Sedimentation of two spheres in active nematic fluids confined in a square tube

We investigate the sedimentation behavior of two identical spheres settling in a square tube filled with active nematic fluids. This study explores a wide range of activity strengths (0 ≤ |ζ| ≤ 0.0025) and viscosities (0.094 ≤ μ ≤ 0.28), encompassing both extensile and contractile fluids with various anchoring conditions. A total of nine distinct sedimentation patterns are identified, including the steady (CLS) and helical (CLH) state along the centerline, steady (CPS) and periodic (CPP) state in the centerline plane, periodic state at the center of the wall (CWP), juxtaposed (DJP) and abreast (DAP) periodic state in the diagonal or reverse-diagonal plane, as well as the movement to the corner (MC) and chaotic state (CS). We introduce two dimensionless parameters, the Ericksen number Er = μVd/2K and the ratio of viscous to active force Rva = 2 μV/|ζ|d, to quantify the boundary of pattern transitions. We find that the maximum relative angle between the two spheres during the oscillation process (DJP) determines whether the pair evolves into a new stable configuration, exhibiting helical trajectories (CLH), or undergoes damped oscillations and aligns more closely with gravity (CLS). Additionally, the equilibrium spacing in the CPS governs whether the spheres can approach sufficiently close, which leads to the defect-mediated elastic interaction, potentially facilitating a transition toward the CLS pattern. Other sedimentation patterns, excluding the MC and CS, are also analyzed. The settling velocities are compared across different patterns. For strong normal anchoring, the settling efficiencies follow the order: CLS > CLH > DJP > CPS. With strong planar anchoring, we observe: CPP ≈ CPS > DAP. For weak anchoring, the trend is CPS > CWP.