Abstract
In this paper, we present a continuum modeling for three-dimensional flows of non-colloidal, non-Brownian suspensions of particles immersed in a Newtonian liquid. Such suspensions exhibit complex behaviors such as jamming, anisotropic normal stresses and shear-induced particle migration. These non-Newtonian effects arise from the solid contact forces between the particles when the suspension is sufficiently concentrated. The modeling consists of a macroscopic one-phase model close to the "Suspension Balance Model" (Nott and Brady, 1994; Mills and Snabre, 1995; Morris and Boulay, 1999; Nott et al., 2011). The particles migration flux is governed by the divergence of the contact stresses tensor. In order to describe the flow in general 3D geometries, a frame-invariant constitutive law for the stresses in flowing suspensions is developed. It is similar to the second-order fluid modeling, which is well-known in polymer rheology, and allows for the presence of anisotropic normal stresses. The material functions are deduced from discrete simulation data from the literature. The behavior of the model in shear and extensional flows is discussed, as well as its limitations when used for a more general flow. To assess the modeling, numerical computations are performed using a finite volume method from the OpenFOAM suite. The implementation of the modeling is first validated by studying particles migration in some classical rheometric flows and then by studying the complex flow of a suspension in a tube through an abrupt expansion.