Particle dynamics and pattern formation in a rotating suspension of positively buoyant particles

Numerical simulations of positively buoyant suspension in a horizontally rotating cylinder were performed to study the formation of radial and axial patterns. The order parameter for the low-frequency segregated phase and dispersed phase is similar to that predicted for the settling suspension by Lee and Ladd [J. Fluid Mech. 577, 183 (2007)], which is the average angular velocity of the particles. The particle density profiles for axial bands in the buoyancy-dominated phase shows an amplitude equivalent to the diameter of the cylinder. Axial density profiles show sinusoidal behavior for the drag-dominant phase and oscillating sinusoidal behavior for the centrifugal-force-dominant phase. Results also indicate that the traveling bands are formed as a consequence of the inhomogeneous distribution of particles arising from a certain imbalance of drag, buoyancy, and centrifugal forces. In the centrifugal limit, particles move towards the center of the cylinder, aggregating to form a dense core of particles with its axis coinciding with that of the rotating cylinder, a behavior which is in contrast to the sedimenting particles. The particle distribution patterns obtained from the simulations are found to be in good agreement with the experiments of Kalyankar et al. [Phys. Fluids 20, 083301 (2008)].

Figure 1

The circles $C_1$ (blue) and $C_2$ (red) describe the locus of zero radial and angular velocities, respectively. Panel (a) corresponds to the drag dominant phase with the point P indicating the dynamical center of the system, and panel (b) corresponds to centrifugal dominant phase; here the point A indicates the stagnation point.

Figure 2

Particle trajectories at different velocities of the cylinder rotating counter-clockwise as indicated by the arrow. Initial position of the particle in all three cases is $(r,\theta ,z)=(0.5,0,0)$. (a) $\mathrm{\Omega }=0.1$: drag-dominant regime, the particle rises to reach the cylinder wall and gets dragged down because of rotation. (b) $\mathrm{\Omega }=1$: balance of gravity, drag, and centrifugal forces imposing a closed trajectory to the particle. (c) $\mathrm{\Omega }=10$: centrifugal-force-dominant regime, the particle swirls inward to the center.

Figure 3

Schematic representation of a rotating suspension in a horizontal cylinder.

Figure 4

Velocity vectors and particle distribution at steady state for a buoyancy-dominated regime. Red particles move downwards, whereas the blue particles move against gravity. (a) $\mathrm{\Omega }=0.09\mathrm{rad}/\mathrm{s}$, particles form a bed at the top (GB); (b) $\mathrm{\Omega }=0.12\mathrm{rad}/\mathrm{s}$, particles which are dragged along the cylinder tend to rise (F1); (c) $\mathrm{\Omega }=0.15\mathrm{rad}/\mathrm{s}$, more particles are dispersed into the cylinder (F1/F2); (d) $\mathrm{\Omega }=0.3\mathrm{rad}/\mathrm{s}$, particles transit into the right half of the cylinder (LT).

Figure 5

Phase evolution for drag-dominated regime. Here $n_r$ represents the number of rotations of the cylinder, red particles move down and blue particles move up. Black particles indicate initial configuration $\left(n_r=0\right)$.

Figure 6

Velocity vectors and particle distribution at steady state for the centrifugal-force-dominating regime. (a) $\mathrm{\Omega }=0.75\mathrm{rad}/\mathrm{s}$, particles are dispersed throughout the cylinder (HR); (b) $\mathrm{\Omega }=15\mathrm{rad}/\mathrm{s}$, increased centrifugal force propels particles towards the rotating axis (DB); (c) $\mathrm{\Omega }=30\mathrm{rad}/\mathrm{s}$, (CL) particles congregate around the center of the cylinder; (d) $\mathrm{\Omega }=35\mathrm{rad}/\mathrm{s}$, (CL) further increased velocity produces no change in the qualitative behavior of the particles. Red particles move downwards, whereas the blue particles move against gravity. Magenta indicates radially inward motion.

Figure 7

Phase evolution for centrifugal-force-dominated regime. Black is used for the initial configuration. Here $n_r$ represents the number of rotations of the cylinder; red particles move down, and blue particles move up. Magenta indicates inward motion towards the center of the cylinder.

Figure 8

Equilibrium concentration profiles in the radial direction for different frequencies of rotation. Here $n_p$ and $N_0$ are the number of particles after reaching steady state and at $t=0$, respectively.

Figure 9

Order parameter $\left(Q\right)$ for various particle concentrations is plotted versus the two dimensionless numbers (a) $\mathrm{\Omega }a/u_f$ and (b) $\mathrm{\Omega }d/u_f$, where $n_0$ is the average particle concentration, $a$ is the particle radius, and $d$ is the mean interparticle separation $\left(d=n_0^{-1/3}\right)$.

Figure 10

Axial patterns observed in the centrifugal-force-dominated regime. Red particles move towards the right, and the blue particles move to the left. (a) Aandom initial configuration of particles suspended in the cylinder; (b) HR phase, $\mathrm{\Omega }=1.25\mathrm{rad}/\mathrm{s}$, uniform distribution of particles throughout the cylinder; (c) LD, $\mathrm{\Omega }=2.5\mathrm{rad}/\mathrm{s}$ (top view, gravity is pointing into the plane of the paper), and particles converge into high-concentration regions while rising and spread out as they reach the top; (d) LD, $\mathrm{\Omega }=2.5\mathrm{rad}/\mathrm{s}$ (front view, gravity is pointing downward).

Figure 11

Concentration profiles at three different instants indicating exchange of particles (traveling bands) along the axial direction for particle density, ${\rho }_p=0.15\mathrm{g}/\mathrm{cc}$ and $\mathrm{\Omega }=2.5\mathrm{rad}/\mathrm{s}$. Here $n_p$ and $N_0$ are the number of particles after reaching steady state and at $t=0$, respectively.

Figure 12

Central core formation in the centrifugal limit (CL) for $\mathrm{\Omega }=30\mathrm{rad}/\mathrm{s}$. Particles congregate around the rotational axis of the cylinder reaching the maximum packing limit for a suspension.

Figure 13

Axial segregation $\left({\rho }_p=0.19\mathrm{g}/\mathrm{cc}\right)$ in the stable band phase for $\mathrm{\Omega }=2.25\mathrm{rad}/\mathrm{s}$: (a) front view at 800 rotations of the cylinder; (b) top view at 800 rotations; (c) front view at 900 rotations; (d) top view at 900 rotations. Red particles move towards the right, and the blue particles move towards the left. Gravity points downward on the plane of the paper in front view and into the plane of the paper in top view.

Figure 14

Equilibrium concentration profiles of particles for ${\rho }_p=0.19\mathrm{g}/\mathrm{cc}$ at $\mathrm{\Omega }=2.25\mathrm{rad}/\mathrm{s}$. There is almost no exchange of particles along the rotating axis over time indicating stable band patterns. Here $n_p$ and $N_0$ are the number of particles after reaching steady state and at $t=0$, respectively.