Tracer limit in a gas mixture under shear flow with repulsive interactions (original) (raw)

Tracer diffusion under shear flow for general repulsive interactions

Physics of Fluids, 1995

Tracer diffusion in a steady shear flow state is analyzed. A kinetic model incorporating a temperature dependence in the collision frequencies is used. This allows for the consideration of a general repulsive intermolecular interaction. A perturbative scheme is applied to get the shear rate dependence of the tracer diffusion tensor in terms of the mass ratio, the force constants ratio, and a parameter characterizing the interaction potential considered. In addition, the heat tlux arising from the concentration gradient of the tracer species is also evaluated. The results are illustrated for the two extreme cases of Maxwell molecules and hard spheres. 0 1995 American Institute of Physics.

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

Physics of Fluids, 2000

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 ͑1996͔͒, 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 constants͒. 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.

Mutual diffusion in a binary mixture under shear flow

Physical Review E, 1998

Mass transport in a dilute binary mixture of Maxwell molecules under steady shear flow is studied. The analysis is made from an exact perturbation solution of the Boltzmann equation through first order in the concentration gradient. The reference state ͑zeroth order approximation͒ corresponds to the exact recent solution ͓Phys. Rev. E 52, 3812 ͑1995͔͒ of the Boltzmann equation in the uniform shear flow, which holds for arbitrary values of the shear rate. The results show that the mass flux obeys a generalized Fick's law where, due to the anisotropy of the problem, a mutual diffusion tensor is defined. This tensor is a highly nonlinear function of the shear rate and the parameters of the mixture ͑particle masses, concentrations, and force constants͒. The calculations presented here extend previous results derived in the limit cases of self-diffusion and tracer particles. ͓S1063-651X͑98͒02601-4͔

Nonlinear mass and momentum transport in a dilute gas

The Journal of Chemical Physics, 1992

Far from equilibrium particle and momentum transport in a binary mixture subject to uniform shear flow is analyzed. Particles of each species are labeled by a "color charge." Mutual diffusion is created by the action of an external field that accelerates particles of different species along opposite directions. For a dilute gas of Maxwell molecules, the set of two coupled Boltzmann equations is seen to be solvable by the moment method. The color conductivity tensor and the shear viscosity coefficient are obtained as nonlinear functions of the shear rate and the color field. The usual choice of the external color field [Cummings et al., J. Chem. Phys. 94,2149 ( 199 1) ] yields a zero-field limit of the color conductivity tensor different from the self-diffusion tensor. In order to avoid the above discrepancy, a different form of the external field is proposed.

Thermodynamics of dilute gases in shear flow

We consider the eeect of shear and normal viscous pressures on the non-equilibrium entropy of ideal gases in Couette ow. These results extend the previous ones (Bidar et al., Physica A 233 (1996) 163), where normal pressure eeects were ignored. Furthermore, we analyze the non-equilibrium contributions to the chemical potential, which may be useful in the analysis of shear-induced eeects on colligative properties and chemical equilibrium. c

Dissipative homogeneous Maxwell mixtures: ordering transition in the tracer limit

Granular Matter, 2011

The homogeneous Boltzmann equation for inelastic Maxwell mixtures is considered to study the dynamics of tracer particles or impurities (solvent) immersed in a uniform granular gas (solute). The analysis is based on exact results derived for a granular binary mixture in the homogeneous cooling state (HCS) that apply for arbitrary values of the parameters of the mixture (particle masses m i , mole fractions c i , and coefficients of restitution α ij). In the tracer limit (c 1 → 0), it is shown that the HCS supports two distinct phases that are evidenced by the corresponding value of E 1 /E, the relative contribution of the tracer species to the total energy. Defining the mass ratio µ ≡ m 1 /m 2 , there indeed exist two critical values µ (−) HCS and µ (+) HCS (which depend on the coefficients of restitution), such that E 1 /E = 0 for µ (−) HCS < µ < µ

Phase-ordering dynamics of binary mixtures with field-dependent mobility in shear flow

The European Physical Journal B, 2000

The effect of shear flow on the phase-ordering dynamics of a binary mixture with field-dependent mobility is investigated. The problem is addressed in the context of the time-dependent Ginzburg-Landau equation with an external velocity term, studied in self-consistent approximation. Assuming a scaling ansatz for the structure factor, the asymptotic behavior of the observables in the scaling regime can be analytically calculated. All the observables show log-time periodic oscillations which we interpret as due to a cyclical mechanism of stretching and break-up of domains. These oscillations are dumped as consequence of the vanishing of the mobility in the bulk phase.

Heat and momentum transport in a multicomponent mixture far from equilibrium

Physica A: Statistical Mechanics and its Applications, 2001

Explicit expressions for the heat and momentum fluxes are given for a low-density multicomponent mixture in a steady state with temperature and velocity gradients. The results are obtained from a formally exact solution of the Gross-Krook model [Phys. Rev. 102, 593 (1956)] of the Boltzmann equation for a multicomponent mixture. The transport coefficients (shear viscosity, viscometric functions, thermal conductivity and a cross coefficient measuring the heat flux orthogonal to the thermal gradient) are nonlinear functions of the velocity and temperature gradients and the parameters of the mixture (particle masses, concentrations, and force constants). The description applies for conditions arbitrarily far from equilibrium and is not restricted to any range of mass ratios, molar fractions and/or size ratios. The results show that, in general, the presence of the shear flow produces an inhibition in the transport of momentum and energy with respect to that of the Navier-Stokes regime. In the particular case of particles mechanically equivalent and in the tracer limit, previous results are recovered.

Thermodynamics of ideal gases under shear: a maximum-entropy approach

The maximum-entropy formalism is used here as a basis of a thermodynamic description of ideal gases under shear. We have obtained expressions for the entropy and the equations of state in nonequilibrium steady states characterized by a given shear viscous pressure and have identified in physical terms the Lagrange multiplier conjugated to the viscous pressure. Our results for those thermodynamic quantities generalize previous expressions which were limited to second order in the shear viscous pressure. These results show a reduction of the shear viscosity at high shear rates and also show how previous results of nonequilibrium molecular dynamics could be compatible with our thermodynamic analysis.

Steady-state shear flows via nonequilibrium molecular dynamics and smooth-particle applied mechanics

Physical Review E, 1995

We simulate both microscopic and macroscopic shear flows in two space dimensions using nonequiIi brium molecular dynamics and smooth-particle applied mechanics. The time-reversible microscopic equations of motion are isomorphic to the smooth-particle description of inviscid macroscopic continu um mechanics. The corresponding microscopic particle interactions are relatively weak and long ranged. Though conventional Green-Kubo theory suggests instability or divergence in two-dimensional flows, we successfully define and measure a finite shear viscosity coefficient by simulating stationary plane Couette flow. The special nature of the weak long-ranged smooth-particle functions corresponds to an unusual kind of microscopic transport. This microscopic analog is mainly kinetic, even at high density. For the soft Lucy potential which we use in the present work, nearly all the system energy is po tential, but the reSUlting shear viscosity is nearly all kinetic. We show that the measured shear viscosi ties can be understood, in terms of a simple weak-scattering model, and that this understanding is useful in assessing the usefulness of continuum simulations using the smooth-particle method. We apply that method to the Rayleigh-Benard problem of thermally driven convection in a gravitational field.