Mean field surrogate model of a dilute and chaotic particle suspension

Abstract

We present a mean field approximation for a dilute suspension of small particles, which accounts for a non-homogeneous particle distribution and small inertial effects of the particle phase. The model consists of the momentum equation for the fluid coupled to a transport equation for the probability density function on the particle phase space. Apart from the wellknown particle extra stress tensor first presented by Batchelor (1970), the fluid equation contains additional force terms resulting from inertia and hydrodynamic interaction of the particles. A brief sketch of the main derivation steps is presented. Mean field modelIn [7] we derived a macroscopic model for the behavior of a dilute particle suspension, which accounts for small inertial particle effects. The model is based on the asymptotic approach for a single inertia-free particle by Junk and Illner [5], where the suspended bodies are characterized by the size ratio 1. By introducing a scaled density ratio ρ = r ρ, r ≥ −1 with ρ = ρ p /ρ f ratio of particle to fluid density, our extended approach allowed for the classification of different particle regimes (light-weighted, normal, heavy). The resulting asymptotic particle model was combined with the averaging framework by Batchelor [1] for the derivation of effective stresses. Because of the assumption of ergodicity, which is inherent in [1], it is unclear whether the description is also reasonable for non-homogeneous particle distributions in space. By applying the kinetic approach of [2] to the asymptotic particle model, we come up with a novel suspension model which covers the one in [7], is valid for non-homogeneous particle distributions and poses an approximation to the full suspension problem.In the following we present and discuss our new mean field model. Consider an open domain Ω ⊂ R 3 with boundary ∂Ω and a time interval (0, T ), T > 0, the macroscopic suspension velocity and pressure are given by (u 0 , p 0 ) : Ω × (0, T ) → R 3 × R and the particle probability density function (pdf) by ψ : Ω × SO(3) × (0, T ) → R + 0 on the particle phase space Ω × SO(3). The mean field description of a dilute particle suspension consists of the momentum balance of the fluid in Ω,with the mass conservation realized by the solenoidity condition for u 0 , coupled with a transport equation for the particle pdfin Ω × SO(3). The model is complemented with suitable initial and boundary conditions on ∂Ω for u 0 , ψ. For a smooth function M : SO(3) → R 3×3 with M(R) skew-symmetric, L[M] denotes the vector field associated with the left Lie derivative on SO(3), [3], i.e., for any smooth function θ : SO(3) → R (L[M](θ)). Given a basis of the skew-symmetric matrices. Moreover, ∇ and ∇· denote the gradient and divergence operators in R 3 , and B : R 3 → R 3×3 is the unique mapping which fulfills B(v) · x = v × x for all v, x ∈ R 3 . By ψ Ω (x, t) = SO(3) ψ(x, R, t) dR we denote the marginal pdf over Ω with the Haar measure dR on SO(3).The model (1) is characterized by the following parameters: Reynolds number Re (rati...