# Chiral propulsion: the method of effective boundary conditions

Leonid Korneev1 Dmitri E. Kharzeev2,3 Alexandre G. Abanov1,4 1Department of
Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA
2Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794, USA
3Department of Physics, Brookhaven National Laboratory, Upton, NY 11973-5000,
USA
4Simons Center for Geometry and Physics, Stony Brook, NY 11794, USA

###### Abstract

We propose to apply an “effective boundary condition” method to the problem of
chiral propulsion. For the case of a rotating helix moving through a fluid at
a low Reynolds number, the method amounts to replacing the original helix (in
the limit of small pitch) by a cylinder, but with a special kind of partial
slip boundary conditions replacing the non-slip boundary conditions on the
original helix. These boundary conditions are constructed to reproduce far-
field velocities of the original problem, and are defined by a few parameters
(slipping lengths) that can be extracted from a problem in planar rather than
cylindrical geometry. We derive the chiral propulsion coefficients for
spirals, helicoids, helically modulated cylinders, and some of their
generalizations using the introduced method. In the case of spirals, we
compare our results with the ones derived by Lighthill and find a very good
agreement. The proposed method is general and can be applied to any helical
shape in the limit of a small pitch. We have established that for a broad
class of helical surfaces the dependence of the chiral propulsion on the
helical angle $\theta$ is universal, $\chi\sim\cos\theta\sin 2\theta$ with the
maximal propulsion achieved at the universal angle
$\theta_{m}=\tan^{-1}(1/\sqrt{2})\approx 35.26^{\circ}$.

###### Contents

  1. I Introduction
  2. II Chiral propulsion of an infinite helicoid
    1. II.1 Chiral propulsion of a solid body moving through viscous fluid
    2. II.2 A helicoidal system
    3. II.3 Chiral propulsion of a helicoid
  3. III Small pitch limit. The method of effective boundary conditions
    1. III.1 Effective boundary conditions for planar geometry
    2. III.2  Chiral propulsion of a cylinder with Navier boundary conditions
  4. IV Examples and applications
    1. IV.1 Example 1: Chiral propulsion of $N$-spiral
    2. IV.2 Example 2: Chiral propulsion of $N$-helicoid
    3. IV.3 Example 3: Chiral propulsion of a modulated cylinder
  5. V Conclusion
  6. A Chiral propulsion of spiral
  7. B Chiral propulsion of $N$-spiral
  8. C A-integrals in the limit of small pitch
  9. D Streamlining the array of parallel wires
    1. D.1 Shear flow parallel to wires
    2. D.2 Shear flow transverse to wires

##  I Introduction

The motion of a solid object in a viscous fluid is conventionally described by
hydrodynamic Navier-Stokes equations with appropriate boundary conditions on
the surface of the object. Typical boundary conditions used are the non-slip
boundary conditions stating that the fluid is not moving relative to the
object’s surface. The problem is notoriously complicated because of the non-
linearity of hydrodynamic equations resulting in turbulence. It is, however,
dramatically simplified at low Reynolds numbers when nonlinear inertial terms
in equations can be neglected. The regime of low Reynolds numbers is very
relevant for micron-scale objects moving in the water and is of great
interest, e.g., for modeling the motion of microorganisms Zhang ; Ghosh ;
Schamel ; Keaveny ; walker ; aif2018chiral ; Purcell . In this regime, the
Navier-Stokes equation is reduced to a linear Stokes equation.

A problem of interest for this work is the transformation of the torque
applied to solid objects into translational motion (propulsion) which occurs
if the chiral symmetry - symmetry between left and right - is broken by the
shape of an object or boundary conditions on its surface. We refer to this
phenomenon as to a chiral propulsion.aif2018chiral  A simple example of a
chiral shape is given by a helicoidal shape characterized by symmetry w.r.t.
the simultaneous rotation by angle $\phi$ and translation by $(\phi/2\pi)d$
along the same axis. We will refer to the length parameter $d$ of the
helicoidal shape as the pitch. It is expected that the application of the
torque along the symmetry axis of a helicoid will result in translational
motion along that axis. It turns out that even for this symmetry, it is hard
to find a non-trivial shape for which Stokes equations can be solved exactly,
and various approximations have been used to find a relation between the
angular velocity and the translational propulsion velocity of a helicoidal
system.

Significant progress can be made for a spiral shape – a helicoidal shape made
out of a thin wire. Lighthill, in his seminal work on a motion of a flagellum
Lighthill  used the cross-section of the flagellum as a small parameter and
related a swimming velocity of an arbitrary shape filament with a constant
cross-section to forces acting on the filament. Singular solutions – known as
Stokeslet and source doublets – placed along the centerline of the filament
produce velocity field satisfying non-slip boundary conditions on the
filament’s surface in the leading order in the diameter of a cross-section.
Using this approximate solution Lighthill was able to compute the velocity of
the propulsion of a spiral made of a very thin wire Lighthill  (see Ref.
Liu_flag,  for review of this and other approaches).

The chiral propulsion has also been computed analytically for few other
geometries (see Ref. li,  for review). For example, the propulsion velocity of
a rotating cylinder with a helical perturbation on its surface was calculated
by using perturbation theory in the amplitude of deviations from a cylindrical
base shape. Such perturbative approach successfully predicts the chiral
propulsion of helicoidal structures with large pitch $d$ but fails in the case
of a small pitch $d\ll R$, where $R$ is the radius of the cylinder.li

This work is based on a simple observation that for a small pitch $d\ll R$,
the outer part of the helicoid looks like a surface of a cylinder modulated at
a scale much smaller than the radius of the cylinder. This suggests replacing
the actual fluid flow outside of that imaginary cylinder with a flow generated
by effective boundary conditions on the corresponding imaginary surface. These
boundary conditions are well-known Navier, or partial slip boundary conditions
(PSBC). We propose to apply the method of effective PSBC to solving chiral
propulsion problems in the limit of small pitch. We will check that the method
produces the same results for the spiral as the ones obtained in Ref.
Lighthill, . The proposed method can be used for other geometries as well.

The paper is organized as follows. In section II we state the chiral
propulsion problem and analyze its geometric nature and symmetries. In section
III we present the main results of the paper and compute the chiral propulsion
using the proposed method of effective PSBC. We discuss possible applications
in section IV and significance of our results in section V. Technical details
are relegated to appendices.

##  II Chiral propulsion of an infinite helicoid

###  II.1 Chiral propulsion of a solid body moving through viscous fluid

Consider a solid body moving in an unbounded incompressible viscous fluid
domain $\mathcal{M}$. We denote the surface of the body by $\Gamma$ . The
fluid incompressibility condition can be written as

$\displaystyle\bm{\nabla}\cdot\bm{v}=0\,,\qquad\bm{x}\in\mathcal{M}\,,$ (1)

where $\bm{v}(\bm{x})$ is the fluid velocity. Since we are interested in the
case of small Reynolds numbers we use the Stokes equation for the velocity of
the fluid

$\displaystyle-{\bm{\nabla}}P+\eta\Delta{\bm{v}}=0\,,\qquad\bm{x}\in\mathcal{M}\,.$
(2)

Here $\eta$ is the shear viscosity of the fluid, and $P$ is the pressure. The
equation (2) is assumed to be valid outside of the domain occupied by the
solid body.

The non-slip boundary conditions on the surface require that the velocity of
the fluid at the surface is equal to the velocity of the surface itself:

$\displaystyle\bm{v}\Big{|}_{\Gamma}\ =\bm{U}+\bm{\Omega}\times\bm{X}\,,$ (3)

Here $\bm{U},\;\bm{\Omega}$ are the velocity and the angular velocity of the
rigid body, respectively. The vector $\bm{X}$ is the position vector of the
element on the surface $\Gamma$. We also require the vanishing of the fluid
velocity at infinity

$\displaystyle\bm{v}(\bm{x})\to 0\,,\quad\mbox{as}\quad\bm{x}\to\infty\,.$ (4)

The hydrodynamic equations (1-4) fully determine the motion of the fluid given
the shape of the solid body $\Gamma$ together with parameters
$\bm{U},\bm{\Omega}$. However, often instead of the velocity $\bm{U}$ and
angular velocity $\bm{\Omega}$ of the body we know the external force
$\bm{F}_{ext}$ and torque $\bm{\tau}_{ext}$ acting on the body. The external
force and torque acting on an object are

$\displaystyle F^{ext}_{i}$ $\displaystyle=-\int_{\Gamma}f_{i}({\bf
X})\,d\sigma\,,$ (5) $\displaystyle\tau^{ext}_{i}$
$\displaystyle=-\epsilon^{ijk}\int_{\Gamma}X_{j}f_{k}({\bf X})\,d\sigma\,,$
(6)

where the integral is taken over the surface of the body, $\bm{X}$ is the
position of the element of the surface. The stress force acting on the element
$d\sigma$ of the surface is defined by the fluid motion and is given by

$\displaystyle f_{i}=-Pn_{i}+\eta(\partial_{i}v_{j}+\partial_{j}v_{i})n_{j},$
(7)

where $\bm{n}$ is a vector outward-normal to the surface. Here and in the
following we will use the Einstein’s convention of summation over repeated
indices.

As the Stokes problem is linear we expect the general linear relationship
between the force and the torque acting on the body and linear velocity and
angular velocity of the body:

$\displaystyle\bm{F}^{ext}/\eta$
$\displaystyle=-K\cdot\bm{U}-C^{\dagger}\cdot\bm{\Omega}\,,$
$\displaystyle\bm{\tau}^{ext}/\eta$
$\displaystyle=-C\cdot\bm{U}-R\cdot\bm{\Omega}\,.$ (8)

Here $K,C,R$ are tensors characterizing the motion of a rigid body in an
unbound fluid which depend only on the geometric shape of the body.Happel1983
111Notice that there is no dependence of $K,C,R$ on the viscosity coefficient,
and these tensors are purely geometric. The Stokes phenomenon can invalidate
this statement. In the presence of the Stokes phenomenon, the inertial terms
of the Navier-Stokes equation cannot be uniformly neglected, and the
dissipative viscosity could appear as a cutoff. $K$ and $R$ are translation
and rotation tensors, respectively. They are both symmetric. $C$ is the so-
called coupling tensor which is not symmetric in general. $C$ represents a
coupling between rotational and translational motion.

The problem of chiral propulsion can then be formulated in the following way.
Consider a body moving through the fluid subject to the condition that the
total external force acting on the body is zero222In a related approach, we
could push the body through liquid with non-zero force and require zero-torque
condition.

$\displaystyle\bm{F}^{ext}=0\,.$ (9)

Then from (8) we immediately find that the velocity $\bm{U}$ and the angular
velocity $\bm{\Omega}$ of the body are related by a linear relation

$\displaystyle\bm{U}=\mathcal{X}\bm{\Omega}\,,\qquad\mathcal{X}=K^{-1}C^{{\dagger}}\,.$
(10)

The tensor $\mathcal{X}_{ij}$ is fully determined by the shape of the solid
body and does not depend on the history of the motion and other parameters
such as shear viscosity $\eta$ as long as we are in the Stokes’ regime. We
refer to the tensor $\mathcal{X}_{ij}$ as the _chiral propulsion tensor_. The
chiral propulsion tensor is generally not symmetric
$\mathcal{X}_{ij}\neq\mathcal{X}_{ji}$.

###  II.2 A helicoidal system

In this work, we are interested in the chiral propulsion of an infinite length
body rotating around its helicoidal symmetry axis $z$. Let us first define
what we mean by the helicoidal shape in general.

Consider a cylindrical coordinate system $(r,\phi,z)$. We refer to the body as
to an infinite length helicoidal system if it possesses mixed translational-
rotational symmetry, i.e., transforms into itself under:

$\displaystyle(r,\phi,z)\to\left(r,\phi+\alpha,z+\frac{\alpha}{2\pi}d\right)\,.$
(11)

Here the angle $\alpha$ is arbitrary, and $d$ is the pitch of the helicoidal
system. 333In mechanics this symmetry results in the conservation of the
combination $\frac{d}{2\pi}P+M=const$, where $P$ is the total momentum of a
system and $M$ is the total angular momentum LANDAU197613 .

The symmetry (11) is also assumed to be a symmetry of the hydrodynamic
boundary conditions on the surface of the helicoidal system. It is lower than
the symmetry of Stokes equations (1,2). In addition to breaking full
translational and full rotational symmetries, a general helicoidal shape also
breaks parity symmetry (reflection with respect to a plane containing
$z$-axis).

In the following, we will also use shapes having an additional discrete
symmetry

$\displaystyle(r,\phi,z)\to\left(r,\phi+\frac{2\pi}{N},z\right)\,.$ (12)

We refer to such shapes as $N$-helicoids. An example of such shape for $N=3$
is shown in Fig.1. Assuming that the typical radius of the $N$-spiral is $R$
we have three length scales in the problem: $R,d,d/N$. It is also convenient
to introduce the distance between spirals $2L$ and the angle $\theta$ formed
by spirals with the axis of the helicoid:

$\displaystyle\frac{L}{\pi
R}=\frac{\cos\theta}{N}\,,\qquad\tan\theta=\frac{2\pi R}{d}\equiv\nu\,.$ (13)

The inverse parameter $\nu^{-1}$ defined in (13) can be regarded as a
dimensionless pitch of the helicoid. A limit of a small pitch corresponds to
$\nu\to\infty$.

It is important to note that in the limit $N\to\infty$ the method of effective
boundary conditions used in this work is exact. Indeed, this method requires
$2L\ll R$. Since $L\sim 1/N$, for sufficiently large $N$ the method applies
for an arbitrary relation between $d$ and $R$.

Figure 1:  Geometry of a $N$-spiral is shown for $N=3$. The angle $\theta$
between spirals and the $z$ axis is related to the pitch $d$ as
$\cot\theta=\frac{d}{2\pi R}$. The distance $2L$ between nearest spirals is
$2L=\frac{d}{N}\sin\theta$.

###  II.3 Chiral propulsion of a helicoid

Another consequence of this symmetry is that if the helicoid’s angular
velocity is aligned with the symmetry axis $z$, the propulsion velocity of the
helicoid is also directed along the $z$-axis. Then instead of the full chiral
propulsion tensor $\mathcal{X}$ one has a _chiral propulsion coefficient_
$\chi$ defined as:

$\displaystyle U_{z}=\chi\,\Omega_{z}\,.$ (14)

The chiral propulsion coefficient is a pseudoscalar changing sign under parity
(reflection with respect to a plane containing $z$-axis) as the velocity
$U_{z}$ and angular velocity $\Omega_{z}$ transform as the components of a
vector and pseudovector, respectively. As both $U_{z}$ and $\Omega_{z}$ are
odd under time-reversal T, the chiral propulsion coefficient $\chi$ must be
T-even and cannot depend linearly on the dissipative T-odd shear viscosity
coefficient $\eta$. This symmetry argument agrees with the cancellation of the
shear viscosity in (10). Therefore, the effect of chiral propulsion is non-
dissipative and geometric in nature.Purcell ; shapere1987self ; Note1

It is hard to find an exact solution for the Stokes problem for a helicoidal
system. Therefore, in the following we use various approximations justified by
the smallness of additional parameters.

In the following we compute the chiral propulsion coefficients for shapes of
helicoidal type using the method of effective boundary conditions. This method
reduces the problem of computation of chiral propulsion for complex surfaces
to the easier one for planar geometries.

##  III Small pitch limit. The method of effective boundary conditions

Let us consider the limit of a small pitch of a helicoid. In this limit we
will substitute the actual surface of the outer boundary of a helicoid by an
effective cylindrical surface with modified boundary conditions. In this
approach the derivation of chiral propulsion is elementary, with the
difficulty shifted to the problem of finding a few parameters of the effective
boundary conditions.

###  III.1 Effective boundary conditions for planar geometry

We start by considering standard non-slip boundary conditions on the $z=0$
surface assuming that the fluid occupies the half-space $z>0$ . These boundary
conditions state that the velocity of the flow is zero on $xy$-plane

$\displaystyle\hat{\bm{z}}\times\bm{v}$ $\displaystyle=0\,,\quad\mbox{at
}z=0\,.$ (15)

Here $\hat{\bm{z}}$ is a unit vector normal to the surface, i.e., in the
positive $z$-direction. In addition, we require that the fluid does not flow
through the surface so that

$\displaystyle\hat{\bm{z}}\cdot\bm{v}$ $\displaystyle=0\,,\quad\mbox{at
}z=0\,.$ (16)

A solution of Stokes equations in the upper half-plane $z>0$ satisfying these
boundary conditions is given by

$\displaystyle\bm{v}$ $\displaystyle=\Omega z\hat{\bm{e}}\,,$ (17)

where $\Omega$ is a constant defining the shear of the flow and $\hat{\bm{e}}$
is a unit vector parallel to the surface $\hat{\bm{e}}\cdot\hat{\bm{n}}=0$.

Let us generalize the non-slip boundary condition (15) to

$\displaystyle v_{i}$ $\displaystyle=\tilde{h}_{ij}\sigma_{jz}\quad\mbox{at
}z=0\,,\quad\mbox{for }i,j=x,y\,.$ (18)

Here $\sigma_{jz}=\eta v_{jz}$ is a viscous shear stress tensor, $\eta$ is a
shear viscosity and $v_{ij}$ is the strain rate defined by

$\displaystyle v_{ij}$ $\displaystyle=\partial_{i}v_{j}+\partial_{j}v_{i}\,.$
(19)

It is convenient to absorb the shear viscosity coefficient $\eta$ into the
definition of the matrix $h_{ij}=\eta\tilde{h}_{ij}$ and write

$\displaystyle v_{i}$ $\displaystyle=h_{ij}v_{jz}\quad\mbox{at
}z=0\,,\quad\mbox{for }i,j=x,y\,.$ (20)

One can easily recognize in (18) the tensorial form bazant_vinogradova_2008
of Navier’s partial slip boundary conditions (PSBC). navier1827lois  The
matrix elements $h_{ij}$ have dimensions of length defining the so-called slip
lengths for a general case of an anisotropic surface. Physically, PSBC mean
that there is a partial slip, i.e., finite velocity of the fluid along the
surface proportional to the shear stress at the surface. The tensor of the
second rank $h_{ij}$ is defined by the properties of the surface.

Assuming that $h_{ij}$ is constant along the surface $z=0$ one can easily find
the shear flow at $z>0$ satisfying (18). Namely, the solution for the flow is
given by

$\displaystyle v_{z}$ $\displaystyle=0\,,$ (21) $\displaystyle v_{i}$
$\displaystyle=\Omega(z\hat{e}_{i}+h_{ij}\hat{e}_{j})\,,$ (22)

where $\hat{e}_{i}$ are the components of a unit vector $\hat{\bm{e}}$ tangent
to the surface. As it is seen from (22), for a given direction $\hat{\bm{e}}$
of a shear flow at $z\to\infty$ there is a chiral (left or right) component of
velocity perpendicular to the direction of the shear. This component $v_{ch}$
is given by

$\displaystyle\chi$
$\displaystyle=\frac{v_{ch}}{\Omega}=\hat{e}_{i}\epsilon_{ij}h_{jk}\hat{e}_{k}\,.$
(23)

We will refer to the coefficient $\chi$ as to a chiral propulsion coefficient
in planar geometry. For a generic two by two real matrix $\hat{h}$ one can
always find a reference frame such that the matrix has a form

$\displaystyle\hat{h}$ $\displaystyle=\left(\begin{array}[]{cc}h_{\perp}&-h\\\
h&h_{\parallel}\end{array}\right)\,.$ (26)

In this (principal) reference frame for a shear direction
$\hat{\bm{e}}=(\cos\theta,\sin\theta)$ we find from (23) the chiral propulsion
coefficient

$\displaystyle\chi$ $\displaystyle=h+\frac{h_{\parallel}-h_{\perp}}{2}\sin
2\theta\,.$ (27)

The matrix of effective slipping lengths (26) is also known as an interfacial
mobility tensor. For passive surfaces, it has to be positive definite as a
requirement that the surface does not transfer energy to the fluid, i.e.,
$h_{\parallel},h_{\perp}\geq 0$ and $h^{2}\leq h_{\perp}h_{\parallel}$. We
refer the reader to Refs. bazant_vinogradova_2008, ; kamrin,  for further
details on the properties and applications of the tensorial slip boundary
conditions.

We note that under the reflection with respect to, say, “parallel” axis
$\theta\to-\theta$ we have $\chi\to-\chi$ and $h\to-h$, while
$h_{\parallel,\perp}\to h_{\parallel,\perp}$. This means that the parameter
$h$ of the Navier’s tensor $h_{ij}$ is odd under parity and represents the
parity violating parameters of the surface.

For example, the plane modulated by identical grooves parallel to each other
has a reflection symmetry with respect to at least one axis and, therefore,
$h=0$ for that surface. In all examples considered in Section IV, $h=0$;
however, it is not prohibited to have non-zero $h$, and we keep it in our main
equations for generality.

In the following, we will use the PSBC (18) as an effective boundary condition
in the following way. Imagine a plane with grooves or some other modulations
of its surface. Even if the actual boundary conditions at the surface are non-
slip, at a distance $z$ from the surface such that $z$ is much larger than the
characteristic length of surface modulations (e.g., the distance between
grooves), the flow asymptotically takes the form of a shear444The next
correction to the flow at large distances from the surface is expected to
decay as $\exp(-kz)$, where $k=$ is a primary wavevector of the surface
modulation. (22). This means that at large distances, the flow perturbed by
the surface can be replaced by the one solving Stokes equation with effective
PSBC (18). The actual shape of the surface will determine the parameters of
the $h$-matrix $h_{\parallel},\;h_{\perp},\;h$.

Before computing the PSBC parameters for few surfaces, let us present a
computation of the chiral propulsion of the cylinder surface with PSBC (18).

###  III.2  Chiral propulsion of a cylinder with Navier boundary conditions

Let us consider an infinite cylinder of radius $R$ having PSBC (18) on its
surface. We assume that the cylinder rotates around its axis coinciding with
the coordinate axis $z$ with angular velocity $\Omega$ and propels with
velocity $U$ along the $z$-axis. We also assume that the effective matrix
$h_{ij}$ is constant at the surface of the cylinder and has the form (26) in
the local $xy$ coordinate system which is rotated by the angle $\theta$
clockwise with respect to the frame given by unit vectors
$\hat{\bm{\phi}},\hat{\bm{z}}$ of cylindrical coordinate system $r,\phi,z$.
Rotating (26) by an angle $\theta$ we obtain

$\displaystyle\hat{h}$
$\displaystyle=\left(\begin{array}[]{cc}h_{\phi\phi}&h_{\phi z}\\\
h_{z\phi}&h_{zz}\end{array}\right)=\left(\begin{array}[]{cc}h_{\perp}\cos^{2}\theta+h_{\parallel}\sin^{2}\theta&-h+\frac{h_{\parallel}-h_{\perp}}{2}\sin(2\theta)\\\
h+\frac{h_{\parallel}-h_{\perp}}{2}\sin(2\theta)&h_{\perp}\sin^{2}\theta+h_{\parallel}\cos^{2}\theta\end{array}\right)\,.$
(32)

On the surface of the cylinder we have from (18)

$\displaystyle({\bm{v}}-\Omega
R\hat{\bm{\phi}}-U\hat{\bm{z}})_{i}=h_{ij}\eta^{-1}\sigma_{jr}\,.$ (33)

Here, in the l.h.s., we have a fluid velocity relative to the surface of the
cylinder; the indices take values $i,j=\phi,z$ and the normal direction to the
surface of the cylinder is $r$. The flow outside of the cylinder $r>R$ is
given by the solution of the Stokes equation as:

$\displaystyle v_{r}=0\,,\qquad v_{\phi}=\frac{\Gamma}{2\pi r}\,,\qquad
v_{z}=0\,.$ (34)

Using the expression for the stress tensor in cylindrical coordinates LL6

$\displaystyle\sigma_{\phi r}$
$\displaystyle=\eta\left(\frac{1}{r}\frac{\partial
v_{r}}{\partial\phi}+\frac{\partial v_{\phi}}{\partial
r}-\frac{v_{\phi}}{r}\right)\,,$ (35) $\displaystyle\sigma_{zr}$
$\displaystyle=\eta\left(\frac{\partial v_{z}}{\partial r}+\frac{\partial
v_{r}}{\partial z}\right)\,,$ (36) $\displaystyle\sigma_{zz}$
$\displaystyle=2\eta\frac{\partial v_{z}}{\partial z}\,,$ (37)

we obtain from (33)

$\displaystyle\frac{\Gamma}{2\pi R}-\Omega R$
$\displaystyle=-h_{\phi\phi}\frac{\Gamma}{\pi R^{2}}\,,$ (38)
$\displaystyle-U$ $\displaystyle=-h_{z\phi}\frac{\Gamma}{\pi R^{2}}\,.$ (39)

This system gives both the chiral propulsion velocity $U$ and the circulation
of the flow at infinity $\Gamma$ in terms of the angular velocity $\Omega$:

$\displaystyle\frac{U}{\Omega R}$
$\displaystyle=\frac{2h_{z\phi}}{R+2h_{\phi\phi}}\,,$ (40)
$\displaystyle\frac{\Gamma}{\Omega R^{2}}$ $\displaystyle=\frac{2\pi
R}{R+2h_{\phi\phi}}\,.$ (41)

The relation (40) and values (32) give for the chiral propulsion coefficient
defined by (14) the expression:

$\displaystyle\frac{\chi}{R}$
$\displaystyle=\frac{2h+(h_{\parallel}-h_{\perp})\sin
2\theta}{R+(h_{\parallel}+h_{\perp})-(h_{\parallel}-h_{\perp})\cos
2\theta}\,.$ (42)

This expression is the main result of this work. It gives the chiral
propulsion coefficient of a cylinder in terms of parameters of Navier PSBC. In
the following we apply (42) to compute the chiral propulsion coefficient for a
few particular helicoidal systems.

The chiral propulsion coefficient given by (42) is exact for a cylinder with
Navier boundary conditions. However, to use these formulas for the realistic
surface with effective Navier boundary conditions, the slip lengths $h_{ij}$
and the period of modulation $2L$ must be much less than the typical radius of
the cylinder, i.e.,

$\displaystyle h_{\perp},\;h_{\parallel},\;h,\;2L\ll R\,.$ (43)

##  IV Examples and applications

In this section, we present several applications of (42) to the computation of
the chiral propulsion coefficient for a few helicoidal shapes.

###  IV.1 Example 1: Chiral propulsion of $N$-spiral

Let us start with the geometry of a thin spiral and its generalizations. The
chiral propulsion coefficient of an infinite spiral made out of a thin wire
was computed by Lighthill Lighthill  using slender-body theory (see also Ref.
Liu_flag, ). In this section, we use a similar technique combined with the
method of effective boundary conditions to compute the chiral propulsion
coefficient and compare it with the one of Ref. Lighthill, .

We consider the array of thin parallel wires placed in the shear flow shown in
Fig. 2. The diameter of the wire $2a$ is assumed to be much smaller than the
inter-wire distance $2a\ll 2L$. We derive the following values for Navier
parameters (see Appendix D for details):

$\displaystyle\frac{h_{\perp}}{L}$
$\displaystyle=\frac{1}{2\pi}\left(-\frac{1}{2}+\ln{\frac{L}{\pi
a}}\right)+O\left(\frac{a^{2}}{L^{2}}\right)\,,$ (44)
$\displaystyle\frac{h_{\parallel}}{L}$
$\displaystyle=\frac{1}{\pi}\ln{\frac{L}{\pi
a}}+O\left(\frac{a^{2}}{L^{2}}\right)\,,\qquad h=0\,.$

Figure 2:  An array of parallel wires with radius $a$ and the period $2L$. The
red arrow points in the $\theta$ direction.

Let us now consider a slight generalization of a spiral shape which we will
refer to as $N$-spiral (see Fig. 1). This shape is made out of $N$ infinite
thin wires of radius $a$ with their centerlines satisfying:

$\displaystyle r$
$\displaystyle=R\,,\quad\frac{z}{d}=\frac{\phi}{2\pi}+\frac{k}{N}\,,\quad
k=0,1,\ldots,N-1\,.$ (45)

Locally, the surface of the $N$-spiral looks like an array of parallel wires.
Assuming that the distance between wires is small, one can replace the
$N$-spiral by a cylinder with Navier boundary conditions on its surface with
parameters (44) (see Appendix D for the derivation). For the chiral propulsion
of such a cylinder we obtain from (42,44) using (13):

$\displaystyle N\frac{\chi_{N}}{R}$
$\displaystyle=\frac{\left(\frac{1}{2}+\ln{\frac{L}{\pi
a}}\right)\cos\theta\sin 2\theta}{2-\frac{L}{2\pi R}(1+\cos
2\theta)+\frac{L}{\pi R}\ln{\frac{L}{\pi a}}(3-\cos 2\theta)}\,.$ (46)

Here $\chi_{N}$ is the chiral propulsion coefficient for $N$-spiral. There
were two essential approximations used in deriving (46). One is of the order
of $a^{2}/L^{2}$ when deriving the parameters (44) for Navier boundary
conditions. Another is the replacement of the spiral by the cylinder with
effective slip boundary conditions. One expects the error of the latter
approximation to scale as $h^{\perp,\parallel}/R$ or $L/R$.

For $N=1$ the result (46) should be valid for $L/(\pi a)\gg 1$ and
sufficiently small pitch $\nu\gg\ln\frac{L}{\pi a}$. In this limit
$\theta\approx\frac{\pi}{2}-\frac{1}{\nu}$ and we further approximate (46) as:

$\displaystyle\frac{\chi_{1}}{R}\approx\frac{L^{2}}{\pi^{2}R^{2}}\frac{\ln{(\frac{L}{\pi
a})}+0.5}{1+\frac{2L}{\pi R}\ln{(\frac{L}{\pi a}})}\,.$ (47)

The Lighthill’s result (65) computed in the limit of a small pitch using
(83-85) gives

$\displaystyle\frac{\chi_{1}}{R}\approx\frac{L^{2}}{\pi^{2}R^{2}}\frac{\ln{(\frac{L}{\pi
a})}+0.5}{1+\frac{2L}{\pi R}\ln{(\frac{L}{\pi a}})}\left(1-\frac{L}{\pi
R}\right)\,.$ (48)

It coincides with (47) to the leading order in $L/R$ as expected.

Another interesting limit allowing for some analytic control is the limit of
large $N$ for the propulsion of $N$-spiral. From the condition (43) and (44)
it follows that for sufficiently large $N\gg\frac{1}{2}\ln\frac{L}{\pi a}$ the
result (46) should be valid for all pitches $\nu$ (i.e. for all angles
$\theta$).

Generalizing Lighthill approach we derive the chiral propulsion for the
$N$-spiral (Appendix B) without the use of effective boundary conditions and
obtain (74). This result should be valid for $\pi a\ll L$. On the other hand,
for sufficiently large $N$ the effective boundary condition result (46)
becomes applicable. This allows for a direct comparison of (74) and (46).

We plot the chiral propulsion given by (46) (solid curves) and by the
Lighthill-type formula (74) (dots) as a function of the pitch parametrized by
$\theta$ and at fixed $a/R=0.01$ in Fig. 3. We see a very good agreement
between the formulas already at $N=5$. When the pitch becomes very small,
i.e., $\theta\to\pi/2$ the distance between wire centers $2L$ becomes
comparable with wire diameters $2a$ and one should not rely on either of
expressions (46,74).

Figure 3:  The chiral propulsion coefficient for $N$-spiral with
$N=1,5,10,15,20$ and thickness $\frac{a}{R}=0.01$ is shown as
$N\frac{\chi}{R}$ vs. $\theta$. The solid curves are given by (46), the dots
are given by (74) multiplied by N, the straight line indicates the pitch when
the distance between wire centers $2L$ becomes twice their diameter $2a$ for
N=20.

###  IV.2 Example 2: Chiral propulsion of $N$-helicoid

To the best of our knowledge, the closed-form solution of the chiral
propulsion velocity of an infinite helicoid is not known. While it is
technically possible to derive such a formula using the slender body theory,
we do not attempt it in this work. Instead, we use the proposed method of
effective boundary conditions to compute the chiral propulsion of a helicoid
with a small pitch. We present it here as a particular case of a chiral
propulsion velocity of $N$-helicoid, a generalization that is similar to the
one we made for a spiral in the previous section. We refer to the shape given
in cylindrical coordinates by

$\displaystyle r$ $\displaystyle\leq
R\,,\quad\frac{z}{d}=\frac{\phi}{2\pi}+\frac{k}{N}\,,\quad
k=0,1,\ldots,N-1\,.$ (49)

as $N$-helicoid. Here $\phi\in(-\infty,+\infty)$. The outer boundary of the
$N$-helicoid (at $r=R$) is the previously introduced $N$-spiral (45). The
shape (49) has vanishing thickness and is essentially a surface. Locally a
helicoid with a small pitch looks like an array of protruding plates (see Fig.
4).

Figure 4: An array of protruding plates. Each plate is specified by $z\leq 0$
and $x=2Ln$, $n\in\mathbb{N}$. The thickness of plates is assumed to be
vanishing.

A planar geometry problem for this case is to find $h_{\perp},h_{\parallel}$
for the system of semi-infinite plates given by equations

$\displaystyle\Gamma=\Big{\\{}z\leq 0,\;x=2Ln,\;\;n=0,\pm 1...\Big{\\}}\,.$
(50)

Exact values of $h_{\perp},h_{\parallel}$ for this geometry can be found in
literature davis ; luchini  (see, e.g., equations 14 and 25 of Ref. luchini
.). Here we give numerical values:

$\displaystyle h_{\perp}$ $\displaystyle\approx 0.17L\,,\qquad
h_{\parallel}\approx 0.44L\,,\qquad h=0\,.$ (51)

Replacing N-helicoid by a cylinder with Navier boundary conditions on its
surface we compute the chiral propulsion coefficient using (42) and that
$2L=\frac{2\pi R}{N}\cos{\theta}$:

$\displaystyle N\frac{\chi_{N-helicoid}}{R}$
$\displaystyle\approx\frac{0.85\cos\theta\sin{2\theta}}{1+N^{-1}\cos\theta(1.92-0.85\cos{2\theta})}\,.$
(52)

The applicability of (52) is limited by the condition that both the distance
between plates $2L$ and the corresponding slipping lengths
$h_{\parallel,\perp}\sim L$ are much smaller than the radius of the helicoid.
In this case the condition is simply $L\ll R$ or equivalently $\cos\theta/N\ll
1$.

In the limit of large $N$, neglecting subleading $1/N$ corrections, we obtain

$\displaystyle N\frac{\chi_{N-helicoid}}{R}\approx\,0.85\cos{\theta}\sin
2\theta\,,\qquad\cos{\theta}\ll N\,.$ (53)

The comparison of this formula with the more precise expression given by (52)
is presented in Fig. 5. For $N=20$ the slip lengths are about 10$\%$ of the
radius and the chiral propulsion is within 10$\%$ of the universal formula
(53).

Remarkably, the angle dependence of the chiral propulsion of the $N$-helicoid
in the considered regime $L\ll R$ is universal and is given by Eq. 53 (shown
as a dashed red curve in Fig. 5). It shows a maximum at
$\theta_{m}=\tan^{-1}(1/\sqrt{2})\approx 35.26^{\circ}\approx 0.196\pi$. This
result is expected to be universal for a broad class of helical surfaces. The
conditions for this are the applicability of the method of boundary conditions
(surface details should be smaller than the object’s radius) and that both
$h_{\perp}$ and $h_{\parallel}$ are of the same order determined by the scale
$L$. In this case, only the numeric coefficient in (53) is surface-dependent
in the leading order in $h_{\perp}/R$.

For the propulsion velocity of a helicoid (N=1) with a small pitch we obtain:

$\displaystyle\frac{\chi_{helicoid}}{R}\approx
0.02\frac{d^{2}}{R^{2}}\,,\qquad d\ll R\,.$ (54)

Figure 5:  The dependence of chiral propulsion coefficient for $N$-helicoid is
shown as $N\frac{\chi}{R}$ vs. $\theta$ for N=5 and 20. The solid curves are
given by (52). The dashed curve is given by (53).

Available experimental results for micro-swimmers with corkscrew shapes of
various lengthswalker ; Schamel  exhibit velocity saturation when the length
of swimmers becomes equal to few helical turns, so we can assume them to be of
infinite length. However, pitches of the micro-swimmers in those experiments
are typically larger than a helical radius, and our method does not apply. A
naive application of Eq. (52) to geometries considered in Refs. walker, ;
Schamel,  produces chiral propulsion which is about twice the observed values.
This disagreement is not surprising, as for large pitches considered in those
references, the approximation of a constant force acting on their
surfaceaif2018chiral  might be more applicable. The method of effective
boundary conditions presented here is applicable only when force is
concentrated at the edges of a helicoidal swimmer.

###  IV.3 Example 3: Chiral propulsion of a modulated cylinder

Now we consider the propulsion of a helical cylinder which cross section
slightly deviates from a circle. The surface equation in cylindrical
coordinates is given by

$\displaystyle r(\phi,z)=R+Hf(\phi-2\pi z/d)\,,$ (55)

where $f(\phi)$ is a $2\pi$-periodic function and the dimensionless
deformation amplitude $\varepsilon\equiv H/R\ll 1$ quantifies the smallness of
the deviation of the cross section from the circle.

When the pitch $d$ is much smaller than the radius or $f(\phi)$ has a period
$2\pi/N$ with $N\gg 1$ the surface can be locally viewed as a periodically
modulated plane with the height profile $y(x,z)=Hf((x-\nu z)R^{-1})$ in planar
geometry. Here the coordinates $x,y,z$ correspond to the directions $\phi,r,z$
in cylindrical geometry, respectively. For slightly deformed surfaces with an
amplitude $\varepsilon$ and a period much larger than the amplitude
$2\pi\nu^{-1}\gg\varepsilon$, we can use a domain perturbation theory and
represent the velocity field as series in $\varepsilon$. Then the solution for
velocity satisfying no-slip boundary conditions on the surface is obtained in
each order in $\varepsilon$. In the second order in $\varepsilon$ (or $H$),
the slipping lengths are given by luchini :

$\displaystyle h_{\perp}$
$\displaystyle=2h_{\parallel}=-4\frac{H^{2}}{R\cos\theta}\sum_{n=1}^{\infty}n|f_{n}|^{2}\,,\quad
h=0\,,$ (56)

where the Fourier coefficients of the modulation $f_{n}$ are defined as
$f(\phi)=\sum_{n}f_{n}e^{in\phi}$, and we assumed $f_{0}=0$ without any loss
of generality. When comparing (56) with the results of Refs. luchini ; kamrin
one should keep in mind that the wavevectors of the modulated plane are
$k_{n}=n/(R\cos\theta)$ with $\tan\theta=\nu$.

The relation $h_{\perp}=2h_{\parallel}$ in (56) is specific for the second
order in the amplitude of the modulation—the higher-order terms in amplitude
break this relation. One might also expect the generation of the anti-
symmetric part $h$ of the slip length matrix. However, it does not happen, at
least in the third and fourth orders of the perturbation theory.

Substituting (56) into (42) we obtain the chiral propulsion coefficient in the
leading order in $h_{\parallel,\perp}/R$

$\displaystyle\frac{\chi}{R}\approx(h_{\parallel}-h_{\perp})\sin{2\theta}=4\varepsilon^{2}\frac{\nu}{\sqrt{\nu^{2}+1}}\sum_{n=1}^{\infty}n|f_{n}|^{2}\,.$
(57)

This result reproduces in the limit of small pitch $\nu\gg 1$ the result of
the computation by Li and Spagnolie li  made directly for the cylinder
geometry:

$\displaystyle\frac{\chi_{L-S}}{R}\approx\left(1-\frac{3}{2\nu}\right)4\varepsilon^{2}\sum_{n=1}^{\infty}n|f_{n}|^{2}\,.$
(58)

Similarly to the case of the $N$-spiral considered in Section IV.1, the result
(57) is based on two approximations. One is the basic approximation of the
method of effective boundary conditions, namely $L\ll R$. The accuracy of the
method is expected to be of the order of $L/R\sim 1/\nu$ and is supported by
the comparison of (57) and (58). Another approximation involves the
computation of slipping lengths for a given surface. Both (57) and (58) use
the expansion in the amplitude of the modulation $H=\varepsilon R$ up to the
second order. However, the actual dimensionless small parameter of the
expansion is $H/L\sim N\varepsilon\nu$, where $2L$ is the wavelength of the
modulation. li

It is interesting to compare the numerical results obtained in Ref. li,  to
the prediction of the universal formula (42). The largest values $\nu=6$ and
$N=4$ used in Ref. li,  give $L/R\approx 1/8$ and make (42) applicable.
However, the value $\varepsilon=0.1$ used in that reference for numerical
computations translates into $H/L\sim N\varepsilon\nu\approx 2$, which is not
small at all. Indeed, it was noticed in Ref. li,  that the results of the
perturbation theory extended to $\varepsilon^{4}$ while improving the
agreement with numerics at small $\nu$ fail to describe the data at large
$\nu$.

In the following, we compare the numerics of Ref. li,  with the predictions of
the universal formula (42) used with the slipping lengths obtained numerically
for planar geometry in Ref. luchini, .

Let us consider a simple cosine profile with amplitude $H$ and the wavelength
$2L$, namely, $y(x,z)=H\cos(\pi x/L)$. On dimensional grounds we expect

$\displaystyle\frac{h_{\parallel}}{2L}$
$\displaystyle=-g_{\parallel}\left(s\right)\,,\qquad\frac{h_{\perp}}{2L}=-g_{\perp}\left(s\right)\,,$
(59)

where $s=H/L$ is the dimensionless amplitude of the modulation. The
asymptotics of the introduced functions $g_{\parallel,\perp}$ are given by

$\displaystyle g_{\parallel}(s)$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{\pi s^{2}}{4}&\mbox{for }s\ll
1\,,\\\ \frac{s}{2}-0.22&\mbox{for }s\gg 1\,,\end{array}\right.\qquad
g_{\perp}(s)=\left\\{\begin{array}[]{ll}\frac{\pi s^{2}}{2}&\mbox{for }s\ll
1\,,\\\ \frac{s}{2}-0.085&\mbox{for }s\gg 1\,.\end{array}\right.$ (64)

Here the asymptotic at small $s$ is computed using the domain perturbation
theory as in (56) (see Refs. luchini, ; kamrin, ). The large $s$ limit
corresponds to infinitely deep grooves $H\gg L$. In this limit, one can think
of cosine modulation as a sequence of equally separated barriers and compute
the asymptotics borrowing the result (51) corrected by the shift of the origin
by $H$ to the maximums of barriers. luchini  In the limit of small pitch or
large $N$ the small $s$ asymptotics (64) used in the formula (42) should give
a satisfactory result for chiral propulsion. However, it turns out that this
approximation is not sufficient for the modulation amplitude values used in
Ref. li  and one needs more precise expressions for $g_{\parallel,\perp}$
interpolating between small and large $s$.

Remarkably the functions $\bar{h}_{\parallel,\perp}\equiv
s/2-g_{\parallel,\perp}(s)$ have been computed numerically and plotted for
$s<1$ in Ref. luchini,  (see Fig. 10 in that reference). We have extracted the
functions $g_{\parallel,\perp}$ from the plot in Ref. luchini,  and used them
as $h_{\parallel}=-2Lg_{\parallel}(\varepsilon R/L)$ with $2L=2\pi
R\cos\theta/N$ and similarly for $h_{\perp}$ in our formula (42). The
resulting plot of $\chi/R$ as a function of $\nu$ is shown for
$\varepsilon=0.1$ and $N=4$ in Fig. 6 on top of the numeric data extracted
from Fig. 4b of Ref. li, .

Figure 6: The dependence of chiral propulsion velocity is shown as
$\frac{\chi}{\varepsilon^{2}R}$ vs. $\nu$, where $\varepsilon=H/R=0.1$. The
curve is given by (42) for the surface $f(\phi)=\cos({4\phi)}$. The slipping
lengths $h_{\parallel,\perp}$ have been extracted from the Fig. 10 of Ref.
luchini . The diamonds represent the $N=4$ data from the Figure 4b of Ref. li,
.

Comparing Fig. 6 with Fig. 4b of Ref. li  we see that the solid curve produced
by the method of effective boundary conditions (42) is in good agreement with
numerical data for chiral propulsion in the whole range of $\nu$ for the
modulation given by $f(\phi)=\cos N\phi$ with $N=4$. This agreement is to be
compared with the Fig. 4b of Ref. li  where it was shown that the small
amplitude approximation fails in the small pitch (large $\nu$) regime.

We expect that (42) becomes asymptotically exact for $N\to\infty$. In this
limit the curve will be taking a shape of a universal curve
$\chi\sim\cos\theta\sin{2\theta}\sim\nu(\nu^{2}+1)^{-3/2}$ with the maximum at
$\nu_{m}=\tan\theta_{m}=1/\sqrt{2}$ corresponding to $\theta_{m}\approx
35.26^{\circ}$. At $N=4$ the curve $\chi(\nu)$ shown in Fig. 6 is still far
from that universal limit and exhibits maximum at $\nu\approx 1.2$ or
$\theta\approx 50^{\circ}$.

##  V Conclusion

To summarize, we applied the method of effective boundary conditions to the
chiral propulsion of helical objects. The method is applicable in the limit of
small pitch. In this limit, the chiral propulsion coefficient $\chi$ defined
in Eq. (14) depends only on three slipping lengths entering as parameters
$h_{\parallel},h_{\perp},h$ (32) of the effective Navier boundary conditions
(33), or equivalently, the surface mobility tensor. The main result of this
work is the formula (42) giving the chiral propulsion in terms of these
parameters.

The result is universal and is valid for any object of helical symmetry in the
limit of small pitch. More precisely, the method is self-consistent when the
effective slipping lengths and surface modulation scale are much smaller than
the typical radius of the helical object. The surface-specific $h$-parameters
can be extracted from the solution of a simpler problem of shear flow over the
surface, i.e., in the half-space geometry. We applied this method to compute
chiral propulsion coefficients for $N$-spiral, $N$-helicoid, and a cylinder
with a modulated surface in Section IV with more technical parts relegated to
appendices.

We have established that for a broad class of helical surfaces (see the
discussion after Eq. 53), the dependence of the chiral propulsion on the
helical angle $\theta$ is universal, $\chi\sim\cos\theta\sin 2\theta$ with the
maximal propulsion achieved at the universal angle
$\theta_{m}=\tan^{-1}(1/\sqrt{2})\approx 35.26^{\circ}$.

For the $N$-spiral shape, the chiral propulsion coefficient can be computed
using the Lighthill approach. The original formula for a spiral with $N=1$ was
derived in Ref. Lighthill,  (see Appendix A). We presented its straightforward
generalization for general $N$ in the Appendix B. We showed that in the limit
of the small pitch, the universal formula (42) agrees with the more precise,
surface-specific results (65,74).

We used the existing numerical data and experiments performed for the
propulsion of helical objects with a small pitch necessary to check the
proposed chiral propulsion formula (42). The parameters entering (42) can be
either independently computed or measured for a flow above the surface
modulated in the same way as the surface of the helical object. In the small
pitch regime, our universal formula (42) is in agreement with Lighthill’s
results (see Section IV.1) that have been thoroughly tested. Liu_flag  We also
found a very good agreement of (42) with the numerical data of Ref. li,  as
discussed in Section IV.3.

Acknowledgements: The authors thank Gregory Falkovich for useful remarks. This
work was supported by the US Department of Energy, Office of Science, Basic
Energy Sciences, under Award No. US DOE DESC-0017662 (LK,DEK,AGA) and by the
US National Science Foundation Grant NSF DMR-1606591 (AGA).

##  Appendix A Chiral propulsion of spiral

In his seminal paper Lighthill Lighthill  represented a velocity field
produced by a moving thin filament as an array of Stokeslets and source
doublets. He applied this general method to derive the chiral propulsion of
the spiral of radius $R$ and pitch $d$ defined by
$\mathbf{r}(\phi)=R(\cos{\phi},\sin{\phi},\epsilon\phi)$,
$\phi\in(-\infty,+\infty)$ with $\epsilon=\frac{d}{2\pi R}=\cot\theta$ and
obtained

$\displaystyle\frac{\chi_{1}}{R}$
$\displaystyle=\frac{-\cos{\theta}\sin{\theta}+\cos{\theta}\csc^{2}{\theta}A_{1}}{\cos^{2}{\theta}+\csc{\theta}(A_{2}+A_{3})},$
(65)

where $A_{1,2,3}$ following Ref. Lighthill,  are given by

$\displaystyle A_{1}$
$\displaystyle=\int_{\phi_{c}}^{\infty}d\phi\,\phi\sin\phi\,\xi^{-3/2}\,,$
(66) $\displaystyle A_{2}$
$\displaystyle=\int_{\phi_{c}}^{\infty}d\phi\,\sin^{2}\phi\,\xi^{-3/2}\,,$
(67) $\displaystyle A_{3}$
$\displaystyle=\int_{\phi_{c}}^{\infty}d\phi\,\cos\phi\,\xi^{-1/2}\,,$ (68)

and

$\displaystyle\xi$
$\displaystyle=\epsilon^{2}\phi^{2}+4\sin^{2}\frac{\phi}{2}\,.$ (69)

These integrals are not independent but related by the following identity (can
be easily derived by integrating (68) by parts)

$\displaystyle\epsilon^{2}A_{1}=A_{3}-A_{2}+\frac{\sin{\phi_{c}}}{\sqrt{4\sin^{2}{\frac{\phi_{c}}{2}}+\epsilon^{2}\phi_{c}^{2}}}\,.$
(70)

The short distance cutoff used in the definition of integrals in (66-68) is
given by

$\displaystyle\phi_{c}$ $\displaystyle=\frac{a\pi\sqrt{e}}{2\pi
R\sqrt{1+\epsilon^{2}}}\,.$ (71)

Here $e=2.718\ldots$ not to be confused with the parameter $\epsilon$. For a
single spiral we use $2L=2\pi R\cos\theta$ as a shortest distance between
neighboring turns and $\epsilon=\cot\theta$ to rewrite conveniently

$\displaystyle\phi_{c}$ $\displaystyle=\frac{\pi
a}{L}\frac{\sqrt{e}}{2}\frac{\epsilon}{1+\epsilon^{2}}\,.$ (72)

We compute the Lighthill integrals (66-68) in the limit of a small pitch
$\epsilon\ll 1$ in Appendix C. Using the results (83-85) we obtain for the
chiral propulsion (65)

$\displaystyle\frac{\chi_{1}}{R}$ $\displaystyle\approx\
\epsilon^{2}\frac{1+\ln(\frac{\epsilon}{2\phi_{c}})}{1+2\epsilon\ln(\frac{\epsilon}{2\phi_{c}})}\left(1-\epsilon+\frac{3\epsilon^{2}}{2}\frac{\ln{\phi_{c}}}{1+\ln{\frac{\epsilon}{2\phi_{c}}}}\right)$
(73)

##  Appendix B Chiral propulsion of $N$-spiral

It is straightforward to generalize the Lighthill’s result (65) for the
propulsion of the $N$-spiral – the shape of a thin filament defined by its
centerline
$\vec{r}(\phi)=R\left(\cos{\phi},\sin{\phi},\epsilon\left(\phi-\phi_{n}\right)\right)$
where $n=0,...,N-1,$, $\phi_{n}=2\pi\frac{n}{N}$, and
$\phi\in(-\infty,+\infty)$. The $N$-spiral consists of $N$ identical copies of
a single spiral arranged equidistantly along the common $z$-axis. Simply
applying the Lighthill’s derivation of (65) for to the $N$-spiral we obtain

$\displaystyle\frac{\chi_{N}}{R}=\frac{-\cos{\theta}\sin{\theta}+\cos{\theta}\csc^{2}{\theta}\sum_{n=0}^{N-1}B_{1,n}}{\cos^{2}{\theta}+\csc{\theta}\sum_{n=0}^{N-1}(B_{2,n}+B_{3,n})}\,,$
(74)

where $n=1,2,\ldots,N-1$,

$\displaystyle B_{1,n}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}d\phi\,\phi\sin\phi\,\xi_{n}^{-3/2}\,,$
(75) $\displaystyle B_{2,n}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}d\phi\,\sin^{2}\phi\,\xi_{n}^{-3/2}\,,$
(76) $\displaystyle B_{3,n}$
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\infty}d\phi\,\cos\phi\,\xi_{n}^{-1/2}\,,$
(77)

and

$\displaystyle\xi_{n}$
$\displaystyle=\epsilon^{2}(\phi-\phi_{n})^{2}+4\sin^{2}\frac{\phi}{2}\,,$
$\displaystyle\phi_{n}$ $\displaystyle=2\pi\frac{n}{N}\,.$ (78)

We assume here that by definition $B_{1,0}=A_{1}$, $B_{2,0}=A_{2}$, and
$B_{3,0}=A_{3}$ with the same cutoff scale $\phi_{c}$ as in (71).

##  Appendix C A-integrals in the limit of small pitch

For $A_{1}$ integral we have

$\displaystyle A_{1}$
$\displaystyle=\int_{\phi_{c}}^{+\infty}d\phi\,\frac{\phi\sin\phi}{\left(4\sin^{2}\frac{\phi}{2}+\epsilon^{2}\phi^{2}\right)^{3/2}}=A_{1}^{\phi_{c}}+A_{1}^{\prime}\,,$
(79)

where for $\epsilon\ll 1$

$\displaystyle A_{1}^{\phi_{c}}$
$\displaystyle=\int_{\phi_{c}}^{\pi}d\phi\,\frac{\phi\sin\phi}{\left(4\sin^{2}\frac{\phi}{2}+\epsilon^{2}\phi^{2}\right)^{3/2}}=\frac{1}{(1+\epsilon^{2})^{3/2}}\ln\frac{\pi}{\phi_{c}}+1-\frac{\pi}{2}+\ln\frac{4}{\pi}+\ldots$
$\displaystyle=1-\frac{\pi}{2}+\ln\frac{4}{\phi_{c}}+\frac{3\epsilon^{2}}{2}\ln\phi_{c}+O\left(\phi_{c}^{2},\epsilon^{2}\right)\,.$
(80)

In the second line of (80) all terms containing $\ln(\phi_{c})$ are included
explicitly.

The remaining part of the integral is given by

$\displaystyle A_{1}^{\prime}$
$\displaystyle=\sum_{n=1}^{\infty}\int_{-\pi}^{+\pi}d\phi\,\frac{(\phi+2\pi
n)\sin\phi}{\left[4\sin^{2}\frac{\phi}{2}+\epsilon^{2}(\phi+2\pi
n)^{2}\right]^{3/2}}\,,$ (81)

where we replaced the integral over $\phi$ to the sum of integrals. Using
Euler-Maclaurin formula we computed

$\displaystyle A_{1}^{\prime}$
$\displaystyle=\ln\frac{\epsilon}{8}+\frac{\pi}{2}+1+C+O\left(\epsilon^{2}\ln\epsilon\right)\,.$
(82)

Here the constant $C$ was obtained numerically and as it vanishes with a good
accuracy we conjecture that $C=0$. In the following we use $C=0$.

Combining the above we have for $A_{1}=A_{1}^{\phi_{c}}+A_{1}^{\prime}$:

$\displaystyle
A_{1}\approx\ln{\frac{\epsilon}{2\phi_{c}}}+2+\frac{3\epsilon^{2}}{2}\ln{\phi_{c}}+O\left(\epsilon^{2}\ln\epsilon,\phi^{2}_{c}\right)\,.$
(83)

The calculation of $A_{2}$ and $A_{3}$ can be done in the same manner giving

$\displaystyle A_{2}$
$\displaystyle\approx\frac{1}{2\epsilon}+\ln{\frac{\epsilon}{2\phi_{c}}}-1+\frac{3\epsilon^{2}}{2}\ln{\phi_{c}}+O\left(\epsilon^{2}\ln{\epsilon},\phi^{2}_{c}\right)\,,$
(84) $\displaystyle A_{3}$
$\displaystyle\approx\frac{1}{2\epsilon}+\ln{\frac{\epsilon}{2\phi_{c}}}-2+\frac{\epsilon^{2}}{2}\ln{\phi_{c}}+O\left(\epsilon^{2}\ln{\epsilon},\phi^{2}_{c}\right)\,.$
(85)

##  Appendix D Streamlining the array of parallel wires

Let us consider an infinite array of parallel round wires. We assume that
wires are thin so that their radii $a$ are much smaller than the distance
between wires $a\ll 2L$. We choose the coordinate system such that the
centerlines of wires are given by equations:

$\displaystyle z=0,\quad x=2nL,\;n=0,\pm 1,\pm 2,\ldots\,.$ (86)

We assume no-slip boundary conditions on the surfaces of the wires. We will
fix the shear flow at $z\to+\infty$ and assume that the flow velocity is
constant as $z\to-\infty$. The corresponding solution of Stokes equation will
determine the slipping lengths $h,h_{\perp},h_{\parallel}$ as defined in
Section III.1.

The symmetry of the problem immediately gives $h=0$. To determine
$h_{\parallel}$ and $h_{\perp}$ we exploit the linearity of Stokes equations
and solve two problems for the parallel and transverse shear flows separately.

###  D.1 Shear flow parallel to wires

Let us assume that the shear flow at $z\to+\infty$ is parallel to the
direction of the wires $y$-axis. We use the symmetry of the problem and look
for the $y$-independent solution with $v_{x}=v_{z}=0$. In this case the Stokes
equations (2) can be reduced to the form

$\displaystyle\Delta v_{y}=0\,,$ (87)

where $\Delta=\partial_{x}^{2}+\partial_{z}^{2}$ is the two-dimensional
Laplace operator. This is the 2d Laplace equation that should be solved
outside of wires with the following boundary conditions:

$\displaystyle v_{y}$ $\displaystyle=0\,,\hskip
42.67912pt(x-2nL)^{2}+z^{2}=a^{2},\;\;n=0,\pm 1...$ $\displaystyle v_{y}$
$\displaystyle\to\Omega(z+h_{\parallel})\,,\hskip 62.59596ptz\to+\infty$
$\displaystyle v_{y}$ $\displaystyle\to\Omega h_{1}\,\hskip
93.89418ptz\to-\infty$ (88)

We will show that the following expression is the solution in the leading
order in $a/L\ll 1$:

$\displaystyle v_{y}$
$\displaystyle=\frac{\Omega}{2}z-\frac{\Omega}{2}\frac{\pi
a^{2}}{2L}\frac{\sinh{\frac{\pi z}{L}}}{\cosh{\frac{\pi z}{L}}-\cos{\frac{\pi
x}{L}}}+\frac{\Omega L}{2\pi}\log\left(\cosh{\frac{\pi z}{L}}-\cos{\frac{\pi
x}{L}}\right)-\frac{\Omega L}{2\pi}\log{\frac{\pi^{2}a^{2}}{2L^{2}}}\,,$
$\displaystyle\tilde{v}_{y}$
$\displaystyle=2\tilde{z}+\left[1-\tilde{a}^{2}\frac{\partial}{\partial\tilde{z}}\right]\log\left[\left(\frac{\sinh\tilde{z}}{\tilde{a}}\right)^{2}+\left(\frac{\sin\tilde{x}}{\tilde{a}}\right)^{2}\right]\,,$
$\displaystyle\tilde{v}_{y}$ $\displaystyle=v_{y}\Big{/}\frac{\Omega
L}{2\pi}\,,\quad\tilde{z}=\frac{\pi z}{2L}\,,\ldots\,.$ (89)

Indeed, it is straightforward to check that it is the solution of the Laplace
equation (87). In fact, every term of (89) separately solves the Laplace
equation. As the solution (89) is $2L$-periodic with respect to $x$, it is
sufficient to apply the no-slip boundary condition only on the surface of the
wire at the origin, i.e., at $\Gamma:\;x^{2}+z^{2}=a^{2}$. Expanding in $z/a$
and $x/a$ we obtain from (89):

$\displaystyle\frac{v_{y}}{\Omega L}\Big{|}_{\Gamma}$
$\displaystyle=\frac{z}{2L}-\frac{1}{2L}\frac{\pi a^{2}}{2L}\frac{\pi
z}{L}\frac{1}{\frac{\pi^{2}(z^{2}+x^{2})}{2L^{2}}}\left(1+O\left(\frac{a^{2}}{L^{2}}\right)\right)+\frac{1}{2\pi}\left(\log\left(\frac{\pi^{2}(z^{2}+x^{2})}{2L^{2}}\right)+O\left(\frac{a^{2}}{L^{2}}\right)-\log{\frac{\pi^{2}a^{2}}{2L^{2}}}\right)=O\left(\frac{a^{2}}{L^{2}}\right)\,.$
(90)

A typical velocity scale around wires is $\Omega L$ and we see that the
solution (89) satisfies no-slip boundary condition with $a^{2}/L^{2}$
accuracy.

In the limit $z\to-\infty$ we have up to exponentially decaying terms

$\displaystyle\frac{v_{y}}{\Omega L}\Big{|}_{z\to-\infty}$
$\displaystyle\approx\frac{z}{2L}+\frac{\pi
a^{2}}{4L^{2}}-\frac{1}{2\pi}\frac{\pi
z}{L}+\frac{1}{2\pi}\log{2}-\frac{1}{2\pi}\log{\frac{\pi^{2}a^{2}}{2L^{2}}}\approx\frac{\pi
a^{2}}{4L^{2}}-\frac{1}{\pi}\log{\frac{\pi a}{L}}$ (91)

so that the shear is absent at $z\to-\infty$ and the last condition of (88) is
satisfied with

$\displaystyle h_{1}=-\frac{L}{2\pi}\left(\log{\frac{\pi
a}{L}}-\frac{\pi^{2}a^{2}}{2L^{2}}+O\left(\frac{a^{4}}{L^{4}}\right)\right)\,.$
(92)

Similarly for $z\to+\infty$

$\displaystyle\frac{v_{y}}{\Omega L}\Big{|}_{z\to+\infty}$
$\displaystyle\approx\frac{z}{2L}-\frac{\pi
a^{2}}{4L^{2}}+\frac{1}{2\pi}\frac{\pi
z}{L}+\frac{1}{2\pi}\log{2}+\log{\frac{\pi^{2}a^{2}}{2L^{2}}}\approx\frac{z}{L}-\frac{\pi
a^{2}}{4L^{2}}-\frac{1}{\pi}\log{\frac{\pi a}{L}}\,.$ (93)

Comparing to (88) we find the slipping length:

$\displaystyle h_{\parallel}=\frac{L}{\pi}\left(\log{\frac{L}{\pi
a}}-\frac{\pi^{2}a^{2}}{4L^{2}}+O\left(\frac{a^{4}}{L^{4}}\right)\right)\,.$
(94)

###  D.2 Shear flow transverse to wires

Now we consider the flow with the shear transverse to wires. Namely, we will
look for a solution of Stokes equations satisfying $v_{y}=0$ everywhere and
the following boundary conditions:

$\displaystyle v_{x}$ $\displaystyle=v_{z}=0\,,\hskip
34.14322pt(x-2nL)^{2}+z^{2}=a^{2},\;\;n=0,\pm 1...$ $\displaystyle v_{x}$
$\displaystyle\to\Omega(z+h_{\perp})\,,\hskip 76.82234ptz\to+\infty$
$\displaystyle v_{x}$ $\displaystyle\to\Omega h_{2}\,\hskip
108.12054ptz\to-\infty\,.$ (95)

The problem is essentially two-dimensional. It is convenient to use a stream
function defined as

$\displaystyle v$ $\displaystyle=2i\partial\psi\,,$ (96)

where we used complex notations $v=v_{x}-iv_{z}$ and
$\partial=\frac{\partial}{\partial w}$ with $w=x+iz$. In terms of the stream
function the Stokes equation (2) for two-dimensional flow reduces to the
biharmonic equation $\Delta^{2}\psi=0$. Our goal is to find the function
$\psi$ as an expansion in $a/L\ll 1$. It can be checked that the following
potential satisfies the boundary conditions (95) in leading orders in $a/L$

$\displaystyle\psi$
$\displaystyle=\frac{\Omega}{4}z^{2}+Az+\psi_{S}+\psi_{SD}+\psi_{D}\,,$ (97)

where $\psi_{S}$, $\psi_{SD}$, $\psi_{D}$ are the periodic Stokeslet, periodic
doublet Stokeslet and periodic free dipole solutions accordingly, given by

$\displaystyle A$ $\displaystyle=-\frac{\Omega
L}{2\pi}\left(\frac{1}{2}+\ln{\frac{\pi a}{2L}}\right)\,,$
$\displaystyle\psi_{S}$ $\displaystyle=\frac{\Omega
L}{4\pi}\frac{w-\bar{w}}{2i}\log\left(\sin\frac{\pi w}{2L}\right)+c.c.\,,$
$\displaystyle\psi_{SD}$ $\displaystyle=\frac{i\pi
a^{2}}{L}\bar{\partial}\psi_{S}\,,$ $\displaystyle\psi_{D}$
$\displaystyle=\frac{i\Omega a^{2}}{16}\cot\frac{\pi w}{2L}+c.c.\,.$ (98)

Expanding the velocity given by (98) in $a/L$ on the surface of the wire
$w=ae^{i\alpha}$ we obtain

$\displaystyle\frac{v}{\Omega L}\approx\frac{\pi
a^{2}}{96L^{2}}\left(4-3e^{-2i\alpha}-e^{-2i\alpha}\right)=O\left(\frac{a^{2}}{L^{2}}\right),$
(99)

so we satisfied boundary conditions in the first two orders in $a/L$
expansion. Adding to the solution the higher-order derivatives of Stokeslet
and dipole solution we can satisfy boundary conditions in any desired order.
For our goals, we need only (98).

The asymptotics of velocity at the infinity $z\to\infty$ is

$\displaystyle v=\Omega z+\frac{\Omega
L}{2\pi}\left(-\frac{1}{2}+\log{\frac{L}{\pi
a}}-\frac{\pi^{2}a^{2}}{2L^{2}}+O\left(\frac{a^{4}}{L^{4}}\right)\right).$
(100)

Thus, for $h_{\perp}$ we obtain

$\displaystyle h_{\perp}=\frac{L}{2\pi}\left(-\frac{1}{2}+\log{\frac{L}{\pi
a}}-\frac{\pi^{2}a^{2}}{2L^{2}}+O\left(\frac{a^{4}}{L^{4}}\right)\right)\,.$
(101)

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