Francisco Vega Reyes

Modified Sonine approximation for granular binary mixtures

We evaluate in this work the hydrodynamic transport coefficients of a granular binary mixture in d dimensions. In order to eliminate the observed disagreement (for strong dissipation) between computer simulations and previously calculated theoretical transport coefficients for a monocomponent gas, we obtain explicit expressions of the seven Navier-Stokes transport coefficients by the use of a new Sonine approach in the Chapman-Enskog (CE) theory. This new approach consists of replacing, where appropriate in the CE procedure, the Maxwell-Boltzmann distribution weight function (used in the standard first Sonine approximation) by the homogeneous cooling state distribution for each species. The rationale for doing this lies in the well-known fact that the non-Maxwellian contributions to the distribution function of the granular mixture are more important in the range of strong dissipation we are interested in. The form of the transport coefficients is quite common in both standard and modified Sonine approximations, the distinction appearing in the explicit form of the different collision frequencies associated with the transport coefficients. Additionally, we numerically solve by the direct simulation Monte Carlo method the inelastic Boltzmann equation to get the diffusion and the shear viscosity coefficients for two and three dimensions. As in the case of a monocomponent gas, the modified Sonine approximation improves the estimates of the standard one, showing again the reliability of this method at strong values of dissipation.

Non-Newtonian Granular Hydrodynamics. What Do the Inelastic Simple Shear Flow and the Elastic Fourier Flow Have in Common?

We describe a special class of steady Couette flows in dilute granular gases admitting a non-Newtonian hydrodynamic description for strong dissipation. The class occurs when viscous heating exactly balances inelastic cooling, resulting in a uniform heat flux. It includes the Fourier flow of ordinary gases and the simple or uniform shear flow (USF) of granular gases as special cases. The rheological functions have the same values as in the USF and generalized thermal conductivity coefficients can be identified. These points are confirmed by molecular dynamics simulations, Monte Carlo simulations of the Boltzmann equation, and analytical results from Grads 13-moment method.

Mass transport of impurities in a moderately dense granular gas

Transport coefficients associated with the mass flux of impurities immersed in a moderately dense granular gas of hard disks or spheres described by the inelastic Enskog equation are obtained by means of the Chapman-Enskog expansion. The transport coefficients are determined as the solutions of a set of coupled linear integral equations recently derived for polydisperse granular mixtures [Garzó, Phys. Rev. E 76, 031304 (2007)]. With the objective of obtaining theoretical expressions for the transport coefficients that are sufficiently accurate for highly inelastic collisions, we solve the above integral equations by using the second Sonine approximation. As a complementary route, we numerically solve by means of the direct simulation Monte Carlo method (DSMC) the inelastic Enskog equation to get the kinetic diffusion coefficient D0 for two and three dimensions. We have observed in all our simulations that the disagreement, for arbitrarily large inelasticity, in the values of both solutions (DSMC and second Sonine approximation) is less than 4%. Moreover, we show that the second Sonine approximation to D0 yields a dramatic improvement (up to 50%) over the first Sonine approximation for impurity particles lighter than the surrounding gas and in the range of large inelasticity. The results reported in this paper are of direct application in important problems in granular flows, such as segregation driven by gravity and a thermal gradient. We analyze here the segregation criteria that result from our theoretical expressions of the transport coefficients.

Steady base states for Navier-Stokes granular hydrodynamics with boundary heating and shear

We study the Navier-Stokes steady states for a low density monodisperse hard sphere granular gas (i.e a hard sphere ideal monatomic gas with inelastic inter-particle collisions). We present a classification of the uniform steady states that can arise from shear and temperature (or energy input) applied at the boundaries (parallel walls). We consider both symmetric and asymmetric boundary conditions and find steady states not previously reported, including sheared states with linear temperature profiles. We provide explicit expressions for the hydrodynamic profiles for all these steady states. Our results are validated by the numerical solution of the Boltzmann kinetic equation for the granular gas obtained by the direct simulation Monte Carlo method, and by molecular dynamics simulations. We discuss the physical origin of the new steady states and derive conditions for the validity of Navier-Stokes hydrodynamics.

Effect of inelasticity on the phase transitions of a thin vibrated granular layer

We describe an experimental and computational investigation of the ordered and disordered phases of a vibrating thin, dense granular layer composed of identical metal spheres. We compare the results from spheres with different amounts of inelasticity and show that inelasticity has a strong effect on the phase diagram. We also report the melting of an ordered phase to a homogeneous disordered liquid phase at high vibration amplitude or at large inelasticities. Our results show that dissipation has a strong effect on ordering and that in this system ordered phases are absent entirely in highly inelastic materials.

Homogeneous steady states in a granular fluid driven by a stochastic bath with friction

The homogeneous state of a granular flow of smooth inelastic hard spheres or disks described by the Enskog–Boltzmann kinetic equation is analyzed. The granular gas is fluidized by the presence of a random force and a drag force. The combined action of both forces, which act homogeneously on the granular gas, tries to mimic the interaction of the set of particles with a surrounding fluid. The first stochastic force thermalizes the system, providing for the necessary energy recovery to keep the system in its gas state at all times, whereas the second force allows us to mimic the action of the surrounding fluid viscosity. After a transient regime, the gas reaches a steady state characterized by a scaled distribution function that depends not only on the dimensionless velocity c ≡ v/v0 (v0 being the thermal velocity) but also on the dimensionless driving force parameters. The dependence of and its first relevant velocity moments a2 and a3 (which measure non-Gaussian properties of ) on both the coefficient of restitution α and the driven parameters is widely investigated by means of the direct simulation Monte Carlo method. In addition, approximate forms for a2 and a3 are also derived from an expansion of in Sonine polynomials. The theoretical expressions of the above Sonine coefficients agree well with simulation data, even for quite small values of α. Moreover, the third order expansion of the distribution function makes a significant improvement in accuracy for larger velocities and inelasticities. Results also show that the non-Gaussian corrections to the distribution function are smaller than those observed for undriven granular gases.

Pattern imaging of primary and secondary electrohydrodynamic instabilities

A little known electrohydrodynamic instability, which we call a rose window, is observed in air/liquid interfaces in electric fields with unipolar space charge distributions. Depending on the liquid properties, the rose window may appear from an initial rest state (primary instability) or on top of another instability, the classical unipolar-injection-induced instability, destroying its pattern (secondary instability). After imaging of the rose window, we use an edge-detection filter to find the instability threshold and study the characteristic pattern as a function of the liquid properties. Results show that the specific properties of the electric field, due to charge injection, are the cause of the rose-window and that the primary and secondary rose windows are essentially different instabilities.

Transport properties for driven granular fluids in situations close to homogeneous steady states

The transport coefficients of a granular fluid driven by a stochastic bath with friction are obtained by solving the inelastic Enskog kinetic equation from the Chapman-Enskog method. The heat and momentum fluxes as well as the cooling rate are determined to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the collisional cooling cannot be compensated locally for the heat produced by the external driving force, the reference distribution

Impurity in a granular gas under nonlinear Couette flow

f^{(0)}

Hydrodynamics of a Granular Gas in a Heterogeneous Environment

(zeroth-order approximation) depends on time through its dependence on temperature. This fact gives rise to conceptual and practical difficulties not present in the undriven case. On the other hand, to simplify the analysis and given that we are interested in computing transport in the first order of deviations from the reference state, the steady-state conditions are considered to get explicit forms for the transport coefficients and the cooling rate. A comparison with recent molecular dynamics simulations for driven granular fluids shows an excellent agreement for the kinematic viscosity although some discrepancies are observed for the longitudinal viscosity and the thermal diffusivity at large densities. Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. As expected, no instabilities are found thanks to the presence of the external bath.