Study of sedimentation characteristics of an elliptical squirmer in a vertical channel

We used a two-dimensional lattice Boltzmann method to simulate the sedimentation motion of an elliptical squirmer in a vertical channel, taking into account the case of a circular squirmer, aiming to more realistically simulate the swimming of microorganisms in nature. The study in this was divided into two phases. The first phase comprised the numerical calculations of an elliptical squirmer with an aspect ratio of c = 2.0 and revealed three typical motion modes: steady inclined motion, wall-attraction oscillation, and large-amplitude oscillation. It was found that the formation of these three motion modes and transitions between modes are related to the pressure distribution formed between the elliptical squirmer and wall. In addition, significant differences exist between the motions of elliptical and circular squirmers. The force generated by the interaction between the elliptical squirmer and wall does not all point towards its center of mass, resulting in an additional torque on the elliptical squirmer; this is not the situation for the circular squirmer. The second phase of the study simulated squirmers with different aspect ratios (c = 1.0, c = 3.0). It was found that for an elliptical squirmer with an aspect ratio c = 3.0, the large-amplitude oscillation mode (among the above three motion modes) no longer exists. By combining the motion modes of a circular squirmer in the channel, it can be observed that as the aspect ratio c increases, the squirmer’s head direction tends to be more vertical, which may reduce the drag force during swimming.


Introduction
In recent years, active particles such as swimming microorganisms and artificial microswimmers have received widespread attention, owing to their relevance in biophysical and industrial technology applications.Microorganisms (e.g., diatoms, Chlamydomonas) can be used for processes such as environmental remediation and water treatment [1][2][3][4][5]; meanwhile, synthetic self-propelled particles (e.g.artificial cells, micro-swimming devices) have been designed for targeted drug delivery [6][7][8][9], precision surgery [10][11][12], etc.Compared to passive particles, self-propelled particles have an intrinsic self-propelling mechanism and display richer and more complex dynamic characteristics.For example, researchers have observed that Escherichia coli (E.coli) has a selfassembly behavior and have successfully applied it for the unidirectional rotation of nanofabricated objects, also known as bacterial ratchet motors [13].Chlamydomonas reinhardtii (C.reinhardtii) performs an unusual 'runand-tumble' motion in the dark [14] and exhibits phototropism in the presence of light sources [15].When Volox is present in a liquid with chemical and nutrient gradients, it exhibits chemotaxis and viscosity, causing it to swim against the flow and develop unique swimming characteristics [16,17].More interestingly, when two Volvox colonies swim near a wall, they exhibit a behavior pattern similar to 'dancing' [18].The two Volvox colonies first attract each other, and then 'waltz' or 'minuet' with each other.In addition, active particles are accompanied by many significant collective behaviors such as bioconvective motion [19][20][21], enhanced transport [22,23], and phase separation [24].
Currently, the most popular approach to simulating the swimming of self-propelled particles is the classical squirmer model proposed by Lighthill [25] and Blake [26,27].This approach is aimed at three-dimensional (3D) spherical squirmers or two-dimensional (2D) circular squirmers.Since its origin, this model has been widely used in numerical and theoretical studies and its feasibility has been confirmed.Li and Ardekani [28] used a squirmer model to numerically simulate the motions of microorganisms with different swimming modes near a wall and found that there were several different swimming behaviors, including moving away from the wall after collision, swimming parallel to the wall, and periodic oscillation along the wall.The numerical simulation results indicated that the wall had an attractive effect on the microorganisms.Li and Ardekani [28] varied the initial distance and orientation angle between the squirmer and wall and found that these two initial variables only affected the specific position between the squirmer and wall and did not affect the swimming behavior of the squirmer near the wall.Zöttl and Stark [29] studied the 3D dynamics of spherical microswimmers in cylindrical Poiseuille flows and found that the swimmers oscillated and tumbled in such flows.Ouyang et al [30] simulated the motion of a circular squirmer in power-law fluids using an immerged-boundary-lattice Boltzmann method and found that the squirmer swam faster in a shear-thinning fluid than in a Newtonian fluid.Samatas and Lintuvuori [31] numerically simulated the motion of large-scale spherical squirmers with chirality in suspension and found that swimmers could spontaneously synchronize their rotations.Burada et al [32] numerically simulated the interactions between a pair of spherical squirmers with chirality and found that the interactions between the two squirmers could lead to attraction or repulsion; this may have been related to the collective motion behavior of the ciliated microorganisms.Qi et al [33] used the lattice Boltzmann method to study the hydrodynamic properties of squirmers in a simple shear flow and showed that there were four different motion modes of the squirmer: horizontal, attractive, oscillation, and chaotic.Zheng et al [34] numerically studied the behavior of squirmer interacting with a circular obstacle, and interestingly found that in most cases the puller always move forward around the obstacle, while the pusher move backward.In addition, they described six swimming modes of the squirmer, namely orbiting forward, ∞-loop, dancing forward, C-loop, orbiting backward and scattering.Li et al [35] used the lattice Boltzmann method to study the flow characteristics of a fixed squirmer in the channel at low Reynolds number, and found that the increase of the self-propelled strength would induce a pair of backflow regions near the channel wall, which would increase the drag coefficient.In addition, the change of Reynolds number does not affect the flow structure, but it has a negative correlation with the drag coefficient.
Approximately 80% of bacterial genera in the natural environment have elongated bodies with an average aspect ratio of approximately 3 [36].Phytoplankton (such as C. reinhardtii, Paramecium, E. coli) also tend to become elongated as their volume increases [37].In addition, as the aspect ratio increases, the mobility of the bacteria also increases; they are positively correlated [36].Therefore, in addition to spherical or circular selfpropelled particles, axisymmetric self-propelled particles have also received attention from many researchers.Ouyang et al [38,39] combined multiple circular squirmers to form a rod squirmer, studied the hydrodynamic behavior of the rod squirmer in different rheological flows, and deduced the velocities of squirmers with different shapes.In addition, Ouyang et al [40] also used the method to study the fluid dynamics of a dumbbellshaped swimmer in a viscoelastic fluid, and found that dumbbell-shaped squirmer expended less energy in a viscoelastic fluid than Newtonian fluid, and neutral squirmer expended more energy than puller or pusher.Zantop and Stark [41] constructed a claviform squirmer by linearly arranging several spherical squirmers and introduced a new velocity model.Liu et al [42] used an immerged-boundary-lattice Boltzmann method based on a circular squirmer model and a conformal-mapping technique to numerically simulate the hydrodynamic characteristics of elliptical squirmers.They found that as the Re increases, the swimming velocity of the pusher increases monotonically, whereas the swimming velocity of the puller first decreases and then increases.In addition, as the aspect ratio of the elliptical squirmer increases, the hydrodynamic efficiency of the neutral squirmer increases, whereas the variations of the pusher and puller are not monotonic.Subsequently, Liu et al [43] numerically simulated the migration and rheological properties of elliptical squirmer in a Poiseuille flow and identified five typical migration modes and three rheological states.The migration modes were stable sliding, periodic tumbling, damped swinging, periodic swimming, and chaotic migrating, and the rheological states were stable, sub-stable, and unstable.These modes also existed for a pair of elliptical squirmers.Jana et al [44] observed through experiments that as the blockage ratio increases, the movement trajectory of Paramecium in a capillary tube achieves a spiral-to-linear transition.Zöttl and Stark [45,46] further modified a theoretical squirmer model to better match elongated body swimmers and determined that the trajectory of an elongated body swimmer was the same as that of the spherical or circular squirmer; the aspect ratio only affected the frequency of the squirmers' swinging and tumbling.Ashtari et al [47] conducted a numerical study on the peristaltic transport of elliptical particles.The results showed that the aspect ratio and initial inclination angle were very important for peristaltic transport.Moreover, they found by controlling the Reynolds number that elliptical particles swim faster than circular particles; this was related to the relationship between the rotational inertia and motion stability.Kyoya et al [48] simulated the collective motion of ellipsoidal squirmer sin a singlemolecule suspension, and found that the swimming mode and the aspect ratio affected various collective motions, such as ordered motion, aggregation motion, and rotational motion.In addition, they also emphasized the importance of particle shapes in collective motion.
Notably, the motion of microorganisms (shear microswimmers, Poiseuille microswimmers, etc) [49] is constrained by the surrounding environment.The most natural external field effect is gravity; almost all microorganisms are affected by gravity, and in some cases, gravity may have a significant effect on the swimming motions of microorganisms.As a result, studies have considered the motion mechanisms of squirmers under gravity.Fadda et al [50] studied the dynamics of chiral swimmers near a flat plate under gravity through 3D simulations and showed that different dynamic effects would occur under different gravity states.When chirality was present, the squirmer would change from a linear motion to a circular motion.Ouyang and Lin [51] recently studied the sedimentation motions of 2D squirmers in narrow channels and summarized four different motion modes: a vertical motion mode, attractive motion mode, oscillatory motion mode, and upward motion mode.An oscillatory motion mode was also reported in Ahana and Thampi's [52] swimming simulation of a single squirmer in a horizontal channel, which exhibited an oscillatory trajectory across the width of the channel.Ouyang and Lin [51] found that both the inertia and density ratios have a competitive effect on a squirmer's motion mode and that a puller swims faster than a pusher at small squirmer-to-fluid density ratios, whereas the opposite is true for large squirmer-to-fluid density ratios.Hill and Pedley [53] found that under the influence of gravity, a suspension of photosynthetic algae slightly denser than water swam upward in still water (on average).In general, an upper region of suspension being denser than a lower region may lead to bioconvection.Kuhr et al [46] numerically simulated the collective sedimentation of a squirmer model under gravity.They found that both the ratio of the squirmer's swimming velocity to the sedimentation velocity and its swimming mode affected the sedimentation profile, whereas the squirmer's average vertical direction was strongly dependent on the height.Kuhr et al [54] utilized a multiparticle collision dynamics method to examine the hydrodynamics of a monolayer of squirmer model microswimmers confined to a boundary under strong gravity and reported that when the densities and types of squirmers were different, a squirmer interacted with the flow field through its generated hydrodynamics before eventually appearing in different states of motion.Ying et al [55] used the lattice Boltzmann method to simulate the sedimentation of two self-propelled particles in a 2D vertical channel and summarized five typical motion modes of a single squirmer.The motion modes of the two squirmers were obtained by combining the different motion modes of a single squirmer.In addition, the angle at which the two squirmers separated from each other and swimming velocity were important for the mutual transitions between the different motion modes.Subsequently, Guan et al [56] obtained similar conclusions in their simulations of the rise of two self-propelled particles in a 2D vertical channel.Qi et al [57] used an immerged-boundary-lattice Boltzmann method to numerically simulate the sedimentation motion of a single 2D bottom-heavy squirmer in a narrow vessel.The study found four settling modes: vertical motion, unilateral oscillation, oscillation, and tilt.They also found that increasing the swimming Reynolds number Re s and swimming strength |β | would lead to a transition between modes.
Despite the significant progress that has been made, studies on the swimming of microorganisms under external forces (e.g., gravity) remain very limited.Understanding the mechanisms of microbial motions under gravity and how they respond to the external environment provides a good approach to designing artificial or engineered microswimmers with gravity-sensing devices.However, almost all of the current studies on the effects of gravity concern spherical or circular swimmers; relatively few descriptions exist of the motions of elliptical swimmers under gravity.In addition, most of the above studies only considered the effect of the wall surface perpendicular to the direction of gravity on the swimmer's dynamics; very few studies exist on the sedimentation of elliptical swimmers in vertical channels.In natural environments, nearly 80% of microorganisms exhibit rod shapes.Accordingly, it is necessary to study the sedimentation of elliptical swimmers in a channel.In view of this, the goal of this study was to comprehensively analyze the motion behaviors of elliptical swimmers with different aspect ratios settling in a 2D vertical channel under gravity.We adopted the elliptical squirmer model modified by Keller [58] based on the 2D squirmer model to determine the self-propulsion of the elliptical squirmer.We investigated the differences between the motion modes of elliptical squirmer settling in the channel and those exhibited by a circular squirmer, and the effect(s) of the aspect ratio on the swimmers' motion behavior(s).
The main focus of this study is the sedimentation characteristics of an elliptical squirmer in a vertical channel under gravity.Note that although we present the 2D results here, the 2D simulations can be used as a convenient starting point for the 3D simulations.This is especially true for lattice Boltzmann method, where, after all, it is a relatively straightforward task to modify the 2D code to handle 3D simulations.In addition, previous studies [38,39,42,43,45,46] have shown that the essence of the phenomenon remains unchanged even when the dimensionality is reduced.Moreover, comparing 2D simulation results with 3D simulation results can also help us to understand the effect of dimensionality.

Numerical method 2.1. Lattice Boltzmann method
In this study, a 2D lattice Boltzmann method with a single-relaxation-time model was used to simulate the sedimentation motion of an elliptical squirmer in a channel.The discrete lattice Boltzmann equations are as follows [59]: In the above, f i (x, t) is the density distribution function for the mesoscopic velocity e i at position x and time t, Δt denotes the unit time step of the simulation.τ = τ 0 /Δt is the relaxation time and is related to the kinematic viscosity ν = c s 2 (τ −0.5)Δt.The solution of equation (1) consists of two parts, namely collision and migration.It can be considered that the fluid node first experiences collision and subsequently experiences the migration process.Moreover, the term on the right of equation (1) can be expressed as Ω i as a whole, which is also known as the BGK collision operator.f i (eq) (x, t) indicates the equilibrium distribution function in the direction of i, and following Qian et al [59], can be expressed as follows: Here, c s denotes the speed of sound determined by c s = c/ 3 , c = Δx/Δt; Δx corresponds to the lattice spacing.w i indicates the weight associated with the lattice model.In the 2D model (i.e., the D2Q9 model), the weight w i and the discrete velocity vector e i are respectively defined as follows: 0, 0 for i 0 1, 0 , 0, 1 for i 1 to 4 1, 1 for i 5 to 8 Using the density distribution function, the fluid density ρ and velocity u can be determined as follows [59]: The macroscopic mass and momentum equations in the low Mach number limit can be obtained using the Chapman-Enskog expansion [60] as follows: For simplicity, both the lattice spacing and time step are set to 1 in this study, i.e., Δx = Δt = 1.This approach is commonly used in lattice Boltzmann simulations.

Elliptical squirmer model
In our previous paper [55], we described the model proposed by Blake [26,27] for spherical or circular squirmer motions, which assumes that the squirmer propagates to its surface boundary in both radial and tangential oscillatory fluctuations to achieve the motion of a microswimmer with an array of cilia.From this, two sets of solutions for the velocity components of the 2D squirmer in the radial and tangential directions in a Newtonian Stokes flow are given, as follows: In the above, r and q denote the radial and polar unit vectors at a given point on the squirmer surface, respectively.A n and B n are the corresponding time-dependent amplitudes, as shown in figure 1.For simplicity in problems related to squirmer motions, tangential fluctuations are usually considered but normal-direction deformations are ignored [26,27].Based on the same assumptions, we consider a simplified squirmer model represented by a second-order truncated tangential velocity (n 2) as follows: Here, the coefficient B 1 contributes to the propulsion of the squirmer and determines the steady-state swimming velocity U s = B 1 /2 in the Stokes flow limit (for a spherical or circular squirmer).In contrast, the coefficient B 2 determines the intensity of the vorticity around the squirmer.θ is the angle between the horizontal direction and direction vector of the squirmer.Accordingly, a key parameter β = B 2 /B 1 is typically defined to represent three types of squirmers: puller (β > 0), pusher (β < 0), and neutral squirmer (β = 0).As indicated by the type names, the puller generates a forward pull by performing a breast-stroke-like motion with its front [19] (e.g., algae such as Chlamydomonas reinhardti).In contrast, the pusher produces a push forward by swinging its cilia at the back [61] (e.g., bacteria such as E. coli and Bacillus subtilis).A neutral squirmer (e.g., Volvox) is associated with a symmetric flow without vorticity.Figure 2 illustrates puller-and pusher-induced flow schematics.
However, the microswimmer model described above is for the motion of a spherical or circular squirmer, as used in most previous microswimmer studies [28][29][30][31][32][33].In this study, we consider microswimmers with elliptical shapes; therefore, we adopt the elliptical squirmer model modified by Keller [58] based on the above squirmer model to simulate the motion of an elliptical squirmer in a fluid.The surface tangential velocity of the elliptical squirmer is given as follows [58]: Here, a and b are the semi-major and semi-minor axes of the elliptical squirmer, respectively.The aspect ratio c is defined as c = a/b.In this study, we take c as 2.0 and 3.0.

Boundary conditions
The lattice Boltzmann method often requires a special treatment of the moving boundary of the squirmer surface to achieve a no-slip boundary condition.In this work, we used the improved bounce-back scheme proposed by Lallemand and Luo [62].Compared to a stair-case discretization scheme with first-order accuracy, this improved bounce-back scheme has second-order accuracy in lattice Boltzmann simulations [62].3, it can be seen that node A will bounce at node F after passing through the boundary link (i.e., e 6 ), and the density distribution function f 6 of node A is obtained by node B through migration, so it is necessary to update f 6 through calculation.To update f 6 , both the nearest node and next nearest-neighboring node along the boundary link (i.e., e 6 ) from node A are considered, i.e., nodes B, C, and D. Accordingly, firstly, a parameter q = |AF|/|AB| is introduced to determine the location of the boundary node F.Then, an interpolation scheme is used to recalculate f 6 based on the surrounding nodes as follows: In the above, u F is the velocity of the moving surface at the boundary node F, as shown in figure 1(b).If we set q = 0.5, then equation (10) is the classical bounce-back boundary.Additional details regarding this scheme can be found in the reference [62].
As a squirmer has a self-propelled velocity, the velocity of the boundary node F needs to be updated by adding the self-propelled velocity u s shown in equation (9) before conducting the interpolation of equation (10).
Here, U and Ω denote the translational and rotational velocity of the squirmer, respectively, and R denotes the position vector from the center of mass of the squirmer to the boundary node F. By scanning every fluid-solid boundary link, the forces and torques exerted on the squirmer boundary nodes can be calculated according to the momentum exchange scheme proposed by Lallemand and Luo [62].Moreover, the method proposed by Aidun, Lu and Ding [63] is adopted in this study to calculate the additional forces and torques generated by the transitions between fluid and solid nodes during the motion of the squirmer.For this method, we describe it in detail in appendix B so that readers can clearly and intuitively understand some of the details of the simulation in this study.Finally, the total force and torque are calculated by integrating the obtained forces and torques over the surface of the squirmer.Accordingly, the motion of the squirmer can be solved by Newton's equation as follows: In the above, x denotes the center of mass of the squirmer, Ω and J denote the angular velocity and moment of inertia of the squirmer, respectively, and F and T correspond to the force and torque exerted on the squirmer, respectively.When the squirmer is neutrally suspended, the density ρ p is equal to the density ρ of the fluid.Note that for F and T in equations ( 12) and (13), it should include the forces and torques on the squirmer's boundary nodes, the additional forces and torques generated by the transitions of the fluid-solid nodes to each other, the elastic forces and torques exerted by the interaction of the squirmer with the wall, and its own gravity.In this study, the Velocity Verlet method was used to solve the above Newtonian equations.

Squirmer-wall interaction
In our simulations, the squirmer inevitably moves close to the wall in certain cases.When the squirmer is in close contact with the wall, collisions occur and high pressures can be generated.To avoid non-physical overlap, we need to introduce additional short-range repulsive forces.Therefore, we adopt the elastic force model proposed by Glowinski et al [64]; this model been widely used to manage particle-wall and particle-particle interaction problems.However, the formulation proposed by Glowinski et al [64] is for spherical or cylindrical particles.Thus, we need to adjust it slightly to account for the effect of the elliptical squirmer.The final expression of the elastic force is as follows: Here, X c is the position of the point on the surface of the elliptical squirmer closest to the wall and X is the position of the corresponding point on the wall.Thus, |X-X c | is the minimum distance from any point on the elliptical squirmer to the wall, as shown in figure 4. Δr is a truncation distance indicating the region where the repulsive force begins to exist in numerical simulations.In this study, Δr = 2Δx.c ij is a force-scaling factor defined as the difference between the gravity and buoyancy of the squirmer in the simulation, i.e., c ij = (ρ p -ρ)gΑ p , where ρ p and ρ denote the density of the solid and fluid, respectively.Α p is the area of the elliptical squirmer.g is the gravitational acceleration.ε is the stiffness coefficient; in this study, ε = 10 −3 .Notably, unlike the case of a circular particle, as the elastic force does not necessarily point towards the center of mass of the elliptical squirmer, the elastic force will generate a torque on the elliptical squirmer.Accordingly, this resulting torque must be added to the calculation.The squirmer started to move under the effects of both gravity and self-propulsion.We used a moving mesh method, when the squirmer moved down more than one mesh distance, the squirmer and the entire flow field information were moved up one mesh to make the squirmer move for a sufficiently long time.In addition, we set the left and right walls of the channel to a no-slip boundary condition (processed in the lattice Boltzmann method using a bounce-back scheme).The bottom of the channel was set to an imperturbation state using non-equilibrium extrapolation and the top was set to a sufficiently developed boundary condition.
As an elliptical squirmer is subject to both gravity and self-propulsion during the sedimentation process, to consider the degrees of influence of both, a non-dimensional parameter α was introduced to represent the selfpropelling strength of the squirmer as follows: In the above U s represents the steady swimming velocity reached by the squirmer under self-propulsion.Both Ouyang et al [51] and our previous paper [55] validated that the circular squirmer eventually reaches a steady velocity U s = B 1 /2 at a low Reynolds number.As there is no exact theoretical solution for the steady swimming velocity of an elliptical squirmer in 2D conditions, we used the steady velocity B 1 /2 of the circular squirmer as a substitute; U g represents the sedimentation velocity of the 2D passive particle under gravity.As there is no Stokes solution for the sedimentation velocity U g of a passive particle in a 2D study, we adopted the analytical formula proposed by Happel and Brenner [65] for the calculation.The final expression was as follows: Here, K is a constant related to the ratio of the channel width and major axis of the elliptical particle (W * = L/d) and is expressed as follows: We can observe the effect of self-propelling strength on the motion mode of the elliptical squirmer by varying the value of α.If α ? 1, the sedimentation of the squirmer is predominantly self-propelled and the state is referred to as the cruising regime.If α = 1, the sedimentation of the squirmer is predominantly owing to gravity and we refer to this state as the strong gravity state.In this study, we chose α to vary from 0.3 to 1.1.Based on the above U g and U s , we defined the corresponding Reynolds number as follows: s s g g In the above, Re g represents the Reynolds number obtained only under gravity, and Re s represents the Reynolds number obtained only under self-propelled action (also known as the swimming Reynolds number).Unless otherwise specified, we fixed certain parameters in the simulation as follows: ρ = 1, ρ p = 1.05,Δx = 1, and Re g = 0.5.The remaining initial parameters were the same as above.

Sedimentation of a non-squirmer elliptical particle in a channel
To validate the correctness of our procedure for analyzing the motion of an elliptical particle, the sedimentation of an elliptical particle in the vertical channel was simulated and compared with previous numerical results.As shown in figure 6, the semi-major and semi-minor axes of the elliptical particle are a = 14 and b = 7, respectively, the width of the vertical channel L = 8a, the height H = 90a, and the density ρ f and kinematic viscosity ν of the fluid are 1.0 and 0.0146, respectively.The center of mass of the particle is located at x = 0.375L, y = 0, at a height away from upstream 40a and downstream 50a, and its initial angle is set as −45°between the major axis and positive direction of the x-axis.After being released from a stationary state, the particle eventually oscillates near the center of the channel under gravity.Figure 6 shows our simulation results and those of Qi [66] and Nie et al [67].We simulated the final steady Reynolds number Re p of the particle to reach 17.04 (taking the long axis of the elliptical particle as the length scale), i.e., quite close to the final Reynolds number Re p obtained by Qi [66], Huang and Joseph [66], and Nie et al [67].The consistency of the calculation results with the available results indicates that the lattice Boltzmann method adopted herein is appropriate for addressing the motions of elliptical particles.

Free-swimming motion of a circular and elliptical squirmers in an unconfined domain
We simulated a circular squirmer swimming freely in an unconfined domain and compare the results with previous numerical results and the theoretical solution B 1 /2 at a low Reynolds number to verify the correctness of the squirmer model.As shown in figure 7, we placed a circular squirmer with its head vertically downward in the center of a 20d × 20d square domain with periodic boundary conditions in all directions, and let it start moving from stationary under self-propulsion, where Re s is set to 0.005 and d represents the diameter of the circular squirmer, d = 40.We can see that each swimming type of squirmer reaches the theoretical swimming velocity B 1 /2 (which has also been verified by Ouyang et al [30] through numerical simulation).This confirms the correctness of the squirmer model used in this study.The choice of d = 40 is to satisfy the resolution requirement.The results also show that the elliptical squirmer we selected is consistent with the resolution requirement, as the semi-minor axis b of the ellipse is set to 20.The requirements for squirmer resolution and mesh independence are additionally described in detail in appendix C, which can be found in appendix C. The velocity U of the circular squirmer is normalized by U * = U/U s and the time t is normalized by t * = U s t/d.
Then, we simulated the velocity evolution of an elliptical squirmer with different swimming types in an unconfined domain and observed and analyzed the motion mechanisms and flow fields of the elliptical squirmers under free-swimming motions.As shown in figure 8, an elliptical squirmer with the head direction vertically downward was placed in the center of a 40a × 40a square domain with periodic boundary conditions in all directions and was allowed to move from stationary under self-propulsion.In the figure, Re s = 0.005 and a represents the semi-major axis of the elliptical squirmer, a = 40.In addition, the elliptical squirmer's velocity U is normalized by U * = U/B 1 and the time t is normalized by t * = U g t/(2a).For the elliptical squirmer, although there is no exact theoretical solution, it can be observed that the final steady velocity of each swimming type of the elliptical squirmer is consistent at Re s = 0.005 and U s ≈ 0.56B 1 , indicating that the swimming type does not affect the final steady velocity of the elliptical squirmer.
Figure 9 shows the streamlines around the elliptical squirmer for each swimming type at Re s = 0.005.It can be seen that the upward streamlines at the tail of the pusher are generated by its 'push' action, whereas the downward streamlines at the head position of the pusher are generated by the downward swimming of the pusher.The puller's streamlines are the opposite, with streamlines pulled back at both the tail and head.The streamline structure around the neutral elliptical squirmer is similar to that of the non-squirmer elliptical particle.
Next, we considered the effect of the swimming Reynolds number on the final steady velocity of the elliptical squirmer, as shown in figure 10.We selected β = −3 and 3 to obtain the velocity evolution of the elliptical squirmer under different values of Re s .It can be seen that the final steady velocity of the elliptical squirmer is consistent at an extremely low Reynolds number of U s ≈ 0.56B 1 , whereas the situation differs when the Reynolds number gradually increases to represent a low but finite fluid inertia (e.g., Re s = 0.5).At this time, the swimming velocity of the pusher will increase whereas the swimming velocity of the puller will decrease.This is owing to pusher being 'pushed' forward by the tail while the puller is 'pulled' forward by the head.With the increase of Re s , the 'trailing negative flow' of the pusher's tail and 'leading negative flow' of the puller's head are enhanced (as seen in figures 9(a) and (c)).Thus, the pusher's velocity is increased while the puller's velocity is reduced, owing to the reverse thrust of the 'leading negative flow.' Figure 11 shows the final steady velocity of the 2D elliptical squirmer at different aspect ratios c.For the sake of observation, we convert the aspect ratio c to the eccentricity e of the ellipse and show it in the abscissa, where the eccentricity e = (1−b 2 /a 2 ) 0.5 .In addition, we qualitatively compare the free swimming of the 3D ellipsoidal squirmer with the different eccentricities simulated by Keller and Wu [58] in an unconfined domain.As the eccentricity e increases, the final steady swimming velocity of the 2D elliptical squirmer is consistent with the steady velocity of the 3D ellipsoidal squirmer in terms of the trend, i.e., the steady swimming velocity increases with the increase of the eccentricity e (aspect ratio c) until it finally becomes needle-shaped.This also shows that the 2D elliptical squirmer model simulated in this study is correct.Notably, at a low Reynolds number, for a 2D circular squirmer (e = 0), the steady swimming velocity U s = B 1 /2, whereas for a 3D spherical squirmer, the steady swimming velocity U s = 2B 1 /3 (as verified by Li and Ardekani [28]).

Results and discussion
This section provides the results of our simulations and related discussions.To better understand the motion mechanisms of an elliptical squirmer in the channel, Part A of this section first describes the motion modes of an elliptical squirmer with an aspect ratio c = 2.0 settling in a vertical channel.Then, in Part B, we discuss the motion modes of squirmers with different aspect ratios c to investigate the effects of the different aspect ratios on the squirmer's motion.It is worth noting that when showing the motion modes of squirmer with different aspect ratios, we did a lot of simulation tests on the aspect ratio c, and finally obtained the same simulation results, and we chose c = 1.0, 2.0 and 3.0 for analysis because it can qualitatively illustrate the transition of squirmer's motion modes with the increase of aspect ratio as well as the difference between the elliptical squirmer and the circular squirmer.

Sedimentation of an elliptical squirmer with aspect ratio c = 2.0 in a vertical channel
As different aspect ratios of the elliptical squirmer may lead to different motion behaviors in a fluid, this section first presents the motion modes of the elliptical squirmer in the vertical channel for the aspect ratio c = 2.0.We simulate the computational model depicted in figure 5 for different values of α and β, where the self-propelling strength factor α ranges from 0.3 to 1.1 (i.e., 0.3 α 1.1) and the swimming type factor β  falls within [−5, 5] (i.e., −5 β 5).In figure 12, we summarize three typical motion modes of the elliptical squirmer, namely, steady inclined motion (SIM), wall-attraction oscillation (WAO), and large-amplitude oscillation (LAO).For the two modes of motion shown in figure 12 (SIM and LAO), the direction of motion of the elliptical squirmer may be upward or downward depending on whether the head or tail of the elliptical squirmer is initially captured by the wall when it is close to the wall; however, this does not affect the elliptical squirmer's motion mechanisms.
As the flow field features and trajectories are completely similar when the same motion are obtained in different cases, we select a few representative cases for the analysis to illustrate the above three motion modes and role of the flow field in detail.Figure 13 shows the trajectory and head direction of an elliptical squirmer at α = 1.1 and β = −1, 3, and −5.For comparison, the results for a non-squirmer elliptical particle are also shown.The figure shows the three motion cases of the elliptical squirmer in which it ultimately moves steadily, whereas for the passive particle, its trajectory shows a LAO in the channel.Notably, although the trajectory of the elliptical squirmer at α = 1.1, β = −5 is similar to that of the passive particle, the passive particle does not collide with the wall, whereas the elliptical squirmer does.Compared to the elliptical squirmer, the trajectory of the passive particle has a smaller vibration amplitude and larger change in the head direction.Figure 13 shows that the motion of the elliptical squirmer with self-propulsion in the channel is more complex than that of the passive particle.
Based on the results shown in figures 13, 14 shows the corresponding instantaneous streamlines at final stabilization.From figures 14(a) and (b), we can see that the head directions of the pusher and puller during the steady motion are different.The pusher's head direction is away from the wall and will move away from the wall under the action of the self-propulsion.In contrast, the fluid in the channel keeps squeezing the pusher so that it generates a force towards the wall.This indicates that the wall has an attractive force on it, thereby achieving force equilibrium.The puller's head direction towards the wall under self-propulsion will be close to the wall; at this time, the wall produces a repulsive force on the puller and the repulsive force is greater than the selfpropulsion.Thus, the puller will form a steady oscillation mode near the wall.The force of the wall on the elliptical squirmer and formation of the squirmer's motion modes will be explained in more detail below.In figures 14(c) and (d), although the motion modes are similar, it can be seen from the streamline diagrams that the motion mechanisms of the two are different.From figure 14(c), we can see that the pusher's head and tail have outward pushing streamlines during its motion consistent with the motion characteristic of the pusher's tail 'pushing forward,' whereas figure 14(d) shows that the non-squirmer elliptical particle does not have such a characteristic.and pusher generate SIM and WAO motion modes, significant differences exist between them.For example, the forces relative to the wall and head direction are inconsistent (as mentioned above, see figures 14(a) and (b)).To understand this phenomenon and the reasons for the formation of the motion modes in more detail, we considered the elliptical squirmer from the perspective of a force analysis.
First, cases with α = 1.1, β = −1 and α = 0.5, β = 1 are selected for analyzing the SIM mode. Figure 16 shows the corresponding trajectories and head directions.Figure 17 shows the pressure distribution of the corresponding elliptical squirmer during the steady motion in the channel (the red dot shown in figure 16 was selected as the sampling time).The pressure p is normalized by p * = (p−p ∞ )/(ρU g 2 ), where the value of p ∞ represents the pressure of the undisturbed flow.All of the forces are marked on the elliptical squirmer in figure 17.For the puller, when it is close to the right wall, there is an evident high-pressure zone in the middle region of the body, so the wall exerts a repulsive force on it.To balance the repulsive force, the puller needs to swim inward towards the wall (figure 17(a)), ultimately achieving a force balance under the action of selfrepulsion, pressure, and gravity and maintaining a SIM mode.In contrast, the pusher has an evident lowpressure zone when it approaches the wall.To balance the attractive force generated by this low-pressure zone, the pusher needs to swim outward away from the wall, ultimately achieving a force balance under the action of self-propulsion, pressure, and gravity.Zöttl and Stark [29] also pointed out that the wall would exert a repulsive  force on the puller to keep puller away from the wall and an attraction force on the pusher to be attracted to the wall when studying the effect of the wall on microbial motions.
Figure 18 shows the corresponding trajectories and head directions for α = 0.7, β = 3 and −3; these exhibit attractive oscillation near the wall.We select one complete period (corresponding to the red dots 1-4 in figure 18) to analyze the pressure distribution of the elliptical squirmer during its motion in the channel, as shown in figures 19 and 20. Figure 19(a) shows the pressure distribution (t = 0) at the furthest distance between the puller and wall during motion.At this time, there is a high-pressure zone between the middle of the puller's body and wall, indicating the repulsive force of the wall on the puller.Meanwhile, there is a low-pressure zone between the puller's head and wall, and the pressure generated in this zone is not towards the puller's center of mass.As a result, the main effect is to make puller produce a clockwise torque.This makes the angle between the puller's head direction and the positive direction of the y-axis smaller, resulting in a larger component of the puller's self-propulsion in the y-direction and ultimately making the puller move inward towards the wall.Figure 19(b) depicts the pressure distribution (t = T/4) of the puller during its motion near the wall.It can be seen that there are several pressure zones between the puller and wall at this time, but the main pressure zone still represents the repulsive force of the wall on the puller, whereas the high-pressure and low-pressure zones existing on the puller's head make it produce a counterclockwise torque.This causes it to rotate counterclockwise, resulting in a larger angle in the puller's head direction, and the self-propulsion force in the y-  direction begins to be smaller than the repulsive force generated by the wall.Accordingly, the puller begins a decelerated motion under the action of inertia.Figure 19(c) shows the pressure distribution (t = T/2) at the closest distance between the puller and wall.At this time, the puller has decelerated to 0. It will continue to rotate counterclockwise under the pressure the wall and will begin to move away from the wall under the repulsive force of the wall.Figure 19(d) depicts the pressure distribution (t = 3T/4) of the puller during its motion away from the wall.At this time, similar to the situation at t = 0, there is only a low-pressure zone between the puller's head and wall.This makes the puller begin to rotate clockwise.The component of the self-propulsion in the y-direction begins to gradually increase; thus, the puller will decelerate away from the wall and ultimately return to the same situation as t = 0, completing a period of motion.The above process forms the puller's WAO mode.Notably, significant differences exist in the motions between elliptical and circular squirmers.For an elliptical squirmer, owing to its elongated shape, certain forces of the wall on the elliptical squirmer will not be directed towards its center of mass, resulting in a torque effect on the elliptical squirmer.In contrast, the circular squirmer does not have this situation, resulting in the different motion modes between the two squirmers.Figure 20 shows the trajectory and head direction of the circular puller at α = 0.7 and β = 3 (i.e., the same case as with the elliptical squirmer) and instantaneous pressure distribution.The circular puller finally reaches a SIM mode where the force equilibrium is achieved under the action of self-propulsion, pressure, and gravity and maintains a steady inclined motion.
The analysis of the WAO motion mode of the pusher is similar to the motion analysis process of the puller described above.The main difference is that the wall generates an attractive force on the pusher, causing the pusher's head direction to be away from the wall (for force balance).Figure 21 depicts the process of the pusher attracting oscillation near the wall (corresponding to the red dots 1-4 in figure 18(b)).As shown in the figure 21, when the pusher is close to the wall, it rotates counterclockwise owing to the high-pressure zone at the tail.At this time, the attractive force of the wall is greater than the component of the self-propulsion force in the ydirection.In contrast, when the pusher moves away from the wall, it will rotate clockwise, and the attractive force of the wall will gradually be smaller than the component of the self-propulsion force in the y-direction.Similarly, as the force of the wall on the circular pusher is entirely pointed towards its center of mass, it does not generate additional torque on the circular pusher, so the motion mode of the circular pusher is not the same as that of the elliptical pusher.The circular pusher finally achieves a LAO motion mode at α = 0.7 and β = −3.Ouyang and Lin [51] also reported the LAO mode of a circular pusher when studying the sedimentation of a single circular squirmer in a 2D narrow channel.
In addition, we select a case with α = 0.7, β = −5 to analyze the LAO motion mode.The trajectories and head directions of the pusher are shown in figure 22.We take the same operation as when analyzing the WAO motion mode, i.e., selecting a complete period (e.g., figures 18(a) and (b)) for analysis.However, we mainly focus on the pressure distribution when the pusher is close to the wall (i.e., at t = 0 and t = T/2), so figure 23 only shows the pressure distribution corresponding to t = 0 and t = T/2 when the pusher moves in the channel.As shown in figure 23(a), when the pusher approaches the left wall, the wall continues to exert an attractive force on the middle region of its body while the high and low pressures at the tail form a counterclockwise torque.This causes the pusher to rotate counterclockwise and to begin to swim towards the center of the channel under selfpropulsion.As shown in figure 23(b), when the pusher approaches the right wall, the high and low pressures at the tail of the pusher form a clockwise torque, making the pusher rotate clockwise.Then, it swims to the center of the channel again under the effect of self-propulsion to form a LAO mode.This type of motion mode can potentially be used to enhance the mixing of suspensions in microfluidic channels at low Reynolds numbers [68][69][70].
Finally, for the motion modes obtained for the elliptical squirmer shown in figure 15 at different values of α and β, we analyze how its motion modes transition.First, by keeping α fixed and observing the pusher while taking figures 21 and 23 as an example, we can observe that when the value of α is fixed, the larger value of β, the greater the pressure strength between the pusher and wall in the channel (this is also the same for the puller).Therefore, the torque generated at the tail of the pusher is also larger, leading to a larger change in the head direction of the pusher (as can be seen by comparing the head direction values of figure 18(b) and figure 22).The attraction force of the wall on the pusher is getting increasingly smaller and the self-propulsion force is getting larger.Ultimately, the squirmer leaves the wall to transition from the WAO mode to the LAO mode.For the pusher, the same reasoning applies to the mode transitions in other cases when the value of α is fixed.Second, for the puller, when the value of α is fixed, the greater the value of β, the greater the repulsive force of the wall on the puller and the more easy it is to break the SIM mode.Accordingly, the puller begins to perform an attraction oscillation near the wall, thereby transitioning from the SIM mode to the WAO mode.
When the value of β is fixed, the pusher's swimming velocity increases with increasing values of α (according to the definition of α).Taking β = −3 as an example, for the pusher with α = 0.5 and 0.7, when the value of α is larger, its swimming velocity is also larger, and its component in the y-direction is also larger.As the pusher's head direction is away from the wall, its SIM mode will be broken at a larger swimming velocity, resulting in a  from the SIM mode to the WAO mode.The same reasoning applies to the transitions of the motion modes for the pusher in other cases.For the puller, taking β = 3 as an example, when the value of β is fixed, as the value of α increases, the swimming velocity gradually increases, and the puller will be closer to the wall.This will lead to a greater repulsive force of the wall on the puller and the SIM mode will be broken, resulting in a transition from the SIM mode to the WAO mode.In addition, figure 24 shows the variation of the nondimensional frequency (Strouhal number, St) of the two motion modes for the WAO and LAO modes of the elliptical squirmer with the value of α.The St value of the WAO mode is much larger than that of the LAO mode and the St values of both increase with increasing values of α.This also illustrates the correctness of the above theoretical analysis, because a larger component of the swimming velocity in the y-direction leads to a larger oscillation frequency of the elliptical squirmer, that is, a larger St value.

Effect of different aspect ratios c on the motion of the squirmer
In this section, we explore the effect of different values of the aspect ratio c on the motion of the elliptical squirmer.As mentioned in Part A, elliptical and circular squirmers have distinct differences in motion.Thus, this section first investigates the motion modes of the circular squirmer (c = 1) under the initial conditions consistent with figure 5, as shown in figure 25.The motion modes of the circular squirmer in figure 25 are obtained in the range 0.5 α 1.1, −5 β 5 and the motion for α = 0.3 is not given because gravity dominates the motion of the squirmer when α is small (whereas we are concerned with the effect of the aspect ratio c on the motion of the squirmer).For the circular squirmer, compared with the aspect ratio c = 2.0 (see figure15), it can be seen that as the aspect ratio c increases from 1.0 to 2.0, the pusher achieves a shift from the LAO mode to the WAO mode at β = −3.This is because as c increases, the pusher's head direction becomes  more inclined to the direction (x-direction) to reduce the drag force of the swimming.As a result, the swimming velocity component in the y-direction decreases, the displacement of the pusher in the y-direction decreases, and the motion mode transitions.Figure 26 shows the changes of the head directions of the corresponding circular and elliptical pushers, where the time t is normalized by t * = tU g /d; the motions of the circular and elliptical pushers at α = 0.7 and β = −3 are selected for analysis.From figure 26, it can be seen that the head direction of the elliptical pusher rotates in the range of approximately −315°to −270°.This is smaller in the vertical direction than that of the circular pusher, i.e., its head direction is more inclined to the vertical direction during swimming.In addition, as c increases, the puller mainly transitions from the SIM mode to the WAO owing to the force exerted by the wall on the circular puller (c = 1), which is only towards the center of the circle and does not generate torque on it.The elliptical puller is different (see figure 19); the wall force will generate torque near the head position of the elliptical puller, making the swimming velocity unsteady in the y-direction.As a result, there will be a transition from the SIM mode to the WAO mode (Part A provides some analysis).
Next, we study the motion modes of the elliptical squirmer with an aspect ratio of c = 3.0 in the vertical channel.Figure 27 shows the final motion modes of the elliptical squirmer at different values of α and β.  with figure 15, it can be observed that the elliptical squirmer with an aspect ratio of c = 3.0 no longer has a LAO mode and the SIM modes are almost all concentrated on the pusher (β < 0).In contrast, the WAO modes are all concentrated on the neutral squirmer (β = 0) and puller (β > 0).We first analyze the motion of the puller.For the puller, most of its motions are WAO modes, and their trajectories and head directions are similar to those shown in figure 18(a).The reason for the formation of WAO mode of the puller is also consistent with that of the aspect ratio c = 2.0 (see figure 19).The motion mode of the puller at α = 0.3, β = 3 and 5 is a SIM, possibly owing to the small α resulting in a small swimming velocity in the y-direction and making the puller achieve force equilibrium under self-propulsion, pressure, and gravity.In contrast, as α increases, the swimming velocity in the y-direction increases and this equilibrium is broken, resulting in a transition in the motion mode.
We select a case with α = 1.1 and β = −5 to analyze the reason for the disappearance of the pusher's LAO mode. Figure 28(a) shows the trajectory and head direction of the corresponding pusher.The pressure distribution of the pusher in the channel is obtained by sampling at the red dots (the force arrows are not marked here because the force situation is dense) shown in figure 28(b).Compared with figure 23, the pusher with an aspect ratio of c = 3.0 is more inclined to the vertical direction (the direction of the pusher sedimentation, i.e., xdirection) in the head direction than with c = 2.0.Moreover, it can be seen from figure 28(b) that there is only a low-pressure zone between the pusher and wall at this time, indicating that the wall has a strong attractive force on the pusher and can cause the pusher to become stably adsorbed near the wall.The low pressure at the tail of  the pusher near the and high pressure at the tail away from the wall cancel each other and do not make the pusher rotate; thus, the pusher moves almost vertically and steadily downward under the action of selfpropulsion, pressure, and gravity.This seems to be explained from a physical perspective as well, because for microorganisms with large aspect ratios (e.g.E. coli), the head direction always follows the fluid flow direction to reduce the drag force caused by swimming.In our simulations, this observation is also confirmed by the fact that for all α and β in the study range at c = 3.0, the head direction in the pusher's SIM mode is almost always close to the vertical direction of motion.Combined with the comparison between circular (c = 1.0) and elliptical squirmers (c = 2.0), we can speculate that as the aspect ratio c continues to increase, the pusher's head direction will be more inclined to the vertical direction until it reaches the completely vertical direction.Similarly, compared to the aspect ratio c = 2.0, the pusher with c = 3.0 no longer has a WAO mode for the same reason and ultimately exhibits a SIM mode.

Conclusions
This study used a 2D lattice Boltzmann method to simulate the sedimentation of an elliptical squirmer in a vertical channel.An improved squirmer model was introduced to analyze the self-propelled motion of the elliptical squirmer and a non-dimensional parameter α = U s /U g was defined to represent the self-propelling   of the squirmer.U s was replaced by the steady swimming velocity of the circular squirmer; U g denoted the sedimentation velocity of a non-squirmer particle with the same diameter in the Stokes flow.In the ranges of 0.3 α 1.1 and −5 β 5, we investigated the motion mechanisms of elliptical squirmers with different aspect ratios.The main conclusions are as follows.
(1) We investigated the motion mechanism of an elliptical squirmer with aspect ratio c = 2.0 in the channel and observed three different motion modes: SIM, WAO, and LAO.Among them, the LAO mode was only present in the motion of the pusher, whereas the SIM and WAO modes were represented in all swimming types of the squirmers.The formation of the different motion modes was related to the pressure distribution around the squirmer.
For squirmers swimming near a wall, differences in swimming types (e.g., pusher, puller, and neutral squirmer) can also lead to significant differences in their interactions with the wall.When a pusher swims near the wall, the wall often exhibits an attractive force on it under pressure, and at this time the pusher's head direction is always away from the wall.In contrast, the puller is the opposite; the wall will exert a repulsive force on the puller to force the puller's head direction towards the wall.In addition, the pressure generated by the interaction between the elliptical squirmer and wall is often not towards its center of mass, resulting in an additional torque for the elliptical squirmer.This makes the motion of the elliptical squirmer more complex than that of the circular squirmer.
(2) When the fixed self-propelling strength α is constant, a larger swimming type factor | β | will result in a larger pressure distribution around the elliptical squirmer and the additional torque on the squirmer will also increase accordingly.When the fixed swimming type factor β is constant, a larger self-propelling strength α will result in a larger swimming velocity of the squirmer (according to the definition of the self-propelling strength α).This is the reason for the transitions of the different motion modes of the squirmer in different ranges of α and β.
(3) Compared with the motion modes at c = 2.0, the elliptical squirmer at c = 3.0 no longer has a LAO motion mode.The SIM modes are almost all concentrated on the pusher, whereas the WAO modes are all concentrated on the neutral squirmer and puller.The disappearance of the LAO mode can be attributed to the fact that the pusher with aspect ratio of c = 3.0 tends to be more vertical (in the direction of fluid flow) in the head direction.At this time, the wall has a strong attractive force on it, causing it to be steadily attracted near the wall.As a result, the pusher moves downward almost vertically and steadily under the effect of selfpropulsion, pressure, and gravity.Combined with the sedimentation motion of a circular squirmer with aspect ratio c = 1.0 in a vertical channel, it can be found that as the aspect ratio c increases, the head direction of the squirmer tends to follow the direction of the fluid flow to reduce the drag force on itself.In addition, when changing from a circular squirmer to an elliptical squirmer, an additional torque is generated by the interaction force with the wall, leading to changes in the motion modes of both.
(4) (At present, the study of elliptical squirmer in this manuscript is carried out in the 2D case.In the follow-up research work, we hope to carry out the study of the motion characteristics of 3D ellipsoidal squirmer, and furthermore, to carry out the simulation studies of the interaction and even the collective behavior of multiple ellipsoidal squirmer. It is worth noting the viscosity ν in the table is obtained by combining equations ( 16) and (18), and the viscosity is different for different values of the aspect ratio.
In addition, the fluid boundary nodes also impart additional momentum to the squirmer, which can be derived using equastions (B1) and (B2).The total force and torque exerted on the squirmer at the moment t 0 + 1/2 is given by [

Appendix C . Squirmer resolution and mesh independence check
Due to the improved bounce-back scheme adopted in this study has certain requirement on the resolution of the squirmer, and to further verify whether the numbers of mesh used in this study meet the accuracy requirement, considered simulating the circular squirmer with different diameters in an unconfined zone to confirm whether it can reach the theoretical solution B 1 /2 (the numbers of mesh in the calculation domain varies with different diameters).Figure 29 shows the final steady swimming velocity of the circular squirmer with different diameters, where all the circular squirmers are selected at Re s = 0.005, β = −3.It can be seen from the figure that the steady swimming velocity of the circular squirmer with diameter d = 40 and d = 60 completely overlap and both reach the theoretical swimming velocity B 1 /2, while the final steady swimming velocity of the circular squirmer with diameter d = 20 does not reach the theoretical swimming velocity and exists a numerical fluctuation, which indicates that d = 20 does not meet the resolution requirement.Considering the cost of time and computation, we finally chose to set the diameter d to 40, and set the minor axis to 40 when simulating the elliptical squirmer.

Figure 1 .
Figure 1.Schematic diagrams of circular and elliptical squirmers with the head direction to the right: (a) circular squirmer, (b) elliptical squirmer.

Figure 2 .
Figure 2. Flows induced by a puller and pusher in the horizontal direction.A puller is driven by pulling the fluid from the front, and a pusher is driven by pushing the fluid from the back.
Figure 3 shows a schematic diagram to briefly introduce the scheme.As shown in figure 3, it can be seen that nodes located inside the squirmer boundary represent solid nodes (represented by black solid squares in the figure), nodes located outside the squirmer boundary represent fluid nodes (represented by gray solid circles in the figure), and nodes located on the squirmer boundary are called boundary nodes (represented by open circles), i.e., the location on the squirmer boundary where the bounce-back occurs.Lallemand and Luo [62] indicated that all fluid and solid nodes have propagation actions after a collision occurs.Therefore, it is necessary to confirm the exact locations of the boundary nodes (i.e., the open circles in figure 3), and then recalculate the equilibrium distribution function for the migration from the solid nodes to the fluid nodes.For a better description, we can take the example of node A in figure 3.As shown in figure

Figure 3 .
Figure 3. Schematic of the bounce-back scheme in the lattice Boltzmann method proposed by Lallemand and Luo [62].Solid squares: solid nodes; solid circles: fluid nodes; open circles: boundary nodes.

Figure 4 .
Figure 4. Schematic diagram of the repulsive force model corresponding to the elliptical squirmer.

2. 5 .
Computational model and parameter definitionsAs shown in figure5, we simulated the sedimentation motion of an elliptical squirmer under the effect of gravity in a 2D vertical channel filled with a fluid of density ρ and viscosity ν.The semi-major and semi-minor axes of the elliptical squirmer were a and b, respectively.The density was set to ρ p .The major axis of the elliptical squirmer was chosen as the length scale.We set the height and width of the channel to H = 45d and L = 4d, respectively.The coordinate origin was established on the left wall corresponding to the center of mass of the elliptical squirmer, where d represented the major axis of the ellipse, i.e., d = 2a.As this study simulated the motions of elliptical squirmers with aspect ratios of 2.0 and 3.0, i.e., c = 2.0 or c = 3.0, we used b = 20Δx, a = 40Δx, or a = 60Δx.We used X and Y to describe the vertical and horizontal positions of the elliptical squirmer, respectively, and θ to represent the swimming direction of the squirmer (the angle with the positive direction of the y-axis, where θ is positive in the counterclockwise direction).The initial position of the elliptical squirmer was placed at Y = 1.5d,X = 0 at a height away from upstream of H u = 20d and downstream H d = 25d, i.e., H = H u + H d .The initial direction θ = −π /4.

Figure 5 .
Figure 5. Schematic diagram of the calculation model.

Figure 6 .
Figure 6.Results of passive elliptical particle settling in the channel: (a) trajectory of the particle, (b) time evolution of the particle direction with respect to the horizontal direction.

Figure 7 .
Figure 7. Schematic of the motion of circular squirmer in the channel and time history of the circular squirmer's velocity for different swimming types at Re s = 0.005.

Figure 8 .
Figure 8. Schematic of the motion of elliptical squirmer in the channel and time history of the elliptical squirmer's velocity for different swimming types at Re s = 0.005.

Figure 9 .
Figure 9. Streamline distribution around the elliptical squirmer for each swimming type at Re s = 0.005: (a) β = −3 (an elliptical pusher), (b) β = 0 (a neutral elliptical squirmer), (c) β = 3 (an elliptical puller).The white ellipse represents the pusher, the green ellipse represents the neutral squirmer, and the black ellipse represents the puller (the part after that is also represented in the same way).

Figure 11 .
Figure 11.Final steady swimming velocity of a 2D elliptical squirmer and a 3D ellipsoidal squirmer at different eccentricities e.

Figure 15
Figure 15 summarizes the final motion modes of the elliptical squirmer in all of the considered ranges of α and β.As shown in figure 15, the LAO motion mode is entirely concentrated on the pusher (β = −5), whereas the other two motion modes are represented on both the puller and pusher.Notably, although both the puller

Figure 15 .
Figure 15.Distribution of motion modes for elliptical squirmer with aspect ratio c = 2.0 at different values of α and β.Square: SIM; Circle: WAO; Diamond: LAO.

Figure 20 .
Figure 20.Kinematic features of the circular puller at α = 0.7, β = 3 and the corresponding instantaneous pressure distribution: (a) the trajectory and head direction of the puller, (b) instantaneous pressure distribution of the puller.

Figure 24 .
Figure 24.Non-dimensional frequency (Strouhal number, St) corresponding to the WAO and LAO modes of the elliptic squirmer changes with the value of α.

Figure 25 .
Figure 25.Distribution of motion modes of circular squirmer at different values of α and β.Square: SIM; Circle: WAO; Diamond: LAO.

Figure 26 .
Figure 26.Evolution of the head directions of the circular and elliptical pushers.

Figure 27 .
Figure 27.Distribution of motion modes for elliptical squirmer with aspect ratio c = 3.0 at different values of α and β.Square: SIM; Circle: WAO.

Figure 28 .
Figure 28.Kinematic features of the pusher at α = 1.1, β = −5 and the corresponding instantaneous pressure distribution: (a) the trajectory and head direction of the pusher, (b) instantaneous pressure distribution of the pusher.

Figure 29 .
Figure 29.Schematic of the motion of circular squirmer in the channel and time history of the circular squirmer's velocity for different diameters at Re s = 0.005, β = −3.
3 (c = 2.0) or ν = 0.45 (c = 3.0) Elliptical squirmer density ρ p 1.05 The semi-major and semi-minor axes of the elliptical squirmer a, b a = 40 or a = 60, b = 20 Characteristic scale d d = 2a The height and width of the channel H, L H = 45d, L = 4d Aspect ratio of elliptical squirmer c c = 2.0 or c = 3.0 The vertical position of the elliptical squirmer X initial position X = 0 The horizontal position of the elliptical squirmer Y initial position Y = 1.5dSwimming direction of the squirmer θ initial direction θ = −π /4 Self-propelling strength α [0.3, 1.1] Swimming type factor β [−5, 5] Steady swimming velocity of the squirmer at very low Reynolds numbers U s Given α and U g , it is calculated by equation (15) Sedimentation velocity of the 2D passive particle U g It is calculated by equation (16) The Reynolds number obtained only under self-propelled action Re s It is calculated by equation (18) The Reynolds number obtained only under gravity Re g 0.5

where
FBN denotes the fluid-boundary nodes, CN corresponds to the covered nodes, and UN signifies the uncovered nodes.Therefore, the total force and torque acting on the squirmer at time t 0 is[63] combining the calculated force with Newton's equation, the motion of the squirmer can be solved.With this analysis, the processes of fluid node destruction and creation are addressed.