# Stability of a Dumbbell Micro-Swimmer

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Basic Equations and Numerical Methods

#### 2.1. Squirmer Model

**u**

^{s}is given by

_{n}is the nth Legendre polynomial,

**e**is the unit orientation vector of the squirmer, and

**r**is the position vector, r = |

**r**|. A solitary squirmer swims at speed U

_{0}= 2B

_{1}/3. As in our previous studies [28,34,35], we omitted squirming modes higher than the second (i.e., B

_{n}= 0 when $n\ge 3$) because the effect of higher modes becomes negligibly small compared to that of the first and second modes in far-field hydrodynamic interactions.

_{2}/B

_{1}. A squirmer with positive β is a puller, analogous to a micro-swimmer for which thrust is mainly generated in front of the body. A squirmer with negative β is a pusher, analogous to a micro-swimmer for which thrust is mainly generated behind the body. A squirmer with β = 0 is a neutral swimmer, analogous to a micro-swimmer for which the thrust and drag centers coincide.

#### 2.2. Basic Equations

_{m}is the surface of squirmer m.

**J**is the Green’s function for unbounded fluid given by ${J}_{ij}={\delta}_{ij}/r+{r}_{i}{r}_{j}/{r}^{3}$, where

**r**=

**x**–

**y**, and

**δ**is the Kronecker delta. The single-layer potential

**q**is found by subtracting the traction force on the inner surface,

**f**

_{in}, from that on the outer surface,

**f**

_{out}, i.e.,

**q**=

**f**

_{out}−

**f**

_{in}. The boundary condition is given by:

**U**

_{m}and

**Ω**

_{m}are the translational and rotational velocities of squirmer m, ${r}_{m}^{c}$ is the center of squirmer m and ${u}_{m}^{s}$ is the squirming velocity of squirmer m, defined by Equation (1).

**F**

_{m}and torques

**T**

_{m}. The equilibrium conditions for squirmer m are:

- (a)
- In Section 3, we assume that the distance between the centers of two far-field squirmers is invariant in time, i.e., l = const (Figure 1). In this case,
**F**_{m}is generated so as to satisfy the constant distance condition.**T**_{m}is assumed to be zero, such that squirmer orientation satisfies the torque-free condition. - (b)
- In Section 4, we connect two near-field squirmers by a dragless rigid rod (Figure 2). In this case, both squirmers exhibit rigid body motion, i.e., ${U}_{2}={U}_{1}+{\Omega}_{1}\wedge \left({r}_{2}^{c}-{r}_{1}^{c}\right)$ and ${\Omega}_{2}={\Omega}_{1}$.
**F**_{m}and**T**_{m}are determined so as to achieve the rigid body motion. - (c)
- In Section 5, we connect two squirmer surfaces by a dragless linear spring (Figure 6). The spring is connected at ${r}_{m}^{s}$ on the surface of squirmer m, and generates force ${F}_{m}^{s}$ given by:$${F}_{1}^{s}=k\left(\left|{r}_{2}^{s}-{r}_{1}^{s}\right|-{l}_{0}\right)\frac{{r}_{2}^{s}-{r}_{1}^{s}}{\left|{r}_{2}^{s}-{r}_{1}^{s}\right|}\text{},\text{}{F}_{2}^{s}=-{F}_{1}^{s}$$
_{0}is the equilibrium length of the spring. The force generates a torque on squirmer m as:$${T}_{m}^{s}=\text{}\left({r}_{m}^{s}-{r}_{m}^{c}\right)\wedge {F}_{m}^{s}$$

#### 2.3. Numerical Methods

## 3. Linear Stability Analysis of a Dumbbell Squirmer in the Far-Field

**e**

_{1}and

**e**

_{2}, respectively. For simplicity, we assumed that both

**e**

_{1}and

**e**

_{2}existed in the same x–y plane. The angle of orientation vector

**e**

_{m}from the x-axis was θ

_{m}, where m is 1 or 2. To force dumbbell motion, the distance between the centers of the squirmers was set as invariant in time, i.e., l = const, by imposing x-direction force F

_{x}to squirmer 1 and −F

_{x}to squirmer 2. We imposed no external torque on the squirmers; therefore, squirmer orientation was determined so as to satisfy the torque-free condition. The torque condition can induce instability in the swimming of the dumbbell squirmer; we further investigated this phenomenon. Squirmer m swims with a translational velocity of

**U**

_{m}by generating surface squirming velocity. The squirming velocities were assumed to be the same for both squirmers, inducing the stresslet

**S**, given by [31]:

**I**is the identity matrix.

**F**and stresslet

**S**are exerted on squirmer 1, rotational velocity ω far from the squirmer can be approximated as:

**ε**is the alternating unit tensor. The kernel function

**K**is given by [41]:

_{x}acting on squirmer 1 generates no rotational velocity at the center of squirmer 2, ${r}_{2}^{c}$. In contrast, the stresslet of squirmer 1 generates the following rotational velocity at ${r}_{2}^{c}$:

**U**

_{2}−

**U**

_{1}, where the x-component is zero by definition. The y-component is non-zero, and can be approximated as ${U}_{0}\mathrm{sin}\left({\theta}_{2}-{\theta}_{1}\right)$ in the leading order. The y-component velocities induce a rotational velocity of ${U}_{0}\left(\mathrm{sin}{\theta}_{1}-\mathrm{sin}{\theta}_{2}\right)/l$ to both squirmers relative to the center-to-center vector. Thus, the orientation change of the squirmers can be expressed as:

_{0}> 0. Thus, dumbbell puller squirmers can achieve stable swimming when they are oriented in opposite directions. In this case, however, the swimming velocity of the dumbbell squirmer is zero. These results indicate that the dumbbell squirmer cannot achieve stable forward swimming under the present problem settings, i.e., two squirmers in the far field with no external torque applied.

## 4. Swimming of a Dumbbell Squirmer Connected by a Short Rigid Rod

**U**

_{m}and

**Ω**

_{m}are the translational and rotational velocities, respectively, of squirmer m. To achieve rigid body motion, the connecting rod imparts forces

**F**

_{m}and torques

**T**

_{m}to the squirmers.

#### 4.1. Two Identical Squirmers Connected Side-by-Side

_{max}increased as β increased. These results indicate that swimming velocity was not significantly increased in the case of the side-by-side dumbbell squirmers.

#### 4.2. Squirmers with Different Modes Connected Fore and Aft

_{1}and β

_{2}, connected fore and aft (Figure 2b). A dragless short rigid rod was again connected at the minimum distance between the spherical surfaces. The two squirmer centers were placed along a line parallel to the y-axis, and both squirmers were oriented in the y-direction. Due to symmetry of the problem along the y-axis, the dumbbell swam in the y-direction with velocity U. The rod exerted y-direction force F, and no torque was generated on the squirmers.

_{1}> 0 and β

_{2}< 0. When β

_{1}= 3 and β

_{2}= −3, velocity increased to become about 1.4 times larger than that of a solitary squirmer. When β

_{1}= −3 and β

_{2}= 3, velocity decreased to become about 0.6 times smaller than that of a solitary squirmer.

_{1}+ β

_{2}= 0. When β

_{1}+ β

_{2}> 0, F was positive and the squirmers tended to attract each other. When β

_{1}+ β

_{2}< 0, F was negative and the squirmers tended to repel each other.

_{1}+ β

_{2}< 0.

## 5. Stability of a Dumbbell Squirmer Connected by a Spring

#### 5.1. Two Identical Squirmers Are Connected Side-by-Side

_{0}= a. The spring constant k was set as k/μU

_{0}= 100, to ensure that the spring was sufficiently strong and its length did not change considerably. The spring force generated torque on the squirmer, as shown in Equation (6). The squirmers had the same swimming mode β.

_{1}= 1° and ϕ

_{2}= 2°. The dumbbell squirmer swam with velocity

**U**. We varied swimming mode β, and examined the stability of side-by-side swimming. We observed that stability was not affected by the magnitude of the initial disturbance, provided that the disturbance was not too large.

_{0}, such that side-by-side swimming did not greatly affect swimming speed (Figure 3).

**F**

_{s}to squirmer 1 and resulting in spring torque

**T**

_{s}(red arrow). This torque counterbalanced hydrodynamic torque

**T**

_{u}, which was generated by the squirming velocity of squirmer 2. Since the torques canceled each other out, the squirmer was able to maintain steady orientation while swimming.

#### 5.2. Squirmers with Different Modes Connected Fore and Aft

_{1}and β

_{2}, connected fore and aft by a dragless linear spring (Figure 6b). The spring connected the bottom of squirmer 1 (${r}_{1}^{s}={r}_{1}^{c}+a{e}_{1}$) and the top of squirmer 2 (${r}_{2}^{s}={r}_{2}^{c}+a{e}_{2}$). The spring generated force and torque as given by Equations (5) and (6), at an equilibrium length of l

_{0}= a and spring constant of k/μU

_{0}= 100.

_{1}= 1° and ϕ

_{2}= 2°. The dumbbell squirmer swam at velocity

**U**. To reduce the number of parameters, we assumed that β

_{1}= 0 and varied swimming mode β

_{2}to examine the stability of fore and aft swimming. We again observed that stability was not affected by the magnitude of the initial disturbance, provided that the disturbance was not too large.

_{2}= −3 was about 1.2U

_{0}, indicating that the fore and aft squirmers were able to increase swimming speed considerably (Figure 5a).

_{1}= 0 and β

_{2}= −3. A large velocity was generated behind the aft squirmer, whereas a small velocity was generated in front of the fore squirmer and between the squirmers. In this case, the orientation vectors were aligned. The velocity field around the dumbbell squirmers is shown schematically in Figure 10b. Since the aft squirmer was a strong pusher, the squirmers tended to repel each other (Figure 5b). The repulsion flow stretched the connecting spring, imparting force

**F**

_{s}to the squirmers. The spring force resulted in spring torques

**T**

_{s}(red arrows), which overwhelmed the hydrodynamic torques

**T**

_{u}generated by the squirming velocities; thus, the two squirmers eventually aligned.

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

- Gao, W.; Wang, J. The Environmental Impact of Micro/Nanomachines: A Review. ACS Nano
**2014**, 8, 3170–3180. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Peyer, K.E.; Zhang, L.; Nelson, B.J. Bio-inspired magnetic swimming microrobots for biomedical applications. Nanoscale
**2013**, 5, 1259–1272. [Google Scholar] [CrossRef] [PubMed] - Feng, J.; Cho, S.K. Mini and Micro Propulsion for Medical Swimmers. Micromachines
**2014**, 5, 97–113. [Google Scholar] [Green Version] - Rao, K.J.; Li, F.; Meng, L.; Zheng, H.; Cai, F.; Wang, W.A. Force to Be Reckoned With: A Review of Synthetic Microswimmers Powered by Ultrasound. Small
**2015**, 11, 2836–2846. [Google Scholar] [CrossRef] [PubMed] - Nain, S.; Sharma, N.N. Propulsion of an artificial nanoswimmer: A comprehensive review. Front. Life Sci.
**2015**, 8, 2–17. [Google Scholar] [CrossRef] - Zarei, M.; Zarei, M. Self-Propelled Micro/Nanomotors for Sensing and Environmental Remediation. Small
**2018**, 14, 1800912. [Google Scholar] [CrossRef] [PubMed] - Chen, H.; Zhao, Q.; Du, X. Light-Powered Micro/Nanomotors. Micromachines
**2018**, 9, 41. [Google Scholar] - Morita, T.; Omori, T.; Ishikawa, T. Passive swimming of a microcapsule in vertical fluid oscillation. Phys. Rev. E
**2018**, 98, 023108. [Google Scholar] [CrossRef] - Shi, J.-M.; Cui, R.-F.; Xiao, J.; Qiao, L.-Y.; Mao, J.-W.; Chen, J.-X. Pair Interaction of Catalytical Sphere Dimers in Chemically Active Media. Micromachines
**2018**, 9, 35. [Google Scholar] [CrossRef] - Manjare, M.; Yang, F.; Qiao, R.; Zhao, Y. Marangoni Flow Induced Collective Motion of Catalytic Micromotors. J. Phys. Chem. C
**2015**, 119, 28361–28367. [Google Scholar] [CrossRef] - Wang, W.; Duan, W.; Ahmed, S.; Sen, A.; Mallouk, T.E. From One to Many: Dynamic Assembly and Collective Behavior of Self-Propelled Colloidal Motors. Acc. Chem. Res.
**2015**, 48, 1938–1946. [Google Scholar] [CrossRef] [PubMed] - Yang, X.; Wu, N. Change the Collective Behaviors of Colloidal Motors by Tuning Electrohydrodynamic Flow at the Subparticle Level. Langmuir
**2018**, 34, 952–960. [Google Scholar] [CrossRef] - Liu, C.; Xu, T.; Xu, L.-P.; Zhang, X. Controllable Swarming and Assembly of Micro/Nanomachines. Micromachines
**2018**, 9, 10. [Google Scholar] - Purcell, E.M. Life at low Reynolds number. Am. J. Phys.
**1977**, 45, 3–11. [Google Scholar] [CrossRef] - Elgeti, J.; Winkler, R.G.; Gompper, G. Physics of microswimmers—Single particle motion and collective behavior: A review. Rep. Prog. Phys.
**2015**, 78, 056601. [Google Scholar] [CrossRef] [PubMed] - Bechinger, C. Active particles in complex and crowded environments. Rev. Mod. Phys.
**2016**, 88, 045006. [Google Scholar] [CrossRef] - Ishikawa, T. Suspension biomechanics of swimming microbes. J. R. Soc. Interface
**2009**, 6, 815–834. [Google Scholar] [CrossRef] [Green Version] - Lauga, E.; Powers, T.R. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys.
**2009**, 72, 096601. [Google Scholar] [CrossRef] [Green Version] - Guasto, J.S.; Rusconi, R.; Stocker, R. Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech.
**2012**, 44, 373–400. [Google Scholar] [CrossRef] - Goldstein, R.E. Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech.
**2015**, 47, 343–375. [Google Scholar] [CrossRef] - Lighthill, M.J. On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Math.
**1952**, 5, 109–118. [Google Scholar] [CrossRef] - Blake, J.R. A spherical envelope approach to ciliary propulsion. J. Fluid Mech.
**1971**, 46, 199–208. [Google Scholar] [CrossRef] - Pedley, T.J.; Brumley, D.R.; Goldstein, R.E. Squirmers with swirl: A model for Volvox swimming. J. Fluid Mech.
**2016**, 798, 165–186. [Google Scholar] [CrossRef] - Würger, A. Self-Diffusiophoresis of Janus Particles in Near-Critical Mixtures. Phys. Rev. Lett.
**2015**, 115, 188304. [Google Scholar] [CrossRef] - Thutupalli, S.; Seemann, R.; Herminghaus, S. Swarming behavior of simple model squirmers. New J. Phys.
**2011**, 13, 073021. [Google Scholar] [CrossRef] [Green Version] - Magar, V.; Goto, T.; Pedley, T.J. Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Math.
**2003**, 56, 65–91. [Google Scholar] [CrossRef] - Magar, V.; Pedley, T.J. Average nutrient uptake by a self-propelled unsteady squirmer. J. Fluid Mech.
**2005**, 539, 93–112. [Google Scholar] [CrossRef] - Ishikawa, T.; Kajiki, S.; Imai, Y.; Omori, T. Nutrient uptake in a suspension of squirmers. J. Fluid Mech.
**2016**, 789, 481–499. [Google Scholar] [CrossRef] - Michelin, S.; Lauga, E. Efficiency optimization and symmetry-breaking in a model of ciliary locomotion. Phys. Fluids
**2010**, 22, 111901. [Google Scholar] [CrossRef] [Green Version] - Michelin, S.; Lauga, E. Unsteady feeding and optimal strokes of model ciliates. J. Fluid Mech.
**2013**, 715, 1–31. [Google Scholar] [CrossRef] [Green Version] - Ishikawa, T.; Simmonds, M.P.; Pedley, T.J. Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech.
**2006**, 568, 119–160. [Google Scholar] [CrossRef] - Götze, I.O.; Gompper, G. Mesoscale simulations of hydrodynamic squirmer interactions. Phys. Rev. E
**2010**, 82, 041921. [Google Scholar] [CrossRef] [Green Version] - Navarro, R.M.; Pagonabarraga, I. Hydrodynamic interaction between two trapped swimming model micro-organisms. Eur. Phys. J. E
**2010**, 33, 27–39. [Google Scholar] [CrossRef] - Ishikawa, T.; Pedley, T.J. Coherent Structures in Monolayers of Swimming Particles. Phys. Rev. Lett.
**2008**, 100, 088103. [Google Scholar] [CrossRef] - Ishikawa, T.; Locsei, J.T.; Pedley, T.J. Development of coherent structures in concentrated suspensions of swimming model micro-organisms. J. Fluid Mech.
**2008**, 615, 401–431. [Google Scholar] [CrossRef] - Zöttl, A.; Stark, H. Hydrodynamics determines collective motion and phase behaviour of active colloids in quasi-two-dimensional confinement. Phys. Rev. Lett.
**2014**, 112, 118101. [Google Scholar] [CrossRef] - Ishikawa, T.; Pedley, T.J. Diffusion of swimming model micro-organisms in a semi-dilute suspension. J. Fluid Mech.
**2007**, 588, 437–462. [Google Scholar] [CrossRef] - Ishikawa, T.; Pedley, T.J. The rheology of a semi-dilute suspension of swimming model micro-organisms. J. Fluid Mech.
**2007**, 588, 399–435. [Google Scholar] [CrossRef] - Pedley, T.J. Spherical squirmers: Models for swimming micro-organisms. IMA J. Appl. Math.
**2016**, 81, 488–521. [Google Scholar] [CrossRef] - Youngren, G.K.; Acrivos, A. Stokes flow past a particle of arbitrary shape: A numerical method of solution. J. Fluid Mech.
**1975**, 69, 377–403. [Google Scholar] [CrossRef] - Kim, S.; Karrila, S.J. Microhydrodynamics: Principles and Selected Applications; Butterworth Heinemann: London, UK, 1992. [Google Scholar]

**Figure 1.**Problem settings for the far-field analysis. Two squirmers (1 and 2) with orientation vectors

**e**

_{1}and

**e**

_{2}, respectively, are shown in the far field. The center of squirmer 1 was placed at the origin of coordinate. We assumed that

**e**

_{1}and

**e**

_{2}were in the x–y plane, where the x-axis was taken as the direction passing through the center of squirmer 2 at ${r}_{2}^{c}=\left(l,0,0\right)$ and l is the distance between the squirmer centers. This distance was assumed to be time-invariant with the application of x-direction force F

_{x}. No external torque was exerted; thus, the squirmer orientation was determined to satisfy the torque-free condition. The orientation angles θ

_{1}and θ

_{1}were defined from the x-axis. The radius of the squirmers was a, which satisfied l >> a. The squirmers swam with velocity

**U**

_{i}, where i was 1 or 2, and exerted a stresslet of

**S**.

**Figure 2.**Problem settings for a dumbbell squirmer connected by a dragless rigid rod. The length of rod ε was set at 0.01a. (

**a**) Two identical squirmers with swimming mode β were placed side-by-side at an angle of ϕ from the y-axis. Due to the symmetry of the problem, the dumbbell swam in the y-direction with velocity U. The rod imparted the x-direction force F and the z-direction torque T to the squirmers. (

**b**) Two squirmers with swimming modes β

_{1}and β

_{2}were placed fore and aft. The squirmers were oriented in the y-direction and swam in the y-direction at velocity U. The rod exerted the y-direction force F.

**Figure 3.**Swimming velocity of a side-by-side dumbbell squirmer connected by a short rigid rod. (

**a**) Contour plot of the velocity in β–ϕ space. (

**b**) Maximum velocity U

_{max}and the angle with maximum velocity ϕ

_{max}at each β value.

**Figure 4.**Force F and torque T induced by the connecting rod of the side-by-side dumbbell squirmers. (

**a**) Contour plot of the force in β–ϕ space, where negative F indicates that the squirmers tended to repel each other without the rod. White lines indicate F = 0. (

**b**) Contour plot of torque in β–ϕ space, where negative T indicates that the squirmers tended to turn away from each other without the rod.

**Figure 5.**Swimming behavior of a fore and aft dumbbell squirmer connected by a short rigid rod. (

**a**) Contour plot of velocity at various swimming modes. White line indicates U/U

_{0}= 1. (

**b**) Contour plot of force at various swimming modes, where negative F indicates that the squirmers tended to repel each other without the rod. White line indicates F = 0.

**Figure 6.**Problem settings for a dumbbell squirmer connected by a spring. The spring was connected between ${r}_{1}^{s}$ and ${r}_{2}^{s}$ on the surface of squirmer 1 and 2, respectively. Squirmer i had orientation vector

**e**

_{i}, squirming mode β

_{i}, angle from the y-axis ϕ

_{i}, and the center position ${r}_{i}^{c}$. (

**a**) Two identical squirmers with swimming mode β were connected side-by-side. (

**b**) Two squirmers connected fore and aft.

**Figure 7.**Effect of β on the trajectories of the squirmer centers of dumbbell squirmers connected side-by-side. To generate a small disturbance, two squirmers were initially placed at ${r}_{1}^{c}$ = (0, 0, 0) and ${r}_{2}^{c}$ = (3a, 0, 0.1a) with ϕ

_{1}= 1° and ϕ

_{2}= 2°. The equilibrium length of the spring was set at a. (

**a**) Circular trajectories obtained in negative β cases. Inset shows the cases β = −1 and −3. (

**b**) Straight trajectories obtained for β = 0, 1, and 3.

**Figure 8.**Mechanism of stable swimming of dumbbell squirmers connected side-by-side. (

**a**) Velocity field around a stable dumbbell squirmer with β = 3. (

**b**) Schematics of the torque balance. Two squirmers tended to move away from each other, and the spring was stretched. The spring force

**F**

_{s}induced spring torque

**T**

_{s}to the squirmer (red arrows). The squirming velocity (blue arrows), generated hydrodynamic torque

**T**

_{u}. The configuration became stable when the two torques canceled each other out.

**Figure 9.**Effect of β

_{2}on the trajectories of the squirmer centers of the dumbbell squirmer connected fore and aft. The squirming mode of squirmer 1 was β

_{1}= 0. The squirmers were initially placed at ${r}_{1}^{c}$ = (0, 0, 0) and ${r}_{2}^{c}$ = (0, −3a, 0.1a) with ϕ

_{1}= 1° and ϕ

_{2}= 2° to generate a small disturbance. The equilibrium length of the spring was a. (

**a**) Circular trajectories obtained with β

_{2}= −0.5, 0, and 1. (

**b**) Straight trajectories obtained with β

_{2}= −1 and −3.

**Figure 10.**Mechanism of the stable swimming in a dumbbell squirmer connected fore and aft. (

**a**) Velocity field around a stable dumbbell squirmer with β

_{1}= 0 and β

_{2}= −3. (

**b**) Schematics of the torque balance. Two squirmers tended to repel each other, and the spring was stretched. Spring force

**F**

_{s}induces spring torque

**T**

_{s}to the squirmer (red arrows). The spring torque stabilized the aligned configuration when it overwhelmed hydrodynamic torque

**T**

_{u}, which was caused by the squirming velocity.

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ishikawa, T.
Stability of a Dumbbell Micro-Swimmer. *Micromachines* **2019**, *10*, 33.
https://doi.org/10.3390/mi10010033

**AMA Style**

Ishikawa T.
Stability of a Dumbbell Micro-Swimmer. *Micromachines*. 2019; 10(1):33.
https://doi.org/10.3390/mi10010033

**Chicago/Turabian Style**

Ishikawa, Takuji.
2019. "Stability of a Dumbbell Micro-Swimmer" *Micromachines* 10, no. 1: 33.
https://doi.org/10.3390/mi10010033