A frictionless microswimmer

We investigate the self-locomotion of an elongated microswimmer by virtue of the unidirectional tangential surface treadmilling. We show that the propulsion could be almost frictionless, as the microswimmer is propelled forward with the speed of the backward surface motion, i.e. it moves throughout an almost quiescent fluid. We investigate this swimming technique using the special spheroidal coordinates and also find an explicit closed-form optimal solution for a two-dimensional treadmiler via complex-variable techniques.

swimmer of length ℓ, thickness d with rounded caps undergoing treadmilling at velocity U . It is reasonable to assume (and the analysis in the following section can be used to justify) that all the dissipation is associated with the rounded ends. Hence, by dimensional analysis, the power dissipated in treadmilling is of the order of µdU 2 where µ is the viscosity coefficient. Let us compare this with the power needed to drag the "frozen" treadmiler. By Cox slender body theory [12] the power needed to drag the tube with velocity U is ∼ µℓU 2 / log (2ℓ/d).
Hence the ratio of power invested in dragging and swimming scales like (ε log 1/ε) −1 and can be made arbitrarily large. Here ε = d/ℓ is the aspect ratio of the swimmer.
One can now ask if there are slender treadmilers that are arbitrarily better than the slender rod-like treadmiler above? Consider now an elongated ellipsoidal microswimmer whose surface is given by z 2 /b 2 +r 2 /a 2 = 1 where r 2 ≡ x 2 +y 2 . Let us assume again, that the viscous dissipation is a result of a tip propulsion, and estimate the position of the tip from the condition |dr/dz| = 1. It can be readily demonstrated that in the case of ε = a/b ≪ 1 the tip is located at a distance of b(1 − 1 2 ε 2 ) from the center and its typical width scales as aε. Therefore, applying the same arguments as before, the dissipation rate should then scale as P ∼ µaεU 2 and the ratio of power expanded in dragging and swimming becomes bµU 2 / log (2b/a) µaεU 2 ∼ 1 ε 2 log 1/ε . We shall see, by a more accurate analysis, that for prolate spheroid the ratio of power in treadmilling to dragging is of the order of (ε log ε) −2 . In the following sections we shall analyze two models of treadmillers in 3 and 2 dimensions, respectively.
[ 2 ] § To whom correspondence should be addressed. E-mail: avron@tx.technion.ac.il c 2006 by The National Academy of Sciences of the USA 1 Alternatively, we may think of a a slender microbot that is topologically equivalent to a toroidal swimmer proposed by Purcell [1], i.e. the surface is not created or destroyed, but rather undergoes a continuous tank-treading motion Since the flow is axisymmetric we introduce the scalar stream-function Ψ (unique up to an additive constant) that satisfies the continuity equation The velocity components are readily obtained from [ 3 ] as where the symbols b H stand for the appropriate Lamé metric coeffi- The vorticity field can be obtained from [ 3 ] as where the operator E 2 is given by Following the standard procedure, the pressure is eliminated from the Stokes equation by applying the curl operator to both sides, with conjunction with [ 4 ] this yields the equation E 4 Ψ = 0 for the streamfunction. The boundary conditions [ 2 ] at the microswimmer surface τ = τa in terms of the stream-function become and the conditions at infinity The solution for Ψ that is regular on the axis and at infinity, and also even in ζ can be derived from a general semiseparable solution [9,10], where Ωm(τ, ζ) is a solution of E 4 Ψ = 0 composed from spheroidal harmonics that decay at infinity, and Gm and Hm are the Gegenbauer functions of the first and the second kind, respectively. The coefficients Am in [ 6 ] can be expressed in terms of Cm and U via the use of the boundary condition Ψ = 0 at τ = τa. Substituting [ 6 ] into [ 5 ] we arrive after some algebra at the tridiagonal infinite system of equations for U and the coefficients Cm , Here C0 = −c 2 U , E Regularity of Ψ implies that the admissible solution of [ 7 ] should satisfy CmHm(τa) → 0 as m → ∞ while the exponentially growing solution with Cm ∼ O((τa + √ τ 2 a − 1) 2m ) should be discarded. The viscous drag force exerted on the prolate spheroid (in the x3direction) is solely determined by the C2-term in [ 6 ] corresponding to a monopole (Stokeslet) velocity term decaying like 1/r far from the particle, F = −(4πµ/c) C2 [11]. Either F or U can be specified in addition to the surface velocity, u(ζ). In the swimming problem F = C2 = 0, and u(ζ) determines the propulsion velocity U .
The problem of the "frozen" spheroid (i.e. u(ζ) = 0) in the uniform ambient flow −U e3 corresponds to substituting bm = 0, m ≥ 2 in [ 7 ]. In this case the equations for Cm can readily be solved yielding the well-known result for Ψ and the drag force [11], Propulsion velocity. In order to determine the velocity of propulsion of the microswimmer freely suspended in the viscous fluid, one must solve Eqs. [ 7 ] with C2 = 0 as the particle is force (and torque) free. Let us consider the following velocity distribution at the boundary where us is a typical velocity of surface treadmilling and G2(ζ) = 1 2 (1 − ζ 2 ). One may verify that for a sphere (c/a → 0) u(ζ) = us sin θ, while for an elongated swimmer u(ζ) ≃ us almost everywhere except the near vicinity of the poles ζ = ±1. More generally it can be readily shown that the solution satisfying [ 5 ], [ 10 ] is given by [ 6 ] with ∀m , Cm = 0; m ≥ 4, Am = 0 and Note that U can be related to the surface motion 2 via the use of the Lorentz reciprocal theorem [12], where (b u, b σ) is the velocity and stress field corresponding to translation of the same shaped object when acted upon by an external force b F . For purely tangential surface motion considered in this work we We calculate the local tangential stress component b σ τ ζ from the solution corresponding to streaming past a rigid prolate spheroid, while 2 In the laboratory frame the velocity at the surface is a superposition of the translational velocity U and purely tangential motions u b F is given by [ 9 ]. Substitution into [ 12 ] with dS = b Hϕ b H ζ dϕ dζ yields after some algebra which holds for an arbitrary tangential boundary velocity u(ζ). In the special case of u(ζ) given by [ 10 ] it can be readily demonstrated that integration yields the propulsion speed [ 11 ]. Also, [ 13 ] actually solves the infinite tridiagonal system [ 7 ], since knowing U, F (i.e. C0, C2) one can iteratively obtain all the other Cm's by direct substitution. The scaled swimming speed of the microswimmer is depicted in Figure 1 as a function of the scaled elongation. The values of the propulsion velocity corresponding to a spherical swimmer (c = 0) and a slender swimmer (c ≫ a) can be determined via [ 12 ] without invoking special spheroidal coordinates. For a sphere, the local traction force b σ · n = − 3µ 2aÛ andF = −6πµaÛ and thus the self-propulsion velocity can be found as [13] where e is the unit vector in the direction of locomotion. Substituting u = us sin θ e θ and e = er cos θ − e θ sin θ, where θ is the spherical angle measured with respect to e, we arrive at U = 1 2 R π 0 us sin 3 θ dθ = 2 3 us in agreement with the result shown in Figure 1.
The drag force exerted on the rod-like microswimmer upon translation along its major axis with velocityÛ || is given byF ≈ −4πµaÛ || /(ε log 1/ε) and the local friction force is b For the 'needle-shaped' microswimmer the surface velocity u = u(ζ)e ζ ≃ −us e over almost the whole surface, it follows that U ≃ us. As seen in Figure 1 the propulsion velocity U/us → 1 as c/a grows and equals to 0.95 already at c ≃ 5.3a. As intuitively expected, the micro-swimmer is self-propelled forward with the velocity of the surface treadmilling , while the boundary velocity in the laboratory frame is (almost) zero. Swimming efficiency. Since the fluid around the elongated microswimmer propelled by continuous surface treadmilling is almost quiescent, except for the near vicinity of the poles, it is natural to expect low viscous dissipation and high hydrodynamic swimming efficiency. Several definitions of hydrodynamic efficiency have been proposed [13,14,19] here we follow the definition δ = F · U /P, where P is the energy dissipated in swimming with velocity U , and the expression in the numerator is the work expanded by dragging the "frozen" swimmer at velocity U upon action of an external force F [13]. δ is dimensionless and compares the self-propulsion with dragging (some authors use the reciprocal efficiency 1/δ). The higher δ the more efficient the swimmer is. For an axisymmetric swimmer propelled along the symmetry axis, F · U = RU 2 , where the scalar R is the appropriate hydrodynamic resistance. The work done by an arbitrary shaped swimmer and dissipated by viscosity in the fluid is given by where E is the rate-of-strain tensor, V is the fluid volume surrounding the swimmer and S its surface. Expressing the product E:E as P ωiωi + 2(∂ivj )(∂jvi) allows re-writing P for microswimmers self-propelled by purely tangential motions u as [13] where κs = −(∂s/∂s)· n is the curvature measured along the path of the surface flow, expressible in terms of the unit tangential and normal vectors, s and n, respectively. Let us now estimate δ of the spheroidal treadmiler described in the previous subsection. For a prolate spheroid s = e ζ , n = eτ , respectively, and κs can be calculated as δ is plotted as a function of the elongation c/a in Figure 2. Evidently, δ grows unbounded as c/a → ∞ does, corresponding, in the limit to a frictionless swimmer. For the spherical treadmiler δ can be calculated from [ 14 ] with u = us sin θ e θ , us = 3 2 U and κs = 1/a, P = 2µ R S 9 4 U 2 sin 2 θ 1 a dS = 12 πµaU 2 . Dividing 6πµaU 2 by P we find δ = 1 2 in agreement with [ 16 ] (see Figure 2) and the theoretical bound (i.e. δ ≤ 3 4 ) in [13] . For the slender swimmer the asymptotic behavior of δ can be estimated from [ 16 ] by expanding δ in a series around τa = 1 and using This result is shown in Figure 2 as a dashed line. For comparison, the efficiency of spherical squirmers self-propelled by propagating surface waves along their surface (the mathematical model of cianobacteria [15]) has the upper bound δ ≤ 3 4 , while numerically calculated values of δ do much worse than dragging and the corresponding swimming efficiency is usually less than 2% [13]. Swimming by surface treadmilling is remarkably more efficient than the rotating helical flagellum [16], beating flexible filament [17], the Percell's "three-link swimmer" [18] or locomotion by virtue of shape strokes [14,19]. The surface treadmilling is probably superior to any inertialess swimming techniques proposed so far.
Also, the swimming efficiency of the ellipsoidal treadmiler is superior by a factor of (ε log 1/ε) −1 over the estimate of δ corresponding to the rod-like treadmiler with rounded ends derived from purely scaling arguments in the introduction. Therefore, the geometry (via κs) plays an important role in minimizing the dissipation in surface treadmilling, which is rather surprising since the drag force on slender nonmotile object does not depend on its shape to the first approximation. Optimal swimming.We can set an upper bound on δ for a spheroidal microswimmer in terms of surface integrals of an arbitrary velocity u(ζ) analogously to [13]. The power dissipated in self-propulsion is bounded from below according to [ 14 ] by where we used the previously derived result for κs. The power expanded in dragging at the same speed is found from [ 9 ] and [ 13 ] as Combining the last two results we obtain an upper bound on δ as The term in the figure brackets can be shown to be bounded from above by 2/3 while its maximum is obtained for u(ζ) = us p 1 − ζ 2 p τ 2 a − ζ 2 corresponding to the 2-term boundary velocity expansion [ 7 ] with b2, b4 = 0 , bm = 0, m ≥ 6 (and, thus, representing a rotational flow). Thus, for an elongated swimmer (τa → 1) we arrive at δ ≤ 1 3ε 2 log 1/ε which does better than [ 17 ] by a factor of (log 1/ε) −1 and also superior over the scaling estimate for the rod-like swimmer by a factor of O(1/ε). Note that the asymptotic behavior δ ∼ 1 ε 2 log 1/ε was derived from simple scaling arguments in the introduction.
It can be demonstrated that the surface velocity [ 10 ] is not optimal, i.e. it does not minimize P for a prescribed propulsion speed U . To see this consider the slightly perturbed boundary velocity such that |u4| ≪ |u2| and u2 ∼ us. The solution of the linear problem [ 7 ] yields |ω| = O(u4) and therefore, the volume integral in  where I2(τa) is given by [ 15 ] and I4 is some other function of τa. The velocity of propulsion can be found in the close form as U = u2F2(τa) + u4F4(τa) from [ 13 ], where F2 is equal to the expression in the figure brackets in [ 11 ] and F4 is some other function of τa. As the propulsion velocity to be fixed, we require U = usF2. This yields dissipation where the function in the brackets can be shown to be positive and bounded for all τa > 1, and vanishes only at τa → 1. Therefore, one can always choose some u4 < 0 such that P < I2u 2 s leading to reduction in the dissipation in [ 15 ]. The above perturbation analysis shows that, quite surprisingly, vorticity production could bring a reduction in viscous dissipation, leading to more efficient swimming.
To address the question of optimal swimming we consider an arbitrary boundary velocity via the expansion, that meets all the above requirements for regularity and evenness in ζ,

umGm(ζ). [ 19 ]
where it follows from [ 8 ] that um = −bm/(2c 2 τa). To find a set of Fourier coefficients um, m = 2, 4, 6, . . . L corresponding to the optimal swimming, one should minimize the dissipation integral, P, while keeping the propulsion speed U fixed. The dissipation integral P = − R S σ τ ζ u dS being bilinear in ui, can be expressed as P = 1 2 P ui Pij uj . Note however that the tangential stress σ τ ζ at the surface of the microswimmer requires the knowledge of the velocity gradient at the surface (rather than velocity along). Alternatively, since the optimal velocity field is rotational, calculation of P from [ 14 ] requires the knowledge of vorticity everywhere.
The propulsion velocity given by [ 13 ] is linear in ui, i.e. U = P j Fj uj . The optimal set of coefficients ui is to be determined from ∂ ∂u i (P − λU ) = 0, or just from P j Pij uj = λ Fi, where λ is a Lagrange multiplier. We found the closed form optimal solution for the two-term boundary velocity [ 19 ], while for L > 4 closed form expressions are cumbersome, and numerical solutions were derived instead. Analogously to the theory for a 2-D swimmer (see the next section), where the explicit optimal solution was shown to acquire an infinite number of harmonics in the expansion for the boundary velocity, increasing the truncation level L in [ 19 ] will further improve the efficiency of swimming, though the enhancement appears to be minor. To illustrate this, we calculate the optimal solution upon varying L. The optimal boundary velocity upon varying L is depicted in Figure  3 for the elongation of c = 2.5a and compared with the one-term expression [ 10 ]. The scaled dissipation integral, P/µau 2 s , is plotted vs. c/a upon varying L in Figure 4. It can be readily seen that the convergence with respect to L is rather fast; the deviation between the results corresponding to L = 8 and 10 is less than 1% for all c/a and it vanishes at both limits c = 0 and c/a → ∞. Thus, the 'intuitive' one-term boundary velocity [ 10 ], that yields δ ∼ (ε log ε) −2 (see Figure 2) is nearly optimal for a wide range of elongations and likely so for all elongations. 2-D microswimmer. The two dimensional Stokes equations is conveniently handled by employing complex variables [14,19,21,22]. This allows explicit solution of the optimization problem for the elliptical treadmiler.
Denoting v = vx + ivy and ∂ = 1 2 (∂x − i∂y), the Stokes equations become 2µ∂∂v =∂p, Re∂v = 0. The most general solution to this (with p real) is v = g +f − zg ′ , p = −4µRe(g ′ ) where g, f are any pair of holomorphic functions [20]. Solutions corresponding to multivalued g, f are also legitimate (provided the resulting v, p are single valued). It can be shown (using [ 20 ] below) that the monodromy of g (and off ) around a closed curve gives the total force exerted by the fluid on the interior of the curve. In particular in swimming problems this force must vanish and g, f are therefore always single valued.
The element of force dF ≡ dFx + idFy acting on a length element dz = dx + idy of the fluid can be expressed in terms of v, P and hence in terms of g, f . Straightforward calculation shows that the relation is [ 20 ] Note that here (dx, dy) is tangent rather then the normal to the segment 4 . We consider a 2-D swimmer shaped as an ellipse of semi-axes b, a = 1 ± α (with 0 ≤ α < 1) situated in the complex z = x + iy plane. It is then convenient to define a new complex coordinate ζ by the relation z = ζ + α/ζ. As ζ ranges over the region |ζ| > 1 the corresponding z ranges over the area outside the swimmer. In particular the swimmer boundary corresponds to the unit circle ζ = e iθ . Note that if we consider g, f as functions of ζ rather then z then the general solution of the Stokes equations becomes In the swimmer frame of reference, the boundary condition at infinity v(∞) = −U (where U is the laboratory-frame swimming speed) implies Laurent expansions where U is arbitrarily appended to g. The boundary condition on the swimmer surface is fulfilled by matching v(ζ) to a prescribed boundary motion v| ζ=e iθ = w(θ) = P ∞ n=−∞ wne inθ . It is useful to express w(θ) as w = w+(ζ) + w−(ζ), ζ = e ıθ where w− = P ∞ n=0 w−nζ −n , w+ = P ∞ n=1 w * n ζ −n are both analytic outside the unit circle. Substituting [ 22 ] into [ 21 ] and matching on the unit circle we find In particular the swimming velocity is determined by the constant term in this expansion U = −w0. The corresponding dissipation is calculated using [ 20 ] as Let us next focus on the case of an ellipse swimming by surface treadmilling. The boundary velocity w(θ) being tangent to the swimmer boundary is expressible as w = dz dθ u(θ) = i(ζ − α/ζ)u(θ) for some real-valued function u(θ). Since we consider only swimmers symmetric with respect to the x−axis, we assume u(θ) to be an odd function allowing to write it as u = P un sin(nθ) = 1 2i P un(ζ n − ζ −n ). In terms of this the swimming velocity turn into U = −w0 = 1 2 (1 + α)u1 while the dissipation takes the form P = 2πµ X n`(1 + α 2 )u 2 n − 2αun−1un+1´.
Which may also be written as P = 1 2 P Pijuiuj for a corresponding tridiagonal matrix Pij. The optimal swimming technique for a given α is the one that minimizes the dissipation while keeping the swimming velocity U = 1 2 (1+α)u1 fixed. The minimizer is the solution of ∂ ∂u i (P −λu1) = 0 for ∀i with λ being a Lagrange multiplier, or just X j Pij uj = λδi,1.

[ 23 ]
It is readily seen that the coefficients u k with even k are not relevant to the optimal swimming and should be set to zero to minimize viscous dissipation. (This is also clear from the fact that u 2k correspond to flows which are antisymmetric with respect to the y−axis.) Denoting b k ≡ u 2k+1 , k ≥ 0 and writing λ = 2usπµ(1 + α 2 ) with us an arbitrary normalization constant having dimensions of velocity we obtain from [ 23 ] the recursion relation Multiplying by x k and summing over k this transforms into a differential equation for the generating function where C is a constant of integration. Requiring the coefficients b k to decay for large k implies that B(x) must be analytic inside the unit disc and hence its potential singularity at x = α must be avoided. This determines the integration constant to be C = α, so that we may write where x+ = 1 α − x and x− = α − x. The corresponding swimming velocity and the dissipation are, respectively, Therefore, combining the last two results yields We recall that in 2-D the dragging problem admit no regular solution within the Stokes approximation 5 . Thus defining the swimming efficiency as δ = (F · U )/P = RU 2 /P makes no sense in the present 2-D context in which F , R are not defined. This may be considered as a mere issue of normalization. We therefore use here an alternative definition of swimming efficiency [19] where δ ⋆ = 4πµU 2 P = (1 + α) 2 2α log [ 24 ] In the slender limit, α → 1, or, a b ≡ 1−α 1+α = ε → 0, as the ellipse degenerates into a needle, the efficiency grows logarithmically unbounded as δ ⋆ ≃ 2 log(1/ε).
It may be of interest to note that truncating our expansion to include any finite number of modes would lead to B(x) which is not only polynomial in x but also algebraic in α. This then implies that the (truncated) efficiency δ ⋆ ∝ B(0) would be algebraic in α implying that [ 24 ] must be modified to a bounded expression. Thus (in contrast to the 3-D case) one cannot obtain the correct asymptotic efficiency without retaining all the modes. The optimal boundary velocity may be found explicitly as "o .
Using the explicit expression we have for B(x) and dz dθ we find the absolute value of w is given by while its direction is known to be tangential. Thus, in the limit ε → 0 one finds (provided ε ≪ θ, |π − θ|) that |w/U | ≃ 1 + log |2 sin θ| log 1/ε , and the boundary velocity approaches the constant U though only at a logarithmic rate. Concluding remarks. In this paper we examined the propulsion of elongated microswimmer by virtue of the continuous surface treadmilling. As the slenderness increases, the hydrodynamic disturbance created by the surface motion diminishes, i.e. the microbot is propelled forward with the velocity of the surface treadmilling, while surface, except the near vicinity of the poles, remains stationary in the laboratory frame. As a result of that, the 'cigar-shaped' treadmiler is self-propelled throughout almost quiescent fluid yielding very low viscous dissipation. The calculation of optimal hydrodynamic efficiency of the 3-D and the 2-D microswimmers reveals that the proposed swimming technique is not only superior to various motility mechanisms considered in the past, but also perform much better than dragging under the action of an external force.