a perturbation method around the shear flow state is proposed. The irreversible fluxes are evaluated explicitly up to first order in the expansion for Maxwell molecules. The transport coefficients obtained are highly nonlinear functions of the shear rate. The results are compared with those previously obtained from the BGK and Boltzmann equations.
The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform... more The high-velocity distribution of a two-dimensional dilute gas of Maxwell molecules under uniform shear flow is studied. First we analyze the shear-rate dependence of the eigenvalues governing the time evolution of the velocity moments derived from the Boltzmann equation. As in the three-dimensional case discussed by us previously, all the moments of degree k >/4 diverge for shear rates larger than a critical value a(~}), which behaves for large k as a~)~k-I. This divergence is consistent with an algebraic tail of the form f(V) ~ V 4 o~a), where a is a decreaging function of the shear rate. This expectation is confirmed by a Monte Carlo simulation of the Boltzmann equation far from equilibrium.
The hierarchy of moments of the Boltzmann equation for a binary mixture of mechanically different... more The hierarchy of moments of the Boltzmann equation for a binary mixture of mechanically different Maxwell molecules is exactly solved. The solution corresponds to a nonequilibrium homogeneous steady state generated by an external force that accelerates particles of each species (or "color") along opposite directions. As a consequence, macroscopic fluxes are induced in spite of the absence of concentration gradients. Explicit expressions for the fluxes of mass and momentum as functions of the field strength, the mass ratio, the molar fractions, and the interaction constant ratio are obtained. In particular, the color conductivity coefficient reduces to the mutual diffusion coefficient in the zero-field limit. Some physically interesting limiting cases are discussed. The maximum-entropy method is used to construct an approximate velocity distribution function from the exact knowledge of the mass and momentum fluxes. This distribution is exact up to second order in the color field and also in the limit of large color field.
A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat max... more A recent hypothesis of D. J. Evans and A. Baranyai according to which the Gaussian thermostat maximizes the average phase-space compression factor A in nonequilibrium steady states is analyzed for a dilute gas under uniform shear flow. Three routes have been followed: (i) an exact solution of the Bhatnagar-Gross-Krook kinetic equation for arbitrary shear rate, (ii) an exact solution of the Boltzmann equation through super-Burnett order, and (iii) a numerical solution of the Boltzmann equation for finite shear rates. The results show that the above hypothesis does not exactly hold for arbitrary shear rates, although the thermostat that maximizes A is close to the Gaussian one. In addition, the influence of the thermostat considered on the nonlinear shear viscosity is also analyzed.
It is shown that an exact solution of the non-linear Boltzmann equation does not verify the varia... more It is shown that an exact solution of the non-linear Boltzmann equation does not verify the variational principle for non-equilibrium steady states proposed by Evans and Baranyai (Phys. Rev. Lett., 67 (1991) 2597). Therefore, the principle does not provide a basis for the understanding of far-from-equilibrium steady states. A few years ago, Evans and Baranyai (EB)[l] proposed a variational principle to characterize non-equilibrium steady states. The EB principle is formulated for systems arbitrarily far from equilibrium and reduces to the principle of minimum-entropy production [2] for near-equilibrium situations. To support their hypothesis, Evans and Baranyai provided simulation data for a thermostatted dense fluid under uniform shear flow. Within statistical uncertainties, the data agreed with the predictions of the principle. Given the essential role that a variational principle would play in the development of a general theory for far-from-equilibrium systems, it is very important to investigate the validity of the EB principle. In a previous paper[3], we used the BGK model kinetic equation [41 to analyse it, and found significant, although small, deviations. Nevertheless, since the BGK equation is a simplified model of the Boltzmann equation, no definitive conclusion about the validity of the principle was obtained. The aim of this letter is to carry out a similar analysis, but now using the non-linear Boltzmann equation, and without introducing any kind of approximation. In this way, the calculations we present here can be considered as exact in the context of kinetic theory. As in ref. [l] and[3], we consider a fluid under uniform shear flow. This state is
The dynamics of an impurity (or tracer particle) immersed in a dilute granular gas under uniform ... more The dynamics of an impurity (or tracer particle) immersed in a dilute granular gas under uniform shear flow is investigated. A non-equilibrium phase transition is identified from an exact solution of the inelastic Boltzmann equation for a granular binary mixture in the tracer limit, where the impurity carries either a vanishing (disordered phase) or a finite (ordered phase) fraction of the total kinetic energy of the system. In the disordered phase, the granular temperature ratio (impurity "temperature" over that of the host fluid) is finite, while it diverges in the ordered phase. To correctly capture this extreme violation of energy equipartition, we show that the picture of an impurity enslaved to the host fluid is insufficient.
A hydrodynamic description for an s-component mixture of inelastic, smooth hard disks (two dimens... more A hydrodynamic description for an s-component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first portion of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth-and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper. I.
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Papers by V. Garzó