Some notes on locomotion at low and

intermediate Reynolds numbers

Stephen Childress

June 1, 2010

1 Outline

• Geometry of locomotion

• The Navier-Stokes fluid

• Vorticity and its creation

• The Stokesian realm and the scallop theorem

• The swimming sheet

• Flagellar and ciliary propulsion

• Intermediate Reynolds numbers [In powerpoint format]

2 Geometry of locomotion

Assumption: The body shape relative to the body frame is a given functionof time B0(t). We call this the standard shape. The current body shape B(t)is obtained under a translation and rotation of the body frame. The latter isdefined by a positional matrix. To say that locomotion has occurred at time tmeans we have realized at time t a nonzero translation.

Example: steady locomotion in two dimensions. Let (xp, yp) be a point onthe standard shape at time t. Let the positional matrix may then be written

P =

cos θ − sin θ Xsin θ cos θ Y

0 0 1

(1)

where (X(t), Y (t)) is the translation and θ(t) the rotation giving the body frameat time t. Them P · v locates the image of this point, where v is the transposeof (xp, yp, 1).

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We choose to define steady locomotion as: at any time t here exists a fixedtime T such that

X(t + T ) = X(t) + ∆X, Y (t+ T ) = Y (t) + ∆Y (2)

for some fixed constants ∆X,∆Y . The mean velocity of locomotion is the(U, V ) = ∆X/T,∆Y/T . In most modeling the body shape will be periodic intime, but it is not necessary that T be the period of the standard shape.

The matrix

Adt = P−1 · dP

dtdt =

0 −dθ dX cos θ − dY sin θdθ 0 −dX sin θ + dY cos θ0 0 0

(3)

is the incremental movement of the point relative to the current frame.

3 Navier-Stokes equations for incompressible fluid

flow

We consider the equations for an incompressible Newtonian viscous fluid. Theseare

∂tu + (u · ∇)u = −1

ρ∇p+ ν∇2

u, ∇ · u = 0, ν = µ/ρ (4)

In our 2D problem, incremental motion ub of the boundary B provides theno-slip boundary condition there: u = ub. Given u(x, t) and p(x, t), the force F

and moment M exerted by the fluid on the body may be computed. Newton’sequations then imply

md2(X, Y )

dt2= F(t),

d

dt

[

Idθ

dt

]

= M(t). (5)

involving the mass (assumed constant) m and moment of inertia I(t). Thefluid mediates the connection of the RHS of each equation to X, Y, θ given thestandard shape B0(t).

4 The Reynolds number

Let u be a typical fluid speed, L a body dimension, T a characteristic time asso-ciated with B0(t). Using these as units we have the dimensionless NS equations

St ∂tu + (u · ∇)u = −∇p+1

Re∇2

u, ∇ · u = 0 (6)

involving the Reynolds number Re = UL/ν and the Strouhal number St =L/UT . Generally in locomotion problems St is of order unity, and we shallbelow choose T to make St = 1. Some typical values of Re:

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• Bacterium 10−5

• Spermatozoan 10−4 − 10−3

• Small wasp 10−1

• Locust 104

• Pigeon, Fish 105

• Michael Phelps 3 × 106

We generally think of Re of orders less than 1 as small, and order 1 to 102 asbeing intermediate.

5 Vorticity and its generation in locomotion

The vorticity vector field ω(x, t) is defined by ω = ∇ × u. The curl of the NSequations yields the vorticity equation

∂tω + u · ∇ω − ω · ∇u − ν∇2ω = 0. (7)

In two dimensions (u = (u, v))the vorticity is the scalar ω = ∂v∂x − ∂u

∂y , satisfying

∂tω + u · ∇ω − ν∇2ω = 0. Vorticity is generally introduced into the fluid bylocomotion of a body in a NS fluid. If Ae(t) is the domain exterior to B(t) wesee that in 2D (disregarding a distant contour integral, and accounting for themotion of the body)

d

dt

∫

Ae(t)

ωdA = −ν∮

B

∂ω

∂nds (8)

where the normal derivative is outward from B.Now suppose the body is fixed and the 2D flow, moves past it satisfying the

no-slip condition. Then on B we have

ν[

nx(∂

∂x(vx − uy) + ny(

∂

∂y(vx − uy)

]

= nxpy − nypx =∂p

∂s, (9)

sod

dt

∫

Ve(t)

ωdt = −∮

B

dp (10)

Thus while no net vorticity is introduced globally, pressure differences along thebody are associated with vortex shedding. The pressure gradient becomes mostacute at sharp corners. In locomotion this shed vorticity, associated with fins,wings, and other appendages and organelles, has important effects:

• A wake may be created, which can propagate far from the body.

• The environment for locomotion may involve the vorticity shed at preced-ing times, i.e. the flow acquires a memory.

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• For body motions which are periodic, e.g., the resulting flow may not beunique. For example the ultimate flow may depend upon initial conditions.

We will look now at low Reynolds number locomotion. Here the vorticity instan-taneously establishes itself thoughout the fluid based upon the current boundaryvelocity. This is due entirely to diffusion of momentum.

6 The Stokesian realm Re 1

Writing the NS equations as

∇2u = Re[St ∂tu + (u · ∇)u + ∇p], ∇ · u = 0 (11)

We may approximate for small Re by the momentum equation ∇2u = 0. We

thus obtain a class of small Re flows of the form u = ∇ × A, ∇2A = 0. For

these flows the viscous stress is the only ingredient in the dynamics, and theviscous force on any fluid element vanishes identically. On the other hand aforce can be applied to the boundary. An example of such a solution in 3D isgiven by A = Ωa3

iz/R, R =√

x2 + y2 + z2 in cylindrical polars, and iz is theunit vector along the z−axis a,Ω are positive constants. We see that on R = athe velocity is u = −Ωaiz × ıR = −Ωa sin θiφ, and so this would correspond tothe no-slip condition on a sphere R = a spinning about the z−axis with angularvelocity −Ω.

The torgue T exerted on the sphere is just ∂R(−Ωa3 sin θR−2) integratedover the surface of the sphere. Using the fact that the viscous force acting onthe sphere surface is µ

R∂R(uφ/R)|R=a, we find a viscous torque acting on thesphere to be T = 8πa3Ω.

But this kind of solution does not involve the pressure, which in our non-dimensionalization was taken to have magnitude or order ρU2. Since viscousforces are of order µU/L, we have not allowed for a pressure force of the samemagnitude as the viscous force. We must replace Re p by a new p reflecting thisordering. This give the reduced equations (and we now return to dimensionalequtions

∇p− µ∇2u = 0,∇ · u = 0 (12)

These are the Stokes equations, describing Newtonian viscous flow at Reynoldsnumber zero. Solutions include not only those already considered (where p = 0,but also a large class with nonzero p.

We now describe a representation suitable for a large class of solutions of theStokes equations. To automatically satisfy the solenoidal constraint, we writethe velocity field in component form as follows:

ui = [∂2χ

∂xi∂xj−∇2χδij ]bj (13)

where b is a constant vector. The corresponding pressure will be

p = µ∂∇2χ

∂xjbj . (14)

4

Here the variable χ satisfies the biharmonic equation ∇2∇2χ = ∇4χ = 0.An example is obtained by solving ∇2χ = − F

4πµRto get a biharmonic term

χ = − F8πµR. This generates the flow

u =F

8πµ

[xixj

R3+ δij

1

R

]

bj, p = − F

4π

xjbjR3

(15)

which corresponds to putting a delta function of magnitude Fb on the RHS ofthe Stokes equations, and is termed a Stokeslet. Since it describes the response ofthe Stokes equations to a point force, it is the fundamental solution for buildingup solutions by superposition.

For example, a sphere of radius a moving with velocity U exerts a force6πµaU on the fluid. The far field of the Stokes flow of a sphere moving withvelocity Ua (as seen by an observed at rest relative to the fluid at infinity) isgiven by the above Stokeslet with F = 6πµaU :

ui =3Ua

4

[xixj

R3+ δij

1

R

]

bj (16)

The finiteness of the sphere is represented by a potential dipole term, so thecomplete solution is

ui =3Ua

4

[xixj

R3+ δij

1

R

]

bj −Ua3

4

[

3xixjbjR5

− bi1

R

]

(17)

equal to U on R = 1. This yields the famous solution of Stokes for flow past asphere.

Two remarks about Stokes flows:

• For the boundary value problem where u is prescribed, the solution existsand is unique for smooth boundaries.

• Time is only a parameter in Stokes flows. The flow is set instantaneouslyby the boundary velocity.

7 The Scallop Theorem

As just noted, “time” has in a sense disappeared from consideration once theStokes equations are adopted. In locomotion at Re = 0, the velocity everywherein the fluid is determined at each instant by the motion of every point on thebody surface. Simultaneously, we have in the Stokes equations a linear system.Thus the equation Adt = P

−1 · dPdtdt allows us to determine the positional

matrix P if we can find the incremental motion A. To do this we need tosolve the Stokes equations instantaneously for the boundary movements of thestandard shape. Call this u1, p1 We combine u1, p1 with a another Stokes flowu2, p2 representing a rigid body motion, such that the force and drag on thebody incurred by u2, p2 cancels that of u1, p1. We emphasize that that thedominance of viscous forces over the inertial reactions of fluid and body in

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Stokes flow implies that locomotion occurs in equilibrium, that is there is nonet force on the body at each instant.

Considering now the symmetric motion of a scallop shell, opening and closingwith period T . The scallop theorem states that such a motion will not lead tosteady locomotion in Stokes flow. The “proof” goes as follows [1]: Such amotion is reciprocal in the sense that the standard shape undergoes the samesequence of shapes under both forward and reversed time, However, if over onecycle time forward the displacement is ∆X, under reversed time it is −∆X,by the linearity of the equations and the unique determination of the fluidflow by the instantaneous motion of the boundary. Since the two motions areindistinguishable it follows that ∆X = −∆X or ∆X = 0.

To explain this a bit more, we introduce a configuration space C whichrecords possible shapes of our body. We will assume the standard shape B0(t)is periodic with respect to time. The motion of the standard shape is then aclosed curve in C, with a direction given by the arrow of time. Suppose nowthat the arrow of time is reversed. The scallop theorem then states that inthe forward and reversed directed curves are indistinguishable, then there isno locomotion. Note that locomotion here refers to the displacement over onecycle. As the cycle repeats, there is an identical rigid body motion relative tothe current shape.

Now the only way that the forward and reversed curves can be indistinguish-able is for it to be an arc, which is traversed twice over a cycle. Thus locomotionis impossible if there is no 2D projection of the curve which encloses area. Thisobservation relates to an interpretation intgroduced by Shapere and Wilczek,which regards A as a gauge potential and Stokesian locomotion as a change inthis potential over a cycle [7].

Some remarks:

• The scallop moves in Stokes flow, back and forth along a line, without anynet displacement.

• Nature breaks the scallop theorem in many ways, notably by passing wavesdown a flagellum, and by the asymmetrical beating of cilia.

• “Locomotion” without further qualification means simply a rigid-bodymovement. For the symmetry of the scallop shell we know locomotion issteady and along a line if it occurs.

• A body can involve non-reciprocal movement and yet not locomote. Forexample a body with two non-reciprocal movements in opposing direc-tions.

• Since time is only a parameter, it is the sequence of configurations thatoccurs over a cycle which is important, not the time that they occur. Ascallop shell which opens fast (but not so fast as to leave the Stokesianrealm) and closes slowly will not locomote.

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7.1 Thrust and drag

There is much concern in the study of steady locomotion about the balanceThrust=Drag. Intuitively, one imagines that the organism’s motions exerts aforce on the fluid. This causes the body to move and then to equilibrate at avelocity where the drag is equal to the produced thrust. In general, that is tosay a a finite Reynolds number, there is no way to make this decompositionprecise in any meaningful way. The nonlinearity of the NS equations precludesthis. It is only in the Stokesian realm that such a decomposition can be madeprecise, and it is worth discussing how this can be done.

Consider a neutrally buoyant organism in 3D in steady locomotion withconstant velocity U. The time-average force exerted by the body on the fluidmust vanish, and because this is Stokes flow this balance is also instantaneous.Relative to an observer fixed relative to the fluid at infinity (so the swimmeris observed to move past and the observer sees the perturbation of the fluidcaused by its passage) we may divide the flow (as we did above) into two parts:(u, p) = (u1, p1) + (u2, p2). Subscript 1 will designate the time-dependent flowwhich satisfies the condition u = U on the moving body, and null conditions atinfinity. Subscript 2 will designate the flow which satisfies u = ub on the bodyand also null conditions at infinity. Recall ub is the instantaneous velocity of apoint on the boundary of the organisms. Together u1 + u2 accounts for the no-slip condition on the boundary. These two time-dependent flows are associatedwith forces F1(t) and F2(t) = −F1(t) exerted by the body on the fluid.

It should be clear that we are now entitled to call −F1 the (instantaneous)drag force (F1 has the same direction as U) , and F2 the (instantaneous) thrust.Taking time averages, −〈F1〉 = D, 〈F2〉 = T are the two constants representingdrag and thrust.

Of course one has a complicated problem in the determination of these flows.Nevertheless it is important to realize the implications of the linearity of theproblem in allowing such a decomposition.

7.2 Efficiency

The preceding decomposition is related to a useful and very plausible way todefine the efficiency in the Stokesian realm. We note first the Froude efficiency,which is often used in problems of animal locomotion. This efficiency is definedby the work done by thrust normalized by total work:

η = ηFroude =U ·TW

(18)

where W is the work done by the organism in its locomotion. (W usual is takenas the mechanical work as opposed to that determined by body metabolism.)The efficiency brings in the above ambiguity.

In Stokes flow, however, the Froude efficiency can be made precise, and can

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be defined at each instant of time. First define the Stokes stress tensor by

σij = −pδij + µ[∂ui

∂xj+∂uj

∂xi

]

, (19)

so momentum conservation is expressed by∂σij

∂xj= 0. The instantaneous work

done by the body on the fluid is then

−∫

B

[uB]iσijnjdS, (20)

where uB = U+ub is the velocity of the body relative to the distant fluid, and n

is an outer normal for the body surface. Using our decomposition u = u1 + u2,and the divergence theorem, this integrates to

∫

VB

∂

∂xj([u1 + u2]iσijdV, (21)

where VB is the region exterior to the body. (We also are using the fact thatu1 and u2 both decay at ∞ like Stokeslets, to eliminate an integral at infinity.)With a little manipulation we obtain

µ

2

∫

VB

[∂ui

∂xj+∂uj

∂xi

]2

dV = Φ(t), (22)

where Φ(t) is the total instantaneous viscous dissipation in the fluid.We thus have an (instantaneous) Froude efficiency in the Stokesian realm

defined by

η =F2 ·U

Φ. (23)

If we define the mixed dissipation

Φmn =µ

2

∫

VB

[∂umi

∂xj+∂umj

∂xi

][∂uni

∂xj+∂unj

∂xi

]

dVB. (24)

We leave it as an exercise for the reader to show, using the zero force condition,that

Φ12 = Φ21 = −Φ11, (25)

and soΦ = Φ11 + 2Φ12 + Φ22 = Φ22 − Φ11 (26)

AlsoF2 · U = −F1 · U = Φ11. (27)

Thus (23) becomes

η =Φ11

Φ22 − Φ11. (28)

Note the theorem which results from (26): Φ11 ≤ Φ22 for any body and anymovement, under the condition of zero force. (I don’t know of an independentproof.) Also, if (27) is a good efficiency, we expect η ≤ 1 or Φ ≤ 1

2Φ22. (I don’tknow of a proof of this.)

We give an example of an efficiency below in our discussion of the Gray-Hancock model of a flagellum.

8

8 The swimming sheet

We now describe a seminal model for locomotion in a viscous fluid, introduced byG.I. Taylor in the 1950s, it has the advantage of providing a unified frameworkfor studying locomotion at finite as well as zero Reynolds number [5]. Taylor’soriginal work was for Stokes flow. The flow is two dimensional and the sheetextends to ±∞ along the x−axis. The mathematical problem is going to reducein the Stokesian case to a biharmonic half-plane problem in y ≥ 0. This isno doubt the simplest setting where a nontrivial locomotion problem can beexplored in an extensive Reynolds number range.

Taylor assumed that the sheet undulated in the y direction, and in so doingstretched slightly, but he also allowed it to directly stretch along its surface.Accordingly we introduce the boundary motion

xB = x+ a cos(kx− ωt − φ)

= x+ β cos ξ + γ sin ξ, ξ = kx− ωt, (29)

yB = b sin ξ, (30)

where β = a cosφ, γ = a sinφ. The special case a = 0 may be termed a wave

of shape, and that for b = 0 a wave of stretching, the sheet staying plane in thelatter case. The boundary conditions corresponding to no slip at the sheet arethen

u(xB, yB, t) = aω sin(ξ − φ) (31)

v(xB , yB, t) = −bω cos ξ. (32)

For the present discussion of the general ideas it will suffice to consider onlythe wave of stretching. The equations to be solved are

u(x+ a cos(kx− ωt), 0, t) = aω sin ξ, v(x, 0, t) = 0, ξ = kx− ωt. (33)

Note that the position at which we impose the condition on u is moving. Thisintroduces a natural expansion in the small dimensionless parameter ak.

We allow for the fact that the dynamic equilibrium of the stretching sheetand the fluid may require that u tends to a finite non-zero number as y → +∞.(We shall consider here only the upper half-plane.) By Galilean invariance theflow can then be interpreted as the swimming of a sheet calculated relative toa co-moving frame. Thus we suppose

u→ U, v → 0, as y → ∞. (34)

Our object is to determine U and thus the swimming velocity caused by themotion of the sheet.

In two dimensions the Stokes equations are

px − µ∇2u = 0, py − µ∇2v = 0, ux + vy = 0. (35)

9

To satisfy the last of (35) we introduce the streamfunction ψ(x, y, t), whereu = ψy, v = −ψx. Elimination of the pressure from the first two equation in(35) shows that ψ is a biharmonic function :

∇4ψ = 0. (36)

Acceptable solutions of (36) have the form

ψm(x, y, t) = (Am + kBmy)e−mky sinm(kx− ωt)

+(Cm + kDmy)e−mky cosm(kx− ωt) + Umy, (37)

where An, Bn, Cn, Dn, Un are arbitrary constants. Thus ψ will have the formof an infinite series of terms of the form (37). We gather these as a series inincreasing powers of ak.

The leading term, ψ1, is or order ωk−2 × ak and is determined by the con-ditions

∂ψ1

∂y(x, 0, t) = aω sin ξ,

∂ψ1

∂x(x, 0, t) = 0. (38)

Thus we find easily A1 = C1 = D1 = U1 = 0, kB1 = aω, so that

ψ1 = yaωe−ky sin ξ. (39)

Since U1 = 0, it can be said that the sheet does not swim to leading order.The second-order terms correct for the displaced location of the boundary

condition on u:

u(x+ a cos ξ, 0, t) = u1(x, 0, t)+∂u1

∂x(x, 0, t)× a cos ξ + u2(x, 0, t) . . .

= aω sin ξ. (40)

Thus u2, v2 satisfy the conditions

u2(x, 0, t) = −∂u1

∂x(x, 0, t)× a cos ξ, v2(x, 0, t) = 0. (41)

The right-hand-side now involves a cos2(ξ)and so the corresponding term ψ2

takes the form

ψ2 = −1

2ωka2ye−2ky cos 2ξ − 1

2ωka2y. (42)

Thus U = U2 = −12ka

2ω determines the leading term of the expansion of theswimming velocity. Assuming k, ω > 0 we see that the flow at infinity is inthe direction of negative x, so that the swimming velocity has the same sign asthe velocity of propagation of the wave of stretching. The physical reason forthis lies in the effect of the boundary conditions on the eddy structure near thesheet. The propagating wave of stretching introduces an asymmetry into theeddy pattern, as we display in Figure 1.

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Figure 1: ψ(x, y, t), in coordinates (ξ, η) = (kx − ωt, ky), for a stretchingplane sheet in Stokes flow, with k = ω = 1, a = .3.

The flow along the top of the figure is from right to left. Note that the flownear the wall is dominated by motion in this direction. The counter-rotatingeddy on the left exerts a thrust to the right at the wall, but this is more thancompensating by the dominant thrust to the left. The reaction of this net forceon the fluid is what drives the sheet through the fluid.

The last remark emphasized our calculation as one of swimming. On theother hand we might just as well have stated that the effect of the sheet is todrive a uniform flow extending to infinity! This seems paradoxical, that a sheetcould drive an infinite expanse of fluid into uniform motion. The resolution ofthis lies in the fact that the sheet is also infinite in expanse. Any finite pieceof the sheet would have not such an effect at infinity, although it would swimrelative to the fluid at infinity. The speed of swimming would presumably bealtered somewhat by the finite size.

We draw attention as well to the relation of our calculation to the problemof pumping in a channel with flexible walls. It is reasonable to assume thatin a channel of width H >> k−1, with two walls executing the movementsconsidered here, fluid would be pumped with a velocity given approximately byU2.

8.1 Some results for the general case

For a, b nonzero the calculation procedes similarly, and there results

U = U2 =1

2kω(b2 + 2ab cosφ− a2). (43)

The pressure field corresponding to ψ may be found from the formula

∂p

∂x= µ∇2∂ψ

∂y, (44)

because the right-hand side is periodic in x with zero mean. To leading orderwe find,

p1 = −2µωke−η [(b+ β) cos ξ + γ sin ξ]. (45)

Despite the fact that the sheet does not swim to leading order, the effort ormechanical work done by the sheet while swimming can be computed from theleading terms. The work done by the sheet, per unit horizontal projected area,is

Ws = −〈u(xB , yB, t)(σ11n1 + σ12n2) + v(xB , yB, t)(σ21n1 + σ22n2)〉, (46)

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where n1(x, t), n2(x, t) is the upward normal vector from the sheet,

n1 = −∆(1 + δ2)−1/2, n2 = (1 + ∆)−1/2,∆ =dyB

dx= bk cos ξ, (47)

and

σ =

(

−p+ 2µψxy µ(ψyy − Ψxx

µ(ψyy − Ψxx −p − µψxy

)

. (48)

From these expressions we may calculate

Ws = µω2k(a2 + b2). (49)

This value is to be doubled if both sides of the sheet are considered.Some other quantities of interest in applications of the swimming sheet are

the constant fluxes of mass and momentum associated with its movements. letfm, fx, fy denote fluxes of mass and (x, y) momentum. Then

fm = −〈ψ(x, yB , t) − UyB〉

∼ −〈ψ1(x, yB, t)〉 = −1

2βb. (50)

Similarly, using (48) we have

fx = 〈∫

∞

yB

σ11dy〉 = −2µωkγb, (51)

fy = 〈∫ ∞

yB

σ21dy〉 = µωkβb. (52)

These fluxes can affect the flow field whenever the parameters of the sheet areslowly modulated in space. Although we do not give details, one application ofthese fluxes is to the envelope model of ciliary propulsion, which is mentionedbelow.

We remark that we cannot discuss the efficiency of the sheet (even a finitesection of it) according to our earlier criteria, since we are working in two di-mensions and the Stokes paradox precludes our computing a drag of a finitebody.

8.2 Finite Reynolds numbers

We now consider the wave of stretching with no special assumption on theReynolds number, still retaining the same form and small amplitude of stretch-ing [6]. It is instructive now to adopt a dimensionless formulation. If U = ω/k,L = 1/k, T= 1/ω are the reference velocity, length, and time, the dimensionlessmomentum equations are now

Re[− ∂

∂ξ+ u

∂

∂ξ+ v

∂

∂η](u, v) + (pξ, pη) −∇2(u, v) = 0, (53)

12

where η = ky and as before ξ = kx− ωt, ∇2 now being in the latter variables.We have assumed here, based upon the boundary conditions, that the variabledepends only upon ξ, η. Since velocities remain small, of amplitude ε ≡ ak, theexpansion of the dimensionless streamfunction ψ is

ψ(ξ, η) = εψ1(ξ, η) + ε2ψ2(ξ, η) + . . . , (54)

where, after eliminating pressure

Re∂∇2ψ1

∂ξ+ ∇4ψ1 ≡ Lψ1 = 0. (55)

We see that we may write

ψ1 = <[Ae−η+iξ + Be−λη+iξ ], λ =√

1 − iRe. (56)

The boundary conditions (38) then yield

ψ1 = Re−1<[(λ+ 1)(e−η+iξ − e−λη+iξ)]. (57)

For ψ2 we now have two contributions ψ2 = ψ21 +ψ22. The first comes fromthe shifted boundary condition and is obtained as in the case of Stokes flow:

ψ21 = −1

4Re−1<[(e−2η − e−2λη)e2iξ(1 + λ)] − η/2. (58)

The second contribution comes from the nonlinear terms and satisfies null con-ditions on η = 0. The equation satisfied by ψ22 is

Lψ22 = −Re∂(ψ1 ,∇2ψ1)

∂(ξ, η). (59)

Since we will focus on the swimming velocity, we take the ξ-average of (59).With 〈·〉 denoting this average, using (56) we obtain

d4〈ψ22〉dη4

= −Re2

2<[AB∗(1 + λ∗)e−(1+λ∗)η + |B|2(λ + λ∗)e−(λ+λ∗)η ]. (60)

Integrating three times with respect to η and using the conditions at infinity,we have

〈u22〉(0) = U22 +Re2

2<[AB∗(1 + λ∗)−2 + |B|2(λ+ λ∗)−2] = 0. (61)

Since A = −B = (1 + λ)/Re,

U22 =1

2<

[ 1 + λ

1 + λ∗− |1 + λ|2

(λ+ λ∗)2

]

. (62)

Since <(λ) = F (Re) ≡ [(1+√

1 + Re2)/2]1/2, this expression can be reduced to

U22 =1

4[1/F − 1]. (63)

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But U21 = −1/2, see (58), so we arrive at the swimming dimensional velocity

U =1

4a2kω[1/F − 3]. (64)

We thus see that as Re increases from 0 to ∞ the swimming speed increasesfrom our earlier result of 1/2, but remains finite in the limit see [6].

Finally, we give the finite Reynolds number result for U in the case a = 0,obtained exactly as for the case b = 0.

U =b2kω

4

(

1 +1

F

)

. (65)

Some remarks:

• At Re = 0 the direction of locomotion is opposite to the phase speed ofthe wave for the wave of shape, but in the same direction for the wave ofstretching.

• In both cases U has a finite limit as Re → ∞, but only for the wave ofstretching does this make sense. It can be shown that for the wave of shapethe correct asymptotics for large Re requires that ε

√Re 1, meaning

the wave amplitude is large compared to the boundary-layer thickness, inorder to avoid separation of the boundary layer [4].

• For the wave of stretching, the infinite Re limit is o.k. and leads to afinite locomotion speed. However if Re = ∞, i.e. if ν = 0, the stretchingwall has no effect on the fluid and no locomotion occurs. This is a goodexample of the difference between the limit to a value in a physical problem(ν = 0), and the physics when the value is actually assumed.

• Let us consider the breaking of the scallop theorem by the wave of stretch-ing, with xB(x, t) = x+F (x, t) = x+a cos(kx−ωt) = a+a cos(ωt) cos(kx)+a sin(ωt) sin(kx).The configuration of the sheet can be fully specified bygiving the Fourier coefficients for the x−variation of F as functions oftime. Thus we have the curve (c1(t), c2(t)) in a 2D configuration space,where

c1(t) = a cos(ωt), c2(t) = a sin(ωt) (66)

This defines an oriented circle and the scallop theorem is broken.The im-plications of time-reversal symmetry should be carefully considered in thelight of this example.It is not simply a matter of material orbits of bound-ary points being indistinguishable from their time reversals. That occursin the above example, where the relative phase of distinct boundary pointsmust be invoked to see that the movement is not reciprocal. In a senseone must consider the entire collection of boundary points in tracking thetime evolution of the body configuration, as captured by the configurationspace.

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9 Flagellar locomotion

Although we shall see below that the swimming sheet will provide a useful modelof ciliary propulsion, it is not in itself representative of a natural organelle forlow Reynolds number locomotion. By far the most common strategy seen inthe natural world of loe Reynolds number is to utilize long slender organelles,such as cilia, flagella, and rigid stalks as the building material for a locomotingbody. We shall attempt in this section to understand the usefulness of this bodygeometry from the mechanical principles of Stokesian fluid dynamics.

Slenderness can usually be handled by linear distributions of the basic build-ing blocks of Stokes flows, Stokeslets, dipoles and so one. Powerful numericalmethods have been developed based upon this approach. In the present discus-sion we try to get a feeling for the undedrlying fluid mechanics.

9.1 Approximate theory for a thin stalk

Since a long slender filament in Stokes flow can presumably apply force to thefluid, forces necessarily concentrated near a curve, it is tempting to representits hydrodynamical effect in terms of a distribution of tensor Stokeslets,

Sij(x, y, z) =1

8πµ

(xixj

r3+

1

rδij

)

. (67)

Let us consider for simplicity a straight stalk whose axis lies along the segment−L1 ≤ z ≤ L2 of the z-axis, and let the boundary of the stalk be the cylinderx2 + y2 = a2,−L1 ≤ z ≤ L2. We assume that a L1 + L2.

Our object now is a see what surface velocities result from a given distribu-tion of forces along the axis of the stalk, and try to adjust the forces so that thesurface velocity amounts to a rigid-body motion of the stalk. If we succeed, thesuperposition of Stokeslets represents the Stokes flow created by the motion ofthe stalk.

Let ft(ζ)δ(x)δ(yδ(z − ζ) be the tangential force at the axis of the stalk, andfn(ζ)δ(x)δ(yδ(z − ζ) be the force normal to the stalk, both at the point z = ζ.Here ft = (fz(ζ), 0, 0) and fn = (fx(ζ), fy(ζ), 0). We seek to evaluate in theneighborhood of the stalk surface the integrals

(ut,un =

∫ L2

−L1

S(x, y, z− ζ) · (ft, fn)dζ (68)

We will show, following Lighthill [2] that a useful approximate theory can bedeveloped by first taking the force distribution as a constant.

We note the following indefinite integrals:

∫

(z2

r3+

1

r

)

dz = −z/r + 2 log(z + r), (69)

∫

z(x, y)/r3dx = −(x, y)/r. (70)

15

Thus if ft 6= 0 were independent of ζ, we would have

8πµf−1z wt =

∫ L2

−L1

(z − ζ)2

(x2 + y2 + (z − ζ)2)3/2+

1

(x2 + y2 + (z − ζ)2)1/2dζ

=(z − L2)

(x2 + y2 + (z − L2)2)1/2− (z + L1)

(x2 + y2 + (z + L1)2)1/2

−2 log(z − L2 +

√

x2 + y2 + (z − L2)2

z + L1 +√

x2 + y2 + (z + L1)2

)

(71)

These terms not involving the logarithm have an interesting structure whenx2 + y2 = a2 and z varies from −L1 to L2. We show graphs in Figure 2.

Figure 2: The terms in wt for the case L1 = 2, L2 = 3, a = .1. The dottedline is 2 log(2400).

The logarithmic terms clearly are not well approximated by a constant, butthat is nevertheless the approximation usually made. In fact the dominant termsin the expansion for small a2 where z is bounded away from the endpoints ofthe stalk are adopted:

wt = fz1

4πµ

[

log(4L1L2

a2

)

− 1]

fz + O(a2L−2), L = (L1L2)1/2. (72)

We also have on x2 + y2 = a2,

8πµf−1z (ut, vt) = (x, y)[(a2 + (z − L2)

2)−1 − (a2 + (z + L1)2)−1]. (73)

Thus, away from the endpoints we have (ut, vt) = O(a/L). Thus, in leadingterms

ut =1

4πµ

[

log(4L1L2

a2

)

− 1]

ft + O(a/L). (74)

Consider now the effect of the x-directed point forces. Again assuming fx isindependent of ζ. The resulting velocity is

8πµf−1x un =

∫ L2

−L1

(2x2 + y2 + (z − ζ)2, xy, x(z − ζ))

(x2 + y2 + (z − ζ)2)3/2dζ. (75)

16

We now need to use, in addition to (69), (70)

∫

r−3dz =z

(x2 + y2)r, (76)

∫

r−5dz =z

3a2r3+

2z

3a4r. (77)

We then find, away from endpoints, that on x2 + y2 = a2 we have

8πµf−1x un = (

2x2

a2+ log

4L1L2

a2,2xy

a2, 0) + O(a/L). (78)

The terms here in x2 and xy are formally O(1) and need to be dealt with. Wetherefore consider a potential flow contribution to the Stokes flow, given bydipole velocity ∇(x/r3). The velocity field for this dipole is

(r−3 − 3x2r−5,−3xyr−5,−3xzr−5). (79)

If we assume a uniform distribution of strength g over the stalk, the resultingflow, when evaluated on the stalk, takes the form

ug = g( 2

a2− 4x2

a4,−4xy

a4, 0) + O(a/L). (80)

Now choosing g = a2

16πµ and combining the two fields, we obtain a compositeun given by

un =1

8πµ

[

log(4L1L2

a2

)

+ 1]

fn + O(a/L), (81)

where we now include the contribution of fy as obtained by a parallel compu-tation.

We can turn the relation between force and velocity around, and view (74)and (81) as relations defining a resistance to motion of the stalk in the tangentialand normal directions respectively. Strictly speaking the resistance so calculatedis exact only in the limit of zero a, and the logarithmic terms are such as to makethe expressions convergence very slowly to this limit. Nevertheless, these resultshave led to the use of an approximate way of studying the resistance of a slender,flexible flagellum of arbitrary shape to tangential and normal movements. Theidea is to introduce tangential and normal resistance coefficients KT , KN , sothat utKT ∆s and unKN∆s are respectively the tangential and normal forcesexerted on the fluid by a small segment of flagellum of length ∆s, in motionthrough the fluid with tangential and normal velocity ut, un. If a is the radiusof the cross section of the flagellum, then the force is intended to be calculatedfor a slender object, so necessarily ∆s a.

In first glance it would seem that we have already, in (74) and (81), a niceapproximate derivation of KT , KN . Note that (74) and (81) suggest that ,because of critical factor of 2, when the logarithm dominates we can expect KN

to be about twice KT . In practice the factor is smaller, but KN/KT may be

17

safely assumed to exceed 1 for any slender smooth flagellum. However there is abasic problem in the use of these formulas. The expressions involve the productbc, and it is unclear what value should be assigned to this constant. Variousways of dealing with this calibration problem have been proposed, which wewill not discuss here. We simply adopt the the very useful premise that suitableKT , KN can be found, and proceed to construct a theory of flagellar propulsion.It is a fact that a purely local theory of flagellar resistance is not mathematicallytenable, so the method is leads to a simple, transparent model rather than anexact asymptotic formulation.

9.2 Gray-Hancock theory of flagellar propulsion

This theory studies the motion of a flagellum idealized as a curve in space whosemotion through the fluid leads to forces derivable from resistance coefficients.The theory, due to Gray and Hancock, is a resistive force theory.

We represent the instantaneous position of the flagellum in terms of its arclength,

r = (x, y, z) = R(s) =(

X(s), Y (s), Z(s))

. (82)

Then dR/ds = t is the local tangent vector in the direction of increasing s. Weassume the wave form to be periodic and the motion to be along the x-axis, sothat there are constants Λ, λ such that

X(s+ Λ) = X(s) + λ, Y (s+ Λ) = Y (s), Z(s + Λ)) = Z(s). (83)

Thus Λ is the distance along the length of the curve needed to travel one wave-length λ along the x-axis:

α =λ

Λ=

wavelength

arc length of one wave. (84)

Thus α−1 is the expansion factor for arc length accounting for waviness.We assume that the flagellum is inextensible. As in the swimming inexten-

sible sheet, an observer moving to the right who sees a standing wave, will seea point of the flagellum moving along the sheet to the left. Because of inexten-sibility, the speed of the point is a constant Q, the velocity along the flagellumbeing −Qt. We then have,

V = αQ. (85)

The observer watching a standing wave thus sees the material points as satisfying

r = R(s+Qt). (86)

If now the flagellum swims with velocity U i, a material point on the flagellummoves relative to the fluid at infinity with velocity w, where

w = (U + V )i−Qt. (87)

At this stage the fundamental assumption of resistive force theory is made. IfKT , KN are the flagellar resistance coefficients discussed above, it follows that

18

w·(KT t+KNn)ds is the force dF exerted on the fluid by a length ds of flagellumat a point with tangent and normal vectors t,n. The total force F exerted onthe fluid by on a length L of flagellum may then be written in a form whichavoids reference to the normal vector explicitly,

F =

∫ L

0

w ·Mds, M = (KT −KN)dR

ds

dR

ds+KN I. (88)

Using (86) in (88) we obtain the total thrust T exerted by the flagellum on thefluid

T = F·i = (KT −KN )(U+V )

∫ L

0

(X′)2ds+KN (U+V )L−KTQ

∫ L

0

X′ds. (89)

Defining β by∫ L

0

(X′)2ds = βL, (90)

and noting that∫ L

0

X′ds = αL =V L

Q, (91)

we obtainT = (V + U)[(KT −KN )βL +KNL] −KTV L. (92)

We can now study how the flagellum can swim. If the organism consistsonly of flagellum, necessarily U is found by setting T as given by (92) equal tozero. This defines the zero thrust swimming velocity U0;

U0

V=

(ρ− 1)(1 − β)

ρβ + 1 − β, ρ =

KT

KN. (93)

Since β < 1, we see that U0/V < 0 if ρ < 1, so that the swimming is oppositeto the phase velocity of the wave when the flagellum exerts greater normalthan tangential force. The calculations of resistance based upon local theoryindicated that .5 < ρ < 1 so that smooth flagella should always swim in adirection opposite the phase velocity of the wave.

If the flagellum is attached to a head, the thrust developed by the tail mustbalance the extra drag of the head, and which we take to be LUKN δ. Thenswimming spped U is less than U0, i.e. the flagellum drifts backward relative tothe zero thrust swimming seed. Thus we may write

LUKNδ = (V + U)[(KT −KN )βL +KNL] −KTV L, (94)

and therefore

LUKN δ = (U − U0)[(KT −KN)βL +KNL], (95)

It follows that (KT −KN )β+KN is a composite resistance for the flagellum asa whole, the waveform information being only in the parameter β.

19

With the head attached, (93) thus becomes

U0

V=

(ρ− 1)(1 − β)

ρβ + 1 − β + δ. (96)

In calculating δ, it is usually acceptable to neglect the interaction of the headwith the tail, given that the tail is very slender.

9.3 Efficiency

We now apply the definitions given earlier to calculate an efficiency of the flag-ellum within the limitations of the Gray-Hancock . The total work done by theflagellum takes the form

W =

∫ L

0

[(KT −KN )(w · t)2 +KNw2]ds

= KTL[(V − U)2β − 2(V − U)Qα+Q2] +KNL(V − U)2(1 − β). (97)

Now W = Φ11 + 2Φ12 + Φ22. The dissipation Φ11 may be identified with thework done by the moving the force UL[(KT −KN )β+KN ], involving the sero-thrust resistance coefficient resistance coefficient (KT −KN )β+KN , at a speedU. Also Φ22 must be the rate of working W when when the swimming speed iszero. Using (97) we then have

Φ11 = U2L[(KT −KN )β +KN ], (98)

Φ12 = −L(KT −KN )(β − 1)UV, (99)

Φ22 = V 2KTL(β − 2 + α−2) +KNLV2(1 − β). (100)

We now see that indeed Φ11 = −Φ12 when U has its zero thrust value given by(93). The efficiency is

η =Φ11

Φ22 − Φ11, (101)

withΦ22

Φ11=

[(ρ− 1)β + 1][(ρ− 1)β − ρ]

β(ρ − 1)2(β − 1). (102)

The minimum of this last ratio occurs when β = 1/2, (ψ = 45o, when the ratiobecomes (ρ+ 1)2(ρ− 1)−2, so that

ηmax =(ρ− 1)2

4ρ. (103)

This gives an efficiency of .032 when ρ = .7. This low efficiency is typical ofStokesian swimming.

20

9.3.1 Helical waves

Helical flagellar waves result from superposition of two orthogonal planar waves,so that, relative to the moving observer seeing backwards tangential motionalong the flagellum,

R(s) =(

α(s+Qt), b cos k(s+Qt), b sink(s+Qt))

. (104)

Here as before αQ is the phase speed V , and α2 + b2k2 = 1 by the definition ofs. Shifting back to the stationary observer, the velocity of a material point onthe flagellum is

(dX/dt− V, dY/dt, dZ/dt) =(

0,−bk sin k(s+Qt), bk cos(k(s+Qt))

, (105)

and is therefore equivalent to a rigid rotation of the helical structure about itsaxis.

We have so far discussed only thrust, but helical waves bring up explicitlythe matter of the balance of torques in flagellar hydrodynamics. Within Gray-Hancock theory the torque balance may be computed from the moments ofresistance forces, but there is in addition the question of rotation of the flagel-lum. If the flagellum is regarding as a rigid structure, it will rotate along withthe helical wave. If on the other hand the surface of the flagellum is not freeto rotate with the wave, torques associated with the finite cross section of theflagellum can be eliminated.1

We remark that within Gray-Hancock theory, simultaneous torque and thrustbalance for a headless flagellum is impossible, see exercise 2.5. Head rotation,as well as restoring torques due to the mass distribution in a gravitational field,can provide the balance. An appealing idea, due to Chwang and Wu, is to userotation of the flagellum itself to balance swimming torque. A circular cylin-der of radius a spinning on its axis with angular velocity Ω in a viscous fluidgenerates a simple steady velocity field

u =a2Ω × r

r2, (106)

where r is the cylindrical polar radius. The resulting torque on the fluid per unitlength is 2πµa2Ω. For a curve of projected length αL, the net torque about theline of swimming is 2πµa2ΩLα, and this is available for canceling the swimmingtorque. The body rotation does not affect (to leading order) the thrust andmoment caculations of resistive force theory.

10 Ciliary propulsion

A second basic swimming mechanism in the Stokesian realm involves the col-lective use of many small hairlike organelles called cilia. Although flagella and

1G.I. Taylor, in a delightful experiment, constructed a device with zero flagellum torque

by rotating a rigid helical wire inside of a flexible tube, the tube being fixed to an essentially

stationary body.

21

cilia of the eukaryotic organisms (e.g. protozoans, algae, and the multicelluarorganisms) are apparently identical in ultrastructure, we have used the termflagellum when there is only one, or a small number of these appedages on acell., and in spermatozoans and the flgellates, and will use cilia to denote largenumbers of them on the cell, as in the opalinates and teh ciliates. The cili-ated organisms tend to have bodies which are larger that the cell bodies of thespermatozoans by an order of magnitude. Cilia tend to be shorter that flagella,however, which puts their hydrodynamics firmly in the Stokesian realm.

Bacteria have organelles which are also called flagella, but they are verydifferent in ultrastructure and physiology. Also, the term cilia is used in con-nection with various ciliated tissues in metazoans (many-celled animals), andare found in the lining of our repiratory tract. But in these cases there is usuallyonly one or a small number of cilia per cell.

Because of the proximity of a cilium to the cell wall, we can expect to findconsiderable interaction between the cilium and the cell wall. Thus we mightexpect to see differences between the movements of cilia and flagella proper.This an the larger size of the cell suggest that ciliary locomotion might havesome adaptive advantages for larger. Both flagellary and ciliary modes arewidespread, so both are evidently successful strategies, and one can speculateon what trade-offs the distinctly different morphology might represent.

In one sense the proliferation of hairlike appendages is natural to the Stoke-sian realm, purely on the basis of the efficient use of material. The movementof a cylindrical rod of radius A and length L will, according to our approximatelocal theory, generate a force of order µUL/ log(L/A). If the cylinder is brokenup into N 1 thin hairs of length L and radius a = A/

√N , the total force

on all the hair is of order NµUL/ log(L√

N/A. This shows that a Stokesianparachute could be made efficiently of hairs. Such a strategy if probabaly seenin seed dispersal by plants such as dandelions, and by the membraneless hairsutilized as wings by certain minute flying insects.

Such considerations alone may not, however, suffice to explain the possi-ble value of ciliary propulsion to the organism. We have seen from the Gray-Hancock theory that the mechanical efficiency of a flagellum depends on resis-tances through their ratio ρ, and the efficency of a single flagellum tends to bein the range it is perhaps not surprising that efficiencies in ciliary locomotiontend to be comparable.

Perhaps the most compelling advantage of the ciliary strategy is its flexi-bility. A large number of esentially identical organelles can be distributed andcoordinated as a sort of standard “ciliary carpet”. Pieces of this carpet can bearranged as needed on a cell to optimally move a cell of given size and shape.A possible disadvantage of the ciliary mode is the clear need to coordinate themotion of a large number of organelles. The degree to which this coordinationis organized by the nervous system, as opposed to being a passive response tohydrodynamics interactions, is a matter of ongoing research.

22

10.1 The motion of the cilia

Typical parameters of a cilium are: length ∼ .001 cm = 10 µm (1 µm = 10−4 cm); diameter ∼ .25 µm; tip speed ∼ 0,2 cm/sec; beat frequency ∼ 30 cycles/sec.A typical beat cycle consists of two parts, an effective stroke and a recovery

stroke, see Figure 3.

Figure 3. The stroke sequence for a single cilium.

The effective stroke may take somewhat less than 1/2 a cycle. A Reynoldsnumber for a single cilium based on length and tip speed is about .02.. TheReynolds number based upon ciliary diameter is smaller by a factor of 1/40.

Cilia may beat in a vertical plane, as we imply in Figure 3, but in some casesthe recovery stroke involves out-of-plane sideways movements. Cilia may moveas a bundle, comprising a compound cilium. Helical waves are also observed.

In any case the basic motion seems sound from a fluid dynamic viewpoint.The ffective stroke tends to be broadside on, developing the larger force associ-ated with the resistance coefficient KN . During the recovery the motion tendsto be tangential, and therefore to involve the somewhat smaller resistance co-efficient KT .But it should be borne in mind that there are many variations ofthe basic stroke and the foregoing remarks have not addressed the interactionof the cilium with the wall. Note that with geometry nature has successfullybroken the scallop theorem.

An important aspect of ciliary locomotion is the manner in which the cyclesof cilia are coordinated as a function of position. Observations of the surface ofa ciliate show metachronal waves of coordination in the beating pattern. Thesewaves, which flow across the cell body with well-defined wave speed and wavelength, have several forms. In one, the cilia tips in the effective stroke move inthe same direction as the metachronal wave. In the other, the tips and the wavemove in opposite directions.

There are many challenging problems connected with cilia movements. Mod-eling of the basic stroke sequence utilizing the cilium structure as a guide should

23

provide insight into how the various strokes can be generated. Also the roleof the fluid dynamics of interacting cilia on the development of the variousmetachronal wave patterns is of great interest.

10.2 The envelope model

We mention briefly a useful simplified model based upon the idea that a densecarpet of cilia should be well represented by an envelope surface swept outby the cilia tips. This allows a direct application of Taylor’s swimming sheetmodel. The modeling thus amounts to relating the geometry of the movementof cilia tips to the parameters of a metachronal wave.Applied to finite bodies,the envelope model effectively dictates the velocity at the outer edge of the ciliacarpet as an effective “slip” velocity, thus altering the boundary c‘ondition onthe body from the no-slip condition. At the same time, effective stresses arecreated, as we mentioned above, and thses need to be incorporated to determinethe swimming velocity of the body. For details see [8].

The envelope model cannot account for the flow field within the carpet ofcilia, although this is accessible within so-called sub-layer models see e.g. [9].

11 Intermediate Reynolds numbers

We have seen that the swimming sheet can be studied at finite Reynolds num-bers only at small amplitudes, and the larger the Reynolds number, the smallermust be the amplitude to avoid the consequences of flow separation. Thereare fundamental obstacles in the way of any mathematical analysis of locomo-tion at intermediate Reynolds numbers. We have already touched on some ofthe problems- the shedding of vorticity, and the possible non-uniqueness (bi-stability) of the flows. Basically the intermediate range describes the transitionfrom the mechanisms of locomotion appropriate to the low Re, linear, Stokesianrealm and the constraints of the scallop theorem, to the high Re, nonlinear,Eulerian realm, where the symmetry breaking and vortex shedding associatedwith the emergence of fluid inertia becomes an important part of the dynamicsof the fluid.

[Remainder of tutorial is in powerpoint format.]

References

[1] Purcell, E. 1977 Life at low Reynolds number. Am. J. Phys. 45. 3–11.

[2] Lighthill, M.J. 1975 Mathematical Biofluiddynamics. Regional conferenceseries in applied mathematics. Society of Industrial and Applied Mathematics,Philadelphia.

[3] Childress, Stephen 1981 Mechanics of Swimming and Flying. Cambridge.

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[4] Childress, Stephen 2008 Inertial swimming as a singular perturbation,Proceedings the ASME 2008 Dynamic Systems and Control Conference, AnnArbor.

[5] Taylor, G.I. 1951 Analysis of the swimming of microscopic organisms Proc.

Roy. Soc. Lond. A209, 447–461.

[6] Tuck, E.O. 1968 A note on the swimming problem J. Fluid Mech. 31,305–308.

[7] Shapere, A. & Wilczek, F. 1989 Geometry of self-propulsion at lowReynolds number. J. Fluid mech. 198, 557–585.

[8] Brennen, C. 1974 An oscillating-boundary-layer theory of ciliary propul-sion.J. Fluid Mech. 65 , 799–824.

[9] Brennen, Christopher $ Winet, H. 1977 Fluid mechanics of propulsionby cilia and flagella. Ann. Rev. Fluid Mech. 9, 339–398.

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