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Register a new userShear-rate dependent transport coefficients for inelastic Maxwell models
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Vicente Garzo
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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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The Boltzmann equation for inelastic Maxwell models is considered to determine the rheological properties in a granular binary mixture in the simple shear flow state. The transport coefficients (shear viscosity and viscometric functions) are {em exactly} evaluated in terms of the coefficients of restitution, the (reduced) shear rate and the parameters of the mixture (particle masses, diameters and concentration). The results show that in general, for a given value of the coefficients of restitution, the above transport properties decrease with increasing shear rate.
The Boltzmann equation for inelastic Maxwell models is considered to determine the velocity moments through fourth degree in the simple shear flow state. First, the rheological properties (which are related to the second-degree velocity moments) are {em exactly} evaluated in terms of the coefficient of restitution $alpha$ and the (reduced) shear rate $a^*$. For a given value of $alpha$, the above transport properties decrease with increasing shear rate. Moreover, as expected, the third-degree and the asymmetric fourth-degree moments vanish in the long time limit when they are scaled with the thermal speed. On the other hand, as in the case of elastic collisions, our results show that, for a given value of $alpha$, the scaled symmetric fourth-degree moments diverge in time for shear rates larger than a certain critical value $a_c^*(alpha)$ which decreases with increasing dissipation. The explicit shear-rate dependence of the fourth-degree moments below this critical value is also obtained.
The exact nonequilibrium steady state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of the distribution function $f(c)$. In this paper we have inverted the Fourier transform to express $f(c)$ in the form of an infinite series of exponentially decaying terms. The dominant high energy tail is exponential, $f(c)simeq A_0exp(-a|c|)$, where $aequiv 2/sqrt{1-alpha^2}$ and the amplitude $A_0$ is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit ($alphato 0$) and in the quasi-elastic limit ($alphato 1$). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high energy tail exists for velocities $|c-c_0|sim 1/sqrt{q}$ around a crossover velocity $c_0simeq ln q^{-1}/sqrt{q}$, where $qequiv (1-alpha)/2ll 1$. In this crossover region the distribution function is extremely small, $ln f(c_0)simeq q^{-1}ln q$.
Non-Newtonian transport properties of an inertial suspension of inelastic rough hard spheres under simple shear flow are determined from the Boltzmann kinetic equation. The influence of the interstitial gas on rough hard spheres is modeled via a Fokker-Planck generalized equation for rotating spheres accounting for the coupling of both the translational and rotational degrees of freedom of grains with the background viscous gas. The generalized Fokker-Planck term is the sum of two ordinary Fokker-Planck differential operators in linear $mathbf{v}$ and angular $boldsymbol{omega}$ velocity space. As usual, each Fokker-Planck operator is constituted by a drag force term (proportional to $mathbf{v}$ and/or $boldsymbol{omega}$) plus a stochastic Langevin term defined in terms of the background temperature $T_text{ex}$. The Boltzmann equation is solved by two different but complementary approaches: (i) by means of Grads moment method, and (ii) by using a Bhatnagar-Gross-Krook (BGK)-type kinetic model adapted to inelastic rough hard spheres. As occurs in the case of emph{smooth} inelastic hard spheres, our results show that both the temperature and the non-Newtonian viscosity increase drastically with increasing the shear rate (discontinuous shear thickening effect) while the fourth-degree velocity moments also exhibit an $S$-shape. In particular, while high levels of roughness may slightly attenuate the jump of the viscosity in comparison to the smooth case, the opposite happens for the rotational temperature. As an application of these results, a linear stability analysis of the steady simple shear flow solution is also carried out showing that there are regions of the parameter space where the steady solution becomes linearly unstable.
This review is a kinetic theory study investigating the effects of inelasticity on the structure of the non-equilibrium states, in particular on the behavior of the velocity distribution in the high energy tails. Starting point is the nonlinear Boltzmann equation for spatially homogeneous systems, which supposedly describes the behavior of the velocity distribution function in dissipative systems as long as the system remains in the homogeneous cooling state, i.e. on relatively short time scales before the clustering and similar instabilities start to create spatial inhomogeneities. This is done for the two most common models for dissipative systems, i.e. inelastic hard spheres and inelastic Maxwell particles. In systems of Maxwell particles the collision frequency is independent of the relative velocity of the colliding particles, and in hard sphere systems it is linear. We then demonstrate the existence of scaling solutions for the velocity distribution function, $F(v,t) sim v_0(t)^{-d} f((v/v_0(t))$, where $v_0$ is the r.m.s. velocity. The scaling form $f(c)$ shows overpopulation in the high energy tails. In the case of freely cooling systems the tails are of algebraic form, $ f(c)sim c^{-d-a}$, where the exponent $a$ may or may not depend on the degree of inelasticity, and in the case of forced systems the tails are of stretched Gaussian type $f(v)simexp[-beta (v/v_0)^b]$ with $b <2$.
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