# Minimum Entropy Production by Microswimmers with Internal Dissipation

Abdallah Daddi-Moussa-Ider  Max Planck Institute for Dynamics and Self-
Organization (MPIDS), 37077 Göttingen, Germany  Ramin Golestanian  Max Planck
Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford
OX1 3PU, United Kingdom  Andrej Vilfan<EMAIL_ADDRESS>Max Planck
Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
Jožef Stefan Institute, 1000 Ljubljana, Slovenia

###### Abstract

The energy dissipation and entropy production by self-propelled microswimmers
differ profoundly from passive particles pulled by external forces. The
difference extends both to the shape of the flow around the swimmer, as well
as to the internal dissipation of the propulsion mechanism. Here we derive a
general theorem that provides an exact lower bound on the total, external and
internal, dissipation by a microswimmer. The problems that can be solved
include an active surface-propelled droplet, swimmers with an extended
propulsive layer and swimmers with an effective internal dissipation. We apply
the theorem to determine the swimmer shapes that minimize the total
dissipation while keeping the volume constant. Our results show that the
entropy production by active microswimmers is subject to different fundamental
limits than the entropy production by externally driven particles.

## Introduction

Microswimmers are microscale objects that move in a self-propelled way through
a fluid medium at low Reynolds numbers where viscous forces dominate over
inertia [1, 2, 3]. They comprise living swimmers such as microorganisms and
sperm cells, which have been a subject of keen interest by pioneers of
twentieth-century fluid physicists such as Ludwig Prandtl [4] and G. I. Taylor
[5], as well as artificially manufactured colloidal microswimmers [6, 7, 8,
9]. A central question in the field of microswimmers is the energetic cost of
their propulsion. The energetic efficiency of a microswimmer is typically
characterized with Lighthill’s efficiency [10], defined as the equivalent
power needed to pull the swimmer through the fluid with an external force,
divided by the actually dissipated power of the active swimmer moving with the
same velocity. In biological swimmers, the question is whether and how the
swimmers have evolved to achieve a high propulsion efficiency and how close
they can come to the theoretical limits set by the laws of hydrodynamics.
Although Purcell [11] concluded that the energetic expenditure for swimming
represent a very small fraction of the total consumption in bacteria, it is
now known that larger microorganisms, like Paramecium can use about half of
their total power for propulsion [12]. The efficiency of artificial swimmers
is still lagging far behind their natural counterparts and its improvement is
one of the key challenges on the way towards future technological or
biomedical applications. Finally, the entropy production in suspensions of
microswimmers is a fundamental question in stochastic thermodynamics and
statistical physics [13, 14, 15]. A common assumption in these works is to
estimate the “housekeeping” work needed to propel active Brownian particles by
representing autonomous propulsion as external forces acting on the particles
[16, 17, 18], which can be coupled to a chemical reaction [19]. An outstanding
question is whether there are other fundamental limits on the entropy
production by active particles – both because of their propulsion mechanism
and because the laws of self-propelled swimming differ from the externally
driven particles. At the core of all these diverse research topics, there is a
common fundamental question: what is the minimum amount of dissipation needed
by a self-propelled microswimmer, how can it be achieved and how does it
compare to the dissipation by a passive object that is pulled through the
fluid by an external force.

Many biological microswimmers achieve self-propulsion by performing
nonreciprocal deformation cycles via periodic beating of cilia or flagella,
slender appendages anchored to their cell body. The waveform assumed by
beating flagella or cilia follows an asymmetric pattern in an irreversible
fashion to generate propulsion. While bacteria and flagellates usually swim
with a small number of long flagella, numerous other biological swimmers such
as Paramecium or Volvox swim by means of thousands of cilia packed on their
surfaces. Because the cilia are usually an order of magnitude shorter than the
size of the body, they can be described as surface-driven and the action of
cilia consists in generating an effective slip velocity along the surface [10,
20, 21, 22, 23, 24].

Figure 1: Five scenarios of internal dissipation in an active swimmer. a An
active droplet with a fluid-fluid interface, driven by tangential forces at
the interface. b A swimmer driven by tangential forces at an outer surface
(dotted line). c A swimmer driven by normal forces at an outer surface. d A
swimmer driven by unconstrained forces, allowing both tangential and normal
components, at an outer surface. e A surface-driven swimmer with internal
dissipation in the surface layer (green). The magenta arrows denote the active
tractions, defined as the forces exerted by the fluid on the swimmer.

For surface-driven microswimmers, the power being dissipated by viscosity can
broadly be split, from a hydrodynamic perspective, into two distinct parts:
internal and external. The internal dissipation accounts for the local losses
occurring at the propulsive layer. For example, this could be the dissipation
within the ciliary layer or within the boundary layer of phoretic swimmers, or
the dissipation inside the droplet in the case of self-propelled droplets. The
inner dissipation plays the dominant role in most microswimmers, notably in
ciliated microorganisms [25, 26, 27] and also in phoretic swimmers [28, 29].
The external dissipation results from the interaction of the swimmer with the
surrounding fluid environment. External dissipation is inevitable as the
swimmer displaces the fluid it moves through and is independent of the
specifics of the propulsion mechanism. The minimum amount of external
dissipation needed by a swimmer of a given shape with a given swimming
velocity therefore represents a fundamental problem of low-Reynolds-number
hydrodynamics that has been solved analytically for spherical [30, 31, 32, 33]
and spheroidal swimmers [25, 34, 35], as well as numerically for arbitrary
axisymmetric shapes [36, 37]. The propulsive motion can either be stationary
[25, 34, 36] or periodic in time, representing a squirming motion or the
motion of the ciliary envelope [30, 33, 37]. More lately, the swimming
efficiency in non-Newtonian fluids has also been investigated [32, 35].

We recently derived a general solution to determine the lower bound on
external dissipation by an active microswimmer, which we could express with
the passive hydrodynamic drag coefficients of two bodies of the same shape:
one with the no-slip and one with the perfect-slip boundary condition [38].
The solution also shows that the flow profile of the optimal swimmer is a
linear superposition of the flow fields induced by these two passive bodies.
The optimal velocity profile, which in principle poses a complex quadratic
optimization problem, is then reduced to the solution of two passive flows. By
means of this theorem, we determined the flow field of an optimal swimmer of
nearly spherical shape using a perturbative analytical approach [39].

In this paper, we show that the approach can be generalized to derive
fundamental limits on the total dissipation by a swimmer, comprising both
external and internal contributions. We derive novel minimum dissipation
theorems for different classes of swimmers, i.e., surface driven droplets,
swimmers with a finite propulsive layer, as well as swimmers with an effective
surface dissipation. The formulations of the active problems we study here are
illustrated in Fig. 1 and the corresponding theorems are summarized in Table
1. We thus demonstrate that the analytical approach can generically be applied
to a broad class of minimum dissipation problems in microswimmers with
realistically modeled propulsion mechanisms.

Table 1: Minimum dissipation theorems Dissipation function |  Motivation | Theorem |  V-problem |  F-problem
---|---|---|---|---
External dissipation (Ref. [38]) |  Ideal swimmer | $P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{PS}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}$ |  Perfect slip |  No-slip
Surface-driven droplet (Fig. 1a) |  Active droplets using Marangoni effect | $P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{Droplet}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}$ |  Droplet |  No-slip
External tangential forces (Fig. 1b) |  Model of cilia | $P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{NSi}}^{-1}-\bm{\mathsf{R}}_{\text{CP}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}$ |  No-slip (inner core) |  No-slip core, no tangential slip shell
External normal forces (Fig. 1c) |  Phoretic swimmers | $P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{NSi}}^{-1}-\bm{\mathsf{R}}_{\text{DC}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}$ |  No-slip (inner core) |  No-slip core, zero normal velocity shell
External unconstrained forces (Fig. 1d) |  Idealized external propulsion | $P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{NSi}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}$ |  No-slip (inner core) |  No-slip (outer shell)
Surface dissipation (Fig. 1e) |  Coarse-grained (cilia, phoretic) | $P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{Navier}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}$ |  Navier slip |  No-slip

## Results

###  .1 Minimum dissipation theorems with internal dissipation

We consider a microswimmer that self-propels through an incompressible viscous
fluid. The fluid velocity $\bm{v}(\bm{x})$ satisfies the Stokes equation
$\bm{\nabla}\cdot\bm{\sigma}=\bm{0}$ together with the continuity equation
$\bm{\nabla}\cdot\bm{v}$ where $\bm{\sigma}=-p\bm{I}+2\mu\bm{E}$ is the stress
tensor, $p$ the pressure, $\mu$ the shear viscosity, and
$\bm{E}=\tfrac{1}{2}\left(\bm{\nabla}\bm{v}+\bm{\nabla}\bm{v}^{\top}\right)$
the rate-of-strain tensor. The swimmer moves with a translational velocity
$\bm{V}_{\text{A}}$ and angular velocity $\bm{\Omega}_{\text{A}}$, which can
be treated together as a rigid body velocity described by the 6-component
vector
$\bm{\mathsf{V}}_{\text{A}}=[\bm{V}_{\text{A}},\bm{\Omega}_{\text{A}}]$.
Likewise the total force $\bm{F}_{\text{A}}$ and torque $\bm{M}_{\text{A}}$
exerted by the fluid on the swimmer can be merged to a generalized force
$\bm{\mathsf{F}}_{\text{A}}=[\bm{F}_{\text{A}},\bm{M}_{\text{A}}]$. In the
absence of external (other than hydrodynamic) forces, the swimmer is force-
and torque-free, $\bm{\mathsf{F}}_{\text{A}}=\bm{\mathsf{0}}$. The swimmer
self-propels by imposing a fluid velocity $\tilde{\bm{v}}$ on its surface.
Here, all velocities in the co-moving frame are denoted as $\tilde{\bm{v}}$
and those in the laboratory frame as $\bm{v}$. Alternatively, depending on the
formulation, the swimmer can also impose an active contribution to the force
(traction) density on its surface $\bm{f}_{\text{A}}$. As the swimmer moves
through the fluid, it dissipates the power $P_{\text{A}}$. The dissipation
consists of an external contribution in the fluid outside the swimmer and an
internal contribution either in fluids inside the swimmer or internally in the
flow generating mechanism.

In the following, we summarize the minimum dissipation theorems that will be
derived in the subsequent sections. All theorems take the generic shape

$P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{V}}^{-1}-\bm{\mathsf{R}}_{\text{F}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\;.$
(1)

The theorem expresses the minimum bound on the dissipation of the active
swimmer $P_{\text{A}}$ with two passive hydrodynamic drag coefficients,
represented by $6\times 6$ matrices. $\bm{\mathsf{R}}_{\text{V}}$ is the drag
coefficient of a passive body with minimum dissipation while fulfilling the
boundary conditions required by the active problem.
$\bm{\mathsf{R}}_{\text{F}}$ is the drag coefficient of another passive body
that appears in the superposition and has a velocity distribution orthogonal
to the active driving forces. The equality in the theorem (1) is satisfied
exactly when the active flow represents a linear superposition of the two
passive problems. We denote them as the V-problem and the F-problem because
the velocity distribution of the optimal active swimmer is determined by the
solution of the V-problem and the active forces by the solution of the
F-problem. Besides giving the lower limit on dissipation, the theorem also
solves the optimization problem by providing the distribution of velocities
and active forces that allow the swimmer to achieve a given velocity with
minimum dissipation.

We now provide a proof of the theorems stated above. The derivation of the
minimum dissipation theorems is based on two crucial steps: first we find the
solution for the minimum dissipation for the motion of a body of a given
shape, driven by an external force. In our previous work, which considered
external dissipation only [38], this was the perfect-slip body, with
properties similar to those of an idealized air bubble in the fluid, as
depicted in Fig. 2a. The second step that makes the problem solvable consists
in finding another passive problem and form a linear superposition of its flow
and that of the active body. Importantly, the dissipation in the superposition
flow needs to be the sum of the dissipations of the two problems each on its
own. Because dissipation is a quadratic function, the latter condition is not
trivial and requires the application of the Lorentz reciprocal theorem [40].
We therefore first generalize the Helmholtz minimum dissipation theorem for
passive bodies and prove that Stokes flows including fluid-fluid interfaces
and surfaces with the Navier slip boundary condition take the form with
minimum dissipation. We subsequently apply the principle of superposition
between an active swimmer and a passive body to derive the theorems for all
types of swimmers listed in Table 1.

###  .2 Generalization of passive minimum dissipation theorems

We first show that flows around passive bodies with several different boundary
conditions (listed in Table 2) satisfy a minimum dissipation theorem, i.e.,
that any other flow satisfying less stringent boundary conditions has a higher
dissipation rate. These are generalizations of the Helmholtz minimum
dissipation theorem, which states that among all flows that satisfy the
prescribed velocity boundary conditions and incompressibility, but not
necessarily the Stokes equation, the actual Stokes flow has the smallest
dissipation [41]. In the derivation of the theorem for external dissipation
[38], we used the statement that among all bodies of a given shape that move
with a given speed $V$ and have zero normal velocity on their surface (i.e.,
the fluid can not cross the body’s surface), the perfect-slip body has the
lowest dissipation. Here, we extend the theorem to two further types of
boundary conditions.

First, we generalize the theorem to shape-preserving fluid-fluid interfaces.
If the body has a fixed shape, but its interior contains another fluid such
that the boundary condition in the co-moving frame reads
$\tilde{v}^{\perp}=\tilde{v}_{i}^{\perp}=0$ and
$\tilde{\bm{v}}^{\parallel}=\tilde{\bm{v}}^{\parallel}_{i}$, then the minimum
dissipation is reached when the tangential stress on the surface vanishes,
$(\bm{I}-\bm{n}\bm{n})\cdot(\bm{\sigma}-\bm{\sigma}_{i})\cdot\bm{n}=\bm{0}$.
The quantities labeled with the index $i$ refer to the internal fluid domain
and those without to the external. $\bm{n}$ denotes the outward pointing
surface normal. The perpendicular and parallel components of the velocity are
defined as $\tilde{v}^{\perp}=\bm{n}\cdot\tilde{\bm{v}}$ and
$\tilde{\bm{v}}^{\parallel}=(\bm{I}-\bm{n}\bm{n})\cdot\tilde{\bm{v}}$,
respectively. The theorem states that the dissipation of any flow around an
interface with velocity continuity, itself moving with velocity
$\bm{\mathsf{V}}$, satisfies the inequality

$P\geq\bm{\mathsf{V}}\cdot\bm{\mathsf{R}}_{\text{Droplet}}\cdot\bm{\mathsf{V}}\;.$
(2)

Here, $\bm{\mathsf{R}}_{\text{Droplet}}$ denotes the generalized drag
coefficient of the body described by the fluid-fluid interface, e.g., an oil
droplet in water. A proof of the statement is given in Supplmenentary
Information, Sect. A.1. A related version of the theorem has been derived in
Ref. [42] for droplet suspensions. An immediate implication of Eq. (2) is that
the dissipation of the flow around a droplet is always smaller than around a
no-slip body of the same shape and the same velocity. Therefore, the matrix
$\bm{\mathsf{R}}_{\text{NS}}-\bm{\mathsf{R}}_{\text{Droplet}}$ is always
positive definite, which naturally also holds for the matrices
$\bm{\mathsf{R}}_{\text{NS}}$ and $\bm{\mathsf{R}}_{\text{Droplet}}$. The
expression
$(\bm{\mathsf{R}}_{\text{Droplet}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1})^{-1}$
can be rewritten as
$\bm{\mathsf{R}}_{\text{Droplet}}\cdot(\bm{\mathsf{R}}_{\text{NS}}-\bm{\mathsf{R}}_{\text{Droplet}})^{-1}\cdot\bm{\mathsf{R}}_{\text{Droplet}}+\bm{\mathsf{R}}_{\text{Droplet}}$
and is therefore positive-definite as well. We have thus proven that the lower
bound given by Eq. (1) is always positive.

Table 2: List of boundary conditions Type | Velocity | Stress
---|---|---
No slip | $\bm{v}=\bm{0}$ | -
Perfect slip | $v^{\perp}=0$ | $\bm{f}^{\parallel}=\bm{0}$
Navier slip | $v^{\perp}=0$ | $\bm{f}^{\parallel}=\frac{\mu}{\lambda}\bm{v}$
Fluid-fluid interface |  $v^{\perp}=v_{i}^{\perp}=0$, | $\bm{f}^{\parallel}-\bm{f}_{i}^{\parallel}=\bm{0}$
| $\bm{v}^{\parallel}=\bm{v}_{i}^{\parallel}$ |
No tangential slip shell |  $\bm{v}=\bm{v}_{i}$, $\bm{v}^{\parallel}=\bm{0}$ | $f^{\perp}-f_{i}^{\perp}=\bm{0}$
$\bm{f}=\bm{\sigma}\cdot\bm{n}$, $\bm{f}_{i}=\bm{\sigma}_{i}\cdot\bm{n}$

In the second generalization, we introduce an energetic cost to the slip
velocity on the surface, such that the total dissipation is given by (note the
distinction between the velocity in the laboratory frame $\bm{v}$ and in the
co-moving frame $\tilde{\bm{v}}$)

$P=\int_{\cal
S}\mathrm{d}S\left(-\bm{f}\cdot\bm{v}+\frac{\mu}{\lambda}\,\tilde{\bm{v}}^{2}\right),$
(3)

with the traction $\bm{f}=\bm{\sigma}\cdot\bm{n}$ defined as the force density
exerted by the fluid on the body. Here the first term represents the power
transferred from the swimmer to the fluid, which is identical to the total
dissipation in the fluid (Supplementary Information, Sect. B). We therefore
refer to it as external dissipation. The second term represents the internal
dissipation, which is the cost of maintaining the velocity on the surface. We
wrote the dissipation density, which can be arbitrary, as $\mu/\lambda$ in
anticipation of the result that follows. The total dissipation as defined in
(3) is minimal when the flow satisfies the Navier slip condition on the
surface

$\tilde{\bm{v}}=\frac{\lambda}{\mu}\,\bm{f}^{\parallel}\;.$ (4)

Here $\lambda$, which we initially introduced as a free parameter, takes the
role of the slip length and $\mu$ denotes the viscosity. We note that
$\lambda=0$ for the no-slip condition and $\lambda=\infty$ for the perfect
slip. The total dissipation in any flow around that body moving with velocity
$\bm{\mathsf{V}}$ satisfies the inequality

$P\geq\bm{\mathsf{V}}\cdot\bm{\mathsf{R}}_{\text{Navier}}\cdot\bm{\mathsf{V}}\;,$
(5)

where $\bm{\mathsf{R}}_{\text{Navier}}$ denotes the generalized drag
coefficient of the rigid body with shape $\mathcal{S}$ and slip length
$\lambda$. The statement is proven in Supplementary Information, Sect. A.2. By
inserting the no-slip flow into the inequality, we see that
$\bm{\mathsf{R}}_{\text{NS}}-\bm{\mathsf{R}}_{\text{Navier}}$ is positive
definite and the lower bound in Eq. (1) is positive.

###  .3 Derivation of active minimum dissipation theorems

In the second step we will derive active minimum dissipation theorems for 5
swimmer types that include internal dissipation, as listed in Table 1. For
that purpose we will apply the derived inequalities to the superposition of an
active swimmer and a passive body of the same shape. The crucial step is
always to find two passive problems that satisfy the boundary conditions of
the active problem whereby the first problem possesses minimum dissipation and
the second problem has a velocity distribution orthogonal to the forces in the
active problem.

Figure 2: Superposition principle used to derive different variants of the
minimum dissipation theorem. The passive body (V-problem, left) can be
represented as a superposition of the optimal active swimmer (center) and
another passive body (F-problem, right). The blue arrows indicate the velocity
$\bm{V}$ (in laboratory frame) of each body and the red arrows the drag force
$\bm{F}$ on passive bodies, all drawn to scale. The cyan arrows indicate the
velocity $\tilde{\bm{v}}$ and its gradient in the co-moving frame (not to
scale). a A perfect-slip body as a superposition between an optimal swimmer
(external dissipation only) and a no-slip body, as shown in Ref. [38]. b A
droplet as a superposition between the optimal active droplet and a no-slip
body. c A no-slip body as a superposition between a swimmer driven by
tangential forces at the outer layer and a composite passive body. d A no-slip
body as a superposition between a swimmer driven by normal forces and a
droplet with a no-slip core. e A no-slip body as a superposition between a
swimmer driven by unconstrained forces and a no-slip body at the outer
surface. f A Navier-slip body as a superposition between a surface-driven
swimmer with internal dissipation and a no-slip body.

####  .3.1 Active surface-driven droplet

We start by deriving the theorem for the active droplet, which is propelled by
an active tangential force on the surface (Fig. 1a). The latter determines the
stress discontinuity,
$(\bm{I}-\bm{n}\bm{n})\cdot(\bm{\sigma}-\bm{\sigma}_{i})\cdot\bm{n}=\bm{f}_{\text{A}}^{\parallel}$.
Such active stress can result from Marangoni effect, where it is caused by a
gradient in the surface tension and is widely used to propel active droplets
[43]. The active power exerted by this force is

$P_{\text{A}}=-\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{A}}\,.$
(6)

To derive the minimum dissipation theorem for the active droplet, we need one
passive body that minimizes the dissipation while fulfilling the boundary
condition on the surface and another passive body with a velocity distribution
that is orthogonal to the active forces (see below) while also satisfying the
boundary condition. The former is represented by a droplet and the latter by a
no-slip body. We first calculate the dissipation in a flow that is a linear
superposition between the active swimmer, moving with velocity
$\bm{\mathsf{V}}_{\text{A}}$, and a passive hollow no-slip body, moving with
velocity $\bm{\mathsf{V}}_{\text{NS}}$, as illustrated in Fig. 2b. The
dissipated power in the superposition flow can be expressed as

$P_{\text{A+NS}}=-\int_{\mathcal{S}}\mathrm{d}S\,(\bm{f}_{\text{A}}+\bm{f}_{\text{NS}})\cdot(\bm{v}_{\text{A}}+\bm{v}_{\text{NS}})\,.$
(7)

We now apply the Lorentz reciprocal theorem (Methods), (44), by integrating
over the whole space, without surface contributions. However, because the
traction is concentrated on the surface $\mathcal{S}$, we write its
contribution in the form of a surface integral. From (44) it follows that the
two mixed terms are identical
$\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{NS}}\cdot\bm{v}_{\text{A}}=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{NS}}$.
By expressing the velocities in the co-moving system as in (45), we also know
that
$\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{NS}}=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{NS}}+\bm{\mathsf{F}}_{\text{A}}\cdot\bm{\mathsf{V}}_{\text{NS}}=0$.
The latter follows from $\tilde{\bm{v}}_{\text{NS}}=0$ and
$\bm{\mathsf{F}}_{\text{A}}=0$. Therefore, the mixed terms vanish and (7)
reduces to

$P_{\text{A+NS}}=P_{\text{A}}-\bm{\mathsf{F}}_{\text{NS}}\cdot\bm{\mathsf{V}}_{\text{NS}}\;,$
(8)

i.e., the sum of the dissipation contributions by the active swimmer and the
no-slip body, each on its own. We have thus a posteriori justified the choice
of the problem used in the superposition. This decomposition is a crucial step
that is decisive for the feasibility of the solution.

Finally, we know that like any flow satisfying the boundary conditions of
continuous tangential and zero normal velocity on the surface, the dissipation
of the superposition flow satisfies the inequality (2), which states that
$P_{\text{A+NS}}\geq(\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}})\cdot\bm{\mathsf{R}}_{\text{Droplet}}\cdot(\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}})$.
The equality is fulfilled exactly when the superposition corresponds to the
flow around a droplet. This implies
$\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}}=\bm{\mathsf{V}}_{\text{Droplet}}$
and $\bm{\mathsf{F}}_{\text{NS}}=\bm{\mathsf{F}}_{\text{Droplet}}$, along with
$\bm{\mathsf{F}}_{\text{Droplet}}=\bm{\mathsf{R}}_{\text{Droplet}}\cdot\bm{\mathsf{V}}_{\text{Droplet}}$
and
$\bm{\mathsf{F}}_{\text{NS}}=\bm{\mathsf{R}}_{\text{NS}}\cdot\bm{\mathsf{V}}_{\text{NS}}$.
These equations are solved to yield

$\displaystyle\bm{\mathsf{V}}_{\text{Droplet}}$
$\displaystyle=\left(\bm{\mathsf{I}}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\cdot\bm{\mathsf{R}}_{\text{Droplet}}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}},$
(9a) $\displaystyle\bm{\mathsf{V}}_{\text{NS}}$
$\displaystyle=\left(\bm{\mathsf{R}}_{\text{Droplet}}^{-1}\cdot\bm{\mathsf{R}}_{\text{NS}}-\bm{\mathsf{I}}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\;.$
(9b)

With these velocities, we finally obtain the minimum dissipation theorem

$P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{Droplet}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\,.$
(10)

Because the equality in the theorem is fulfilled exactly if the superposition
gives the flow around a droplet, we also know the distribution of velocities
and active forces that minimize the dissipation while maintaining the swimming
speed $\bm{\mathsf{V}}_{\text{A}}$. The optimal slip velocity is given by the
flow around the droplet
$\tilde{\bm{v}}_{\text{A}}=\tilde{\bm{v}}_{\text{Droplet}}$ and the tangential
traction is opposite-equal to that of a no-slip body
$\bm{f}_{\text{A}}^{\parallel}=-\bm{f}_{\text{NS}}^{\parallel}$.

As a simple example, we apply the theorem to a spherical viscous droplet with
external viscosity $\mu$, internal viscosity $\mu_{i}$ and radius $a$.
According to the Hadamard-Rybczynski equation, the drag coefficient of the
droplet is [41]

$R_{\text{Droplet}}=6\pi\mu
a\cdot\frac{\mu_{i}+\frac{2}{3}\mu}{\mu_{i}+\mu}\;,$ (11)

and we obtain

$P_{\text{A}}\geq 6\pi(2\mu+3\mu_{i})aV_{\text{A}}^{2}\;.$ (12)

For equal viscosities, $\mu_{i}=\mu$, the ratio of internal vs. external
dissipation is $3:2$. The minimum dissipation then becomes $30\pi\mu
aV_{\text{A}}^{2}$. The simple additivity of external and internal dissipation
only holds for a sphere. In general, the presence of internal dissipation will
influence the optimal external velocity profile and vice versa.

####  .3.2 Swimmer with an extended propulsive layer and tangential forces

The second problem we solve involves a swimmer with a no-slip surface
$\mathcal{S}_{i}$ that is propelled by tangential forces on a closed outer
surface $\mathcal{S}$ (Fig. 1b), also called control surface [25]. Such a
model has been proposed to describe the ciliary layer, for example in Volvox
[44]. We now apply the inequality for a no-slip body with the shape of the
inner surface ($\mathcal{S}_{i}$). The superposition is formed with a
composite passive body that has a no-slip boundary
$\tilde{\bm{v}}_{\text{CP}}=\bm{0}$ at $\mathcal{S}_{i}$ (Fig. 2c). At the
outer surface ($\mathcal{S}$), the boundary condition is zero tangential
velocity $(\bm{I}-\bm{n}\bm{n})\cdot\tilde{\bm{v}}_{\text{CP}}=\bm{0}$ and
continuity of normal stress $\bm{n}\cdot\bm{\sigma}\cdot\bm{n}$, i.e., zero
normal traction $\bm{n}\cdot\bm{f}_{\text{CP}}=0$. The definition of the
composite body ensures the orthogonality between active forces and the passive
flow, which is required in the derivation below. At the same time the choice
of tangential only forces at the outer surface ensures that it is always
possible to find an active swimmer that exactly satisfies the superposition
condition.

We can express the dissipation in the superposition flow as

$P_{\text{A+CP}}=-\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,(\bm{f}_{\text{A}}+\bm{f}_{\text{CP}})\cdot(\bm{v}_{\text{A}}+\bm{v}_{\text{CP}})\;.$
(13)

We now employ the Lorentz reciprocal theorem on the fluid domain outside
$\mathcal{S}_{i}$ with an additional surface integral of the traction on
$\mathcal{S}$. By combining the forms (44) and (45) we obtain
$\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,\bm{f}_{\text{CP}}\cdot\bm{v}_{\text{A}}=\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{CP}}=\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{CP}}+\bm{\mathsf{F}}_{\text{A}}\cdot\bm{\mathsf{V}}_{\text{CP}}=0$.
On the outer surface, the traction $\bm{f}_{\text{A}}$ is by definition
tangential and $\tilde{\bm{v}}_{\text{CP}}$ normal to the surface, therefore
their product is zero. This also holds for the inner surface where
$\tilde{\bm{v}}_{\text{CP}}=0$. Together with $\bm{\mathsf{F}}_{\text{A}}=0$,
all terms are identically zero. Therefore, the mixed terms in (13) integrate
to zero and we have proven the additivity of the dissipated power

$\displaystyle P_{\text{A+CP}}$
$\displaystyle=P_{\text{A}}-\bm{\mathsf{F}}_{\text{CP}}\cdot\bm{\mathsf{V}}_{\text{CP}}\;.$
(14)

From here, the new version of the minimum dissipation theorem follows in
complete analogy to the derivation of (10):

$P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{NSi}}^{-1}-\bm{\mathsf{R}}_{\text{CP}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\;,$
(15)

where $\bm{\mathsf{R}}_{\text{NSi}}$ is the drag coefficient of a no-slip body
described by the inner surface $\mathcal{S}_{i}$, and
$\bm{\mathsf{R}}_{\text{CP}}$ is the drag coefficient of a composite passive
body imposing a no-slip boundary condition at $\mathcal{S}_{i}$, with zero
tangential velocity and zero normal traction at $\mathcal{S}$. At the surface
$\mathcal{S}$ we have $\bm{f}_{\text{NSi}}$ and
$\tilde{\bm{v}}_{\text{CP}}^{\parallel}=0$. The optimal swimmer therefore has
a propulsive force $\bm{f}_{\text{A}}=-\bm{f}_{\text{CP}}$ and velocity
$\tilde{\bm{v}}_{\text{A}}^{\parallel}=\tilde{\bm{v}}_{\text{NSi}}^{\parallel}$
at the outer surface.

For a spherical body, the drag coefficient of the composite passive body with
the outer radius $a$ and the inner radius $b$ is (Sect. C.1 in the
Supplementary Information)

$R_{\text{CP}}=8\pi\mu\cdot\frac{20a^{5}+11a^{4}b+11a^{3}b^{2}+a^{2}b^{3}+ab^{4}+b^{5}}{28a^{4}+13a^{3}b+13a^{2}b^{2}+3ab^{3}+3b^{4}}\;.$
(16)

Together with the drag coefficient of the no-slip body $R_{\text{NS}}=6\pi\mu
b$, we obtain the limit on dissipation

$P_{\text{A}}\geq 24\pi\mu
bV_{\text{A}}^{2}\cdot\frac{20a^{5}+11a^{4}b+11a^{3}b^{2}+a^{2}b^{3}+ab^{4}+b^{5}}{5(a-b)\left(4a^{2}+ab+b^{2}\right)^{2}}\;.$
(17)

The dissipation as a function of the ratio of the inner to outer radius $b/a$
is shown in Fig. 3. If the propulsive layer is thick compared to the particle
size, $a\gg b$, the dissipation bound is $6\pi\mu bV_{\text{A}}^{2}$, which is
the dissipation by an externally driven sphere. This is also the lowest
possible dissipation in a flow around a no-slip sphere. In the limit $b\to a$,
if the propulsive layer becomes thin, the lowest dissipation is $6\pi\mu
V_{\text{A}}^{2}b/(a-b)$.

Figure 3: Dissipation by spherical swimmers with an extended propulsive layer.
Lower bound on dissipation by a spherical swimmer (radius $b$) with an
extended propulsive layer, such that the active forces act on a concentric
sphere with radius $a$. The red solid lines show the case of tangential
propulsive forces (17), the blue solid lines the case of normal forces (21)
and the green dashed lines the case of unconstrained forces (25).

####  .3.3 Swimmer propelled by normal forces

An additional problem that can be solved is the dissipation by a swimmer that
is also propelled by forces acting at outer surface ($\mathcal{S}$), but this
time the active forces have a direction normal to the surface, i.e.,
$\bm{f}_{\text{A}}\parallel\bm{n}$ (Fig. 1c). The motivation originates from
phoretic swimmers, where the fluid is set in motion by a potential (normal
force) gradient in the boundary layer. Other boundary conditions are
$\tilde{\bm{v}}_{\text{A}}=0$ at the inner surface $\mathcal{S}_{i}$ and the
continuity of velocity at the outer surface.

As in the previous case, we apply the inequality to a body with the no-slip
boundary and the shape of the inner surface $\mathcal{S}_{i}$. We form the
superposition between the active swimmer and a passive body consisting of a
no-slip core at $\mathcal{S}_{i}$ and a fluid-fluid interface at $\mathcal{S}$
(Fig. 2d). Again, we verify the additivity of dissipation for this
superposition, which can be expressed as

$P_{\text{A+DC}}=-\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,(\bm{f}_{\text{A}}+\bm{f}_{\text{DC}})\cdot(\bm{v}_{\text{A}}+\bm{v}_{\text{DC}})\;.$
(18)

Again we apply the Lorentz reciprocal theorem ((44) and (45)) on the fluid
domain outside $\mathcal{S}_{i}$ with additional tractions on $\mathcal{S}$ to
show
$\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,\bm{f}_{\text{DC}}\cdot\bm{v}_{\text{A}}=\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{DC}}=\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{DC}}+\bm{\mathsf{F}}_{\text{A}}\cdot\bm{\mathsf{V}}_{\text{DC}}=0$.
Again, we have $\bm{f}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{DC}}=0$ at both
integration surfaces. At the outer surface $\mathcal{S}$ this is because the
traction $\bm{f}_{\text{A}}$ is normal to the surface and the velocity
$\tilde{\bm{v}}_{\text{DC}}$ is tangential. At the inner surface
$\mathcal{S}_{i}$ their product also vanishes because
$\tilde{\bm{v}}_{\text{DC}}=0$. Therefore, the integrals of the mixed terms in
(18) vanish, proving the additivity
$P_{\text{A+DC}}=P_{\text{A}}-\bm{\mathsf{F}}_{\text{DC}}\cdot\bm{\mathsf{V}}_{\text{DC}}$.
The corresponding minimum dissipation theorem reads

$P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{NS}}^{-1}-\bm{\mathsf{R}}_{\text{DC}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\;.$
(19)

The superposition also states that the optimal swimmer has the active forces
$\bm{f}_{\text{A}}=-\bm{f}_{\text{DC}}$ and normal velocity
$\tilde{\bm{v}}_{\text{A}}^{\perp}=\tilde{\bm{v}}_{\text{NSi}}^{\perp}$ at the
outer surface.

As an example, we calculate the dissipation by a spherical swimmer with inner
radius $b$ and radius of the propulsive layer $a$. The drag coefficient of the
droplet with a no-slip core is (Sect. C.2 in the Supplementary Information)

$R_{\text{DC}}=24\pi\mu
b\cdot\frac{5a^{3}+6a^{2}b+3ab^{2}+b^{3}}{3(8a^{2}b+9ab^{2}+3b^{3})}$ (20)

Together with the $R_{\text{NS}}=6\pi\mu b$, we obtain

$P_{\text{A}}\geq 6\pi\mu
bV_{\text{A}}^{2}\cdot\frac{4(5a^{3}+6a^{2}b+3ab^{2}+b^{3})}{5(2a+b)^{2}(a-b)}\;.$
(21)

The above dependence is shown by the blue line in Fig. 3. In the limit $b\to
0$, the lower bound is $6\pi\mu bV_{\text{A}}^{2}$, which corresponds to a
sphere pulled by an external force. For $b\to a$, the result is $8\pi\mu
b^{2}/(a-b)$. While the expression is similar as for tangential propulsion,
the dissipation is higher by a factor $4/3$.

####  .3.4 Swimmer propelled by unconstrained forces

We now relax the constraint from the last two cases where we required the
forces to be tangential or normal and allow any distribution of forces forces
acting at outer surface ($\mathcal{S}$) (Fig. 1d). As illustrated in Fig. 2e,
we use a superposition between the active swimmer and a no-slip body described
by the outer surface and apply the inequality to the inner shape.

The dissipation of the superposition now reads

$P_{\text{A+NS}}=-\int_{\mathcal{S}+\mathcal{S}_{i}}\mathrm{d}S\,(\bm{f}_{\text{A}}+\bm{f}_{\text{NS}})\cdot(\bm{v}_{\text{A}}+\bm{v}_{\text{NS}})\;.$
(22)

We apply the Lorentz reciprocal theorem ((44) and (45)) on the volume outside
the surface $\mathcal{S}_{i}$ with additional traction at $\mathcal{S}$.
Because the velocity field is also needed in the space between the to
surfaces, we treat the no-slip body as a hollow fluid-filled shell such that
$\tilde{\bm{v}}_{\text{NS}}=0$ inside $\mathcal{S}$. We again see that the
mixed terms disappear because $\tilde{\bm{v}}_{\text{NS}}=0$ and show the
additivity

$P_{\text{A+NS}}=P_{\text{A}}-\bm{\mathsf{F}}_{\text{NS}}\cdot\bm{\mathsf{V}}_{\text{NS}}\;.$
(23)

Now the inequality reads
$P_{\text{A+NS}}\geq(\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}})\cdot\bm{\mathsf{R}}_{\text{NSi}}\cdot(\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}})$.
Combined, these two equations give the minimum dissipation theorem

$P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{NSi}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\;,$
(24)

where $\bm{\mathsf{R}}_{\text{NS}}$ and $\bm{\mathsf{R}}_{\text{NSi}}$ are the
drag coefficients of no-slip bodies with shapes $\mathcal{S}$ and
$\mathcal{S}_{i}$, respectively. The active traction on the outer surface of
the optimal swimmer is $\bm{f}_{\text{A}}=-\bm{f}_{\text{NS}}$ and the
velocity $\tilde{\bm{v}}_{\text{A}}=\tilde{\bm{v}}_{\text{NSi}}$.

In the case of a spherical swimmer with $R_{\text{NS}}=6\pi\mu a$ and
$R_{\text{NSi}}=6\pi\mu b$, the minimum dissipation is

$P_{\text{A}}\geq 6\pi\mu bV_{\text{A}}^{2}\cdot\frac{a}{a-b}\;,$ (25)

which is naturally always lower than the dissipation in the more restrictive
cases of tangential or normal forces (Fig. 3). The advantage over tangential
propulsion is small, however, and never exceeds $\lesssim 1\,\%$ for spherical
swimmers.

####  .3.5 Swimmer with surface dissipation

The last problem for which we can calculate the efficiency limit is a surface-
driven swimmer for which the maintenance of the slip velocity comes with an
energetic cost, namely with the power density $\zeta_{\rm
s}\tilde{\bm{v}}^{2}$, where $\zeta_{\rm s}$ is a constant that characterizes
the efficiency of the propulsion mechanism (Fig. 1e). The total dissipated
power is defined as:

$P_{\text{A}}=\int_{\cal
S}\mathrm{d}S\left(-\bm{f}_{\text{A}}\cdot\bm{v}_{\text{A}}+\zeta_{\rm
s}\tilde{\bm{v}}_{\text{A}}^{2}\right)\;.$ (26)

In this case, we form the superposition with a no-slip body of the same shape
and derive an expression for the dissipation of the superposition flow,
including internal dissipation (Fig. 2f),

$P_{\text{A+NS}}=\int_{\cal
S}\mathrm{d}S\left(-(\bm{f}_{\text{A}}+\bm{f}_{\text{NS}})\cdot(\bm{v}_{\text{A}}+\bm{v}_{\text{NS}})+\zeta_{\rm
s}\tilde{\bm{v}}^{2}_{\text{A}}\right)\;.$ (27)

In the second term we took into account that $\tilde{\bm{v}}_{\text{NS}}=0$
and therefore only $\tilde{\bm{v}}_{\text{A}}$ contributes to the internal
dissipation. From the reciprocal theorem ((44) and (45)) it follows that
$\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{NS}}\cdot\bm{v}_{\text{A}}=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{NS}}=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{NS}}+\bm{\mathsf{F}}_{\text{A}}\cdot\bm{\mathsf{V}}_{\text{NS}}=0$,
proving the additivity

$P_{\text{A+NS}}=P_{\text{A}}-\bm{\mathsf{F}}_{\text{NS}}\cdot\bm{\mathsf{V}}_{\text{NS}}\;.$
(28)

The choice of the V-problem here differs from the previous cases where it only
needed to fulfill the boundary condition, i.e., zero normal velocity on the
surface. Here, we need a problem that minimizes the dissipation including the
internal contribution. This is the case for a body with the Navier slip
condition on the surface. Therefore, the superposition flow satisfies the
inequality (5)

$P_{\text{A+NS}}\geq(\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}})\cdot\bm{\mathsf{R}}_{\text{Navier}}\cdot(\bm{\mathsf{V}}_{\text{A}}+\bm{\mathsf{V}}_{\text{NS}})\;.$
(29)

In analogy with the previous cases, we derive the minimum dissipation theorem
which reads

$P_{\text{A}}\geq\bm{\mathsf{V}}_{\text{A}}\cdot\left(\bm{\mathsf{R}}_{\text{Navier}}^{-1}-\bm{\mathsf{R}}_{\text{NS}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}\,,$
(30)

where $\bm{\mathsf{R}}_{\text{Navier}}$ is the drag coefficient of a body with
the Navier-slip boundary with slip length $\lambda=\mu/\zeta_{\rm s}$. In the
limit $\zeta_{\rm s}\to 0$, we obtain the perfect slip body, as in the
original theorem for external-only dissipation [38].

As in the other cases, the superposition states that the slip velocity of the
optimal swimmer is $\tilde{\bm{v}}_{\text{A}}=\tilde{\bm{v}}_{\text{Navier}}$.
However, the internal dissipation model differs presents an exception with
regard to the optimal active forces. Among the cases discussed here, it is the
only one where both passive problems that form the superposition have non-zero
tractions. Therefore, the optimal swimmer has the traction
$\bm{f}_{\text{A}}^{\parallel}=\bm{f}_{\text{Navier}}^{\parallel}-\bm{f}_{\text{NS}}^{\parallel}$.

In the simplest case of a spherical swimmer, the drag coefficient of the
sphere with Navier slip boundary is [45]

$R_{\text{Navier}}=6\pi\mu a\cdot\frac{1+2\lambda/a}{1+3\lambda/a}\;.$ (31)

The resulting active dissipation is

$P_{\text{A}}\geq 6\pi\mu
aV_{\text{A}}^{2}\left(2+\frac{a}{\lambda}\right)=6\pi
aV_{\text{A}}^{2}\left(2\mu+a\zeta_{\rm s}\right)\;.$ (32)

Again, the additive nature of the external and internal contributions is
limited to spherical geometry.

In the limit $\lambda\ll a$, the leading term in the dissipation agrees with
that with external propulsion if we use $a$ for the inner radius and
$a+\lambda$ for the outer.

Figure 4: Optimal shapes of active surface-driven droplets. a The optimal
shapes of active droplets with three different ratios of the internal to
external viscosity $\mu_{i}/\mu$ and their flow fields. b The dissipation
$\hat{P}_{\text{A}}=P_{\text{A}}/(6\pi(2\mu+3\mu_{i})aV_{\text{A}}^{2})$ of
the optimal active surface-driven droplet, relative to the equivalent droplet
of a spherical shape (12), as a function of the viscosity ratio. The red dot
indicates a topological bifurcation above which a toroidal shape (solid line)
has a dissipation lower than the optimal swimmer with spherical topology
(dashed line). The insets show the optimal shapes at viscosity ratios marked
with squares. Figure 5: Optimal shapes of swimmers with surface dissipation. a
Numerically obtained optimal shapes that minimize the dissipation as a
function of the internal dissipation density $\zeta_{\rm s}$ while keeping the
total volume of the swimmer fixed ($V=4\pi a^{3}/3$). The shapes are
restricted by the minimum curvature radius $r_{\rm min}=\hat{r}_{\rm min}a$. b
Streamlines in the co-moving frame and propulsion velocity $\tilde{v}$ (color
coded) for three optimal shapes, obtained with $a\zeta_{\rm s}/\mu=4$. c The
dissipation by optimal swimmers as a function of the prescribed minimum
curvature radius for a set of internal dissipation densities $a\zeta_{\rm
s}/\mu$. The dissipation is normalized by that of a spherical swimmer,
$\hat{P}_{\text{A}}=P_{\text{A}}/(6\pi aV_{\text{A}}^{2}(2\mu+a\zeta_{\rm
s}))$.

###  .4 Swimmers with optimal shapes

The minimum dissipation theorem provides us with the lowest value of
dissipation rate by a swimmer of a given shape, moving with a given velocity.
We can now make one step further and ask the question about the lowest
possible dissipation by a swimmer of a given volume, regardless of its shape.
In the following we apply the newly derived theorems to determine the optimal
shapes of microswimmers with combined internal and external dissipation. The
optimization problem consists of minimizing the dissipation while keeping
volume and the swimming speed constant. It is well known that the optimal
shape with external dissipation only is not well defined because the
dissipation vanishes in the limit of an infinitely elongated needle [34]. In
the opposite limit, when the internal dissipation, modeled with the term
$\zeta_{\rm s}\tilde{\bm{v}}^{2}$, dominates, the optimal swimmer shape
consists of a body with two thin elongated protrusions along the symmetry axis
[24]. Also those shapes could only be determined by additionally restricting
the allowed curvature of the surface with a dimensionless minimum radius
$\hat{r}_{\text{min}}$, normalized such that $\hat{r}_{\text{min}}=1$
restricts the shape to a spherical body.

####  .4.1 Optimal shape of the active droplet

In the following we numerically determine the optimal shape of a fluid-filled
surface driven swimmer of the type of a self-propelled droplet. For each
axisymmetric shape, we numerically determine the drag coefficients of the
passive droplet and the no-slip body using a custom written boundary element
method (based on Green’s functions from BEMLIB [46]) and use the theorem to
calculate the minimum dissipation by the active droplet. We parameterize the
shape as a chain of segments of equal length, linearly rescale it to impose a
fixed volume, and then use a numerical optimization procedure to determine the
shape with the minimum dissipation.

The resulting shapes obtained for different ratios of the internal to external
viscosity ($\mu_{i}/\mu$) are shown in Fig. 4a. Figure 4b shows the obtained
minimum dissipation, relative to that of a spherical shape (12), along with a
selection of optimal shapes. As anticipated, with a vanishing internal
viscosity, $\mu_{i}=0$, only external dissipation remains relevant and the
optimal shape becomes that of an infinitely thin needle. In the case of equal
viscosities, $\mu_{i}=\mu$, the optimal shape is close to a prolate spheroid
with an aspect ratio of 1.16. The dissipated power is 0.991 that of a
spherical droplet, indicating that the advantage over spherical shape is tiny.
On the other hand, if the internal viscosity dominates, the optimal shapes
first become oblate and eventually transition to a toroid. The discontinuous
topological transition takes place at a viscosity ratio $\mu_{i}/\mu\approx
4$. The propulsion by a toroid rotating inside-out has been studied in the
literature for a long time [47, 11, 48], but only recently a flagellate with a
swimming mode using the same principle has been reported [49].

####  .4.2 Optimal shape of the swimmer with surface dissipation

The second class of swimmers for which we determine the optimal shapes are the
swimmers with internal surface dissipation. Here we face the problem that the
mathematically optimal shapes contain infinitely long axial protrusions. In
order to perform the optimization among realistic shapes, we need to
additionally restrict the radius of curvature. For any point on the surface,
we demand that both principal curvatures $\kappa_{1,2}\leq 1/r_{\rm min}$,
where the minimum radius is determined by its dimensionless value as $r_{\rm
min}=\hat{r}_{\rm min}a$, where $a=\sqrt[3]{3V/4\pi}$ is the radius of the
sphere with the equivalent volume. Again, we use a custom written axisymmetric
boundary element solver to determine the drag coefficients for a given shape
(first rescaled to unit volume) and run the shape parametrization through a
constrained minimization routine.

Examples of optimal shapes for different internal dissipation densities
$\zeta_{\rm s}$ and curvature radii $\hat{r}_{\rm min}$ are shown in Fig. 5a,
with some of the flows shown in Fig. 5b. The minimum dissipated power
$P_{\text{A}}$, scaled by the minimum dissipation of a spherical swimmer (32),
as a function of both parameters is shown in Fig. 5c. We have previously shown
that in the case of dominant internal dissipation [24], the advantage over a
spherical swimmer is $\lesssim 20\,\%$ over a wide range of realistic shapes.
Here, we show that the optimal shape has a much larger influence on the
dissipation when the combination of internal and external dissipation is taken
into account.

###  .5 Swimmer under external force

In addition to the active propulsion, the swimmer can be subject to an
external force, for example when moving in the presence of gravity. Here, we
will determine how velocity and dissipation will be affected in the presence
of an external force $\bm{\mathsf{F}}_{\text{ext}}$. The assumption we make is
that the active driving force density $\bm{f}_{\text{A}}$ remains constant,
whereas the velocity and the dissipated power are affected by the external
force. For example, in the case of the active droplet, we assume that the
tangential driving force on the surface remains constant, but the surface
velocity $\tilde{\bm{v}}_{\text{A}}$ is reduced when the swimmer is pulling
(or pushing) against a resistive load. The assumption of a fixed driving force
is less straightforward in the case of internal dissipation, where we assume
that the additional load affects the slip velocity in the same way as it would
on a boundary with the Navier slip and see that the expression derived above
remains valid.

Because the solution with an external, but without active forces, represents
exactly the V-problem, the swimmer under load can be represented as a
superposition of the unloaded active swimmer and the body from the V-problem
(e.g., passive droplet or no-slip core). The velocity response to the applied
force is therefore determined as

$\bm{\mathsf{V}}_{\text{A}}=\bm{\mathsf{V}}_{\text{A}}^{0}+\bm{\mathsf{R}}_{\text{V}}^{-1}\cdot\bm{\mathsf{F}}_{\text{ext}}\;.$
(33)

Here, $\bm{\mathsf{V}}_{\text{A}}^{0}$ is the unperturbed swimming velocity
and $\bm{\mathsf{R}}_{\text{V}}$ is the drag coefficient of the V-problem.
Thus, the mobility of the passive body from the V-problem
($\bm{\mathsf{R}}_{\text{V}}^{-1}$) acts as the velocity response function of
the active swimmer.

Under the same assumption, the power produced by the active swimmer, which
corresponds to the dissipated power reduced by the power contributed by the
external force, is (see Sect. D in the Supplementary Information for a
derivation)

$P_{\text{A}}(\bm{\mathsf{F}}_{\text{ext}})=P_{\text{A}}^{0}+\bm{\mathsf{F}}_{\text{ext}}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}\;,$
(34)

with $P^{0}_{\text{A}}=P_{\text{A}}(\bm{\mathsf{F}}_{\text{ext}}=0)$ denoting
the dissipated power in the absence of external force (Fig. 6). Interestingly,
the optimal force distribution does not change with the applied load: the
force distribution that minimizes dissipation when moving freely with a given
velocity will still be optimal under load and always follows the force
distribution from the F-problem. The dissipated power is the sum of the power
produced by the swimmer and that of the external force

$\displaystyle P_{\text{diss}}(\bm{\mathsf{F}}_{\text{ext}})$
$\displaystyle=P_{\text{A}}(\bm{\mathsf{F}}_{\text{ext}})+\bm{\mathsf{F}}_{\text{ext}}\cdot\bm{\mathsf{V}}_{\text{A}}$
$\displaystyle=P_{\text{A}}^{0}+2\,\bm{\mathsf{F}}_{\text{ext}}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}+\bm{\mathsf{F}}_{\text{ext}}\cdot\bm{\mathsf{R}}_{\text{V}}^{-1}\cdot\bm{\mathsf{F}}_{\text{ext}}\;,$
(35)

or expressed with the velocity

$P_{\text{diss}}(\bm{\mathsf{V}}_{\text{A}})=P_{\text{diss}}(\bm{\mathsf{V}}_{\text{A}}=0)+\bm{\mathsf{V}}_{\text{A}}\cdot\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}\;.$
(36)

The second term is the dissipation that is expected for an object driven by an
external force (Fig. 6). The first term represents the dissipation rate of a
swimmer that is brought to a stall by an external force. Its lower bound is

$P_{\text{diss}}(\bm{\mathsf{V}}_{\text{A}}=0)\geq\bm{\mathsf{V}}_{\text{A}}^{0}\cdot\left[\left(\bm{\mathsf{R}}_{\text{V}}^{-1}-\bm{\mathsf{R}}_{\text{F}}^{-1}\right)^{-1}-\bm{\mathsf{R}}_{\text{V}}\right]\cdot\bm{\mathsf{V}}_{\text{A}}^{0}\;.$
(37)

In other words, the dissipation can be decomposed into one component that is
required to run the force-generation mechanism
$P_{\text{diss}}(V_{\text{A}}=0)$ and another one that is identical to the
dissipation by a passive body from the V-problem (i.e., a perfect-slip body or
a passive bubble).

Figure 6: Force-dependent dissipation. Dissipation by an active microswimmer
subject to an external force $\bm{\mathsf{F}}_{\text{ext}}$. The dissipated
power ($P_{\text{diss}}$, blue) is the sum of the power output of the swimmer
($P_{A}$, green) and the rate of work by the external force (brown). The
dissipated power differs from the expression
$\bm{\mathsf{V}}_{\text{A}}\cdot\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}$
(grey) by a constant offset (red arrow). The diagram is calculated with a
microswimmer efficiency of $\eta_{m}=0.4$.

###  .6 Rate of entropy production

By taking into account the thermal noise acting on the swimmer as well as an
additional force $\bm{\mathsf{F}}$, the equations of motion are

$\displaystyle\dot{\bm{X}}$
$\displaystyle=\left[\bm{\mathsf{V}}_{\text{A}}^{0}+\bm{\mathsf{R}}_{\text{V}}^{-1}\cdot\bm{\mathsf{F}}+\bm{\xi}\right]_{[1,2,3]}$
(38a) $\displaystyle\dot{\bm{n}}_{\alpha}$
$\displaystyle=\left[\bm{\mathsf{V}}_{\text{A}}^{0}+\bm{\mathsf{R}}_{\text{V}}^{-1}\cdot\bm{\mathsf{F}}+\bm{\xi}\right]_{[4,5,6]}\times{\bm{n}}_{\alpha}$
(38b)

where $\bm{X}$ is the spatial coordinate of the particle and
${\bm{n}}_{\alpha}$ one of the three vectors describing the orientation of the
swimmer. The product in the expression for rotational motion is carried out
using the Stratonovich interpretation. The Brownian noise is characterized by
its autocorrelation function

$\langle\bm{\xi}(t)\bm{\xi}(t+\tau)\rangle=2\bm{\mathsf{D}}\delta(\tau)$ (39)

where the diffusion tensor follows from the Stokes-Einstein relationship as
$\bm{\mathsf{D}}=k_{\text{B}}T\,\bm{\mathsf{R}}_{\text{V}}^{-1}$. Here we
disregard any additional active sources of noise resulting from the
stochasticity of the propulsion mechanism.

The entropy production of the above particle in steady state is commonly
referred to as “housekeeping” entropy [50]. In the classical picture that
replaces the self-propulsion velocity $\bm{\mathsf{V}}_{\text{A}}^{0}$ with
the motion due to an active force,
$\bm{\mathsf{R}}_{\text{V}}^{-1}\cdot\bm{\mathsf{F}}_{\text{A}}^{0}$, the mean
entropy production can be expressed from the statistical definition as in
Refs. [16, 17, 18]. For translational motion, it can be written as
$T\dot{S}_{\text{hk}}=\langle\dot{\bm{X}}\cdot(\bm{F}+\bm{R}_{\text{V}}\cdot\bm{V}_{0})\rangle$
where $\dot{\bm{X}}$ is the stochastic velocity (38a) and
$\bm{F}+\bm{R}_{\text{V}}\cdot\bm{V}_{0}$ the deterministic total force,
consisting of the external and the active contribution. The brackets indicate
averaging over noise realizations, as well as over particle positions and
orientations. By carrying out the noise average,
$\langle\dot{\bm{X}}\rangle=(\bm{R}_{\text{V}}^{-1}\cdot\bm{F}+\bm{V}_{0})$
and switching to our 6-component notation, the “housekeeping” entropy
production reads

$T\dot{S}_{\text{hk}}=\langle\bm{\mathsf{V}}_{\text{A}}\cdot\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}\rangle$
(40)

with the brackets indicating averaging over particle positions.

At this point we apply the finding (36) that the minimum dissipation rate by
the active swimmer can be written as a sum of a constant and a term
proportional to the square of the actual velocity (swimming velocity plus
drift caused by external forces). The second term is identical to the
expression in Eq. (40). By adding the zero-velocity dissipation (37), the
total “housekeeping” entropy production increases to

$T\dot{S}_{\text{hk}}^{\text{tot}}\geq\bm{\mathsf{V}}_{\text{A}}^{0}\cdot\left[\left(\bm{\mathsf{R}}_{\text{V}}^{-1}-\bm{\mathsf{R}}_{\text{F}}^{-1}\right)^{-1}-\bm{\mathsf{R}}_{\text{V}}\right]\cdot\bm{\mathsf{V}}_{\text{A}}^{0}+\langle\bm{\mathsf{V}}_{\text{A}}\cdot\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}\rangle$
(41)

For free-swimming particles, the housekeeping entropy production can be
expressed by means of the microswimmer efficiency, as defined in Ref. [38],

$\eta_{m}=\frac{\bm{\mathsf{V}}_{\text{A}}^{0}\cdot\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}}{\bm{\mathsf{V}}_{\text{A}}^{0}\cdot\left(\bm{\mathsf{R}}_{\text{V}}^{-1}-\bm{\mathsf{R}}_{\text{F}}^{-1}\right)^{-1}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}}\;.$
(42)

Its lower bound is then given by

$T\dot{S}_{\text{hk}}^{\text{tot}}\geq\frac{1}{\eta_{m}}\bm{\mathsf{V}}_{\text{A}}^{0}\cdot\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}\;.$
(43)

Our expression points to a fundamental limit on dissipation by active
particles that is stricter than obtained by treating them as if they were
passive particles with an external driving force
$\bm{\mathsf{R}}_{\text{V}}\cdot\bm{\mathsf{V}}_{\text{A}}$ and a drag
coefficient $\bm{\mathsf{R}}_{\text{V}}$, because the housekeeping dissipation
needs to be offset by the contribution of (37).

## Discussion

In this study we derived fundamental limits on dissipation by several classes
of microswimmers with internal dissipation. The scenarios we discussed
describe the major propulsion mechanisms by different microswimmers: active
droplets driven by the Marangoni effect [43], ciliated microswimmers that can
be approximated with an extended force-generating layer [44], phoretic
swimmers with a position dependent zeta potential, and finally a coarse-
grained model with an arbitrary local dissipation density.

Determining a lower bound on the dissipation requires finding the distribution
of forces that minimize dissipation while maintaining a given swimming speed
for a given type and shape of swimmer. In principle, this presents a complex
PDE constrained optimization problem [36]. However, it can be solved by
generalizing a very powerful approach previously derived for swimmers with
external dissipation alone, i.e., swimmers with an idealized, lossless,
propulsion mechanism [38]. The original minimum dissipation theorem allowed us
to express the minimum external dissipation by a microswimmer with two passive
drag coefficients – one of a no-slip and one with a perfect-slip body of the
same shape. The derivation of the theorem was based on two properties of the
Stokes flow: the Helmholtz minimum dissipation theorem [51] and the Lorentz
reciprocal theorem [40]. Here, we show that under certain conditions analogous
theorems can be derived for other swimmer models that do take into account
internal dissipation. Specifically, one needs to find two passive problems
that satisfy the velocity boundary conditions of the original problem. One of
them needs to represent the flow with minimum dissipation under these boundary
conditions and the other one a passive problem with velocities orthogonal to
the active forces. The lower bound on dissipation can then be expressed with
the reciprocal difference between the two drag coefficients. Although we
restricted our discussion to five scenarios, we expect that other minimum
dissipation problems can be solved with the same method. A straightforward
example is a microswimmer with a propulsive layer covering only a part of its
surface. We also expect that the theorem can be generalized to other
propulsion mechanisms, for example surface tension around a swimmer embedded
in a liquid-air interface (“Marangoni surfer” [52, 53]). Note that besides
determining the bound on dissipation, the theorem, by means of linear
superposition, also provides a distribution of forces that exactly achieves
this bound, thus fully solving the optimization problem.

We subsequently expanded the optimality to a class of swimmers with a fixed
volume, but different shapes. The optimal shape of a surface-driven droplet
depends on the ratio between the internal external viscosity. When the
internal viscosity is small, the optimal shape unsurprisingly becomes prolate
and eventually needle-like, in order to minimize the external dissipation. On
the other hand, if the internal viscosity is large, the optimal shape
undergoes a topological bifurcation and takes the shape of a toroid rotating
inside-out, resembling some swimmer models studied by Taylor [47] and Purcell
[11]. Swimmers with an effective surface dissipation always become elongated
and, like in the case of surface dissipation alone, take the shape of a body
with two protrusions along the symmetry axis if sufficient curvature is
permitted. These shapes bear a remarkable resemblance with many ciliates found
in nature [24].

When the swimmer is moving in the presence of an external force, the total
dissipation can be decomposed into a constant term and a term that corresponds
to the drag of a passive body moved with the same velocity. The dissipation
therefore reaches its minimum when the swimmer is stalled. The drag
coefficient that appears in the expression for dissipation is the same drag
coefficient that describes the velocity response to an external force, and
also determines the noise amplitude through the fluctuation-response theorem.
In contrast with the commonly used picture in which the propulsion of a
microswimmer is replaced by an external “active force” [16, 14, 17], we find
that the dissipation and entropy production need to be corrected for a
contribution describing the energetic cost of force generation. The same holds
for studies that determine the entropy production from the statistical
definition [54, 18, 55]. Our work therefore complements the previous studies
on fundamental limits on entropy production in active microswimmer suspensions
by adding an unavoidable contribution from internal dissipation and also from
the fact that swimming usually generates more dissipation than pulling by a
force. It needs to be stressed that these results hold under the assumption
that the presence of an external load does not affect the active driving
forces on the swimmer. In principle, other types of response are also
conceivable. For example, the opposite limit would be a mechanism maintaining
a prescribed slip velocity regardless of the stress on the surface. In such
cases, the velocity dependent entropy production rate can have a different
form. Furthermore, the entropy production can be influenced by interactions
between swimmers, in particular when they are non-conservative [56].

Finally, while our study can provide a complete hydrodynamic picture of the
problem, it does not take into account the dissipation by the mechanism of
force production, for example a chemical reaction in a phoretic swimmer. The
energetics of the latter was addressed in several studies [29, 19], but a
complete solution providing a dissipation limit for the combined chemical and
hydrodynamic problem still presents an open challenge. Likewise, a major open
question is whether our approach can be used to provide a dissipation limits
for swimmers that move by periodically changing their shape, which was thus
far possible only under limited constraints (e.g., the three sphere swimmer
[57]).

## Methods

### Lorentz reciprocal theorem

In the following we recapitulate the Lorentz reciprocal theorem [58, 40],
which provides us a relationship between two different flow problems sharing
the same geometry and fluid medium. The main problem (here denoted as A) and
the auxiliary problem (B) are then connected through the following integral
relationship:

$\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\bm{v}_{\text{B}}+\int_{\mathcal{V}}\mathrm{d}V\,\bm{\mathfrak{f}}_{\text{A}}\cdot\bm{v}_{\text{B}}\\\
=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{B}}\cdot\bm{v}_{\text{A}}+\int_{\mathcal{V}}\mathrm{d}V\,\bm{\mathfrak{f}}_{\text{B}}\cdot\bm{v}_{\text{A}}\,.$
(44)

Here $\bm{f}$ denotes the traction on the integration surface and
$\bm{\mathfrak{f}}$ any additional body force in the integration volume.

If problem A consists of a body moving with the rigid body velocity
$\bm{\mathsf{V}}_{\text{A}}$ and problem B of a body with the same shape
moving with $\bm{\mathsf{V}}_{\text{B}}$, the reciprocal theorem can also be
expressed with co-moving velocities

$\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{B}}+\int_{\mathcal{V}}\mathrm{d}V\,\bm{\mathfrak{f}}_{\text{A}}\cdot\tilde{\bm{v}}_{\text{B}}+\bm{\mathsf{F}}_{\text{A}}\cdot\bm{\mathsf{V}}_{\text{B}}\\\
=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{B}}\cdot\tilde{\bm{v}}_{\text{A}}+\int_{\mathcal{V}}\mathrm{d}V\,\bm{\mathfrak{f}}_{\text{B}}\cdot\tilde{\bm{v}}_{\text{A}}+\bm{\mathsf{F}}_{\text{B}}\cdot\bm{\mathsf{V}}_{\text{A}}\,.$
(45)

A classical application is to apply the reciprocal theorem in this form to an
active force-free swimmer ($\bm{\mathsf{F}}_{\text{A}}=0$) and a no-slip body
($\text{B}\equiv\text{NS}$) of the same shape, which satisfies
$\tilde{\bm{v}}_{\text{B}}=\tilde{\bm{v}}_{\text{NS}}=0$. Then the left-hand
side of Eq. (45) is zero and the right-hand side yields
$\bm{\mathsf{F}}_{\text{NS}}\cdot\bm{\mathsf{V}}_{\text{A}}=-\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}_{\text{NS}}\cdot\tilde{\bm{v}}_{\text{A}}$,
which is an elegant and frequently used way of determining the velocity of the
active swimmer $\bm{\mathsf{V}}_{\text{A}}$ if one knows its surface velocity
$\tilde{\bm{v}}_{\text{A}}$ [30, 59, 40].

## Acknowledgements

We thank Benoît Mahault for insightful comments on the manuscript and Ray
Goldstein for drawing our attention to the core-shell model of Volvox which
inspired the tangentially driven swimmer model. We acknowledge support from
the Max Planck Center Twente for Complex Fluid Dynamics, the Max Planck School
Matter to Life, and the MaxSynBio Consortium, which are jointly funded by the
Federal Ministry of Education and Research (BMBF) of Germany and the Max
Planck Society. A.V. acknowledges support from the Slovenian Research Agency
(grant no. P1-0099).

## References

  * [1] Lauga, E. & Powers, T. R.  The hydrodynamics of swimming microorganisms.  _Rep. Prog. Phys._ 72, 096601 (2009).
  * [2] Elgeti, J., Winkler, R. G. & Gompper, G.  Physics of microswimmers–single particle motion and collective behavior: a review.  _Rep. Prog. Phys._ 78, 056601 (2015).
  * [3] Bechinger, C. _et al._ Active particles in complex and crowded environments.  _Rev. Mod. Phys._ 88, 045006 (2016).
  * [4] Prandtl, L.  Aufgaben der Strömungsforschung (lecture delivered at the inauguration of the Kaiser Wilhelm Institute for Flow Research in Göttingen, 16 July 1925).  _Naturwissenschaften_ 14, 335–338 (1926).
  * [5] Taylor, G. I.  Analysis of the swimming of microscopic organisms.  _Proc. R. Soc. Lond. A_ 209, 447–461 (1951).
  * [6] Najafi, A. & Golestanian, R.  Simple swimmer at low Reynolds number: Three linked spheres.  _Phys. Rev. E_ 69, 062901 (2004).
  * [7] Golestanian, R.  Phoretic Active Matter.  In Gompper, G., Marchetti, M. C., Tailleur, J., Yeomans, J. M. & Salomon, C. (eds.) _Active matter and nonequilibrium statistical physics_ , vol. 112 of _Lecture Notes of the Les Houches Summer School_ , 230–293 (Oxford University Press, Oxford, UK, 2022).
  * [8] Sharan, P., Nsamela, A., Lesher‐Pérez, S. C. & Simmchen, J.  Microfluidics for microswimmers: Engineering novel swimmers and constructing swimming lanes on the microscale, a tutorial review.  _Small_ 17, 2007403 (2021).
  * [9] Sharan, P., Daddi-Moussa-Ider, A., Agudo-Canalejo, J., Golestanian, R. & Simmchen, J.  Pair interaction between two catalytically active colloids.  _Small_ 2300817 (2023).
  * [10] Lighthill, M. J.  On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers.  _Comm. Pure Appl. Math_ 5, 109–118 (1952).
  * [11] Purcell, E. M.  Life at low Reynolds number.  _Am. J. Phys._ 45, 3–11 (1977).
  * [12] Katsu-Kimura, Y., Nakaya, F., Baba, S. A. & Mogami, Y.  Substantial energy expenditure for locomotion in ciliates verified by means of simultaneous measurement of oxygen consumption rate and swimming speed.  _J. Exp. Biol._ 212, 1819–1824 (2009).
  * [13] Pietzonka, P. & Seifert, U.  Entropy production of active particles and for particles in active baths.  _J. Phys. A: Math. Theor._ 51, 01LT01 (2018).
  * [14] Shankar, S. & Marchetti, M. C.  Hidden entropy production and work fluctuations in an ideal active gas.  _Phys. Rev. E_ 98, 020604(R) (2018).
  * [15] Tociu, L., Fodor, E., Nemoto, T. & Vaikuntanathan, S.  How dissipation constrains fluctuations in nonequilibrium liquids: Diffusion, structure, and biased interactions.  _Phys. Rev. X_ 9, 041026 (2019).
  * [16] Speck, T.  Stochastic thermodynamics for active matter.  _EPL_ 114, 30006 (2016).
  * [17] Dabelow, L., Bo, S. & Eichhorn, R.  Irreversibility in active matter systems: Fluctuation theorem and mutual information.  _Phys. Rev. X_ 9, 021009 (2019).
  * [18] Szamel, G.  Stochastic thermodynamics for self-propelled particles.  _Phys. Rev. E_ 100, 050603(R) (2019).
  * [19] Speck, T.  Active Brownian particles driven by constant affinity.  _EPL_ 123, 20007 (2018).
  * [20] Blake, J. R.  A spherical envelope approach to ciliary propulsion.  _J. Fluid Mech._ 46, 199–208 (1971).
  * [21] Najafi, A. & Golestanian, R.  Propulsion at low Reynolds number.  _J. Phys.: Condens. Matter_ 17, S1203–S1208 (2005).
  * [22] Jülicher, F. & Prost, J.  Generic theory of colloidal transport.  _Eur. Phys. J. E_ 29, 27–36 (2009).
  * [23] Osterman, N. & Vilfan, A.  Finding the ciliary beating pattern with optimal efficiency.  _Proc. Natl. Acad. Sci. U.S.A._ 108, 15727–15732 (2011).
  * [24] Vilfan, A.  Optimal shapes of surface slip driven self-propelled microswimmers.  _Phys. Rev. Lett._ 109, 128105 (2012).
  * [25] Keller, S. R. & Wu, T. Y.  A porous prolate-spheroidal model for ciliated micro-organisms.  _J. Fluid Mech._ 80, 259–278 (1977).
  * [26] Ito, H., Omori, T. & Ishikawa, T.  Swimming mediated by ciliary beating: comparison with a squirmer model.  _J. Fluid Mech._ 874, 774–796 (2019).
  * [27] Omori, T., Ito, H. & Ishikawa, T.  Swimming microorganisms acquire optimal efficiency with multiple cilia.  _Proc. Natl. Acad. Sci. U.S.A._ 117, 30201–30207 (2020).
  * [28] Sabass, B. & Seifert, U.  Efficiency of surface-driven motion: Nanoswimmers beat microswimmers.  _Phys. Rev. Lett._ 105, 218103 (2010).
  * [29] Sabass, B. & Seifert, U.  Dynamics and efficiency of a self-propelled, diffusiophoretic swimmer.  _J. Chem. Phys._ 136, 064508 (2012).
  * [30] Stone, H. A. & Samuel, A. D. T.  Propulsion of microorganisms by surface distortions.  _Phys. Rev. Lett._ 77, 4102–4104 (1996).
  * [31] Ishimoto, K. & Gaffney, E. A.  Swimming efficiency of spherical squirmers: beyond the Lighthill theory.  _Phys. Rev. E_ 90, 012704 (2014).
  * [32] Nganguia, H. & Pak, O. S.  Squirming motion in a Brinkman medium.  _J. Fluid Mech._ 855, 554–573 (2018).
  * [33] Michelin, S. & Lauga, E.  Efficiency optimization and symmetry-breaking in a model of ciliary locomotion.  _Phys. Fluids_ 22, 111901 (2010).
  * [34] Leshansky, A. M., Kenneth, O., Gat, O. & Avron, J. E.  A frictionless microswimmer.  _New J. Phys._ 9, 145–145 (2007).
  * [35] van Gogh, B., Demir, E., Palaniappan, D. & Pak, O. S.  The effect of particle geometry on squirming through a shear-thinning fluid.  _J. Fluid Mech._ 938, A3 (2022).
  * [36] Guo, H., Zhu, H., Liu, R., Bonnet, M. & Veerapaneni, S.  Optimal slip velocities of micro-swimmers with arbitrary axisymmetric shapes.  _J. Fluid Mech._ 910, A26 (2021).
  * [37] Guo, H., Zhu, H., Liu, R., Bonnet, M. & Veerapaneni, S.  Optimal ciliary locomotion of axisymmetric microswimmers.  _J. Fluid Mech._ 927, A22 (2021).
  * [38] Nasouri, B., Vilfan, A. & Golestanian, R.  Minimum dissipation theorem for microswimmers.  _Phys. Rev. Lett._ 126, 034503 (2021).
  * [39] Daddi-Moussa-Ider, A., Nasouri, B., Vilfan, A. & Golestanian, R.  Optimal swimmers can be pullers, pushers or neutral depending on the shape.  _J. Fluid Mech._ 922, R5 (2021).
  * [40] Masoud, H. & Stone, H. A.  The reciprocal theorem in fluid dynamics and transport phenomena.  _J. Fluid Mech._ 879, P1 (2019).
  * [41] Happel, J. & Brenner, H.  _Low Reynolds Number Hydrodynamics_ (Springer Netherlands, 1983).
  * [42] Keller, J. B., Rubenfeld, L. A. & Molyneux, J. E.  Extremum principles for slow viscous flows with applications to suspensions.  _J. Fluid Mech._ 30, 97–125 (1967).
  * [43] Maass, C. C., Krüger, C., Herminghaus, S. & Bahr, C.  Swimming droplets.  _Annu. Rev. Condens. Matter Phys._ 7, 171–193 (2016).
  * [44] Ishikawa, T., Pedley, T. J., Drescher, K. & Goldstein, R. E.  Stability of dancing Volvox.  _J. Fluid Mech._ 903, A11 (2020).
  * [45] Lauga, E., Brenner, M. & Stone, H.  _Microfluidics: The No-Slip Boundary Condition_ , chap. 19, 1219–1240 (Springer, Berlin, Heidelberg, 2007).
  * [46] Pozrikidis, C.  _A Practical Guide to Boundary Element Methods with the Software Library BEMLIB_ (CRC Press, Boca Raton, 2002).
  * [47] Taylor, G. I.  The action of waving cylindrical tails in propelling microscopic organisms.  _Proc. R. Soc. Lond. A_ 211, 225–239 (1952).
  * [48] Leshansky, A. M. & Kenneth, O.  Surface tank treading: Propulsion of Purcell’s toroidal swimmer.  _Phys. Fluids_ 20, 063104 (2008).
  * [49] Hess, S., Eme, L., Roger, A. J. & Simpson, A. G. B.  A natural toroidal microswimmer with a rotary eukaryotic flagellum.  _Nat. Microbiol._ 4, 1620–1626 (2019).
  * [50] Sekimoto, K.  Langevin equation and thermodynamics.  _Prog. Theor. Phys. Suppl._ 130, 17–27 (1998).
  * [51] Guazzelli, E. & Morris, J. F.  _A Physical Introduction to Suspension Dynamics_ (Cambridge University Press, 2009).
  * [52] Lauga, E. & Davis, A. M. J.  Viscous Marangoni propulsion.  _J. Fluid Mech._ 705, 120–133 (2012).
  * [53] Dietrich, K., Jaensson, N., Buttinoni, I., Volpe, G. & Isa, L.  Microscale Marangoni surfers.  _Phys. Rev. Lett._ 125, 098001 (2020).
  * [54] Dal Cengio, S., Levis, D. & Pagonabarraga, I.  Linear response theory and Green-Kubo relations for active matter.  _Phys. Rev. Lett._ 123, 238003 (2019).
  * [55] Ferretti, F. _et al._ Signatures of irreversibility in microscopic models of flocking.  _Phys. Rev. E_ 106, 034608 (2022).
  * [56] Loos, S. A. M. & Klapp, S. H. L.  Irreversibility, heat and information flows induced by non-reciprocal interactions.  _New J. Phys._ 22, 123051 (2020).
  * [57] Nasouri, B., Vilfan, A. & Golestanian, R.  Efficiency limits of the three-sphere swimmer.  _Phys. Rev. Fluids_ 4, 073101 (2019).
  * [58] Lorentz, H. A.  Eene algemeene stelling omtrent de beweging eener vloeistof met wrijving en eenige daaruit afgeleide gevolgen.  _Zittingsverslag van de Koninklijke Akademie van Wetenschappen te Amsterdam_ 5, 168–175 (1896).
  * [59] Elfring, G. J.  A note on the reciprocal theorem for the swimming of simple bodies.  _Phys. Fluids_ 27, 023101 (2015).

## Supplementary Notes

##  A Proof of generalized passive minimum dissipation theorems

The Helmholtz minimum dissipation theorem states that among all incompressible
flows that satisfy a prescribed fixed-velocity boundary condition, the Stokes
flow has the minimum dissipation. In other words, any perturbation to the
Stokes flow that satisfies the same boundary condition leads to an increase in
the total dissipation. In the following, we generalize the minimum dissipation
theorem to two configurations. The first generalization allows the presence of
fluid-fluid interfaces (Eq. (2) in the main text) and the second includes
surfaces with the Navier-slip boundary condition (Eq. (5) in the main text).
Our derivation is adapted from the proof of the original theorem as formulated
in the textbook by Guazzelli and Morris [51].

###  1 Fluid-fluid interface

We first prove the statement that among all flows around a fluid-fluid
interface of a fixed shape, the flow without tangential stress on the surface
has the minimum dissipation. We label the exterior volume as $\mathcal{V}$ and
the interior as ${\mathcal{V}_{i}}$. The solution in both domains with zero
tangential traction at the interface is described with the velocity field
$\bm{v}$. The total dissipation in both domains is determined as

$P=\left[2\mu\int_{\mathcal{V}}+2\mu_{i}\int_{\mathcal{V}_{i}}\right]\mathrm{d}V\,\bm{E}:\bm{E}\,,$
(S.1)

with $\bm{E}$ denoting the rate-of-strain tensor and
$\bm{E}:\bm{E}=\sum_{\alpha,\beta}E_{\alpha\beta}E_{\alpha\beta}$ the
Frobenius inner product. The expression in brackets is used as a shortcut for
the sum of multiple integrals with the same integrand. We now introduce a
perturbation $\bm{v}^{\prime}$ that satisfies the condition
$\bm{v^{\prime}}\cdot\bm{n}=0$ at the interface. The corresponding
perturbation of the strain rate is
$\bm{E}^{\prime}=\frac{1}{2}\left(\bm{\nabla}\bm{v}^{\prime}+\bm{\nabla}\bm{v}^{\prime\top}\right)$.
The perturbation alters the dissipation by

$\displaystyle\Delta P$
$\displaystyle=\left[2\mu\int_{\mathcal{V}}+2\mu_{i}\int_{\mathcal{V}_{i}}\right]\mathrm{d}V\left[\left(\bm{E}^{\prime}+\bm{E}\right):\left(\bm{E}^{\prime}+\bm{E}\right)-\bm{E}:\bm{E}\right]$
$\displaystyle=\left[2\mu\int_{\mathcal{V}}+2\mu_{i}\int_{\mathcal{V}_{i}}\right]\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}^{\prime}+\left[4\mu\int_{\mathcal{V}}+4\mu_{i}\int_{\mathcal{V}_{i}}\right]\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}\,.$
(S.2)

We first show that the second term vanishes if the flow $\bm{v}$ satisfies the
continuity of tangential stress at the interface. Since both $\bm{E}^{\prime}$
and $\bm{E}$ are traceless symmetric tensors, and
$\bm{\nabla}\cdot\bm{v}^{\prime}=0$, we have
$2\mu\bm{E}^{\prime}:\bm{E}=\bm{\nabla}\bm{v}^{\prime}:\bm{\sigma}$, where
$\bm{\sigma}=-p\bm{I}+2\mu\bm{E}$ is the stress field for the unperturbed flow
and $p$ is the corresponding pressure field. Using
$\bm{\nabla}\cdot\bm{\sigma}=\bm{0}$, one obtains
$\bm{\nabla}\bm{v}^{\prime}:\bm{\sigma}=\bm{\nabla}\cdot\left(\bm{v}^{\prime}\cdot\bm{\sigma}\right)$.
Thence, by using the divergence theorem, we find
$4\mu\int_{\mathcal{V}}\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}=-2\int_{\mathcal{S}}\mathrm{d}S\,\bm{n}\cdot\bm{\sigma}\cdot\bm{v}^{\prime}$
and analogously for the internal domain
$4\mu_{i}\int_{\mathcal{V}_{i}}\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}=2\int_{\mathcal{S}}\mathrm{d}S\,\bm{n}\cdot\bm{\sigma}_{i}\cdot\bm{v}^{\prime}$.
The sum of the two terms vanishes at the interface, because
$(\bm{I}-\bm{n}\bm{n})\cdot(\bm{\sigma}-\bm{\sigma}_{i})\cdot\bm{n}=\bm{0}$
and $\bm{n}\cdot\bm{v}^{\prime}=0$.

Now, noting that the first term in Eq. (S.2),
$\int\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}^{\prime}$, is positive-definite, we
can conclude that the minimum dissipated power can be only achieved when
$\bm{E}^{\prime}=\bm{0}$, thereby indicating that the flow that minimizes
dissipation is that with vanishing tangential traction at the interface. This
way we have proven that the dissipation of any flow around an interface with
velocity continuity, itself moving with velocity $\bm{\mathsf{V}}$, satisfies
the inequality

$P\geq\bm{\mathsf{{V}}}\cdot\bm{\mathsf{{R}}}_{\text{Droplet}}\cdot\bm{\mathsf{{V}}}\,,$
(S.3)

which is Eq. (2) in the main text.

###  2 Surface with Navier slip

In the second case, we consider the dissipation in the surrounding fluid
together with surface dissipation as stated in Eq. (3). A flow perturbation
$\bm{v}^{\prime}$ changes the total dissipation by

$\Delta
P=2\mu\int_{\mathcal{V}}\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}^{\prime}+4\mu\int_{\mathcal{V}}\mathrm{d}V\,\bm{E}^{\prime}:\bm{E}+\frac{\mu}{\lambda}\int_{\mathcal{S}}\mathrm{d}S\,{\bm{v}^{\prime}}^{2}+2\,\frac{\mu}{\lambda}\int_{\mathcal{S}}\mathrm{d}S\,\tilde{\bm{v}}\cdot\bm{v}^{\prime}\,.$
(S.4)

Here, the first two terms represent the change of external dissipation (S.2)
and the last two the change of internal dissipation. Again, the first and the
third term are positive for any non-vanishing $\bm{v}^{\prime}$. The second
term evaluates to
$-2\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}\cdot\bm{v}^{\prime}$. Together with
the fourth term, it yields $0$ when
$\bm{f}^{\parallel}=(\mu/\lambda)\,\tilde{\bm{v}}$ at the surface. We have
thus demonstrated that the combined external and surface dissipation of a flow
(3) is minimal if the flow satisfies the Navier-slip condition with slip
length $\lambda$. The inequality for the dissipation in any flow around that
body moving with velocity $\bm{\mathsf{V}}$ is

$P\geq\bm{\mathsf{{V}}}\cdot\bm{\mathsf{{R}}}_{\text{Navier}}\cdot\bm{\mathsf{{V}}}\,,$
(S.5)

proving Eq. (5) from the main text. $\bm{\mathsf{{R}}}_{\text{Navier}}$
denotes the generalized drag coefficient of the rigid body with shape
$\mathcal{S}$ and slip length $\lambda$.

##  B Dissipation rate

Here we recapitulate the relationships for the total dissipation in the flow
around active or passive bodies in an otherwise quiescent Stokes fluid (e.g.,
Ref. [51]). The total dissipation rate in a fluid can be expressed as the
integral of the dissipation density

$P=2\mu\int_{\mathcal{V}}\mathrm{d}V\,\bm{E}:\bm{E}\,.$ (S.6)

Because of energy conservation, the dissipation rate has to be equal to the
flux of work from the object to the fluid. This can be shown by applying the
divergence theorem to Eq. (S.6) and obtaining

$P=-\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}\cdot\bm{v}\,.$ (S.7)

Here $\bm{f}=\bm{\sigma}\cdot\bm{n}$ denotes the traction on the surface of
the particle. If the particle exerts forces on the fluid anywhere else outside
its surface, they can be included in the form of a volume integral
$-\int_{\mathcal{V}}\mathrm{d}V\bm{\mathfrak{f}}\cdot\bm{v}$. We can also
express the velocity field $\bm{v}$ in terms of the velocity field in the co-
moving frame $\tilde{\bm{v}}$ and the rigid-body velocity
$\bm{\mathsf{V}}=[\bm{V},\bm{\Omega}]$ as
$\bm{v}=\tilde{\bm{v}}+\bm{V}+\bm{\Omega}\times\bm{x}$. Then the dissipated
power can be decomposed into the work of the active forces on the body and the
work of the external forces:

$P=-\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}\cdot\tilde{\bm{v}}-\bm{\mathsf{F}}\cdot\bm{\mathsf{V}}\,.$
(S.8)

Here $\bm{\mathsf{F}}=[\bm{F},\bm{M}]$ is the generalized force on the body
with $\bm{F}=\int_{\mathcal{S}}\mathrm{d}S\,\bm{f}$ and
$\bm{M}=\int_{\mathcal{S}}\mathrm{d}S\,\bm{x}\times\bm{f}$.

For a passive body, the force $\bm{F}$ and torque $\bm{M}$ on the body are
linear functions of the translational and rotational velocities and can
therefore be expressed with a generalized drag coefficient $\bm{\mathsf{R}}$
as $\bm{\mathsf{F}}=-\bm{\mathsf{R}}\cdot\bm{\mathsf{V}}$, or with components:

$\begin{bmatrix}\bm{F}\\\
\bm{M}\end{bmatrix}=\begin{bmatrix}\bm{R}_{TT}&\bm{R}_{RT}^{\top}\\\
\bm{R}_{RT}&\bm{R}_{RR}\end{bmatrix}\cdot\begin{bmatrix}\bm{V}\\\
\bm{\Omega}\end{bmatrix}\,.$ (S.9)

The rate of work by the external force is
$\bm{\mathsf{V}}\cdot\bm{\mathsf{R}}\cdot\bm{\mathsf{V}}$. According to Eq.
(S.8), it equals the dissipation in the fluid plus the dissipation in the
boundary $\bm{f}\cdot\tilde{\bm{v}}$ if the latter is not zero (for the Navier
slip boundary).

##  C Drag coefficients of composite passive bodies with spherical shape

We determine the viscous flow field past composite passive bodies of spherical
geometry. The body consists of an outer shell with the radius $a$ and an inner
core with the radius $b$. Following the notation used in the main body of the
paper, we use the subscript $i$ for the flow variables in the interior fluid
domain bounded by the inner and outer spherical surfaces, such that $b\leq
r\leq a$. Absence of subscript indicates the flow variables in the exterior
fluid domain, such that $r\geq a$. Moreover, we assume that the dynamic
viscosity is the same everywhere in the fluid medium. We formulate the
solution of the Stokes equation in the reference frame attached to the
particle. The solution for the stream function satisfying the regularity
condition at infinity can be expressed in spherical coordinates as [41]

$\displaystyle\psi_{i}(r,\theta)$
$\displaystyle=b^{2}V\left(A\,\frac{r}{b}+B\left(\frac{r}{b}\right)^{2}+C\left(\frac{r}{b}\right)^{4}+D\,\frac{b}{r}\right)\sin^{2}\theta\,,$
(S.10a) $\displaystyle\psi(r,\theta)$
$\displaystyle=b^{2}V\left(E\,\frac{r}{b}+G\left(\frac{r}{b}\right)^{2}+H\,\frac{b}{r}\right)\sin^{2}\theta\,,$
(S.10b)

where $A$, $B$, $C$, $D$, $E$, $G$, and $H$ are constants to be determined
from the boundary conditions prescribed at the inner/outer radius or at
infinity. The corresponding hydrodynamic pressure field reads

$\displaystyle p_{i}(r,\theta)$ $\displaystyle=-\frac{2\mu
V}{b}\left(A\left(\frac{b}{r}\right)^{2}+10C\,\frac{r}{b}\right)\cos\theta\,,$
(S.11a) $\displaystyle p(r,\theta)$ $\displaystyle=-\frac{2\mu
V}{b}\,E\left(\frac{b}{r}\right)^{2}\cos\theta\,.$ (S.11b)

In the limit $r\to\infty$, the stream function has to converge to that of a
uniform flow with the velocity $-V\hat{\bm{e}}_{z}$,
$\psi_{2}=\frac{V}{2}\,r^{2}\sin^{2}\theta$ [41]. Accordingly, we obtain
$G=1/2$.

Figure S1: Streamlines and velocities of the flow around two composite passive
bodies. (a) Body with a no-slip boundary condition at the inner sphere and
zero tangential velocity at the outer sphere. (b) Body with a no-slip inner
core and a non-deformable fluid-fluid interface at the outer surface.

###  1 Composite body with zero tangential velocity

We determine in the following the flow around a passive body consisting of an
inner sphere with a no-slip boundary and an outer sphere which imposes zero
tangential velocity and zero normal traction, as shown in Fig. S1a. At $r=b$,
we set $\bm{v}=\bm{0}$. In addition, we require at $r=a$ that $v_{\theta}=0$
and assume that both $\bm{v}$ and $\sigma_{rr}$ are continuous. Defining
$\alpha=b/a\in[0,1]$, we obtain

$\displaystyle A$
$\displaystyle=-2\beta\alpha^{3}\left(6+6\alpha+\alpha^{2}+\alpha^{3}+\alpha^{4}\right)\,,$
(S.12a) $\displaystyle B$
$\displaystyle=\beta\alpha^{3}\left(8+8\alpha+8\alpha^{2}+3\alpha^{3}+3\alpha^{4}\right)\,,$
(S.12b) $\displaystyle C$
$\displaystyle=-\beta\alpha^{5}\left(4+\alpha+\alpha^{2}\right)\,,$ (S.12c)
$\displaystyle D$
$\displaystyle=2\beta\alpha^{3}\left(2+2\alpha-\alpha^{2}\right)\,,$ (S.12d)
$\displaystyle E$
$\displaystyle=-2\beta\alpha^{2}\left(20+11\alpha+11\alpha^{2}+\alpha^{3}+\alpha^{4}+\alpha^{5}\right)\,,$
(S.12e) $\displaystyle H$
$\displaystyle=2\beta\left(8+2\alpha+2\alpha^{2}+2\alpha^{3}+2\alpha^{4}-\alpha^{5}\right)\,,$
(S.12f)

where we have defined the abbreviation
$\beta^{-1}=2\alpha^{3}\left(28+13\alpha+13\alpha^{2}+3\alpha^{3}+3\alpha^{4}\right)$.

The hydrodynamic drag force exerted on the composite can be determined from
the monopole term as $F=8\pi\mu bE$ [41]. Then, the drag coefficient
$R_{\text{CP}}\equiv-F/V$ can be expressed in a scaled form as

$\frac{R_{\text{CP}}}{6\pi\mu
a}=\frac{4}{3}\cdot\frac{20+11\alpha+11\alpha^{2}+\alpha^{3}+\alpha^{4}+\alpha^{5}}{28+13\alpha+13\alpha^{2}+3\alpha^{3}+3\alpha^{4}}\in\left[\frac{20}{21},1\right]\,.$
(S.13)

In particular, by setting $\alpha=1-\epsilon$, it follows that
$R_{\text{CP}}=6\pi\mu
a\left(1-\frac{1}{12}\,\epsilon^{3}+\mathcal{O}\left(\epsilon^{4}\right)\right)$.

###  2 Droplet with a no-slip core

The second passive body consists of a no-slip inner sphere surrounded by a
fluid-fluid interface with equal viscosities on both sides, which imposes zero
normal velocity, zero tangential traction and the continuity of tangential
velocity (Fig. S1b). At $r=b$, we impose $\bm{v}=\bm{0}$. We require that
$v_{r}=0$ at $r=a$ and that both $\bm{v}$ and $\sigma_{r\theta}$ are
continuous at $r=a$. We obtain

$\displaystyle A$
$\displaystyle=\beta\alpha^{3}\left(3+6\alpha+4\alpha^{2}+2\alpha^{3}\right)\,,$
(S.14a) $\displaystyle B$
$\displaystyle=-\beta\alpha^{3}\left(2+4\alpha+6\alpha^{2}+3\alpha^{3}\right)\,,$
(S.14b) $\displaystyle C$
$\displaystyle=\beta\alpha^{5}\left(2+\alpha\right)\,,$ (S.14c) $\displaystyle
D$ $\displaystyle=-\beta\alpha^{3}\left(1+2\alpha\right)\,,$ (S.14d)
$\displaystyle E$
$\displaystyle=-2\beta\alpha^{2}\left(1-\alpha\right)\left(5+6\alpha+3\alpha^{2}+\alpha^{3}\right)\,,$
(S.14e) $\displaystyle H$
$\displaystyle=\beta\left(1-\alpha^{2}\right)\left(2+\alpha+2\alpha^{2}\right)\,,$
(S.14f)

wherein
$\beta^{-1}=2\alpha^{3}\left(1-\alpha\right)\left(8+9\alpha+3\alpha^{2}\right)$.
The corresponding drag coefficient is obtained as

$\frac{R_{\text{DC}}}{6\pi\mu
a}=\frac{4}{3}\cdot\frac{5+6\alpha+3\alpha^{2}+\alpha^{3}}{8+9\alpha+3\alpha^{2}}\in\left[\frac{5}{6},1\right]\,.$
(S.15)

For $\alpha=1-\epsilon$, we obtain $R_{\text{DC}}=6\pi\mu
a\left(1-\frac{1}{4}\,\epsilon+\mathcal{O}\left(\epsilon^{2}\right)\right)$.

##  D Response of the swimmer to an external force

In the following we derive the expressions for the power expenditure by an
active swimmer when it is subject to an additional external force [Eqs. (34)
and (.5) in the main body of the paper]. The assumption is that the active
forces produced by the swimmer remain unaffected, but all flow velocities
change in response to the applied force. The flow of the active swimmer takes
the form

$\bm{v}_{\text{A}}=\bm{v}_{\text{A}}^{0}+\bm{v}_{\text{V}}^{\prime}$ (S.16)

Where $\bm{v}_{\text{V}}^{\prime}$ is the flow of the solution of the
V-problem (Droplet, No-slip, etc.) when pulled by the force
$\bm{\mathsf{F}}_{\text{ext}}$. The rate of work by the swimmer (without the
work of the external force) evaluates to

$P_{\text{A}}=P_{\text{A}}^{0}-\int\mathrm{d}S\bm{f}_{\text{A}}^{0}\cdot\tilde{\bm{v}}_{\text{V}}^{\prime}$
(S.17)

Now we apply the Lorentz reciprocal theorem as formulated in Eq. (45) and
obtain

$P_{\text{A}}=P_{\text{A}}^{0}-\int\mathrm{d}S\bm{f}_{\text{V}}^{\prime}\cdot\tilde{\bm{v}}_{\text{A}}^{0}+\bm{\mathsf{F}}_{\text{A}}^{0}\cdot\bm{\mathsf{V}}_{\text{V}}^{\prime}-\bm{\mathsf{F}}_{\text{V}}^{\prime}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}$
(S.18)

The product in the integral is zero in all problems studied except for the
surface dissipation, for which the assumption of a constant force density
needs to be adapted. In addition, $\bm{\mathsf{F}}_{\text{A}}^{0}=0$ for the
unperturbed active swimmer, and the drag on the passive body is opposite equal
to the external force acting on it,
$\bm{\mathsf{F}}_{\text{V}}^{\prime}=-\bm{\mathsf{F}}_{\text{ext}}$. We
therefore obtain

$P_{\text{A}}=P_{\text{A}}^{0}+\bm{\mathsf{F}}_{\text{ext}}\cdot\bm{\mathsf{V}}_{\text{A}}^{0}\,,$
(S.19)

which is Eq. (34) from the main text.