From discrete particles to continuum fields in mixtures

We present a novel way to extract continuum fields from discrete particle systems that is applicable to flowing mixtures as well as boundaries and interfaces. The mass and momentum balance equations for mixed flows are expressed in terms of the partial densities, velocities, stresses and interaction terms for each constituent. Expressions for these variables in terms of the microscopic quantities are derived by coarse-graining the balance equations, and thus satisfy them exactly. A simple physical argument is used to apportion the interaction forces to the constituents. Discrete element simulations of granular chute flows are presented to illustrate the strengths of the new boundary/mixture treatment. We apply the mixture formulation to confirm two assumptions on the segregation dynamics in particle simulations of bidispersed chute flows: Firstly, the large constituent supports a fraction of the stress that is higher than their volume fraction. Secondly, the interaction force between the constituents follows a drag law that causes the large particles to segregate to the surface. Furthermore, smaller particles support disproportionally high kinetic stress, which is a prediction of the theory on shear-induced segregation.


The Discrete Particle Method
Particles have position r i , velocity v i , angular velocity ω i , diameter d i , mass m i Governed by Newtonian mechanics: Contact forces and body forces:

Coarse graining Objective
Define continuous macroscopic fields such as mass density ρ, velocity V, stress σ, based on particle data The fields should satisfy mass and momentum balance exactly.
Example: A static system of 5 fixed and 5 free particles.
Coarse graining: density ρ and mass balance A) We define the macro-density using a coarse-graining function φ, Coarse graining: density ρ and mass balance A) We define the macro-density using a coarse-graining function φ, B) We define the velocity s.t. mass balance, Coarse graining: momentum balance C) We define stress σ and boundary interaction force density t such that momentum balance, is satisfied (t can be modelled as a boundary condition): Weinhart, Thornton, Luding, Bokhove, GranMat (2012) 14:289 A static system of 5 fixed and 5 free particles.  Magnitude of stress |σ| 2 and boundary interaction force |t| 2 .

Satisfying mass and momentum balance
Mass balance in a static system is trivial.

Satisfying mass and momentum balance
Mass balance in a static system is trivial. Satisfying the momentum balance in a static system requires Magnitudes of stress divergence |∇ · σ| 2 (left), boundary interaction force density |t| 2 (centre), and weight density |ρg| (right).

Important result
Mass/mom. bal. is satisfied locally for any coarse-graining function.
How do we define stress and interspecies drag in mixtures?

Coarse graining in mixtures
For both species ν = s, l, we define

Coarse graining in mixtures
For both species ν = s, l, we define
A static system of 10 small and 5 large particles. Discrete Description: small particles fall easier through holes Continuum description: Large particle phase supports more downward stress than small particles phase ("overstress"):

Segregation equation
We define large particle volume fraction φ l = ρ l ρ and large particle stress fraction f l = σ l zz σzz , and describe the overstress by

Segregation equation
We define large particle volume fraction φ l = ρ l ρ and large particle stress fraction f l = σ l zz σzz , and describe the overstress by We further assume a drag law, with drag coefficient c,

Segregation equation
We define large particle volume fraction φ l = ρ l ρ and large particle stress fraction f l = σ l zz σzz , and describe the overstress by We further assume a drag law, with drag coefficient c, Then momentum balance for shallow flow yields  Mean relative velocity of large phase, w l −w , over time t.

Observation
Segregation kinetics can be measured accurately.  Figure : Kinetic stress fraction f ν = σ k,ν zz /σ k zz in steady state as a function of volume fraction φ ν = ρ ν /ρ, for each constituent ν = s, l and fit f l,fit for B = −0.38. Observation f l,kin < φ l , as required for shear-induced segreg. (Fan Hill 2011).

Conclusions
1 Coarse-graining formulation for mixtures allows measuring the species' stress fraction and the interspecies drag.
2 Segregation kinetics are much slower than equilibration.
3 Large particles support a stress fraction (slightly) higher than their volume fraction, as postulated for gravity-driven segregation. Segregation velocity roughly follows the drag law used in (Gray Thornton 2005).