C. Marín

Nonlinear Transport in a Dilute Binary Mixture of Mechanically Different Particles

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.

Kinetic models for diffusion generated by an external force

Two kinetic models are used to study the homogeneous color diffusion problem in a dilute binary mixture. Both kinetic models incorporate a temperature dependence in the collision frequencies, which allows for the consideration of a general repulsive molecular interaction. The main transport properties as well as the velocity distribution functions are explicitly obtaines in terms of the field strength and the parameters characterizing the mixture. The results are illustrated for the two extreme cases of Maxwell molecules and hard spheres. A comparison between both models and with previous results derived from the Boltzmann equation for Maxwell molecules is carried out.

UNIFORM SHEAR FLOW IN A BINARY MIXTURE WITH GENERAL REPULSIVE INTERACTIONS

A kinetic model for a binary mixture under uniform shear flow is exactly solved. The model incorporates a temperature dependence of the collision frequencies that allows the consideration of general repulsive interactions. The rheological properties of the mixture are obtained as functions of the shear rate, the parameters of the mixture (particle masses, concentrations, and force constants), and a parameter characterizing the interaction considered. In addition, the velocity distribution functions are explicitly obtained. While the transport coefficients are hardly sensitive to the interaction potential, the distribution functions are clearly influenced by the interaction parameter. In the tracer limit, a transition to an alternative state recently found in the context of Boltzmann equation is exactly identified in the case of Maxwell molecules. For non‐Maxwell molecules, preliminary results suggest that this transition is also present although the phenomenon is less significant. A comparison with previo...

Exact solution of the Gross–Krook kinetic model for a multicomponent gas in steady Couette flow

The Gross–Krook (GK) kinetic model of the Boltzmann equation for a multicomponent mixture is exactly solved in a steady state with velocity and temperature gradients (Couette flow). The hydrodynamic fields, heat and momentum fluxes, and the velocity distribution functions are determined explicitly in terms of the shear rate and the thermal gradient. The description applies for conditions arbitrarily far from equilibrium and is not restricted to specific values of the mechanical parameters of the mixture. This work completes a previous study (Physica A 289 (2001) 37) based on a formal series representation of the velocity distribution function.

Shear-rate dependent transport coefficients in a binary mixture of Maxwell molecules

Mass and heat transport in a dilute binary mixture of Maxwell molecules under steady shear flow are studied in the limit of small concentration gradients. The analysis is made from the Gross–Krook kinetic model of the Boltzmann equation. This model is solved by means of a perturbation solution around the steady shear flow solution [Phys. Fluids 8, 2756 (1996)], which applies for arbitrary values of the shear rate. In the first order of the expansion the results show that the mass and heat fluxes are proportional to the concentration gradient but, due to the anisotropy of the problem, mutual diffusion and Dufour tensors can be identified, respectively. Both tensors are explicitly determined in terms of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). A comparison with the results derived from the exact Boltzmann equation at the level of the diffusion tensor shows a good agreement for a wide range of values of the shear rate.

Electrical current density in a sheared dilute gas

Electrical current density of charged particles across a rarefied gas of neutral particles under shear flow is analyzed in the limit of small electric fields. The concentration of the charged species is assumed to be much smaller than that of the neutral species so that the interactions of charged–neutral and neutral–neutral type are the dominant ones. The study is made from the exact Boltzmann equation for Maxwell molecules as well as from a kinetic model for general repulsive interactions. By performing a perturbation expansion around a nonequilibrium state, the current density is explicitly evaluated in the first order of the external field. We get a generalized Ohms law, where an electrical conductivity tensor can be identified. The nonzero elements of this tensor are nonlinear functions of the shear rate, the mass ratio, the force constant ratio, and the interaction parameter.

On the validity of a variational principle for multicomponent systems

The validity of a variational principle for nonequilibrium steady states proposed by Evans and Baranyai [Phys. Rev. Lett. 67, 2597 (1991)] is investigated in the case of a dilute binary mixture described by the well-known Groos–Krook kinetic model. We construct a perturbation solution around the unconstrained shear flow state and evaluate the phase-space compression factor, the temperature ratios, and the nonlinear shear viscosity up to the first-order approximation. All these quantities are nonlinear functions of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). It is shown that this principle does not hold exactly, although deviations from it are small in some situations for not very large shear rates. The calculations presented here extend previous results derived for a single dilute gas.