Dynamics of bidensity particle suspensions in a horizontal rotating cylinder

We report Stokesian dynamics simulations of bidensity suspensions rotating in a horizontal cylinder. We studied the phase space and radial and axial patterns in settling as well as floating systems. Each system was composed of particle mixtures of two different densities. As many as eight unique phases are identified for each system along the radial plane. The bidensity system shows similarity to the monodisperse case only when the radial distribution of the particles is completely uniform. Characteristic behavior of the bidensity systems is identical at low rotation rates and contrasting when centrifugal force dominates. Expressing the phase boundaries in terms of dimensionless parameters $U_s/\left(\mathrm{\Omega }R\right)$ and $g/\left({\mathrm{\Omega }}^{2}R\right)$ gives a linear fit unifying the data in the gravity-dominated regime. At high rotation rates, the behavior is opposing for either system though linear in nature. In the axial direction, number density profiles of both systems affirm the phenomenon of band formation. Location of the axial bands remains the same for heavy and light particles in both systems. We have also reestablished that an inhomogeneous particle configuration in the radial plane induces growing instabilities in the axial plane which amplify to form particle bands similar to monodisperse suspensions.

Figure 1

Schematic sketch of the rotating cylinder system.

Figure 2

Initial configuration in (a) radial direction and (b) axial direction; Particles are color coded as red—light, blue—heavy. The number of particles in the radial and axial simulations were 2300 and 12465 respectively.

Figure 3

Phases in the gravity dominated regime for the settling system with a density ratio 2. (a) Sedimenting phase at $\mathrm{\Omega }$ = 0.1 rad/s, (b) formation of the heavy particle core at $\mathrm{\Omega }$ = 0.13 rad/s, (c) heavy particles segregated from the light particles at $\mathrm{\Omega }$ = 0.6 rad/s, (d) ascending values of $\mathrm{\Omega }$ (at 0.9 rad/s) cause the segregated particles in the previous phases to mix partially.

Figure 4

Number density plots in the gravity dominated regime for (a) heavy and (b) light particles; here, $n_p$ and $N_0$ are the number of particles at steady state and $t=0$, respectively.

Figure 5

Centrifugal force dominated regime for settling system with density ratio 2: (a) complete mixing of heavy and light particles due to balance in the forces ($\mathrm{\Omega }$ = 1.2 rad/s), (b) PM-II phase at $\mathrm{\Omega }$ = 2.5 rad/s heavy particles guided towards the wall, (c) S-II occurs at $\mathrm{\Omega }$ = 7.5 rad/s segregating the heavy particles from the light ones due to increased centrifugal force, (d) complete centrifugation of either of the particles at $\mathrm{\Omega }$ = 12 rad/s.

Figure 6

Number density plots in the centrifugal force dominated regime for (a) heavy particles, (b) light particles; here, $n_p$ and $N_0$ are the number of particles at steady state and $t=0$, respectively.

Figure 7

Final particle concentration contours for the settling system with a density ratio 2 in gravity dominant regime. (a) SD at $\mathrm{\Omega }$ = 0.1 rad/s, (b) CF at $\mathrm{\Omega }$ = 0.13 rad/s, (c) S-I at $\mathrm{\Omega }$ = 0.6 rad/s, (d) PM-I at $\mathrm{\Omega }$ = 0.9 rad/s.

Figure 8

Fianl particle concentration contours for the settling system with a density ratio 2 in centrifugal force dominant regime. (a) M at $\mathrm{\Omega }$ = 1.2 rad/s, (b) PM-II at $\mathrm{\Omega }$ = 2.5 rad/s, (c) S-II at $\mathrm{\Omega }$ = 7.5 rad/s, (d) CL at $\mathrm{\Omega }$ = 12 rad/s.

Figure 9

Axial banding particle configurations of heavy (column I) and light (column II) particles after 1050 rotations of the cylinder. Direction of gravity is into the plane of the paper in (a) and (b) implying a top view and is downward in (c) and (d) which represent front view. Blue and orange particles have rightward motion while silver and gray particles move leftwards. $\mathrm{\Omega }$ = 3.5 rad/s and density ratio is 2.

Figure 10

Number density plots for axial bands in the settling system (a) heavy particles, (b) light particles; here, $n_p$ and $N_0$ are the number of particles at steady state and $t=0$, respectively.

Figure 11

Phases exhibited by the floating system with density ratio 2 when gravity dominates (a) Floating at $\mathrm{\Omega }$ = 0.1 rad/s, (b) CF which occurs at $\mathrm{\Omega }$ = 0.125 rad/s is also mirror image to CF of the settling system, (c) S-I at $\mathrm{\Omega }$ = 0.5 rad/s with undisturbed heavy particle pool, (d) PM-I with almost fully dispersed particles at $\mathrm{\Omega }$ = 1.0 rad/s.

Figure 12

Number density plots in the gravity dominated regime for (a) light particles and (b) heavy particles; here, $n_p$ and $N_0$ are the number of particles at steady state and $t=0$, respectively.

Figure 13

Phases exhibited with density ratio 2 when centrifugal force dominates (a) M at $\mathrm{\Omega }$ = 1.0 rad/s complete mixedness, (b) PM-II at $\mathrm{\Omega }$ = 2.75 rad/s inhomogeneous distribution of light particles, (c) S-II at $\mathrm{\Omega }$ = 7 rad/s, (d) CL at $\mathrm{\Omega }$ = 12 rad/s maximum packed radial core.

Figure 14

Number density plot for phases in the centrifugal force dominated regime (a) heavy particles (b) light particles, here $n_p$ and $N_0$ are the number of particles at steady state and $t=0$, respectively.

Figure 15

Final particle concentration contours for the floating system with a density ratio 2 in gravity dominant regime. (a) FT at $\mathrm{\Omega }$ = 0.1 rad/s, (b) CF at $\mathrm{\Omega }$ = 0.125 rad/s, (c) S-I at $\mathrm{\Omega }$ = 0.5 rad/s, (d) PM-I at $\mathrm{\Omega }$ = 0.9 rad/s.

Figure 16

Final particle concentration contours for the floating system with a density ratio 2 in gravity dominant regime. (a) M at $\mathrm{\Omega }$ = 1.0 rad/s, (b) PM-II at $\mathrm{\Omega }$ = 2.75 rad/s, (c) S-II at $\mathrm{\Omega }$ = 7.0 rad/s, (d) CL at $\mathrm{\Omega }$ = 12.0 rad/s.

Figure 17

Axial configuration for floating system after 1050 rotations. Direction of gravity is into the plane of the paper in (a) and (b) implying a top view and is downward in (c) and (d) which represent front view. Blue and orange particles have rightward motion while silver and gray particles move leftwards.

Figure 18

Number density profiles for the floating system in the axial direction at 850 and 1050 rotations of the cylinder for (a) light and (b) heavy particles at $\mathrm{\Omega }$ = 4.25 rad/s; here, $n_p$ and $N_0$ are the number of particles at steady state and $t=0$, respectively.

Figure 19

Transition boundaries for settling and floating systems in the gravity dominated regime. Open circles and triangles denote the floating and settling system, while the dotted line is a linear fit. (a) SD(FT)—CF, (b) CF—S-I, (c) S-I—PM-I.

Figure 20

Transition boundaries for the systems in centrifugal force dominated regime S-II—CL. Open circles and triangles denote the floating and settling system, while the dotted lines are linear fits.