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Register a new userTransport coefficients for granular suspensions at moderate densities
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The Enskog kinetic theory for moderately dense granular suspensions is considered as a model to determine the Navier-Stokes transport coefficients. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. The suspension model is solved by means of the Chapman--Enskog method conveniently adapted to dissipative dynamics. The momentum and heat fluxes as well as the cooling rate are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the cooling terms (arising from collisional dissipation and viscous friction) cannot be compensated for by the energy gained by grains due to collisions with the interstitial gas, the reference distribution (zeroth-order approximation of the Chapman--Enskog solution) depends on time through its dependence on temperature. On the other hand, to simplify the analysis and given that we are interested in computing transport properties in the first order of deviations from the reference state, the steady-state conditions are considered. This simplification allows us to get explicit expressions for the Navier--Stokes transport coefficients. As expected, the results show that the dependence of the transport coefficients on both inelasticity and density is clearly different from that found in its granular counterpart (no gas phase). Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. In contrast to the granular case (no gas-phase), no instabilities are found and hence, the homogeneous steady state is (linearly) stable.
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The Navier--Stokes transport coefficients of multicomponent granular suspensions at moderate densities are obtained in the context of the (inelastic) Enskog kinetic theory. The suspension is modeled as an ensemble of solid particles where the influence of the interstitial gas on grains is via a viscous drag force plus a stochastic Langevin-like term defined in terms of a background temperature. In the absence of spatial gradients, it is shown first that the system reaches a homogeneous steady state where the energy lost by inelastic collisions and viscous friction is compensated for by the energy injected by the stochastic force. Once the homogeneous steady state is characterized, a emph{normal} solution to the set of Enskog equations is obtained by means of the Chapman--Enskog expansion around the emph{local} version of the homogeneous state. To first-order in spatial gradients, the Chapman--Enskog solution allows us to identify the Navier--Stokes transport coefficients associated with the mass, momentum, and heat fluxes. In addition, the first-order contributions to the partial temperatures and the cooling rate are also calculated. Explicit forms for the diffusion coefficients, the shear and bulk viscosities, and the first-order contributions to the partial temperatures and the cooling rate are obtained in steady-state conditions by retaining the leading terms in a Sonine polynomial expansion. The results show that the dependence of the transport coefficients on inelasticity is clearly different from that found in its granular counterpart (no gas phase). The present work extends previous theoretical results for emph{dilute} multicomponent granular suspensions [Khalil and Garzo, Phys. Rev. E textbf{88}, 052201 (2013)] to higher densities.
The Boltzmann kinetic equation is considered to evaluate the first-order contributions $T_i^{(1)}$ to the partial temperatures in binary granular suspensions at low density. The influence of the surrounding gas on the solid particles is modeled via a drag force proportional to the particle velocity plus a stochastic Langevin-like term. The Boltzmann equation is solved by means of the Chapman--Enskog expansion around the local version of the reference homogeneous base state. To first-order in spatial gradients, the coefficients $T_i^{(1)}$ are computed by considering the leading terms in a Sonine polynomial expansion. In addition, the influence of $T_i^{(1)}$ on the first-order contribution $zeta^{(1)}$ to the cooling rate is also assessed. Our results show that the magnitude of both $T_i^{(1)}$ and $zeta^{(1)}$ can be relevant for some values of the parameter space of the system.
The objective of this study is to assess the impact of a dense-phase treatment on the hydrodynamic description of granular, binary mixtures relative to a previous dilute-phase treatment. Two theories were considered for this purpose. The first, proposed by Garzo and Dufty (GD) [Phys. Fluids {bf 14}, 146 (2002)], is based on the Boltzmann equation which does not incorporate finite-volume effects, thereby limiting its use to dilute flows. The second, proposed by Garzo, Hrenya and Dufty (GHD) [Phys. Rev. E {bf 76}, 31303 and 031304 (2007)], is derived from the Enskog equation which does account for finite-volume effects; accordingly this theory can be applied to moderately dense systems as well. To demonstrate the significance of the dense-phase treatment relative to its dilute counterpart, the ratio of dense (GHD) to dilute (GD) predictions of all relevant transport coefficients and equations of state are plotted over a range of physical parameters (volume fraction, coefficients of restitution, material density ratio, diameter ratio, and mixture composition). These plots reveal the deviation between the two treatments, which can become quite large ($>$100%) even at moderate values of the physical parameters. Such information will be useful when choosing which theory is most applicable to a given situation, since the dilute theory offers relative simplicity and the dense theory offers improved accuracy. It is also important to note that several corrections to original GHD expressions are presented here in the form of a complete, self-contained set of relevant equations.
We introduce and study a simple and natural class of solvable stochastic lattice gases. This is the class of emph{Strong Particles}. The name is due to the fact that when they try to jump to an occupied site they succeed pushing away a pile of particles. For this class of models we explicitly compute the transport coefficients. We also discuss some generalizations and the relations with other classes of solvable models.
The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to analyze transport properties in spatially inhomogeneous states close to the simple shear flow. A normal solution is obtained via a Chapman--Enskog--like expansion around a local shear flow distribution f^{(0)} that retains all the hydrodynamic orders in the shear rate. The constitutive equations for the heat and momentum fluxes are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the reference state and the corresponding generalized transport coefficients are {em exactly} determined in terms of the coefficient of restitution alpha and the shear rate a. Since f^{(0)} applies for arbitrary values of the shear rate and is not restricted to weak dissipation, the transport coefficients turn out to be nonlinear functions of both parameters a and alpha. A comparison with previous results obtained for inelastic hard spheres from a kinetic model of the Boltzmann equation is also carried out.
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