Abstract:The hydrodynamics and rheology of a sheared dilute gas-solid suspension, consisting of inelastic hard-spheres suspended in a gas, are analysed using anisotropic Maxwellian as the single particle distribution function. The closed-form solutions for granular temperature (T ) and three invariants of the second-moment tensor are obtained as functions of the Stokes number (St), the mean density (ν) and the restitution coefficient (e). Multiple states of high and low temperatures are found when the Stokes number is … Show more
“…As said in the Introduction, this state is macroscopically defined by a constant density n, a spatially uniform temperature T (t), and a flow velocity U i = a ij r j , where a ij = aδ ix δ jy , a being the constant shear rate. In addition, as usual in uniform sheared suspensions [17][18][19]39], the average velocity of particles follows the velocity of the fluid phase and so, U = U g . One of the main advantages of the USF at a microscopic level is that in this state all the space dependence of the one-particle velocity distribution function f (r, v, t) occurs through its dependence on the peculiar velocity V = v − U(r) [40].…”
The Boltzmann kinetic equation for low-density granular suspensions under simple shear flow is considered to determine the velocity moments through the fourth degree. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. Two independent but complementary approaches are followed to achieve exact results. First, to keep the structure of the Boltzmann collision operator, the so-called inelastic Maxwell models (IMM) are considered. In this model, since the collision rate is independent of the relative velocity of the two colliding particles, the forms of the collisional moments can be obtained without the knowledge of the velocity distribution function. As a complement of the previous effort, a BGK-type kinetic model adapted to granular gases is solved to get the velocity moments of the velocity distribution function. The analytical predictions of the rheological properties (which are exactly obtained in terms of the coefficient of restitution α and the reduced shear rate a * ) show in general an excellent agreement with event-driven simulations performed for inelastic hard spheres. In particular, both theoretical approaches show clearly that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening effect). With respect to the fourth-degree velocity moments, we find that while those moments have unphysical values for IMM in a certain region of the parameter space of the system, they are well defined functions of both α and a * in the case of the BGK kinetic model. The explicit shear-rate dependence of the fourth-degree moments beyond this critical region is also obtained and compared against available computer simulations. * Electronic address: ruben@unex.es † Electronic address: vicenteg@unex.es;
“…As said in the Introduction, this state is macroscopically defined by a constant density n, a spatially uniform temperature T (t), and a flow velocity U i = a ij r j , where a ij = aδ ix δ jy , a being the constant shear rate. In addition, as usual in uniform sheared suspensions [17][18][19]39], the average velocity of particles follows the velocity of the fluid phase and so, U = U g . One of the main advantages of the USF at a microscopic level is that in this state all the space dependence of the one-particle velocity distribution function f (r, v, t) occurs through its dependence on the peculiar velocity V = v − U(r) [40].…”
The Boltzmann kinetic equation for low-density granular suspensions under simple shear flow is considered to determine the velocity moments through the fourth degree. The influence of the interstitial gas on solid particles is modeled by a viscous drag force term plus a stochastic Langevin-like term. Two independent but complementary approaches are followed to achieve exact results. First, to keep the structure of the Boltzmann collision operator, the so-called inelastic Maxwell models (IMM) are considered. In this model, since the collision rate is independent of the relative velocity of the two colliding particles, the forms of the collisional moments can be obtained without the knowledge of the velocity distribution function. As a complement of the previous effort, a BGK-type kinetic model adapted to granular gases is solved to get the velocity moments of the velocity distribution function. The analytical predictions of the rheological properties (which are exactly obtained in terms of the coefficient of restitution α and the reduced shear rate a * ) show in general an excellent agreement with event-driven simulations performed for inelastic hard spheres. In particular, both theoretical approaches show clearly that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress-strain rate (discontinuous shear thickening effect). With respect to the fourth-degree velocity moments, we find that while those moments have unphysical values for IMM in a certain region of the parameter space of the system, they are well defined functions of both α and a * in the case of the BGK kinetic model. The explicit shear-rate dependence of the fourth-degree moments beyond this critical region is also obtained and compared against available computer simulations. * Electronic address: ruben@unex.es † Electronic address: vicenteg@unex.es;
“…Moreover, it has also been confirmed [39] that the present Padé-approximants provide a useful guide (or, benchmark) to decide the "order" of series solutions that must be retained to obtain accurate solutions for the related granular Poiseuille flow -these results will be discussed in a future publication. Another interesting future work would be to determine the solutions of 13-moment extended hydrodynamic equations [40][41][42] with appropriate boundary conditions for Poiseuille flow and compare them with the present Padé-approximants in the bulk region of the channel.…”
The perturbation expansion technique is employed to solve the Boltzmann equation for the acceleration-driven steady Poiseuille flow of a dilute molecular gas flowing through a planar channel. Neglecting wall effects and focusing only on the bulk hydrodynamics and rheology, the perturbation solution is sought around the channel centerline in powers of the strength of acceleration. To make analytical progress, the collision term has been approximated by the Bhatnagar-Gross-Krook kinetic model for hard spheres, and the related problem for Maxwell molecules was analyzed previously by Tij and Santos [J. Stat. Phys. 76, 1399 (1994)JSTPBS0022-471510.1007/BF02187068]. The analytical expressions for hydrodynamic (velocity, temperature, and pressure) and rheological fields (normal stress differences, shear viscosity, and heat flux) are obtained by retaining terms up to tenth order in acceleration, with one aim of the present work being to understand the convergence properties of the underlying perturbation series solutions. In addition, various rarefaction effects (e.g., the bimodal shape of the temperature profile, nonuniform pressure profile, normal-stress differences, and tangential heat flux) are also critically analyzed in the Poiseuille flow as functions of the local Froude number. The hydrodynamic and rheological fields evaluated at the channel centerline confirmed the oscillatory nature of the present series solutions (when terms of increasing order are sequentially included), signaling the well-known pitfalls of asymptotic expansion. The Padé approximation technique is subsequently applied to check the region of convergence of each series solution. It is found that the diagonal Padé approximants for rheological fields agree qualitatively with previous simulation data on acceleration-driven rarefied Poiseuille flow.
“…In addition to this, a normal damping force acts over the duration of each collision and is given by (Rapaport 2004), where is the normal component of the relative velocity of two colliding particles, denotes the mass of a particle and is the damping coefficient. This particle-level damping is akin to (i) a ‘velocity-dependent’ normal restitution coefficient in a granular gas (Goldhirsch 2003; Brilliantov & Pöschel 2004) and/or (ii) a drag force in a gas–solid suspension (Jackson 2000; Saha & Alam 2017) at large Stokes numbers; the specific role of this damping mechanism on both primary and secondary bifurcations and the emerging patterns are discussed in appendix A. Figure 1. Schematic and the domain binning of the TC cell; is the height of the cylinders, and are the inner and outer radii, and is the gap width; the rotational speeds of the inner and outer cylinders are and , respectively.…”
Section: Molecular Dynamics Simulation Of Taylor–couette Flowmentioning
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