Section: New Results

A Local Lubrication Model for Spherical Particles within an Incompressible Navier-Stokes Flow

The lubrication effects are short-range hydrodynamic interactions essential to the suspension of the particles, and are usually underestimated by direct numerical simulations of particle laden flows.

A local lubrication correction model for particle laden flow of spherical solid particles has been presented and validated. Interactions between a particle and an obstacle (another particle or a wall) can be decomposed into three types: long range hydrodynamics, short range hydrodynamics also called lubrication effects, and mechanical solid-solid contacts.

Long range hydrodynamic interaction are fully resolved by the Volume Penalization method (VP). The incompressible Navier-Stokes equations have been discretized in time using a scalar projection method and in space with a fully second order penalty method.

Due to unresolved scales associated with the grid, short range hydrodynamic interactions are only partially captured by the numerical approach. We thus introduce a local lubrication model. This correction is based on asymptotic expansions of analytical solutions of particle-particle or particle-wall interactions, assuming that the flow within the gap between the particle and the obstacle is in the Stokes regime. Lubrication forces and torques are corrected in a neighborhood of the contact point of two interacting particles where lubrication is poorly captured, as long as the normalized gap width $ϵ$ is smaller than a critical length ${ϵ}_{\text{lub}}$ (a model parameter).

Finally, a linear soft-sphere collision model is used for solid-solid contacts. This model, widely used in the literature [Costa15,Izard14], represents mechanical contacts as two spring-dashpot systems connected at the contact point. The model allows stretching the collision time, to avoid computational overhead in the calculation of the collision force, making the method particularly efficient.

Our local lubrication correction model have been validated on several benchmarks. First, we considered a single particle falling onto a wall at various approach velocities. The comparison with experimental results [Harada01,Joseph01] enables us to validate the dominant lubrication component resulting from the squeezing of the fluid in the gap. The lubrication force and the torque created by the shearing of the fluid in the gap have been validated on oblique particle-wall collisions in dry and wet systems proposed by Joseph and Hunt [Joseph04] . Since lubrication corrections are made locally, our lubrication model does not required tabulation and is compatible to non-spherical particles. The model will be tested for polydisperse flow of ellipsoidal particles in future works.