The distribution of capsules and rheological properties of suspensions in curved tubes are investigated by using an immersed-boundary lattice Boltzmann method. We mainly focus on the effective suspension viscosity and equilibrium positions of capsules as functions of Reynolds number $\left(\text{Re}\right)$, Capillary number $\left(\text{Ca}\right)$ and volume fraction of capsules $\left(\varphi \right)$. We found that at limited inertia ($\mathrm{Re}O\left(10\right)$), the effective viscosity decreases with increasing $\mathrm{Re}$, which is different from the variation trend in straight tubes. Dean's vortices play an important role. When the fluid inertia increases, the vortices are strengthened. They greatly promote the capsules' circumferential transportation by trapping the capsules into their centers and making the location of maximum azimuthal velocity close to them. The curvature effect of the torus vessel is also investigated. When the curvature $\kappa$ is large enough, e.g., $\kappa >0.2$, a scaling law for the effective viscosity as a function of a redefined $\mathrm{Re}$ is proposed. Furthermore, the distribution feature of multiple capsules in the torus vessel is revealed. Generally, for the semidilute regime, the capsules concentrate on the symmetrical plane at low $\mathrm{Re}$ but the center of Dean's vortex at high $\mathrm{Re}$. In addition, for both the dilute and semidilute regime, the scaling law connecting the effective viscosity and the average location of capsules is proposed. Our data support the scaling well. This study may be useful in the design of tubes for capsule transportation.

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• Fluid-particle interactions
• Suspensions
• Rheological properties
• Viscosity
• Immersed boundary methods
• Lattice-Boltzmann methods