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fluid dynamics; Stokes swimming; metaboly
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\begin{document}
\title{Pushmepullyou: An efficient micro-swimmer}

\author{ J.E. Avron, O. Kenneth and D.H. Oaknin\\
\tenrm\!  Department of
Physics, Technion, Haifa 32000, Israel\\
}%
%\email{avron@physics.technion.ac.il} \twelverm
%\date{\today}%
\maketitle
\begin{abstract} The swimming of a pair of spherical bladders that
can change their shape is elementary at small
Reynolds numbers. If large strokes are allowed then the swimmer
has superior efficiency to other models of artificial swimmers.
The change of shape resembles the wriggling motion known as {\it
metaboly} of certain protozoa.
\end{abstract}

Swimming at low Reynolds numbers can be remote from common
intuition because of the absence of inertia
\cite{childress,purcel}. In fact, even the direction of swimming
may be hard to foretell. At the same time, and not unrelated to
this,  it does not require elaborate designs: Any stroke that is
not self-retracing will, generically, lead to some swimming
\cite{wilczek}. A  simple model that illustrates these features is
the  three linked spheres \cite{najafi}, Fig.~\ref{fig:swimmer}
(right), that swim by manipulating the distances $\ell_{1,2}$
between neighboring spheres. The swimming stroke is a closed, area
enclosing, path in the $\ell_1-\ell_2$ plane.

Swimming efficiently is an issue for artificial micro-swimmers. As
we have been cautioned by Purcell \cite{purcel} not to trust
common intuition at low Reynolds numbers, one may worry that
efficient swimming may involve unusual and nonintuitive swimming
styles. The aim of this letter is to give an example of an
elementary and fairly intuitive swimmer that is also remarkably
efficient provided it is allowed to make large strokes.

The swimmer is made of two spherical bladders,
Fig.~\ref{fig:swimmer} (left).  It swims by cyclically changing
the volume of the bladders and the distance between them. For the
sake of simplicity and concreteness we assume that the total
volume is conserved: The bladders exchange volumes in each stroke.
The swimming stroke is a closed path in the $ v-\ell$ plane where
$v$ is the volume of, say, the left bladder and $\ell$ the
distance between them. We shall make the further simplifying
assumption that the viscosity of the fluid in the bladders is
negligible compared with the viscosity of the ambient fluid
\footnote{Saffman pointed out \cite{saffman} that a bladder {\em
with inertia} can swim also in a superfluid.}. For reasons that
shall become clear below we call the swimmer pushmepullyou.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5


Like the three linked spheres, pushmepullyou is mathematically
elementary only in the limit that the distance between the spheres
is large, i.e. when $\e=a/\ell\ll 1$. ($a$ stands for the sphere
radii and $\ell$ for the distances between the spheres.) When
$\e=O(1)$ Stokes equations do not seem to have an explicit
solution in terms of elementary functions. We assume that the
Reynolds number $R={\rho av/\mu}\ll 1$, and also that the
distances between the balls are not too large: ${\ell v}\ll \mu$.
The second assumption is not essential and is made for simplicity
only. (To treat large $\ell$ one needs to replace the Stokes
solution, Eq.~(\ref{stokes}), by the more complicated, but still
elementary, Oseen-Lamb solution \cite{batchelor}.)



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Pushmepullyou is simpler than the three linked spheres: It
involves two spheres rather than three; it is more intuitive and
is easier to solve mathematically. It is also remarkably more
efficient if large strokes are allowed. It can even outperform 
conventional models of
biological swimmers that swim by beating a flagellum
\cite{lighthill}. If only small strokes are allowed then
pushmepullyou, like all squirmers \cite{agk}, becomes rather
inefficient.



\begin{figure}[htb]
%\vskip 4 cm
\hskip 2
cm\includegraphics[width=5cm]{swimmer-z.eps}\includegraphics[width=5cm]{swimmer-i.eps}
\caption{\em Five snapshots of the pushmepullyou swimming stroke
(left) and the corresponding strokes of the three linked spheres
(right). Both figures are schematic. After a full cycle the
swimmers resume their original shape but are displaced to the
right. Pushmepullyou is both more intuitive and more efficient
than the three linked spheres.}\label{fig:swimmer}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The swimming velocity is defined by $\dot X= (U_1+U_2)/2$ where
$U_i$ are the velocities of the centers of the two spheres. To
solve a swimming problem one needs to find the (linear) relation
between the response, $dX$, and the (shape) controls $(d\ell,dv)$.
This relation, as we shall show, takes the form:
\begin{equation}\label{swim}
2\, \dbar X=  \frac {a_1-a_2}{a_1+a_2}\ d\ell\ + \frac 1
{2\pi\ell^2} \ d v,
\end{equation}
where $a_1,a_2$ are the radii of the left and right spheres
respectively and $v$ is the volume of the left bladder. ($\dbar X$
stresses that the differential describing the change in the
location of the swimmer, does not integrate to a function on shape
space. Rather $X(\gamma)$ depends on the path $\gamma$ in shape
space.) The first term says that increasing $\ell$ leads to
swimming in the direction of the small sphere. It can be
interpreted physically as the statement that the larger sphere
acts as an anchor while the smaller sphere does most of the
motion. The second term says that the swimming is in the direction
of the contracting sphere: The expanding sphere acts as a source
pushing away the shrinking sphere which acts as a sink to pull the
expanding sphere. This is why the swimmer is dubbed pushmepullyou.


Certain protozoa and species of {\it Euglena} perform a wriggling
motion known as {\it metaboly} where, like pushmepullyou, body
fluids are transferred from a large spheroid to a small spheroid
\cite{metaboly}. Metaboly is, at present not well understood and
while some suggest that it plays a role in feeding \cite{triemer}
others argue that it is relevant to locomotion \cite{theriot}. The
pushmepullyou model shows that at least as far as fluid dynamics is
concerned, metaboly is a viable method of locomotion. 
Racing tests made by Triemer \cite{triemer} show that
Euglenoids swim 1-1.5 their body length per stroke. This is in
agreement with Eq.~(\ref{swim}) and (\ref{step}) for reasonable choices of 
stroke parameters. Since Euglena resemble deformed pears
--- for which there is no known solution to the flow equations ---
more than disconnected spheres the theory, if correct, has limited
predictive value. Pushmepullyou is therefore, at best, a biological
over-simplification. It has the virtue that admits complete
analysis.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To gain further insight into the pushmepullyou swimmer consider
the special case of small strokes near equal bi-spheres. Using
Eq.~(\ref{swim}) one finds, dropping sub-leading terms in
$\e=a_i/\ell$:
\begin{equation}\label{curvature}
\delta X= \,\frac{1}{6}\, d\log v\wedge d\ell
\end{equation}
Note that the swimming distance {\em does not} scale to zero with
$\e$. This is in contrast with the three linked spheres where the
swimming distance of one stroke is proportional to $\e$.  For
small cycle in the $\ell_1-\ell_2$ plane Najafi et. al. find
(Eq.~(11) in \cite{najafi}):
\begin{equation}\label{curvature-iran}
\delta X=0.7  \e \,d\log \ell_2\wedge d\ell_1
\end{equation}
When the swimmer is elementary, (=when $\e$ is small), it is also
poor.


The second step in solving a swimming problem is to find the
dissipation rate $P$. By general principles, $P$ is a quadratic
form in the velocities in the control space and is proportional to
the (ambient) viscosity $\mu$. The problem is to find this
quadratic form explicitly. We find
\begin{equation}\label{metric}
\frac P {6\pi\mu}=  \left(\frac 1 {a_1} +\frac 1 {a_2}\right)^{-1}
\, \dot\ell^2 +\frac {2} {9\pi}\  \left(\frac 1 {v_1} +\frac 1
{v_2}\right) \dot{v}^2\end{equation} Note that the dissipation
associated with  $\dot\ell$, is dictated by the {\em small} sphere
and decreases as the radius of the small sphere shrinks. (It can
not get arbitrarily small and must remain much larger than the
atomic scale for Stokes equations to hold.) The moral of this is
that pushing the small sphere is frugal. The dissipation
associated with $\dot v$ is also dictated by the small sphere.
However, in this case, dilating a small sphere is expensive.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The drag coefficient is a natural measure to  compare different
swimmers. It measures the energy dissipated in swimming a fixed
distance at fixed speed. (One can always decrease the dissipation
by swimming more slowly.) Choosing the stroke period as the unit
of time the drag is formally defined by \cite{lighthill,samuel}:
\begin{equation}\label{delta}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\delta(\gamma)=\frac { \int_0^1 P dt}{6\pi\mu X^2(\gamma)}\,.
\end{equation}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$X(\gamma)$ is the swimming distance of the stroke $\gamma$. The
smaller $\delta$ the more efficient the swimmer. $\delta$ has the
dimension of length (for swimmers in three dimensions) and is normalized
so that dragging of a sphere of radius $a$ with an external force has
$\delta=a$.

Consider now a large stroke associated with the closed rectangular
path enclosing the box $\ell_s\leq\ell\leq\ell_L,\; v_s\leq v_1,
v_2 \leq v_L\equiv v_0-v_s$, where $v_1 = v$ and $v_2$ are,
respectively, the volumes of the left and right bladders.  If
$a_s\ll a_L$ then from Eq.~(\ref{swim}), $X(\gamma)$ is
essentially  $\ell_L-\ell_s$:
\begin{equation}\label{step}
X(\gamma) = \left(
\frac {a_L-a_s}{a_L+a_s}\right)\ (\ell_L- \ell_s) \left(1+O(\e^3)\right)
%\simeq(\ell_L-\ell_s)\left(1+O\left(\frac
%{v_s}{v_L}\right)\right)
\end{equation}
%(Recall that we assume $\e\ll 1$ and $v_s\ll v_L$.)

To compute the dissipation for the rectangular path we need to
choose rates for traversing it.  The optimal rates are constant on
each leg provided the coordinates are chosen as
$(\ell,\arcsin\sqrt {v\over v_0})$.  This can be seen from the
fact that if we define $x=\arcsin\sqrt{v\over v_0}$,  then
$4v_0\dot{x}^2= \left(\frac 1 {v_1} +\frac 1 {v_2}\right)
\dot{v}^2$ and the Lagrangian associated with Eq.~(\ref{metric})
is quadratic in $(\dot \ell,\dot x)$, like the ordinary kinetic
Lagrangian of non relativistic mechanics. It is a common fact that
the optimal path of quadratic Lagrangian has constant speed. From
Eq.~(\ref{metric}) we find, provided also $\ell_L^2\gg\ell_s^2, \
\ell_L/a_s\gg \sqrt{v_L/v_s}$
\begin{equation}\label{dissipation}
\frac 1 {6\pi\mu} \int P dt\approx \frac{2
a_s\ell_L^2}{T_\ell}\left(1+O\left(\e^2 \frac {v_L}{v_s} \,\frac
{T_\ell}{T_v}\right)\right),\quad T_\ell+T_v=1/2\,
\end{equation}
where $T_\ell$ ($T_v$) is the time for traversing the horizontal
(vertical) leg. (Here $\e^2$ is actually $(a_s/\ell_L)^2$ rather
then the much larger $(a_L/\ell_s)^2$. Also note that the second
term in Eq.~(\ref{metric}) contributed $O(v_L/T_\ell)$ rather then
$O(v_L^2/(v_sT_\ell))$ as one may have expected from
Eq.~(\ref{metric}) which is dominated by the small volume.)

The optimal strategy, in this range of parameters, is to spend most
of the stroke's time on extending $\ell$. By Eq.~(\ref{delta})
this gives the drag
\begin{equation}\label{delta-best}
\delta \approx 4 a_s
\end{equation}  where $a_s$ is the
radius of the small bladder. {\em This allows for the transport of
a large sphere with the drag determined by the small sphere.} To
beat dragging, we need $a_s=a/4$, which means that most of the
volume, $63/64$, must be shuttled between the two bladders in each
stroke.

It is instructive to compare this mode of swimming with the
swimming efficiency of  models of (spherical) micro-organisms that
swim by beating flagella. These have been extensively studied by
the school of Lighthill and Taylor \cite{lighthill,blake} where
one finds $\delta \ge 100\, a$. This is much worse than dragging.
(We could not find estimates for the efficiency $\delta$ for
swimming by ciliary motion \cite{cilia}, but we expect that they
are rather poor, as for all squirmers \cite{agk}.) For models of
bacteria that swim by propagating longitudinal waves along their
surfaces Stone and Samuel \cite{samuel} established the
(theoretical) lower bound $\delta \ge \frac{4}{3} a$. (Actual
models of squirmers do much worse that the bound.) If the
pushmepullyou swimmer is allowed to make large strokes, it can
beat the efficiency of all of the above \footnote{A mechanical
model that has been carefully studied is Purcell's two hinge model
\cite{blades}. One can not make a meaningful comparison between it
and pushmepullyou since their shapes are so different.}.

%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

It is likely that some artificial micro-swimmers will be
constrained to make only small (relative) strokes. Small strokes
necessarily lead to large drag \cite{agk}, but it is still
interesting to see how large. Suppose $\delta\log
\ell\sim\delta\log v,\; a_1\sim a_2$. The dissipation in one stroke is
then\begin{equation}\label{dissipation-small-stroke} \frac {\int P
dt} {6\pi
\mu}=(\delta\ell)^2\left(\frac{a}{T_\ell}\right)
\left(1+O\left(\e^2\frac{T_\ell}{T_v}\right)\right)
\end{equation}
From Eq.~(\ref{curvature}) one finds
\begin{equation}\label{delta-squirmer}
\delta\approx \frac{72}{(\delta\log v)^2}\ a\ .
\end{equation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We shall now outline how the key results,
Eqs.~(\ref{swim},\ref{metric}), are derived. A solution to the
Stokes equation for a single sphere of radius $a$ dragged by a
force $f$ and dilated at rate $\dot v$ is
\begin{equation}\label{stokes}\pi \vec{u}({\vec x};a,f,\dot v) = \frac 1
{6\mu|x|}\left(\left(3+\frac{a^2}{x^2}\right)\vec{f}+\left( 1
-\frac {a^2}{x^2}\right) 3(\vec{f}\cdot\hat x)\hat x\right)+ \frac
{\dot v}{ x^2} \hat x.\end{equation} $\vec{u}(\vec{x};a,f,\dot v)$
is the velocity field at a position $\vec{x}$ from the center of
the sphere. The left term is the known Stokes solution. (A
Stokeslet, \cite{batchelor}, is defined as the Stokes solution for
$a=0$.) The term on the right is a source term.

Since Stokes equations are linear, a superposition of the
solutions for two dilating spheres is a solution of the
differential equations. However, it does not quite satisfy the
no-slip boundary condition on the two spheres: There is an error
of order $\e $. The superposition is therefore an approximate
solution provided the two spheres are far apart.

The (approximate)  solution determine the velocities $U_i$ of the
centers of the two spheres:
\begin{equation}\label{U}
U_i= \vec{u}(a_i
\hat{f};a_i,(-)^jf,0)+\,\vec{u}((-)^i\ell\hat{f};a_j,(-)^if,(-)^i\dot
v),\quad i\neq j\in\{1,2\}
\end{equation}
The first term on the right describes how each sphere moves relative to the
fluid according to Stokes law as a result of the force $\vec f$ acting on it.
The second term (which is typically smaller) describes the velocity of the
fluid surronding the sphere (at distances $\gg a$ but $\ll\ell$) as a result
of the movement of the other sphere.
By symmetry,the velocities and forces are parallel to the axis connecting
the centers of the two spheres, and can be taken as scalars.
To leading order in $\e$ Eq.~(\ref{U}) reduces to
\begin{equation}2\pi U_i=(-)^j \frac f \mu \left(\frac 1 {3 a_i}-\frac 1 {2
\ell}\right) +\frac {\dot v}{2 \ell^2}
\end{equation}
Using $ \dot \ell =-U_1+U_2$, and dropping sub-leading terms in
$\e $, gives the force in the rod
\begin{equation}\label{force}
  f =-6\pi\mu \left(\frac 1 {a_1} +\frac 1{a_2}\right)^{-1} \  \dot \ell
\end{equation}
and Eq.~(\ref{swim}).

We now turn to Eq.~(\ref{metric}). Consider first the case $\dot
v=0$. The power supplied by the rod is $-f(U_2-U_1)=-f\dot \ell$
which  gives the first term. Now consider the case $\dot\ell=0$.
The stress on the surface of the expanding sphere is given by
\begin{equation}\label{stress}
\sigma=-\frac{2\mu \dot v}{4\pi}\, \left(\frac 1
{x^2}\right)^\prime=\frac{\mu\dot v}{\pi a^3}
\end{equation}
The power requisite to expand one sphere is then
\begin{equation}\label{dilating}
4\pi a^2\sigma\dot a=\sigma \dot v =\frac{4\mu}{3 v} (\dot v)^2
\end{equation}
Since there are two spheres, this give the second term in
Eq.~(\ref{metric}). Finally, we note that there can not be mixed
terms in the dissipation proportional to $\dot\ell\dot v$. This is
because the velocity field and the field of force on the surface
of each sphere generated by $\dot \ell$ are constants (parallel to
$\hat{f}$) while the components generated by $\dot v$ are radial.
The two can not be coupled to give a scalar.

{\bf Acknowledgment} This work is supported in part by the EU
grant HPRN-CT-2002-00277. We thank Dana Mackenzie who urged us to
come with numbers comparing micro-swimmers with micro-organisms
and thank H. Berg, H. Stone, and especially Richard Triemer
for useful correspondence and for the Euglena racing tests.%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{10}
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University Press, Cambride, 1981).
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numbers}, Am. J. Physics {\bf 45}, 3-11 (1977).
\bibitem{wilczek}  A. Shapere and F. Wilczek, {\it Geometry of
self-propulsion at low Reynolds numbers}, J. Fluid Mech., {\bf
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Reynolds numbers}, J. Fluid Mech., {\bf 198}, 587-599 (1989).
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(2004) 062901, cond-mat/0402070
\bibitem{triemer} R.E. Triemer, private communication.
\bibitem{theriot} D.A. Fletcher and J.A. Theriot, {\it An
introduction to cell motility for the phsyical scientist},
Physical Biology {\bf 1}, T1-T10 (2004).
\bibitem{metaboly} Beautiful movies of metaboly can be viewed at
the web site of Richard E. Triemer at
http://www.plantbiology.msu.edu/triemer/Euglena/Index.htm
\bibitem{saffman} P.G.
Saffman, {\it The self-propulsion of a deformable body in a
perfect fluid}, 28, 385 - 389, (1967).


\bibitem{batchelor} G.K. Batchelor, {\it An
Introduction to Fluid Dynamics}, (Cambridge University Press,
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\bibitem{lighthill} J. Lighthill, {\it On the squirming motion of
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\bibitem{agk}J. Avron, O. Kenneth and O. Gat,
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\bibitem{samuel} H.A. Stone and A.D. Samuel, {\it Propulsion of
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\bibitem{blake} J.R. Blake, {\it A spherical envelops approach to ciliary
propulsion},  J. Fluid. Mech. {\bf 46}, 199-208 (1971)



%\bibitem{koiller} J. Koiller, R. Montgomery and K. Ehlers,
%{\it Problems and progress in microswimming}, J. Nonliear Sci.
%{\bf 6}, 507-541 (1996); J, Koiller, M. Raup, M. Delgado, J.
%Ehlers and K.M. Montgomery, Comm. App. Math {\bf 17} 3 (1998).



\bibitem{cilia}C. Brennen and H. Winet, {\it Fluid
Mechanics of Propulsion by Cilia and Flagella}, Annual Review of
Fluid Mechanics, (1977), Vol. 9: Pages 339-398

\bibitem{blades} L.E. Becker, S.A. Koehler,
and H.A. Stone, J. Fluid Mech. 490 , 15 (2003); E.M. Purcell,
Proc. Natl. Acad. Sci. 94 , 11307-11311 (1977).


\end{thebibliography}
\end{document}
 [2] J. Happel and H. Brenner, Low Reynolds Number Hydro-
dynamics, (Prentice-Hall, Englewood Cliffs, New Jersey, 1965). [5]
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[10] H.A. Stone and A.D.T. Samuel, Phys. Rev. Lett. 77 , 4102
(1996).
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