Lift force in chiral, compressible granular matter

TL;DR

The paper develops a micropolar continuum model for parity-breaking, compressible granular matter that includes internal rotations and odd viscosity $(\eta_o)$ and analyzes the lift on a bead embedded in such a medium. It combines a linearized Brinkman-type Stokes analysis in Fourier space with nonlinear finite-element simulations in finite domains to access non-equilibrium steady states and oscillatory responses. A key result is that a transverse lift coefficient $M_l$ arises from $\eta_o$ and microrotation, persisting in nonlinear regimes but vanishing in incompressible or zero-odd-viscosity limits; the FEM results corroborate the analytical predictions and reveal characteristic density and microrotation patterns around the bead. This symmetry-informed continuum description provides a pathway for experimental tests in vibrated granular layers and offers insights into active and passive chiral granular flows.

Abstract

Micropolar fluid theory, an extension of classical Newtonian fluid dynamics, incorporates angular velocities and rotational inertias and has long been a foundational framework for describing granular flows. We propose a macroscopic model of granular matter based on micropolar fluid dynamics, which incorporates internal rotations, couple stresses, and broken parity through odd viscosity. Our framework extends traditional micropolar theory to describe chiral granular flows driven far from equilibrium, where energy is continuously injected and dissipated. In particular, we focus on steady states and explicitly neglect energy conservation, reflecting the dissipative nature of granular systems maintained in non-equilibrium by external forcing. Within this setup, we study the lift force experienced by a test bead embedded in a compressible, parity-breaking granular flow. We analyze how odd viscosity and microrotation modify the transverse forces, using both analytical results in the linearized Stokes regime and nonlinear finite element simulations. Our results demonstrate that micropolar fluids provide a physically consistent and symmetry-informed continuum description of chiral granular matter, capable of capturing lift forces that emerge uniquely from odd transport effects.

Figures (4)

• Figure 1: Steady state analytical solutions for (a) drag $M_d$ and lift $M_l$ coefficients, (b) correction to the lift coefficient $\Delta M_l$ due to microrotation. Unless otherwise specified, the parameters take the following values: $\bar{\tau}=1$, $\bar{\eta}_b=1$, $\bar{\mu}_r=0.4$, $\bar{I}=0.1$, $\bar{c}_1=2$, $\bar{\alpha}=0.5$.
• Figure 2: Finite element method results for a compressible odd fluid coupled to a micropolar field flowing through a bead disk in a finite domain are presented. Panels (a,b) show the computational domains (grid is also shown) and boundary conditions: (a) the velocity field with its boundary condition (b.c.) $\bm{v}_0$ (marked by green square), and (b) the pressure field with pressure b.c. $P_0$ applied along the left wall (green segment). Example solution (velocity, pressure, density, and microrotation) fields are displayed for fluids with (c) positive oddity, (d) zero oddity, and (e) negative oddity. For comparison, panel (f) uses the same parameters as (c) but for an incompressible fluid.
• Figure 3: Comparison of forces calculated using the finite element method and obtained via the shell localization. (a) Drag $M_d$, lift $M_l$ force coefficients, and corrections to lift $\Delta{}M_l$ force coefficients due to coupling with the micropolar field are presented. Lift correction as a function of odd $\bar{\eta}_o$ and microrotation $\bar{\mu}_r$ couplings obtained in FEM (b) are compared with the shell localization analytical results (c).
• Figure 4: Finite element method results for Newtonian compressible fluid (without microrotation degree) flowing through a bead disk. Example solution (velocity, pressure, and density) fields are displayed for fluids with (a) zero oddity, (b) positive oddity, and (c) negative oddity. In panels (b,c) instead of the velocity field, we show the difference between the current velocity field and the field in case (a), i.e. without odd viscosity.