Generalized Langevin equation with shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian

Sara, Pelargonio,; Zaccone, Alessio

doi:10.1103/physreve.107.064102

Cited by 9 publications

(3 citation statements)

References 58 publications

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“…In fact, in the Brownian limiting case (m/m g → ∞), the expressions of the rheological properties derived here reduce to those previously obtained [26] from a coarsegrained description based on the Fokker-Planck operator. This agreement justifies the use of this latter approach to analyze the DST effect in dilute granular suspensions [27]. It is important to remark that the DST effect found here occurs in a system with a simple structure (low-density granular gas immersed in a dilute molecular gas).…”

Section: Introductionsupporting

confidence: 82%

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Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model

Gómez González,

Garzó

2024

Entropy

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The Boltzmann kinetic equation for dilute granular suspensions under simple (or uniform) shear flow (USF) is considered to determine the non-Newtonian transport properties of the system. In contrast to previous attempts based on a coarse-grained description, our suspension model accounts for the real collisions between grains and particles of the surrounding molecular gas. The latter is modeled as a bath (or thermostat) of elastic hard spheres at a given temperature. Two independent but complementary approaches are followed to reach exact expressions for the rheological properties. First, the Boltzmann equation for the so-called inelastic Maxwell models (IMM) is considered. The fact that the collision rate of IMM is independent of the relative velocity of the colliding spheres allows us to exactly compute the collisional moments of the Boltzmann operator without the knowledge of the distribution function. Thanks to this property, the transport properties of the sheared granular suspension can be exactly determined. As a second approach, a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to granular suspensions is solved to compute the velocity moments and the velocity distribution function of the system. The theoretical results (which are given in terms of the coefficient of restitution, the reduced shear rate, the reduced background temperature, and the diameter and mass ratios) show, in general, a good agreement with the approximate analytical results derived for inelastic hard spheres (IHS) by means of Grad’s moment method and with computer simulations performed in the Brownian limiting case (m/mg→∞, where mg and m are the masses of the particles of the molecular and granular gases, respectively). In addition, as expected, the IMM and BGK results show that the temperature and non-Newtonian viscosity exhibit an S shape in a plane of stress–strain rate (discontinuous shear thickening, DST). The DST effect becomes more pronounced as the mass ratio m/mg increases.

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Section: Introductionsupporting

confidence: 82%

“…From Equations (27), one can identify the (dimensionless) non-Newtonian shear viscosity η * g = ν M g η g /p g = −P * g,xy / a and the (dimensionless) normal stress difference Ψ * g = P * g,xx − P * g,yy as…”

Section: Rheological Properties Of the Molecular Gasmentioning

confidence: 99%

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model

Gómez González,

Garzó

2024

Entropy

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“…We note that it has recently been shown 31 that the white noise formulation described here is valid only for shear rates that are small compared to the effective vibrational temperature of the bath. However, for the present work we focus on the Markovian case above, and postpone treatment on the generalized, non-Markovian Langevin equation to future work.…”

Section: Nonequilibrium Langevin Dynamicsmentioning

confidence: 92%

Dynamic density functional theory with inertia and background flow

Mills-Williams,

Goddard,

Archer

2024

The Journal of Chemical Physics

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We present dynamic density functional theory (DDFT) incorporating general inhomogeneous, incompressible, time-dependent background flows and inertia, describing externally driven passive colloidal systems out of equilibrium. We start by considering the underlying nonequilibrium Langevin dynamics, including the effect of the local velocity of the surrounding liquid bath, to obtain the nonlinear, nonlocal partial differential equations governing the evolution of the (coarse-grained) density and velocity fields describing the dynamics of colloids. In addition, we show both with heuristic arguments, and by numerical solution, that our equations and solutions agree with existing DDFTs in the overdamped (high friction) limit. We provide numerical solutions that model the flow of hard spheres, in both unbounded and confined domains, and compare with previously derived DDFTs with and without the background flow.

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Lopsided rigid dumbbell rheology from Langevin equation: A graduate tutorial

Phan-Thien,

Pan,

Kanso

et al. 2024

Physics of Fluids

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The modeling of symmetric rigid dumbbell particles suspended in a Newtonian fluid, as a model of a rigid-rod polymeric solution, has been accomplished exclusively through the diffusion equation, detailed elegantly by Bird et al. [Dynamics of Polymeric Liquids: Kinetic Theory, 2nd ed. (Wiley, NY, 1987), Vol. 2]. In this Tutorial, a straightforward approach for modeling a lopsided rigid dumbbell particle in suspension is presented by the Langevin analysis. The connector force between the dumbbell beads is obtained through the rigidity constraint of the center-to-center vector of the dumbbell using its Langevin equation. By directly averaging via the Langevin equation, the evolution of the center-to-center vector and the configuration tensor are derived. The stress expressions for the dumbbell from the Langevin equation and the diffusion equation for the orientation distribution function of the center-to-center vector of the dumbbell are also derived, and the final expressions agree with the established results from other methods.

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Generalized Langevin equation with shear flow and its fluctuation-dissipation theorems derived from a Caldeira-Leggett Hamiltonian

Cited by 9 publications

References 58 publications

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model

Exact Results for Non-Newtonian Transport Properties in Sheared Granular Suspensions: Inelastic Maxwell Models and BGK-Type Kinetic Model

Dynamic density functional theory with inertia and background flow

Lopsided rigid dumbbell rheology from Langevin equation: A graduate tutorial

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