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DOI: 10.1039/D0SM00616E
(Paper)
Soft Matter, 2020, Advance Article

Arne W. Zantop* and
Holger Stark*

Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany. E-mail: a.zantop@tu-berlin.de; holger.stark@tu-berlin.de

Received
8th April 2020
, Accepted 9th June 2020

First published on 25th June 2020

Microswimmers or active elements, such as bacteria and active filaments, have an elongated shape, which determines their individual and collective dynamics. There is still a need to identify what role long-range hydrodynamic interactions play in their fascinating dynamic structure formation. We construct rods of different aspect ratios using several spherical squirmer model swimmers. With the help of the mesoscale simulation method of multi-particle collision dynamics we analyze the flow fields of these squirmer rods both in a bulk fluid and in Hele-Shaw geometries of different slab widths. Based on the hydrodynamic multipole expansion either for bulk or confinement between two parallel plates, we categorize the different multipole contributions of neutral as well as pusher-type squirmer rods. We demonstrate how confinement alters the radial decay of the flow fields for a given force or source multipole moment compared to the bulk fluid.

Artificial microscopic swimmers with various locomotion mechanisms have recently been constructed to study the principal properties of their individual and collective motion. Examples mostly include spherical microswimmers.^{6,18–21} Elongated and flexible active constituents are realized by polar biofilaments in motility assays^{8} or at a fluid–fluid interface,^{7} while bacteria provide a natural realization of active rods.^{4}

Combining elements of the Toner–Tu^{22} and Swift–Hohenberg^{23} theories, continuum models for bacterial microswimmers have been developed that are able to reproduce pattern formation such as vortices and active turbulence.^{4,24–26} Alternative continuum theories for active matter based on liquid-crystal hydrodynamics reproduce the creation, annihilation, and motion of liquid-crystal defects^{27} or are able to describe the dynamics of the intracellular cytoskeleton gel.^{2,28}

In contrast, particle-based simulations are employed to study the collective dynamics of active rods. In the famous Vicsek model coarse-grained aligning interactions are able to reproduce dynamical states such as flocking and swarming.^{29–31} Langevin dynamics simulations of active rods^{4,32,33} or active filaments^{34,35} suggest that many properties might already emerge from short-ranged steric interactions. However, also implicit hydrodynamic pair interactions are included in such simulations to provide more realistic models with novel dynamic states.^{36,37} An extension to single active filaments exists.^{38,39} Explicit hydrodynamic simulation schemes have been applied, as well. For example, the lattice Boltzmann method was used to investigate the properties of microswimmers with various shapes^{40–42} while with the method of multi-particle collision dynamics collective phases of spherical^{43–45} and ellipsoidal microswimmers^{46,47} were studied, as well as realizations of pusher and puller-type swimmers.^{48,49}

In this work, we introduce and characterize single squirmer rods with the idea to model elongated microswimmers in their fluid environment. They then can be used to disentangle the effect of direct steric and long-range hydrodynamic interactions in their collective behavior. The squirmer model swimmer^{45,50,51} is a good model for spherical artificial microswimmers such as Janus particles^{52–54} and also biological organisms such as Volvox.^{55} Squirmers have in common that they propel themselves by an axisymmetric surface–velocity field, which acts on the surrounding fluid.^{45,56,57} In biological systems this is realized by cilia, located all over the cell surface, which perform synchronized collective non-reciprocal motions. The squirmer has frequently been used in hydrodynamic simulations of the collective behavior of microswimmers^{42,58,59} and also by our own group,^{43–45,60–62} where we rely on the mesoscale method of multi-particle collision dynamics.^{45,63,64}

In this article we use squirmers to build rigid rod-shaped microswimmers of different aspect ratios and perform large-scale simulations with multi-particle collision dynamics. We explore their hydrodynamic flow fields both in a bulk fluid and in a Hele-Shaw geometry with varying cell width, where we keep the squirmer rod in the midplane to mimic the fluid interface in the experiments of ref. 7 (cf. Fig. 1).

Fig. 1 Perspective view of a squirmer rod moving in the midplane (in light gray) of a Hele-Shaw cell with slab width Δz. Periodic boundary conditions along the x and y directions are used. |

In our analysis, we rely on the hydrodynamic multipole expansion both in the bulk fluid^{10,40,50,65} and between two parallel plates.^{12–14} This will enable us to categorize the different multipoles of both neutral as well as pusher-type squirmer rods and to determine the different radial decays of their multipole flow fields.

The paper is organized as follows. In Section 2 we introduce the squirmer rod and the methods to generate and analyze its flow fields. We present the results of our analysis in Section 3 and conclude in Section 4.

η∇^{2}u − ∇p + f = 0 ∇·u = 0.
| (1) |

v_{s} = B^{s}_{1}[(ê·_{s})_{s} − ê]
| (2) |

Fig. 2 (a) Schematic of a single squirmer of radius R and with orientation given by the unit vector ê. The surface slip-velocity field of a neutral squirmer is indicated by blue arrows. (b) Schematic of the squirmer rod model. Always, n_{sq} = 10 spherical squirmers are placed on a straight line with distance d to form active rods. All squirmer orientations ê are aligned with the rod axis. (c) Implementation of a pusher type squirmer rod. The surface slip velocity on the rod surface is multiplied by the envelop function f(x_{s}*·ê) of eqn (3) such that the slip-velocity field is concentrated on the rear of the rod. |

Having the surface velocity field distributed over the whole squirmer rod [cf. Fig. 2(b)], our model resembles ciliated microorganisms such as Paramecium. In contrast, bacteria like E. coli propel themselves with a bundle of rotating flagella that pushes fluid backwards while the cell body does not have any surface velocity field. To implement squirmer rods of such a pusher or also of puller type, we use the envelope function

(3) |

(4) |

(5) |

(6) |

u_{r,n}(r) = A_{n}r^{−n} + B_{n}r^{−n−2}.
| (7) |

Relevant for the squirmer rod will be the flow fields of the source dipole (u_{SD}), force dipole (u_{FD}), force quadrupole (u_{SD}), and source octupole (u_{SD}):†

u(r) = u_{SD}(r) + u_{FD}(r) + u_{FQ}(r) + v_{SO}(r) + ….
| (8) |

(9) |

It immediately becomes obvious that in contrast to the bulk fluid one cannot distinguish the flow fields of force and source multipoles with the same angular dependence (same order n) by the radial decay. We note that the coefficients scale differently with the slab width Δz. In Appendix A we motivate _{n} ∝ Δz and _{n} ∝ 1/Δz. However, our simulated data are not always sufficiently accurate to discriminate both cases. So, we assign the relevant multipole assuming that it is preserved from the bulk fluid. The multipole expansion captures the distribution of force and source multipoles in leading order. This distribution generated by the rod surface should remain the same in the Hele-Shaw geometry. However, since we deal with a voluminous rod in contrast to a point-like source, higher-order terms enter to fulfill the boundary conditions on both the rod surface and the bounding plates. The flow fields of these terms will decay faster than the leading bulk moments at some distance from the rod.

Finally, exploiting the orthogonality relations of cosnφ = T_{n}(cosφ), we extract the multipole moments from eqn (9) by projecting ũ_{ρ}(ρ,φ) on the Chebyshev polynomials,

(10) |

(11) |

During the streaming step (i) the point particles with masses m_{0}, positions x_{i}(t), and velocities v_{i}(t) move ballistically during time Δt,

x_{i}(t + Δt) = x_{i}(t) + v_{i}(t)Δt.
| (12) |

For the collision step (ii), the simulation volume is divided by a cubic lattice and the fluid particles are grouped into the cubic unit cells of linear size a_{0} and centered around ξ. First, for the n_{ξ} particles in each cell with volume _{ξ} the mean velocity v_{ξ} and center-of-mass position x_{ξ} are determined. Then, in the center-of-mass frame the fluid particles are assigned new random velocities δv_{i} from a Boltzmann-distribution with temperature T_{0}. To restore overall momentum conservation, the total change in linear momentum, , has to be subtracted from the new velocities, while an additional term containing the difference of the angular momentum before the collision , and the change in angular momentum, , is added to preserve angular momentum. Here x_{i,c} denotes the particle position relative to the center-of-mass position x_{ξ}. Thus, the collision step can be summarized by

v^{new}_{i} = v_{ξ} + δv_{i} − Δv_{ξ} − x_{i,c} × I_{ξ}^{−1}(L_{ξ} − ΔL_{ξ}),
| (13) |

Given the center-of-mass and angular momentum of the rigid squirmer rod, its location and orientation are updated 10 times during each collision step using a standard leapfrog algorithm.^{67} The rod is treated as single rigid body the position and orientation vector of which is updated following ref. 67. To keep the squirmer rod in the midplane of the Hele-Shaw cell, we only use the x and y component of the fluid force acting on the rod to integrate its motion in time throughout the simulation. Due to the symmetry about the midplane only Brownian forces are acting on the squirmer normal to the midplane. Note due to this constraint we do not observe any oscillatory trajectories between the plates as observed in ref. 68.

We have implemented the MPCD model fluid together with the squirmer-rod model in C++ and CUDA to enable the use of graphic cards. Because the main performance bottleneck in MPCD is memory access, we integrate a sorting algorithm and lookup table following ref. 69. Additionally, we use variable size cooperative thread groups to add dynamic load balancing to our MPCD collision routines in CUDA.

To determine how the hydrodynamic moments of the squirmer flow field in a bulk fluid varies with the aspect ratio α in Section 3.1, we simulate single rods in a cubic box of linear size L = 100a_{0} using periodic boundary conditions in all three dimensions. For all other bulk simulations in Sections 3.1 and 3.3, we use box sizes of L = 180a_{0} with periodic boundary conditions. We begin by simulating the system for a time 10^{4}Δt to equilibrate the MPCD fluid flow and then average the flow fields over the time interval from 10^{4}Δt to 5 × 10^{6}Δt. To determine the radial component v_{r}(r,θ) of the velocity field, we also exploit the rotational symmetry of the flow fields by averaging about the rod axis. Using eqn (6), we then obtain the expansion coefficients u_{r,n}(r) for different multipole order n and extract the strengths of the hydrodynamic moments by fitting u_{r,n}(r) from eqn (7) to the curves determined from the simulated flow fields.

For the simulations in Hele-Shaw geometry in Sections 3.2 and 3.3, we use box sizes of L = 200a_{0} with no-slip walls and periodic boundary conditions along the x and y directions. For the slab width we investigate the three values Δz/R = 2.7, 6.0, and 9.3. When determining the flow fields in the Hele-Shaw geometry, the total simulation time is increased up to 10^{7}Δt to compensate for the fact that the flow field cannot be averaged about the rod axis.

To provide a quantitative analysis, we apply eqn (6) and decompose the flow field into its different angular contributions with the nth-order Legendre coefficients u_{r,n}(r). As an example, we show in Fig. 4(a) the measured u_{r,n}(r) as data points for a squirmer rod of α = 4 and also include fitted polynomials in r^{−1} as solid lines.

Fig. 4 (a) Radial Legendre coefficients u_{r,n}(r) of the simulated flow field of a neutral squirmer rod with aspect ratio α = 4.0 calculated from eqn (6) (symbols) and with fitted power laws (solid lines). SD and FQ stand for source dipole and force quadrupole, respectively. (b) Hydrodynamic multipole moments in units of the squirmer parameter B^{s}_{1}R^{3} plotted versus aspect ratio α. (c) Swimming speed v_{rod} of a squirmer rod plotted versus α. v_{rod} is normalized by the swimming speed v_{0} = 2/3B^{s}_{1} of a single squirmer at α = 1. Simulation data (blue dots) and analytical values as given by eqn (15) (dashed black line). |

All radial velocity components u_{r,n}(r) either show distinct power law behavior as for n = 1 and 3 or vanish. The first-order coefficient u_{r,1}(r) clearly shows a pure 1/r^{3} decay, indicating a zero force monopole moment A_{1} while the source dipole moment B_{1} is present, as expected. This is consistent with the fact that we consider microswimmers free of external forces. Consistent with the head–tail symmetry of the forces, the squirmer rods exert on the fluid, we do not observe a force dipole moment, A_{2} = 0, since the second-order coefficient u_{r,2}(r) vanishes. Furthermore, the third-order contribution u_{r,3}(r) is also present and decays with 1/r^{3}, which indicates an additional force quadrupole moment A_{3}. We elaborate on its origin further below. Both curves u_{r,1}(r) and u_{r,3}(r) fall off from the theoretically predicted power law at large r due to the finite size of the simulation box. They also show small deviations from the power law, which we attribute to the flow fields of the image swimmers introduced by the use of periodic boundary conditions. All other coefficients u_{r,n}(r) are too small to be distinguished from noise and hence can be neglected in the following. Thus, the flow field of a neutral squirmer rod is a superposition of a source dipole moment (B_{1} ≠ 0), and a force quadrupole moment (A_{3} ≠ 0), which both decay as 1/r^{3}. In Fig. 4(b) we plot the two moments normalized by the squirmer parameter B^{s}_{1} versus the aspect ratio α. They both increase roughly linearly in α. While the source-dipole moment B_{1} shows a modest increase in α starting from B_{1} = B^{s}_{1}R^{3} for α = 1, the force quadrupole moment A_{3} increases much more strongly from zero and clearly dominates beyond α = 2.

The additional force quadrupole does not break the head–tail symmetry of the squirmer rod, which we therefore denote as neutral. However, the corresponding flow field clearly affects the shape of the overall flow field as we saw in Fig. 3. Since its angular dependence is governed by the third-order Legendre polynomial P_{3}(cosθ) compared to P_{1}(cosθ) of the source dipole, it will influence the hydrodynamic interactions with other squirmer rods, also because its flow field shows the same radial decay, 1/r^{3}.

To shed some light on the origin of the force quadrupole, we determined the local force F_{hyd}(y_{i}), with which the spherical squirmer component i placed at y_{i} along the rod axis acts on the surrounding fluid. Of course, the rod is force free, thus all the forces F_{hyd}(y_{i}) add up to zero, . The forces from the terminal squirmers i = 1 and i = N_{sq} are mainly determined by fluid pressure. They cancel each other up to some remaining force, F_{hyd,p} = F_{hyd}(y_{1}) + F_{hyd}(y_{Nsq}). In the following, we refer the force of the other squirmers on this remaining force, δF_{hyd}(y_{i}) = F_{hyd}(y_{i}) − F_{hyd,p}/(N_{sq} − 2), so that they sum up to zero: . In Fig. 5(a) we plot the relative force δF_{hyd}(y_{i}), the schematic in Fig. 5(b) rationalizes the different signs of δF_{hyd}(y_{i}). In the center of the rod (y = 0), the relative local force is negative due to the surface velocity of the squirmer rod pushing fluid backwards, while at both ends of the rod the fluid is dragged with the rod. This generates the observed force quadrupole.

Finally, as we show in Fig. 4(c), the swimming speed v_{rod} of the rod increases with α and then saturates. With increasing aspect ratio the swimming speed is more and more determined by the maximum surface velocity around the equator of the constituent squirmers and thus the speed increases. At large α the cap regions of the squirmer rod become irrelevant and the swimming speed saturates. To calculate an analytic expression for the swimming speed, we average the surface slip-velocity field v_{s} of the squirmer rod over the whole surface with area S_{rod}.^{70} For the component along the squirmer axis ê we obtain

(14) |

(15) |

In the lower panel of Fig. 6, further away from the rod (|x/l_{S}| > 1) the streamlines are approximately parallel to the bounding plates in agreement with the Poiseuille flow profile of the single force and source multipoles as detailed in Appendix A. However, in the direct vicinity of the rod, |x/l_{S}| < 3/4, near-field flow along the z direction occurs, which is due to terms with exponential decay in the full hydrodynamic solution in a slab geometry.

We now present a more quantitative analysis of the simulated flow fields in Fig. 7 and 8. As for the bulk fluid, we decompose the flow fields into the different angular contributions given by the Chebyshev polynomials T_{n}(cosφ) using eqn (10). Since from the radial decay of the expansion coefficients ũ_{ρ,n}(ρ) we cannot distinguish between the force multipole of nth order and the source multipole of n + 1th order, we are guided by the analysis in the bulk fluid from Section 3.1 and attribute the expansion coefficients ũ_{ρ,1}(ρ) and ũ_{ρ,3}(ρ), plotted in Fig. 7 and 8, to a source dipole (SD) and force quadrupole (FQ), respectively.

Fig. 8 Multipole analysis of the simulated flow field of a neutral squirmer rod confined in a Hele-Shaw cell with slab width Δz = 2.7R for different aspect ratios: (a) α = 1.75, (b) α = 3.25, and (c) α = 4.0. Otherwise, the same description as in Fig. 7 is used. |

In Fig. 7 we plot ũ_{ρ,1/3}(ρ) averaged over the cell height for different slab widths Δz. For a small width Δz = 2.7R = 0.34l_{S} the coefficients are in very good agreement with the expected power-law decay: 1/ρ^{2} for the source dipole and 1/ρ^{4} for the force quadrupole. Both power laws fit well over the whole range of the radial distance. For slab width Δz = 6R = 0.75l_{S} the measured coefficients ũ_{ρ,n}(ρ) approach the predicted power-law decays at a radial distance of approximately ρ/l_{S} = 1. This is larger than the slab width, where we expect the far-field solutions of the different multipoles to become valid. At this distance the corresponding streamlines shown in the lower panel of Fig. 6 are parallel to the bounding plates as predicted by the multipole flow fields. Finally, increasing the slab width further to Δz = 9.3R = 1.2l_{S} the measured coefficients ũ_{ρ,n}(ρ) show clear deviations from the expected power-law decay over the whole range of the radial distance ρ. All in all, when we vary the slab width, the force quadrupole moment stays the same, while the source dipole moment increases with decreasing Δz.

In Fig. 8 we choose the smallest slab width Δz/R = 2.7, where we obtained the best agreement in the previous figure and plot the coefficients ũ_{ρ,1/3}(ρ) for different aspect ratios α. Again there is very good agreement between the coefficients and the expected power law decay. Only at radial distance smaller than l_{S} one realizes deviations, as expected. This is, in particular, visible in plot (a).

Fig. 9 Absolute values of the radial Legendre coefficients |u_{r,n}(r)| of the simulated flow field of a pusher-type squirmer rod with aspect ratio α = 4.0 calculated from eqn (6) (symbols) and with fitted power laws (solid lines). FD, FQ stand for force dipole and quadrupole, while SD, SO mean source dipole and octupole, respectively. |

In Fig. 10 we analyze the flow field of the pusher-type squirmer rod in the Hele-Shaw geometry for different slab widths. In addition to the findings for the neutral squirmer rod (cf. Fig. 7), we also observe the force dipole. Compared to the bulk fluid, it now has a stronger radial decay, ũ_{ρ,2}(ρ) ∝ 1/ρ^{3}, as described by theory. At distances ρ/l_{S} < 1.3 the strong but shorter ranged force-dipole flow field dominates, while for ρ/l_{S} > 1.3 the slower decay of the source-dipole field takes over. Increasing the slab width, the flow field of the force dipole always dominates in the given radial range [cf. Fig. 10(b) and (c)]. The reason is the green curve shifts upwards while the blue curve shifts downwards with increasing Δz. Thus, the dipole-force moment increases with Δz in qualitative agreement with _{n} ∝ Δz (cf. Section 2.3), while the source-dipole moment decreases with Δz again in qualitative agreement with _{n} ∝ 1/Δz.

Fig. 10 Multipole analysis of the simulated flow field of a pusher-type squirmer rod with aspect ratio α = 4 for different slab widths of the Hele-Shaw cell: (a) Δz = 2.7R, (b) Δz = 6.0R and (c) Δz = 9.3R. In addition to the source dipole (n = 1, blue dots) and force quadrupole/source octupole (n = 3, red pluses), the force dipole (n = 2, green crosses) is observed. Otherwise, the same description as in Fig. 7 is used. |

The third-order coefficient ũ_{ρ,3}(ρ) ∝ 1/ρ^{4} does not change very significantly in one direction with increasing Δz. We take this as an indication that both the force quadrupole and source octupole contribute to the flow field. A more detailed analysis of _{3} and _{3} is not feasible with the hydrodynamic MPCD method since due to thermal fluctuations it requires long averaging in order to obtain smooth flow lines at large distances. Therefore, larger system sizes would require an immense computational effort.

The flow field of the neutral squirmer rod in the bulk fluid shows the expected source dipole, while a force quadrupole moment develops linearly with increasing aspect ratio and becomes dominant beyond α = 2. It is due to a non-uniform distribution of the force, with which the rod acts on the fluid. By taking an average of the surface slip-velocity field over the rod surface, the actual swimming velocity is overestimated. In the Hele-Shaw geometry the radial decay of the multipole flow fields changes as predicted by theory. Especially at low slab width Δz we find a good match with our simulations. The flow field of the source dipole now decay as 1/ρ^{2} and dominates at radial distances ρ > Δz for all α and Δz over the field of the force quadrupole, which now decays as 1/ρ^{4}.

For the pusher-type squirmer rod with noticeable elongation we observe that the flow field is composed of four hydrodynamic moments: force dipole, source dipole, force quadrupole, and source octupole. In bulk the force dipole completely dominates the flow field and determines the radial decay with 1/r^{2}. However, in the Hele-Shaw geometry the radial decay changes to 1/ρ^{3} and is less long-ranged. Nevertheless, for larger slab widths and the recorded radial distances it dominates the flow field, while for small slab widths we see a cross over to the longer-ranged source-dipole field. This is in qualitative agreement with the expected behavior of the strength of the multipole moments with increasing slab width: the force dipole becomes stronger while the source dipole weakens. Finally, in the Hele-Shaw geometry the flow fields of force quadrupole and source octupole have the same radial decay. Varying slab width suggests that they both contribute.

Our work shows how elongated microswimmers generate additional hydrodynamic multipole moments compared to swimmers of spherical shape, which leads to a more complex appearance of the generated flow field. Since rods experience additional torques in a non-uniform flow field via the strain-rate tensor^{71} this will generate different behavior in suspensions of squirmer rods compared to their pure steric interactions. Furthermore, the source dipole is predicted to be the dominant hydrodynamic moment in a Hele-Shaw geometry,^{13} which we confirm for neutral squirmer rods for varying slab width. However, for pusher and puller rods the dominance of the force dipole depends on the slab width and radial distance.

Therefore, in the continuation of this work, we plan to investigate the collective dynamic behavior of the squirmer rods introduced in this work. Thereby we will gain an understanding how hydrodynamic interactions through the self-generated flow fields contribute to different types of dynamic behavior such as swarming in active nematics or active turbulent phases. As a further extension of the presented model, we plan to introduce bending rigidity between the different components of the squirmer rod to model active flexible filaments used, for example, in ref. 7 and 8.

(16) |

In the Stokes flow regime any flow field surrounding a solid body can be written as the sum of hydrodynamic multipoles or singularities, which can be derived from the Stokeslet in eqn (16). For the flow field of the squirmer rod we list the relevant terms:

u(r) = u_{FD}(r) + u_{FQ}(r) + u_{SD}(r) + u_{SO}(r)….
| (17) |

Following ref. 14 and 65 we derive the flow fields of force multipoles with uniaxial symmetry from the Stokeslet in eqn (16) by applying the directional derivative ê·∇_{p}n times on _{ij}ê_{j}, where ∇_{p} acts on the location of the point force. For n = 1 and 2 we thus obtain the respective flow fields of a hydrodynamic force dipole,

(18) |

(19) |

(20) |

(21) |

(22) |

ũ_{ρ}(ρ,φ) = ũ_{ρ,FD}(ρ,φ) + ũ_{ρ,FQ}(ρ,φ) + ũ_{ρ,SD}(ρ,φ) +ũ_{ρ,SO}(ρ,φ) + ….
| (23) |

(24) |

(25) |

(26) |

(27) |

In each of the constituent parts of the squirmer rods, we use cylindrical coordinates (ρ,φ,z), where z = 0 is the center of each spherical squirmer. The constituent parts are sections of spheres of radius R that extend from z = h_{min} to z = h_{max}. Integration over the enclosed volume with constant mass density ρ_{0}, gives the mass

(28) |

Based on the known formula for the moment-of-inertia tensor for a solid body with uniform mass density ρ_{0}, , we first calculate the I_{zz} components of the constituent parts. Integrating ρ_{0}ρ^{2} over the enclosed volume yields

(29) |

(30) |

(31) |

We now use the previous results to calculate the relevant moments of inertia I_{rod,ij} for the squirmer rod. For the moment I_{rod,zz} the relevant rotational axis along ẑ goes through all the centers of mass of the n_{s} segments. Thus, we can just add up their moments of inertia:

I_{rod,zz} = 2I_{cap,zz}+ (n_{s} − 2)I_{mid,zz}.
| (32) |

(33) |

(34) |

(35) |

To provide an estimate for the swimming velocity of the squirmer rod, we apply the approximation that the surface force density is constant along the surface of the passive rod, when pulled by force F′ along its long axis. The estimate for the swimming velocity is thus the average of the surface slip velocity v_{s} as given in eqn (14).

To determine the swimming velocity, we calculate the surface of the squirmer rod,

(36) |

v_{s}·ê = B^{s}_{1}(cos^{2}θ − 1)
| (37) |

(38) |

(39) |

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## Footnote |

† Although the source quadrupole shares the same radial dependence as the force dipole in the Hele-Shaw geometry, it is not considered in this list for two reasons. First, we solely observe a force dipole in the bulk fluid but no source quadrupole. Second, the strength of ũ_{ρ,2}(ρ) grows with the slab width Δz in our simulations, which identifies it as a force dipole rather than a source quadrupole. We will elaborate on this further below. |

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