The classic experiments of Segre and Silberberg showed that a dilute suspension of spherical particles flowing down a pipe concentrates in a narrow ring [1]. Particles migrate across streamlines but eventually settle to an equilibrium radial position, approximately 0.6R (R is the pipe radius), where the inertial forces and wall forces balance. Recent experiments [2] suggest that there may be a second equilibrium position at high Reynolds numbers, but there is no theory for this as yet. Our investigation has so far focused on single particles and dilute suspensions in a square duct. Here we show the trajectories of individual particles from numerical simulations based on the lattice-Boltzmann model at a relatively low Reynolds number, Re = 100.
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Trajectories of individual particles released near the center of the duct.A quarter of the pipe cross-section is shown in each case. The thin solid lines are the fluid stream lines and the heavy line is the particle trajectory. |
Particles migrate normal to the fluid streamlines until they get close enough to interact with the wall. In the first case (left) the particle is released slightly offset from the center of the duct; it migrates directly to its equilibrium position. Similarly particles released along the diagonals migrate to the nearest corner (center). Particles placed in non-symmetric positions again cross the stream lines normally, but abruptly change direction when interactions with the wall become significant. In most cases particles land up in one of the corners, although there is region of stability near the center of the face at Reynolds numbers below 500. But individual particles do not migrate to positions near the center of the duct.
In multiparticle suspensions, an initially random distribution (a), evolves into three different steady-state distributions depending on Reynolds number.
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Snapshots of particle configurations in a duct flow at different Reynolds numbers: (a) Initial configuration (b) Re = 100 (c) Re = 500 (d) Re = 1000.The flow is into the plane of the page. |
At Re = 100 (b), particles are gathered around the eight equilibrium positions and strongly aligned in the direction of the flow, making linear chains of more or less uniformly spaced particles. Similar trains of particles were observed in recent laboratory experiments [2]. At Re = 500 (c), the particles are gathered in one of four stable positions near each corner. By a Reynolds number of 500, the trains are unstable and the spacing between the particles is no longer uniform. Instead transient aggregates of closely-spaced particles are formed, again near the corners of the duct. However at still higher Reynolds number, Re = 500 (d), there is another change in particle configuration, where particles appear in the center of the duct. A central band was first observed in experiments in a cylindrical flow [2], but its origin remains unclear. Since there are no single-particle equilibrium positions at the duct center, the presence of particles in the inner region must be due to multi-body interactions.
Even though the suspensions are dilute and there are no attractive forces, clusters of particles are observed at Reynolds numbers in excess of 300 (c). To understand the migration towards the center at high Re, we have investigated the migration of isolated pairs of particles (dumbbells). A dumbbell was constructed by connecting two particles with a stiff spring; the force-free separation of the pair was 1.05d, where d is the particle diameter. At Re = 100 the equilibrium position of a dumbbell is similar to that of a single particle, less than 0.2H from the wall.
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Equilibrium distance of a single particle (open circles) and a dumbbell (closed circles) from the nearest wall. Note the additional equilibrium positions near the center of the duct for dumbbells in high Reynolds number (Re > 700) flows. |
The dumbbell moves slightly closer to the wall as the Reynolds number increases, just like a single particle. However, at Re > 700 there is a sudden onset of additional equilibrium positions near the center of the duct. Thus particle clusters remain near the corner at Re < 500, but can move to the inner region when Re > 700 [3].