J. Fluid Mech., page 1 of 12. c© Cambridge University Press 2012 1doi:10.1017/jfm.2012.177

Unsteady swimming of small organisms

S. Wang and A. M. Ardekani†

Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

(Received 1 February 2012; revised 14 March 2012; accepted 1 April 2012)

Small planktonic organisms ubiquitously display unsteady or impulsive motion toattack a prey or escape a predator in natural environments. Despite this, the role ofunsteady forces such as history and added mass forces on the low-Reynolds-numberpropulsion of small organisms, e.g. Paramecium, is poorly understood. In this paper,we derive the fundamental equation of motion for an organism swimming by meansof the surface distortion in a non-uniform background flow field at a low-Reynolds-number regime. We show that the history and added mass forces are important as theproduct of Reynolds number and Strouhal number increases above unity. Our resultsfor an unsteady squirmer show that unsteady inertial effects can lead to a non-zeromean velocity for the cases with zero streaming parameters, which have zero meanvelocity in the absence of inertia.

Key words: micro-organism dynamics, propulsion, swimming

1. IntroductionPlanktonic organisms play an important role in tropic dynamics, marine ecosystem

and biomedical applications. They use contractile elements (cilia and flagella) andappendages to propel themselves. They undergo oscillatory motions or impulsivejumps to displace, change their swimming direction, attack a prey or escape apredator (Daniel 1984; Jiang & Kiørboe 2011). Ciliates, such as Paramecium, swim bymetachronal beating of cilia on the surface. Recently, Hamel et al. (2011) showed thatParamecium uses synchronized beating of the cilia to rapidly accelerate away from anintermediate aggression, which is followed by a large oscillation in velocity. For moresevere aggression, Paramecium shows jumping behaviour to escape. Other planktonicprotists, such as the ciliate Balanion comatum, also escape from hydromechanicalsignals by jumps. During these jumps, the swimming velocity of the organismincreases more than a five-fold in the period of time required to move one bodylength(Jakobsen 2001). Jumping is common among protozoan and metazoan plankton (Jiang& Kiørboe 2011). Copepods, the most abundant metazoans in the ocean, ‘swim bysequentially sticking the swimming legs posteriorly’ (Jiang & Kiørboe 2011), whichresults in sequences of small jumps for the purpose of ambush feeding. Since jumpingorganisms respond to hydromechanical signals rather than direct contact, ambient fluidmotion (e.g. turbulent flows) can also trigger micro-organisms’ acceleration in additionto flow disturbances generated by a prey or a predator (Magar & Pedley 2005).Unsteady motion has also been reported for flagellated organisms. Recently, Gollub

† Email address for correspondence: aardekan@nd.edu

2 S. Wang and A. M. Ardekani

and coworkers (Guasto, Johnson & Gollub 2010) resolved the oscillatory velocityfield, induced by synchronous and asynchronous beating flagella of a free swimmingalgal cell. In order to better understand these cases, it is essential to explore thephysics of fluids that lies behind the motion of accelerating micro-organisms. Despitethe widespread implications of unsteady swimming on micro-organisms in nature, therole of unsteady hydrodynamic forces on their propulsion has not been explored inthe literature. In this manuscript, we derive the fundamental equation of motion forunsteady swimming of small organisms.

Even though unsteady swimming of micro-organisms is poorly understood, the studyof unsteady motion of rigid particles has a long history (Ardekani et al. 2009).Boussinesq (1885), Basset (1888) and Oseen (1927) examined unsteady motion ofa spherical particle in an unbounded stagnant flow of an incompressible Newtonianfluid. The hydrodynamic forces acting on a spherical particle consist of three terms:Stokes drag, history (Basset) force and added mass force (Basset 1888). The role ofhistory force on the motion of particles is stronger when their density is comparableto the density of ambient fluid environments, ρp/ρf ∼ O(1), where ρp is the particledensity (van Aartrijk & Clercx 2010). This condition holds for swimming organismsand thus the history force cannot be neglected for unsteady bio-locomotion. Arminski& Weinbaum (1979) studied the Basset–Langevin equation for different unsteadyexcitation forces of different waveforms. The equation of motion for an isolated sphere(Maxey & Riley 1983 and, independently, Gatignol 1983) and interacting spheres(Ardekani & Rangel 2006) in an unsteady non-uniform background flow have beendeveloped in the limit of small particle Reynolds number. The Reynolds number,Re = ρf Ua/η, is the ratio of convective to viscous forces, where U is the particlevelocity, a the particle radius, ρf the fluid density and η is the fluid viscosity. Thesestudies include unsteady inertia due to the fluid and particle but neglect convectiveinertial effects. Consequently, the Basset force’s integration kernel decays as t−1/2,where t is the time. Mei & Adrian (1992) included the convective term in thelimit of a small Strouhal number Sl Re 1 and found that the Basset force’sintegration kernel decays as t−2 instead of t−1/2. The Strouhal number, Sl = a/Uτ ,is a measure of the convective time scale to the time scale of unsteadiness, τ .When the vorticity generated at the particle boundary has not diffused out of theOseen distance, the unsteady inertia dominates the hydrodynamic forces, while thecorrection of convective inertial effects to the hydrodynamic forces is O(Re) and canbe neglected as noted by Lovalenti & Brady (1993). Unsteady hydrodynamic forceshave been used to determine transport of small particles in turbulent flows, particledispersion and deposition (Lovalenti & Brady 1993). For swimming micro-organisms,the unsteadiness arises from unsteady propulsion mechanisms (e.g. unsteady beatingof flagella or cilia), turbulent velocity fluctuations of the ambient fluid and/or velocitydisturbances generated by a prey or predator. In this paper, we assume unsteady inertiais large compared with convective inertia, Sl Re, which is the case for the examplesdiscussed above since micro-organisms live in a low-Reynolds-number regime. Forexample, a swimming algal cell has Reynolds and Strouhal numbers of 0.001 and 1,respectively (Guasto et al. 2010).

We first derive the fundamental equation of motion for a spherical unsteadyswimmer in a non-uniform flow. Then, in order to determine the effect of unsteadyhydrodynamic forces on a swimming micro-organism, we use an archetypal modelwidely used in the literature to study low-Reynolds-number swimming called thesquirmer model. The squirmer model consists of a spherical shaped body withwavelike deformations of the outer surface (Lighthill 1952) and describes the

Unsteady swimming of small organisms 3

self-propulsion of ciliates moving by synchronized metachronal beating of cilia ontheir surface (Blake 1971), such as cyanobacteria, colonies of flagellates such asVolvox (Drescher et al. 2009) and Paramecium. The model has been recently usedto study the hydrodynamic interaction between swimmers (Ishikawa, Simmonds &Pedley 2006), mixing due to propulsion of micro-organisms (Lin, Thiffeault &Childress 2011), bio-locomotion in complex fluids (Zhu et al. 2011) and stratifiedfluids (Doostmohammadi, Stocker & Ardekani 2012). Unsteady squirmers have beenstudied by Magar & Pedley (2005) to explore the nutrient uptake of micro-organisms.Later, Giacche & Ishikawa (2010) examined interaction of two unsteady squirmerssince the beating time scale is comparable to the time scale of near-field interaction.Yeomans and coworkers (Pooley, Alexander & Yeomans 2007; Alexander & Yeomans2010) considered a different unsteady swimming model using the Stokes equations,neglecting unsteady inertia, to explore hydrodynamic interaction between three sphereswimmers of Najafi & Golestanian (2004). Even though micro-organisms in natureundergo unsteady swimming, the role of unsteady forces (history and added massforces) on their swimming behaviour, to the best of our knowledge, has not beenexplored previously and is the focus of this paper.

2. Analytical description of unsteady forcesHere we consider the locomotion of an unsteady spherical swimmer that propels

itself by generating tangential surface motion, vs, swimming with velocity V(t) ina Newtonian fluid. The swimmer’s centre of mass is instantaneously located at Y(t).The undisturbed flow, in the absence of the swimmer, is described by u(x, t) and themodified velocity field due to presence of the swimmer is v(x, t), where x is theposition vector. The Navier–Stokes equations in the frame of reference translating withthe velocity of the swimmer, V(t), are given as

ρfDwDt=−∇p+ η∇2w+ ρf

(g− dV

dt

), ∇ ·w= 0, (2.1)

w= vs +Ω × z at |z| = a, (2.2)w= u− V as |z| →∞, (2.3)

where w(z, t)= v(x, t)− V(t) is the velocity field in the swimmer’s frame of reference,z = x − Y(t) is the position vector instantaneously measured from the centre of theswimmer, Ω is the angular velocity of the organism, p is the pressure field and gis the gravitational acceleration. The last term in the momentum equation (2.1) is thed’Alembert force per unit volume, which arises due to the acceleration of the frameof reference. The boundary condition, (2.2), enforces the local fluid to move withthe time-dependent tangential surface velocity of the swimmer, vs. The last boundarycondition, (2.3), indicates that the flow field approaches the background flow field atinfinity. The derivative D/Dt and d/dt are used here to denote the time derivativefollowing a fluid element and the swimmer, respectively. The total hydrodynamic forceacting on an unsteady swimmer can be written as

F=∮

Sσ ·n dS, (2.4)

where the surface integral is over the surface of the spherical swimmer. Here n isthe unit outward normal to S, σ = −pI + η(∇w + ∇wT) is the stress tensor and I isthe identity tensor. Following Maxey & Riley (1983), the force can be decomposed to

4 S. Wang and A. M. Ardekani

the hydrodynamic forces due to the undisturbed flow field, wu = u − V , and disturbedflow field, wd = v − u, referred to as Fu and Fd, respectively. The force due to theundisturbed flow is identical to the one for a rigid particle of a size much smaller thanthe length scale of the variation of the undisturbed flow and the equation of motioncan be written as

msdVdt= (ms − mf )g+ mf

DuDt|Y(t) + Fd, (2.5)

where ms is the swimmer mass and mf is the mass of fluid displaced by the swimmer.The detailed derivation of (2.5) is given in Maxey & Riley (1983, equations (20)–(26))and is not repeated here. Next, we shall consider the hydrodynamic forces acting onthe swimmer due to the disturbed flow, wd, which is different from that correspondingto the rigid sphere due to the surface disturbances generated by the swimmer. Sincethe micro-organisms’ Reynolds number is small (Re 1 and Sl Re), convectiveinertia generated by the micro-organism can be neglected. Thus, the governingequations for the disturbed flow field can be described by unsteady Stokes equations

ρf∂wd

∂t= −∇pd + η∇2wd, (2.6a)

wd = V − u+Ω × z+ vs at |z| = a, (2.6b)

wd→ 0 |z| →∞, (2.6c)

where superscript d corresponds to disturbed flow variables. Using the Lorentzreciprocal theorem, which has also been used to calculate the hydrodynamic forcedue to the disturbed flow of a rigid sphere (Maxey & Riley 1983), we can derive theequation of motion for a swimming micro-organism. Consider the flow field around animpulsively started sphere, Vaux = δ(t)e(1), generating the velocity and stress fields ofvaux and σ aux , respectively:∫

Sn · σ d · vaux dS−

∫Sn · σ aux · wd dS

=∫

Vρf

[wd(z, 0) · vaux(z, s)− vaux(z, 0) · wd(z, s)

]dV, (2.7)

where ˆ denotes the Laplace transform operator and s is the Laplace variable. Thevelocity and stress fields of an impulsively started sphere, the chosen auxiliaryproblem, is given by Burgers (1938) after solving unsteady Stokes equations

vaux = (e(1) ·∇)∇ψ − e(1)∇2ψ, (2.8a)vaux(s)= e(1) at r = a, (2.8b)

ψ = Q1

r+ Q2

re−λr, (2.8c)

n · σ aux =−12ρf sz(z · e(1))/a− 3

2η/a(1+ λa)e(1) at r = a, (2.8d)

where e(1) is a unit vector along the swimming direction (without loss ofgenerality e(1) = (1, 0, 0)), r is the radial distance from the centre of the sphere,λ2 = ρf s/η and the coefficients are defined as Q1 = 3aη(1 + λa)/2ρf s + a3/2 andQ2 = −3aη exp(λa)/2ρf s. For a swimmer and background flow field that are initially

Unsteady swimming of small organisms 5

zero, the force from disturbed flow can be written, in the Laplace space, as

Fd1 = e(1) ·

∫Sn · σ d dS=

∫Sn · σ aux · wd dS. (2.9)

Recall that the swimmer is much smaller than the length scale corresponding to thevariation of the background flow. Thus, wd in the neighbourhood of the swimmer canbe approximately represented by its Taylor expansion at the origin. After satisfying theboundary condition listed in (2.6) on the surface of the spherical swimmer, the velocitywd can be written as

wdi = Vi − ui[Y , s] + vs

i (zS, s)+ zjCij + 12

zjzkDijk, (2.10a)

Cij =−∂ ui

∂zj|Y(t) − εijkΩk, Dijk =− ∂2ui

∂zj∂zk|Y(t), (2.10b)

where ε is the permutation tensor and zS corresponds to the position vector on thesurface of the swimmer. After combining the results of (2.8)–(2.10), the hydrodynamicforces acting on the swimmer due to the disturbed flow can be written as

Fdi =−(Vi − ui)

[6πaη(1+ λa)+ 1

2mf s

]− a2

[(Dijj + 2Djji)πηa(1+ λa)+ 1

20mf sDijj

]− 3

2η(1+ λa)/a

∫Svs

i dS. (2.11)

Finally, we can calculate the Laplace inverse of (2.11) and evaluate the fundamentalequation of motion for an unsteady spherical swimmer in a non-uniform backgroundflow field

msdVdt= (ms − mf )g+ mf

DuDt

∣∣∣∣Y(t)

− 6πaη

V(t)− u(Y(t), t)− 1

6a2∇2u|Y(t)

− 1

2mf

ddt

V(t)− u(Y(t), t)− 1

10a2∇2u

∣∣∣∣Y(t)

− 6πa2η

∫ t

0dτ

dV(τ )− u(Y(τ ), τ )− 1

6a2∇2u|Y(τ )

/dτ

[πν(t − τ)]1/2

− 32η/a

∫Svs dS− 3

2η

∫S

∫ t

0

dvs/dτ

[πν(t − τ)]1/2 dτ dS. (2.12)

where ν is the kinematic viscosity of fluid. Here, we have derived the unsteady Faxenequation for a self-propelled particle in a non-uniform flow. The first and second termson the right-hand side of (2.12) are the gravity force and pressure gradient of theundisturbed flow, respectively. The next three terms correspond to the Stokes dragcontribution, the added or virtual mass forces and the Basset force or history termdue to the translation of the swimmer in a non-uniform background flow; the lasttwo terms correspond to the Stokes drag contribution and the Basset forces due tothe surface distortion of the swimmer. The acceleration of the organism imposes anacceleration of the flow field which leads to inertial forces acting on the organism,referred to as added mass force. It should be noted that the added mass force ofa spherical swimmer that propels by surface oscillation tangential to its surface is

6 S. Wang and A. M. Ardekani

identical to that for a rigid sphere. Equation (2.12) can be used for an acceleratingswimmer moving in an unsteady non-uniform background flow.

3. Unsteady squirmerIn this section, we apply the fundamental equation of motion to a specific swimming

model called a ‘squirmer’. The unsteady squirmer is first introduced by Blake (1971).The metachronal beating of cilia on the surface of micro-organisms can be modelledwith a tangential surface velocity on the surface of the organisms. Here we neglectsmall radial displacements of cilia (Giacche & Ishikawa 2010). The tangential surfacevelocity on the surface of the squirmer can be described as

vsθ =

∞∑n=1

Bn(t)Vn(µ), (3.1)

where Vn is defined as

Vn = 2√

1− µ2

n(n+ 1)P′n(µ), (3.2)

where µ= cos θ and Pn is the Legendre polynomial of the first kind of degree n. Herer, θ and φ denote spherical polar coordinates, where θ = 0 is along the swimmingdirection. The surface tangential velocity is axisymmetric and independent of φ.Following Magar & Pedley (2005), we retain the first three terms of the tangentialsurface velocity whose coefficients can be defined as a function of three modalamplitudes b1, b2 and b3 (Blake 1971; Magar & Pedley 2005; Giacche & Ishikawa2010)

B1 = B10 + B11 cosωt + B12 cos 2ωt, (3.3a)B2 = B21 sinωt + B22 sin 2ωt, (3.3b)B3 = B30 + B31 cosωt + B32 cos 2ωt. (3.3c)

where ω is the frequency of surface disturbances and is equal to cilia beatingfrequency for ciliates (Blake 1971); B1, B2 and B3 are functions of time and representwave of tangential displacements of cilia on the surface of the micro-organism:

B10 = aωε2 2b2

5

(3b3

7− b1

), B11 = aωεb1, B12 = aωε2 b2

5

(b1 + 2b3

7

), (3.4a)

B21 =−aωεb2, B22 =−aωε2

(b2

1

2− 3b1b3

7+ b2

2

14− b2

3

28

), (3.4b)

B30 = aωε2 b2

5

(2b1 + b3

3

), B31 = aωεb3, B32 = aωε2 2b2

5

(b3

3− 3b1

). (3.4c)

Equations (3.3) and (3.4) are obtained under the assumption that a material point thatis initially at θ0, at time t will move to the location (Blake 1971)

θ = θ0 + ε(b1V1 + b2V2) sinωt + b3V3 cosωt, (3.5)

where ε is the amplitude of oscillation of cilia wave which is a small parameter. Formicro-organisms usually ε ∼ 1/20− 1/10, swimming speeds are about 10 body lengthsper second and surface motion frequency ω ∼ 20–1000 s−1 (Blake 1971; Stone &Samuel 1996). We consider a squirmer that is stationary for t < 0 while it accelerates

Unsteady swimming of small organisms 7

for t > 0 due to surface disturbance as described in (3.1). Equations (2.5) and (2.11)can be simplified for a squirmer in a stagnant background flow as

mssV =−6πηa

[V − 2

3B1

]− 6πηλa2

[V − 2

3B1

]− 1

2mf sV. (3.6)

Here, we neglect the contribution of the gravitational force compared withhydrodynamic forces due to self-propulsion of the swimmer. The first, second andthird terms correspond to the quasi-steady hydrodynamic force, Basset force and addedmass force, respectively. The swimming velocity of the squirmer can be expressed inLaplace space as

V(s)= 23

B1 −23

B1

(12

mf + ms

)s

6πηa (1+ λa)+(

12

mf + ms

)s. (3.7)

Only the first term of the tangential surface velocity, (3.1) (parameter B1), contributesto the propulsion of the organisms, while other terms affect the flow field and nutrientuptake but not propulsion. If one solves quasi-steady Stokes equations (Blake 1971),instead of the unsteady Stokes equation as done here, the swimming velocity is simplyV = (2/3)B1 and its time average can be described as 〈V〉 = (2/3)〈B1〉 = (2/3)B10,which is referred to as a streaming parameter in the literature (Magar & Pedley2005). For a non-zero streaming parameter, the micro-organism swims around with anon-zero mean velocity. When the streaming parameter is zero but b1b2, referred to asa hovering parameter, is non-zero, the organism stirs the fluid around it but its meanvelocity is zero. In order to include the effect of unsteady forces, we need to calculatethe inverse Laplace transform of (3.7) using the contour integration technique. Let usdefine

VB(s)= V − 23

B1

=−23

(12

mf + ms

)B10 + B11s2/(s2 + ω2)+ B12s2/(s2 + 4ω2)

6πηa

(1+

√a2s

ν

)+(

12

mf + ms

)s

. (3.8)

The inverse Laplace transform of the function VB(s) can be written as

VB(t)= 12πi

∫ γ+i∞

γ−i∞estVB(s) ds, (3.9)

where i =√−1. Since the function VB is analytic in the complex domain except for afinite number of isolated singularities inside the contour shown on figure 1, its inverseLaplace transform can be calculated using the residue theorem. The function V(s) hasfour poles at ±iω and ±2iω, a branch point at s = 0 and a branch cut from s = 0to infinity along the negative real axis as shown in figure 1. The inverse Laplacetransform can be calculated after deforming the inversion contour into a keyholecontour in the complex s-plane, cut along the negative real axis and summing the

8 S. Wang and A. M. Ardekani

R

D

E

C

F

B

A

i

i

i

i

i

i

FIGURE 1. Complex plane contour for the evaluation of the inverse Laplace transform.

residues at the poles ±iω and ±2iω:

VB(t)=− 12πi

[∫BC,FA+∫

CD,EF+∫

DE

]× estVB(s)ds+

2∑k=1

res(±ikω). (3.10)

The integrals on paths BC, FA and DE are zero. The sum of residues leads to

I1(t, ω)=2∑

k=1

−23

B1kkSlReA2 cos kωt − A1 sin kωt

A21 + A2

2

, (3.11)

where

A1 = 1+√

3kSlRe

2C, A2 =

√3kSlRe

2C+ kSlRe, (3.12a)

SlRe=(

12

mf + ms

)ω/6πηa, C =

(12

mf + ms

)32

mf

, (3.12b)

where C is unity for a neutrally buoyant swimmer. The integral on paths CD and EFcan be written as

I2(t, ω)= 12πi

∫ ∞0

e−xt[VB(xe−iπ)− VB(xeiπ)] dx

=2∑

k=0

− 23π

B1kSlRe∫ ∞

0

√3SlRex

Ce−xωt x2/A3

k2 + x2dx, (3.13)

where

A3(x)= (1− SlRex)2+3SlRex

C. (3.14)

Unsteady swimming of small organisms 9

Organism Size(2a µ m)

Speed(mm s−1)

ω/2π(Hz)

Re Sl Sl·Re Reference

Algal cell 10 0.3 33 0.0015 3.5 0.005 Guasto et al. (2010)Opalina 350 0.1 4 0.02 44 0.8 Brennen (1974)Paramecium 250 1 30 0.13 24 3 Brennen (1974)Pleurobrachia 15 000 — 10 — — 3500 Brennen (1974)

TABLE 1. Physical parameters for representative organisms

Consequently, the squirmer velocity can be calculated as

V = 23 B1(t, ω)+ I1(t, ω)+ I2(t, ω). (3.15)

Here I1 is a harmonic function of time while I2 decays to zero with time. The transientbehaviour of the micro-organism cannot be captured using quasi-steady Stokesequations unless SlRe 1. Moreover, the long time unsteady swimming velocityof the micro-organism would be incorrectly predicted, and one needs to use theunsteady Stokes equation instead, as done in this manuscript. Figure 2(a) shows thedimensionless velocity of the swimmer, V∗ = V/aεω, as a function of dimensionlesstime, t∗ = ωt, and it is compared with that predicted by quasi-steady Stokes equations(Blake 1971). The mean velocity of the squirmer deviates from the prediction of quasi-steady Stokes equations at early times (e.g. first three periods of oscillation). Thisdeviation is due to the function I2. The long time behaviour of the swimmer is alsomodified due to the history and added mass forces (due to the function I1). The Bassetand added mass forces are larger than the hydrodynamic forces predicted by quasi-steady solution as shown in figure 2(b), where the dimensionless force is defined asf ∗ = F/6πηa2εω. The long time behaviour of (3.15) can be simplified for two differentlimits, SlRe 1 and SlRe 1. In the limit of SlRe 1, the long time velocity can bewritten as V ∼ (2/3)B1 +

∑2k=1B1kkSlRe sin kωt. Thus, the contribution of the history

and added mass forces to the swimming velocity compared with one due to the quasi-steady Stokes force scales with SlRe. In this limit, the Basset and added mass forcescan be neglected since SlRe is small. On the other hand, in the limit of SlRe 1, thelong time velocity can be written as V ∼ (2/3)B1 −

∑2k=1(2/3)B1k cos kωt ∼ (2/3)B10.

Thus, in this limit, the contribution of the history and added mass forces leads to asteady swimming velocity even though cilia motion generates an oscillatory surfacedisturbance. Table 1 shows the physical parameters for representative organisms.Based on the value of SlRe, the contribution of unsteady forces is important forParamecium and Pleurobrachia and it should be taken into account. Figure 2(c) showsthe amplitude of oscillation of the organism’s velocity, normalized by that predicted byquasi-steady Stokes equations for the two different squirmers described in the legendof figure 2 with an identical mean velocity. It approaches unity as SlRe approacheszero, but it can be reduced to 10 % of the prediction of quasi-steady solution as SlReincreases to O(100) and reduces as SlRe−1/2 at large values of SlRe. For a neutrallybuoyant micro-organism, SlRe−1/2 = a

√ω/3ν, which represents the ratio of organism

size to the distance over which the vorticity generated at the organism surface diffusesduring the time scale of unsteadiness. The amplitude of oscillation of Basset, addedmass and Stokes forces are plotted in figure 2(d). At SlRe ∼ O(1), these forces arecomparable, while the Basset force is much larger than the quasi-steady Stokes forceas SlRe increases.

10 S. Wang and A. M. Ardekani

2

1

2

Blake solutionPresent solution Stokes force

Basset forceAdded mass force

10–2

100

1

t (ms)

–1

0

1

2

3

4

5

0 20 40 60

0

10

–100 20 40 60

20

40

60

80

100

0 50 100 150 20010–2 102100

10

20

30

40

50

0 5 10 15 20 25 30

Present resultsExperimental set IExperimental set IIExperimental set IIIExperimental set IV

Stokes forceBasset forceAdded mass force

0

(b)

(c) (d)

(e)

(a)

FIGURE 2. (Colour online available at journals.cambridge.org/flm) Unsteady hydrodynamicforces (i.e. Basset and added mass forces) affect the swimming velocity of small organisms.(a) The dimensionless swimming velocity of the organism calculated in this manuscript iscompared with the Blake solution where unsteady hydrodynamic forces are neglected. (b) TheBasset and added mass forces acting on the organism are larger than the hydrodynamic forcespredicted by quasi-steady Stokes equations. The data in (a) and (b) are shown for a squirmerwith B10 = 2.39, B11 = 3.5, B12 = 0.16, and SlRe = 10. (c) The ratio of velocity amplitudepredicted by the present analysis normalized by that predicted by the Blake solution is plottedfor two different squirmers with an identical mean velocity. The solid line corresponds toB10 = 2.39, B11 = 3.5, B12 = 0.16 and square symbols correspond to B10 = 2.39, B11 = −1,B12 = 3.13. (d) The amplitude of oscillation of Basset, added mass and Stokes forces areplotted for the two organisms explained in (c). (e) The velocity of a jumping copepodexperimentally measured by Jiang & Kiørboe (2011) is compared with the present results.The plot corresponds to B1 = 2.82 mm s−1, C = 1, a= 300 µm and ν = 10−6 m2 s−1.

Unsteady swimming of small organisms 11

In order to mathematically model jumping behaviour, we calculate the time-dependent swimming velocity for a spherical organism with a surface velocity ofvsθ = B1δ(νt/a2) sin θ where B1 is constant. Accordingly, the swimming velocity decays

as

V = 23π

B1

∫ ∞0

Cx3/2e−νxt/a2

3(1− Cx/3)2+x dx. (3.16)

Figure 2(e) shows the velocity of the swimmer as a function of time. The velocitydecay of a jumping organism calculated in (3.16) shows good agreement with theexperimental measurements of Jiang & Kiørboe (2011) for a jumping copepod.

4. ConclusionsIn this paper, we have derived the fundamental equation of motion for an unsteady

micro-organism swimming in a non-uniform flow field. We showed that the unsteadyinertial effects, the history and added mass forces, are not negligible for finite SlRe.The mean velocity of the squirmer deviates from the prediction of quasi-steady Stokesequations due to the contribution of history and added mass forces (function I1).Consequently, our results show that unsteady inertial effects can lead to propulsion ofan unsteady squirmer even for the cases with zero streaming parameters. These resultsare compatible with the recent experimental results by Hamel et al. (2011) wherethey showed that a Paramecium escapes an aggression by generating time-reversibleeffective and recovery strokes. This type of stroke does not generate any effectivepropulsion if all of the inertial effects are neglected (Lauga 2011) as predicted by thePurcell theorem (Purcell 1977). However, the unsteady inertial effects discussed in thispaper explain the propulsion observed in the experiment.

AcknowledgementThis work is supported by NSF grant CBET-1066545.

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