This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segre-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration.

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Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows

An experimental study of the migration of dilute suspensions of particles in Poiseuille flow at Reynolds numbers from the entrance, changes from one centred at the annulus predicted by the theory to one with the particles primarily on the inner annulus. The case of slightly non-neutrally buoyant particles was also investigated. A particle trajectory simulation based on asymptotic theory was performed to facilitate the comparison of theory and the experimental observations.

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Inertial migration of rigid spherical particles in Poiseuille flow

The observation of inhomogeneous radial distributions of particles in tube flow dates from the work of Poiseuille (1836) who was mainly concerned by the flow of blood and the behavior of the red and white corpuscles it carries. These results were then generalized to non-biological flows and experiments on pipe flow of suspensions also indicated that significant deviations from ideal Poiseuille flow could occur in the presence of particles. We will consider systems where the fluid flow in the absence of particles is unidirectional. We will first present how fluid-particle interactions can induce lateral migration in the case of a single rigid particle in a shear flow, as a function of the Reynolds number. While the focus is upon inertial migration, a brief discussion of lateral migration in polymeric and viscoelastic fluids, where the nonlinearity results from the non-Newtonian behavior of the suspending fluid, will be presented at the conclusion of this Section. The role of interparticle interactions in a sheared fluid will be considered in the third section in the case of Stokes flow. The last section will briefly present how sedimentation can affect lateral motion.

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Lateral Forces on a Sphere

In the mechanics of multiphase (or multicomponent) mixtures, one of the outstanding issues is the formulation of constitutive relations for the interaction force. In this paper, we give a brief review of the various relations proposed for this interaction force. The review is tilted toward presenting the works of those who have used the mixture theory (or the theory of interacting continua) to derive or to propose a relationship for the interaction (or diffusive) force. We propose a constitutive relation which is general and frame-indifferent and thus suitable for use in many flow conditions. At the end, we provide an alternative approach for finding the drag force on a particle in a particulate mixture. This approach has been used in the non-Newtonian fluid mechanics to find the drag force on surfaces.

Constitutive relations for the interaction force in multicomponent particulate flows

Philip Saffman made valuable theoretical contributions to different areas of low-Reynolds-number hydrodynamics. Three themes are selected for discussion here: (i) the lift force on a sphere in a shear flow at small, but finite Reynolds number, (ii) Brownian motion in thin liquid films, and (iii) particle motion in rapidly rotating flows. In addition, brief descriptions are given of some of Saffman's other contributions including dispersion in porous media, the average velocity of sedimenting suspensions, and compressible low-Reynolds-number flows.

Philip Saffman and viscous flow theory

In laminar flow of suspensions through a circular tube, particles migrate into a concentric annular region with the radius about 0.6 of the tube radius. This phenomenon is well-known as the tubular pinch effect. To clarify the mechanism, the influences of relative particle-fluid velocity, Reynolds number and particle diameter on radial particle displacement (lateral migration) were investigated by measuring the motion of a suspended spherical particle in laminar flow through a circular tube. Consequently, it was found that the motions of particles could be classified by the particle Reynolds number. And, especially, the relation between the particle Reynolds number and the equilibrium radial position of particles and the existence of critical particle Reynolds number for shifting from unilateral migration to bilateral migration were clarified.

Hiroshi AOKl

Papers

Study on the Tubular Pinch Effect in a Pipe Flow : I. Lateral migration of a single particle in laminar poiseuille flow