The hydrodynamic behavior of a squirmer swimming in the flow of power-law fluid is studied numerically with an immersed boundary-lattice Boltzmann method. The power-law fluids with three typical power-law indexes (n = 0.5, 1, and 1.5) are selected for embodying the characteristics of non-Newtonian fluid. The results show that, at zero Reynolds number, the squirmers swim slower in the shear-thinning fluid (n = 0.5) than in the Newtonian fluid (n = 1). The speed of squirmer swimming increases monotonically with the Reynolds number in the shear-thickening fluid (n = 1.5). When the fluid inertia is taken into account, the speed of a pusher (a kind of swimmer type) increases monotonically with the Reynolds number, while the speed of a puller (another kind of swimmer type) shows a non-monotonic increasing tendency. Some critical points, beyond which the pullers swim with lateral displacement, are obtained. The flow field and force distribution around the squirmer are calculated and analyzed. The power expenditure of squirmers in different kinds of flow is also studied. It is found that, as the Reynolds number increases, the power expenditure generally decreases in shear-thinning fluid, which is different from the case in Newtonian fluid and shear-thickening fluid. Finally, the hydrodynamic efficiency is introduced to compare the power expenditure of squirmers with that of a counterpart particle towing in the same type of fluid. Results show that the hydrodynamic efficiency of pushers increases with the Reynolds number, while the hydrodynamic efficiency of pullers shows non-monotonic behavior when the Reynolds number is varied.

We simulated the sedimentations of two unequal spheres with different densities in a square tube for Galileo numbers (Ga) from 5 to 25, resulting in a Reynolds number range of $0.8\leqslant Re_{T}\leqslant 17.3$ based on the terminal settling velocity. The sedimentation of spheres with different densities is dynamically more complex than that of identical spheres. At high Ga the spheres oscillate in the centreline plane of the tube, where they are initially released from rest. By contrast, the spheres move to the diagonal or reverse–diagonal plane of the tube at low Ga, reaching a steady or periodic state depending on the density difference between them. A phase diagram illustrates the transitions between different sedimentation behaviours depending on Ga and the density difference. A possible mechanism for these behaviours is also presented. Furthermore, we compare two-dimensional (2-D) and three-dimensional computations for our system to attain a better understanding of the hydrodynamic interactions between two unequal spheres at low but finite Reynolds number. Comparing relative trajectories, periods of oscillation and flow features shows that 2-D circular cylinders oscillate much more strongly and frequently than spheres under the same flow conditions. In particular, spheres do not have the discontinuity in period that arises in the 2-D case from the change in rotation sign of a heavy particle.

We simulated the motions and interactions of circular squirmers under gravity in a two-dimensional channel at finite fluid inertia, aiming to provide a comprehensive analysis of the dynamic features of swimming microorganisms or engineered microswimmers. In addition to a squirmer-type factor (β), another control parameter (α) was introduced, representing ratio of the self-propelling strength to the sedimentation strength of squirmers. Simulations were performed at 0.4 ≤ α ≤ 1.2 and −5 ≤ β ≤ 5. We first considered the sedimentation of a single squirmer. Five patterns were revealed, depending on both α and β: steady downward falling, steady inclined falling or rising and small-scale or large-scale oscillating. Compared with a pusher (β < 0, gaining thrust from rear), a puller (β > 0, gaining thrust from front) is more likely to break down its symmetrical structure and subsequently lose stability, owing to the high-pressure regions on its lateral sides. Typically, a pusher settles faster than a puller, whereas a neutral squirmer (β = 0) settles in between. This is related to the ‘trailing negative flow’ behind a pusher and ‘leading negative flow’ before a puller. We then placed two squirmers in line with the gravity direction to study their interactions. Results show pullers attract each other and come into contact as a result of the low-pressure regions between them, whereas the opposite is observed for pushers. The interactions between two pullers are illustrated by their respective patterns. In contrast, pushers never come into contact and maintain distance from each other with increasing separation. We finally examined how a puller interacts with a pusher.