Momentum balance of a laminar flow over a bed of particles

TL;DR

This paper introduces a new framework using immersed boundary methods to analyze the momentum balance in laminar particle-laden flows, explicitly considering fluid inside particles and evaluating stress balances in multiple directions.

Contribution

It develops a novel approach to close the momentum balance for particle-laden flows by explicitly accounting for fluid within particles and analyzing stress balances in both streamwise and vertical directions.

Findings

Stresses remain in equilibrium during unsteady flow conditions.

Vertical stress balances reveal the roles of collisions and hydrodynamic drag.

A correlation between shear rate and mixture stress gap suggests a way to close the mixture momentum balance.

Abstract

We develop a framework for analyzing the momentum balance of laminar particle-laden flows based on immersed boundary methods, which solve the Navier-Stokes equations and resolve the particle surfaces. This framework differs from previous studies by explicitly accounting for the fluid inside the particles, which is a by-product of the immersed boundary method, allowing us to close the momentum balance for the flow around a single rolling sphere. We then compute a momentum balance of a laminar Poiseuille flow over a dense bed of particles, finding that the stresses remain in equilibrium even during unsteady flow conditions. While previous studies have focused on stresses for the streamwise momentum balance, the present approach also allows us to evaluate stress balances in the vertical direction, which are necessary to understand the role that collisions and hydrodynamic drag play during dilation and contraction of particle beds. While our analysis accounts for the fluid and particle phases separately, we attempt to establish a momentum balance for the fluid/particle mixture, but find that it does not completely close locally due to collision stresses not being resolved across the particle diameter. However, we find a correlation between the local shear rate and the gap in the mixture balance, which can potentially be used to close the balance for the mixture.