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.

The mechanical and thermal behavior of nonisothermal fiber-filled composites in a three-dimensional printing process is studied numerically with a smoothed particle hydrodynamics method. A classical microstructure-based fiber suspension model with a temperature-dependent power-law viscosity model and a microstructure constitutive model is implemented to model a fiber-filled system. The fiber microstructure is described by a second-order tensor A₂ which describes the spatially averaged orientation of the fibers. Two benchmark cases are presented to validate the reliability of the present implementation. Three typical printing modes are tested to assess the characteristics of printed layers. The results show that the printed layer becomes thicker, and the fiber alignment in the printing direction is enhanced in the bottom half of the layer and reduced in the top half due to the existence of nonisothermal effects in the process. The variation in fiber orientation becomes larger with increasing fiber concentration. By increasing the Peclet number, the deposited layer thickness reduces and the fiber alignment in the printing direction is enhanced in the top half and reduced in the bottom half. The evolution of the orientation and the velocity gradient tensors projected along several streamlines are discussed to illustrate the effects of the temperature and different printing modes on the deposited layer.

We study the hydrodynamics of a spherical and dumbbell-shaped microswimmer in a tube. Combined with a squirmer model generating tangential surface waves for self-propulsion, a direct-forcing fictitious domain method is employed to simulate the swimming of the microswimmers. We perform the simulations by considering the variations of the swimming Reynolds numbers (Re), the blockage ratios (κ) and the relative distances (d_s) between the squirmers of the dumbbell. The results show that the squirmer dumbbell weakens the inertia effects of the fluid more than an individual squirmer. The constrained tube can speed up an inertial pusher (propelled from the rear) and an inertia pusher dumbbell; a greater distance d_s results in a slower speed of an inertial pusher dumbbell but a faster speed of an inertial puller (propelled from the front) dumbbell. We also illustrate the swimming stability of a puller (stable) and pusher (unstable) swimming in the tube at Re = 0. At a finite Re, we find that the inertia and the tube constraint competitively affect the swimming stability of the squirmers and squirmer dumbbells. The puller and puller dumbbells swimming in the tube become unstable with increasing Re, whereas an unstable–stable–unstable evolution is found for the pusher and pusher dumbbells. With increasing κ, the puller and puller dumbbells become stable while the pusher and pusher dumbbells become unstable. In addition, we find that a greater d_s yields a higher hydrodynamic efficiency η of the inertial squirmer dumbbell.

The deformation and breakup of viscoelastic drops in simple shear flows of Newtonian liquids are studied numerically. Our three-dimensional numerical scheme, extended from our previous two-dimensional algorithm, employs a diffusive-interface lattice Boltzmann method together with a lattice advection–diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids and the latter to describe the polymer dynamics modeled by the Oldroyd-B constitutive model. A block-structured adaptive mesh refinement technique is implemented to reduce the computational cost. The multiphase model is validated by a simulation of Newtonian drop deformation and breakup under an unconfined steady shear, while the coupled algorithm is validated by simulating viscoelastic drop deformation in the shear flow of a Newtonian matrix. The results agree with the available numerical and experimental results from the literature. We quantify the drop response by changing the polymer relaxation time λ and the concentration of the polymer c. The viscoelasticity in the drop phase suppresses the drop deformation, and the steady-state drop deformation parameter D exhibits a non-monotonic behavior with the increase in Deborah number De (increase in λ) at a fixed capillary number Ca. This is explained by the two distribution modes of the polymeric elastic stresses that depend on the polymer relaxation time. As the concentration of the polymer c increases, the degree of suppression of deformation becomes stronger and the transient result of D displays an overshoot. The critical capillary number for unconfined drop breakup increases due to the inhibitive effects of viscoelasticity. Different distribution modes of elastic stresses are reported for different De.

We investigate the hydrodynamics of a spherical and dumbbell-shaped swimmer in a viscoelastic fluid, modelled by the Giesekus constitutive equation. The ‘squirmer’, a model of a micro-swimmer with tangential surface waves at its boundaries, is simulated utilizing a direct-forcing fictitious domain method. We consider the competitive effects of the fluid inertia and elasticity on the locomotion of the swimmers. For the neutral squirmer, its speed increases monotonically with increasing Reynolds number in the Giesekus fluid, in contrast to holding a constant speed in a Newtonian one. Meanwhile, the speed of the finite inertial squirmer increases monotonically with fluid elasticity, as measured by the Weissenberg number. Regarding the dumbbell-shaped swimmers in the Giesekus fluid, we find the pusher–puller (the puller is in front of the pusher) dumbbell swimming to be significantly faster than other dumbbells, providing practical guidance in assembling the complex micro-swimming devices. We further consider the squirmer's (or the dumbbell-shaped squirmer's) energy expenditure and hydrodynamic efficiency, finding that swimming in viscoelastic fluids expends less energy than its Newtonian counterpart, and the neutral squirmer expends more energy than a pusher or puller. The energy expenditure and hydrodynamic efficiency are relevant to steady swimming speeds, in which a faster counterpart squirmer (dumbbell-shaped squirmer) expends less energy but has higher hydrodynamic efficiency.

The hydrodynamic behavior of a two-dimensional elongated micro-swimmer (squirmer rod) in a bulk fluid is studied numerically with an immersed boundary-lattice Boltzmann method. The swimming Reynolds numbers, the number of the squirmers (squirmer is a self-propelled model to assemble the elongated micro-swimmer), and the relative distance between two adjacent squirmers' centers are, respectively, set being 0.05 ≤ Re_s ≤ 5, 2 ≤ i ≤ 8, and 0.75d ≤ s ≤ 1.5d (d is the diameter of the squirmer) to investigate the swimming speed, the power expenditure, and the hydrodynamic efficiency of the micro-swimmer. The results show that the swimming speed of the puller rod (a type of elongated micro-swimmers propelled from the front) increases monotonically with enhancing i. The pusher rod (a type of elongated micro-swimmers propelled from the rear) with more pushers i swims faster at Re_s ≤ 0.1, whereas it swims slower at Re_s > 0.1. The speed of the pusher rod increases monotonously with the decrease in the distance s, whereas a non-monotonous variation is found for the puller rod. Meanwhile, the more i or smaller s for the puller and pusher rods, the more power P they expend, and no appreciable distinction is found for their P at 0.05 ≤ Re_s ≤ 1. By continuing to increase Re_s (Re_s > 1), P increases monotonically for both the puller and pusher rods, and it is more appreciably for the pusher rods. Finally, the hydrodynamic efficiency η of the pusher rods increases monotonically with the translational Reynolds number Re_U, in contrast to that of the puller rods, which decreases (Re_U ≤ 1) and then increases (Re_U > 1) with Re_U. For the pusher rods at Re_U > 1, the increasing ratio of η is proportional to ${\text{Re}}_{\mathrm{U}}^{0.7}$ . The higher η is found for the squirmer rods with smaller i or greater s.