Navier–Stokes transport coefficients for driven inelastic Maxwell models

Chamorro, Moisés G.; Garzó, Vicente; Reyes, Francisco Vega

doi:10.1088/1742-5468/2014/06/p06008

Cited by 6 publications

(16 citation statements)

References 72 publications

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“…However, in order to correctly describe the velocity dependence of the original IHS collision rate, one usually assumes that ν 0 is proportional to T q with q = 1 2 . Here, as in previous works on IMM [12,20], we take ν 0 ∝ nT q , with 0 ≤ q ≤ 1 2 . The case q = 0 is closer to the original Maxwell model of elastic particles while the case q = 1 2 is closer to hard spheres.…”

Section: Boltzmann Kinetic Theory For Inelastic Maxwell Models Ofmentioning

confidence: 99%

“…The expressions of η * s , κ * s and µ * s for IHS have been recently derived by considering the first Sonine approximation [12]. Their explicit forms are displayed in the Appendix B.…”

Section: B Steady Statesmentioning

confidence: 99%

“…In this paper, we consider a simplified version of the model of suspensions used in Refs. [4,6,12] where only the drag force term is accounted for. As mentioned before, this situation could correspond to a gas-solid flow where the mean velocity of the particles follows the velocity of the fluid (such as in the case of the simple shear flow [14]).…”

Section: Introductionmentioning

confidence: 99%

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Inelastic Maxwell models for monodisperse gas–solid flows

Kubicki¹,

Garzó²

2015

J. Stat. Mech.

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The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to analyze transport properties for monodisperse gas-solid suspensions. The influence of the interstitial gas phase on the dynamics of solid particles is modeled via a viscous drag force. The Chapman-Enskog method is applied to solve the inelastic Boltzmann equation to first order in the deviations of the hydrodynamic fields from their values in the homogeneous cooling state. Explicit expressions for the Navier-Stokes transport coefficients are exactly obtained in terms of both the coefficient of restitution and the friction coefficient characterizing the amplitude of the external force. The conditions under which a hydrodynamic regime independent of the initial conditions is reached are widely discussed. Finally, the results derived here are compared with those previously obtained for inelastic hard spheres in steady state conditions by using the so-called first Sonine approximation. * vicenteg@unex.es; http://www.eweb.unex.es/eweb/fisteor/vicente/ 2

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Section: Boltzmann Kinetic Theory For Inelastic Maxwell Models Ofmentioning

confidence: 99%

“…The expressions of η * s , κ * s and µ * s for IHS have been recently derived by considering the first Sonine approximation [12]. Their explicit forms are displayed in the Appendix B.…”

Section: B Steady Statesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inelastic Maxwell models for monodisperse gas–solid flows

Kubicki¹,

Garzó²

2015

J. Stat. Mech.

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“…Three different values of the mass ratio are considered. As occurs in monocomponent granular gases [78,79], we observe that in general the qualitative dependence of the Navier-Stokes shear viscosity of the mixture (for driven and undriven systems) on inelasticity of IHS is well captured by IMM: η * increases with decreasing α. On the other hand, this increase is faster for IMM and so, the IMM predictions 2 , m1/m2 = 2, and σ1/σ2 = 1.…”

Section: B Comparison With the Transport Coefficients Of Ihsmentioning

confidence: 55%

Unified hydrodynamic description for driven and undriven inelastic Maxwell mixtures at low density

Khalil¹,

Garzó²

2019

Preprint

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A hydrodynamic description for inelastic Maxwell mixtures driven by a stochastic thermostat with friction is derived by solving the set of Boltzmann kinetic equations by means of the Chapman-Enskog method. Contrary to previous works where the constitutive relations for the fluxes were restricted to states near the homogeneous steady state, the expressions of the Navier-Stokes transport coefficients are obtained by considering a more general time-dependent reference state. This allows us to describe the system under wider physical conditions. The transport coefficients are given in terms of the solutions of a set of nonlinear differential equations. These differential equations (which must be in general numerically solved) involve the derivatives with respect to the scaled parameter of the thermostat ξ * . Exact expressions for the Navier-Stokes transport coefficients are obtained in the special cases of undriven mixtures (ξ * = 0) and driven mixtures under steady conditions (ξ * = ξ * st , where ξ * st is the value of the reduced noise strength at the steady state). While the transport coefficients display a weak dependence on ξ * for small inelasticity (and so, driven and undriven systems have similar transport properties), this dependence significantly increases with increasing inelasticity. As a complement, the results for inelastic Maxwell models (IMM) are compared against known results for inelastic hard spheres (IHS) obtained in the leading Sonine approximation. While the IMM predictions for the diffusion transport coefficients compare very well with those derived for IHS, significant quantitative differences are found in the cases of the shear viscosity coefficient and the heat flux transport coefficients.

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“…As in previous papers pertaining to IMM [21,23,24], the velocity moments of the Boltzmann collision operator are given in terms of a collision frequency ν 0 . This parameter can be seen as a free parameter of the model that can be chosen to optimize the agreement with the properties of interest of the original Boltzmann equation for IHS.…”

Section: Introductionmentioning

confidence: 99%

Generalized transport coefficients for inelastic Maxwell mixtures under shear flow

Garzó¹,

Trizac²

2015

Phys. Rev. E

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The Boltzmann equation framework for inelastic Maxwell models is considered to determine the transport coefficients associated with the mass, momentum and heat fluxes of a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. The Boltzmann equation is solved by means of a Chapman-Enskog-like expansion around the (local) shear flow distributions f (0) r for each species that retain all the hydrodynamic orders in the shear rate. Due to the anisotropy induced by the shear flow, tensorial quantities are required to describe the transport processes instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled equations, which can be analytically solved as functions of the shear rate a, the coefficients of restitution αrs and the parameters of the mixture (masses, diameters and composition). Since the reference distribution functions f (0) r apply for arbitrary values of the shear rate and are not restricted to weak dissipation, the corresponding generalized coefficients turn out to be nonlinear functions of both a and αrs. The dependence of the relevant elements of the three diffusion tensors on both the shear rate and dissipation is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coefficients from their forms for vanishing shear rates is in general significant. A comparison with the previous results obtained analytically for inelastic hard spheres by using Grad's moment method is carried out showing a good agreement over a wide range of values for the coefficients of restitution. Finally, as an application of the theoretical expressions derived here for the transport coefficients, thermal diffusion segregation of an intruder immersed in a granular gas is also studied.

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Navier–Stokes transport coefficients for driven inelastic Maxwell models

Cited by 6 publications

References 72 publications

Inelastic Maxwell models for monodisperse gas–solid flows

Inelastic Maxwell models for monodisperse gas–solid flows

Unified hydrodynamic description for driven and undriven inelastic Maxwell mixtures at low density

Generalized transport coefficients for inelastic Maxwell mixtures under shear flow

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