Formation and stability of band patterns in a rotating suspension-filled cylinder

We explore the phenomenon of segregation and pattern formation in the complex system of a rotating horizontal cylinder completely filled with a dilute suspension of non-Brownian particles. A general dimensionless analysis is presented, which reveals the importance of the different dimensionless parameters involved. A detailed account of the mechanism of segregation and formation of axial bands for the case of low viscosity fluids is given. According to the analysis the axial pressure gradient associated with an inertial-mode excitation within the bounded fluid is responsible for the formation of bands in interleaving nodal planes of the excitation. The question of stability of the band patterns is addressed and a phase diagram in the appropriate dimensionless space is presented.

Figure 1

Graphically designed illustration of different stages observed for a suspension of heavy particles completely filling a rotating horizontal cylinder. (a) For very high frequencies the suspension forms a thin coating on the inner cylinder surface. (b) Just above the critical frequency $\left({\Omega }_H\right)$ the suspended particles form an off-centered tube. (c) In the interval ${\Omega }_L<\Omega <{\Omega }_H$ stable, periodically spaced, bands are observed. (d) Lowering the rotation frequency below ${\Omega }_L$ results in the breakdown of the suspended bands.

Figure 2

(a) Numerical (solid line) and theoretical (circles) trajectories for the low Reynolds number regime $({\Pi }_5=0.05)$. (b) Numerical (solid line) and experimental (circles) results for the projection of the trajectory of a 1.59 mm Nylon ball in a water-filled rotating horizontal cylinder. The inner tube diameter is $55.8\mathrm{mm}$ and the rotation rate is $4.9\mathrm{rad}∕\mathrm{s}$.

Figure 3

(Color online) (a) Projection of the velocity field in the vertical plane through the axis of rotation, onto that plane. (b) Perturbation pressure in the horizontal plane through the axis of rotation.

Figure 4

Scheme of a circular trajectory of a suspended particle in a flow which consists of a superposition of solid body rotation and the flow generated by the $(1,1,n)$ inertial mode. (a) The axial component of the velocity field is indicated. The corresponding drag is in the same direction. The net impulse per revolution in this case is zero. (b) Same as (a), but for the axial component of the pressure gradient.

Figure 5

Normalized net axial impulse acting on the suspended particle as a function of trajectory radius. The particle is assumed to undergo a circular uniform motion. Positive values correspond to stable bands for positively buoyant particles and vice versa.

Figure 6

Numerical simulation of the migration tendencies of a 1.59 mm Nylon ball placed in a flow field which consists of a superposition of solid body rotation and the $(1,1,n)$ inertial mode. Different initial positions result in migration to the same nodal plane ($z^{*}\cong 2$ in this instance) where the gravity-induced trajectory correlates the theoretical flow pattern. The Rossby number in this case is $\mathrm{Ro}=0.05$.

Figure 7

Phase diagram in the ${\Pi }_1\text{−}{\Pi }_3$ plane depicting the different possible trajectory scenarios of a single particle in a flow field which consists of a rigid rotation and the flow due to the $(1,1,n)$ inertial mode. These are the pinned to wall (I), the stationary and axially stable (II), the stationary and axially unstable (III), the stuck to bottom (IV), and the spiral motion toward a fixed point (V).