# Self-diffusiophoretic propulsion of a spheroidal particle in a shear-
thinning fluid

Guangpu Zhu1111These authors contribute equally to this work. Brandon van
Gogh2,41 Lailai Zhu1  On Shun Pak2,3 222Email addresses for correspondance:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Yi Man52 1Department of Mechanical Engineering,
National University of Singapore, 117575, Singapore 2 Department of Mechanical
Engineering, Santa Clara University, Santa Clara, CA 95053, USA 3 Department
of Applied Mathematics, Santa Clara University, Santa Clara, CA 95053, USA 4
Department of Energy Science and Engineering, Stanford University, Stanford,
CA 94305, USA 5Department of Mechanics and Engineering Science at College of
Engineering, and State Key Laboratory for Turbulence and Complex Systems,
Peking University, Beijing 100871, PR China

###### Abstract

Shear-thinning viscosity is a non-Newtonian behaviour that active particles
often encounter in biological fluids such as blood and mucus. The fundamental
question of how this ubiquitous non-Newtonian rheology affects the propulsion
of active particles has attracted substantial interest. In particular,
spherical Janus particles driven by self-diffusiophresis, a major physico-
chemical propulsion mechanism of synthetic active particles, were shown to
always swim slower in a shear-thinning fluid than in a Newtonian fluid. In
this work, we move beyond the spherical limit to examine the effect of
particle eccentricity on self-diffusiophoretic propulsion in a shear-thinning
fluid. We use a combination of asymptotic analysis and numerical simulations
to show that shear-thinning rheology can enhance self-diffusiophoretic
propulsion of a spheroidal particle, in stark contrast to previous findings
for the spherical case. A systematic characterization of the dependence of the
propulsion speed on the particle’s active surface coverage has also uncovered
an intriguing feature associated with the propulsion speeds of a pair of
complementarily coated particles not previously reported. Symmetry arguments
are presented to elucidate how this new feature emerges as a combined effect
of anisotropy of the spheroidal geometry and nonlinearity in fluid rheology.

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##  1 Introduction

Due to their small sizes, swimming microorganisms such as bacteria and
spermatozoa live in a low-Reynolds-number world, where viscous forces dominate
inertial forces. They use a variety of strategies to overcome the challenge of
generating self-propulsion at low Reynolds number (Purcell, 1977). Extensive
studies have elucidated the hydrodynamics of these biological propulsion
mechanisms and shed light on their profound roles in various biological
processes (Fauci & Dillon, 2006; Lauga & Powers, 2009). In recent decades,
there are also growing interests in developing synthetic active particles that
can self-propel like living microorganisms for biomedical and microfluidic
applications, including self-assembly (Schwarz-Linek et al., 2012; Wensink et
al., 2014), drug delivery (Gao & Wang, 2014), and motion-based microsensing
(Kagan et al., 2009). Some synthetic active particle designs are inspired by
biological systems, such as artificial helical propellers (Zhang et al., 2009;
Ghosh & Fischer, 2009), which mimic the helical structure of bacterial
flagella (Lauga, 2016). Other novel designs exploit different physical or
physico-chemical mechanisms to achieve self-propulsion (Schweitzer & Farmer,
2003; Bechinger et al., 2016; Patteson et al., 2016; Moran & Posner, 2017).

In particular, a major class of synthetic active particles converts chemical
energy into motility by asymmetric chemical reactions on the particle surface.
A variety of novel synthetic active colloids has been developed (Patiño et
al., 2018; Buttinoni et al., 2012; Zhou et al., 2018). For instance,
microspheres half-coated in platinum, also known as Janus particles, can self-
propel via catalytic decomposition of hydrogen peroxide on the platinum-coated
surface (Howse et al., 2007; Sánchez et al., 2015). While the exact mechanism
underlying the resulting motion is still under debate (Brown & Poon, 2014;
Ebbens et al., 2014; Eloul et al., 2020), it has been hypothesised that the
motion is diffusiophoretic as a result of the gradients of molecular oxygen
produced by the catalytic decomposition on the half-coated surface
(Golestanian et al., 2005, 2007; Moran & Posner, 2017). Since the solute
concentration gradient is self-generated, the motion of these active particles
is also referred to as self-diffusiophoresis. To model the self-
diffusiophoretic motion, a common approach is to separate the fluid domain
into outer (the bulk fluid) and inner (the interaction layer) regions, where
the short-range solute-particle interaction is assumed to be confined in the
interaction layer (Anderson, 1989; Jülicher & Prost, 2009). When the
interaction layer is thin relative to particle size, the phoretic effects can
be represented by a distribution of effective slip velocities at the particle
surface, analogous to the squirmer model (Lighthill, 1952; Blake, 1971;
Pedley, 2016) proposed for swimming ciliates such as Paramecium and Volvox.
While the slip velocity in the squirmer model is determined by the beating
motion of short cilia covering the cell, the slip velocity of a self-
diffusiophoretic particle is proportional to the solute concentration gradient
and phoretic mobility calculated from the interaction potential in the
interaction layer (Anderson, 1989; Jülicher & Prost, 2009). As a remark,
recent studies have indicated that the standard self-diffusiophoretic
framework described may become ineffective when the reactive species are
charged (Brown et al., 2017; De Corato et al., 2020; Asmolov et al., 2022).

Extensive studies have elucidated various interesting features of self-
diffusiophoretic motion in a Newtonian fluid (Moran & Posner, 2017). However,
most biological fluids such as blood and mucus display non-Newtonian (complex)
rheological behaviors including viscoelasticity and shear-thinning viscosity
(Hwang et al., 1969; Baskurt & Meiselman, 2003). Since these synthetic active
particles will invariably encounter biological fluids in their biomedical
applications, a fundamental question is how different non-Newtonian
rheological behaviors impact the propulsion of these active particles
(Patteson et al., 2016). While many previous theoretical and experimental
studies focused on swimming in viscoelastic fluids (Sznitman & Arratia, 2014;
Elfring & Lauga, 2015; Li et al., 2021; Spagnolie & Underhill, 2023; Natale et
al., 2017; De Corato et al., 2015; Zöttl & Yeomans, 2019; Saad & Natale, 2019;
Bechinger et al., 2016), recent studies have begun to address the effect of
shear-thinning viscosity (Montenegro-Johnson et al., 2013; Vélez-Cordero &
Lauga, 2013; Gagnon et al., 2014; Li & Ardekani, 2015; Park et al., 2016;
Gómez et al., 2017). A shear-thinning fluid loses its viscosity with applied
shear due to changes in its microstructure. Such a non-Newtonian behaviour was
found to impact the propulsion of various low-Reynolds-number swimmers in
qualitatively different manners (Datt et al., 2015, 2017; van Gogh et al.,
2022; Qu & Breuer, 2020; Demir et al., 2020; Qin et al., 2021). In particular,
Datt et al. (2015) considered a general spherical squirmer model in a shear-
thinning fluid and demonstrated how shear-thinning rheology can both enhance
and hinder its propulsion, depending on specific details of the slip velocity.
Interestingly, in a latter study (Datt et al., 2017), spherical self-
diffusiophoretic particles were found to always swim slower in a shear-
thinning fluid than in a Newtonian fluid for any level of active surface
coverage. This also prompts the question to what extent the conclusion of
hindered swimming continues to hold for non-spherical self-diffusiophoretic
particles.

Swimmers with non-spherical shapes are commonly found in both nature and
engineered systems. For instance, ciliates such as Paramecium and Tetrahymena
have approximately prolate spheroidal body shapes. Keller & Wu (1977)
considered a spheroidal squirmer model, which was extended by later studies to
probe the effect of geometrical shape upon ciliary locomotion (Ishimoto &
Gaffney, 2013; Theers et al., 2016; Poehnl et al., 2020). Furthermore,
synthetic active particles of non-spherical shapes, including prolate
spheroids and general slender bodies, were also fabricated and studied
experimentally and theoretically (Champion & Mitragotri, 2006; Champion et
al., 2007; Glotzer & Solomon, 2007; Shemi & Solomon, 2018; Poehnl et al.,
2020; Zhu & Zhu, 2023; Katsamba et al., 2022; Poehnl & Uspal, 2021; Yariv,
2019). In particular, Poehnl et al. (2020) analysed the self-diffusiophertic
motion of spheroidal particles in a Newtonian fluid. However, much less is
known about these spheroidal active particles in non-Newtonian fluids. A
recent study has suggested that shear-thinning rheology can indeed enhance the
propulsion of a squirming spheroid (van Gogh et al., 2022). However, it
remains unclear whether or not a spheroidal self-diffusiophoretic particle can
swim faster in a shear-thinning fluid than in a Newtonian fluid, which was
shown impossible for the spherical case (Datt et al., 2017). In this work, we
fill in this knowledge gap by analysing the self-diffusiophoretic motion of a
spheroidal particle in a shear-thinning fluid. We use asymptotic analysis and
numerical simulations to reveal how shear-thinning viscosity impacts the
propulsion speed of a prolate spheroidal self-diffusiophoretic particle with
different eccentricities and levels of active surface coverage. Our results
have uncovered some propulsion behaviours not observed in the spherical case
and we present symmetry considerations to help elucidate the emergence of
these new features as a combined effect of particle anisotropy and nonlinear
fluid rheology.

##  2 Problem formulation

###  2.1 Geometrical setup

Figure 1: Geometric configuration of a spheroidal Janus particle. The model is
presented in prolate spheroidal coordinates $(\tau,\zeta,\phi)$. The
coordinate grid is indicated by dashed lines, and the basis vectors are
denoted $\bm{e}_{\tau}$ and $\bm{e}_{\zeta}$. The active cap of the particle,
depicted in grey, spans from $\zeta=-1$ to $\zeta_{0}$. The rest of the
surface is inert.

We examine a prolate spheroidal particle characterised by a major axis, $a$,
and a minor axis, $b$, as illustrated in figure 1. The prolate spheroidal
coordinates $(\tau,\zeta,\phi)$, where $\tau\in[1,\infty)$, $\zeta\in[-1,1]$,
and $\phi\in[0,2\pi)$, are employed in this work. The prolate spheroidal
coordinates can be related to the cylindrical coordinates $(r,z,\phi)$ as

$\displaystyle r$
$\displaystyle=c_{f}\sqrt{\tau^{2}-1}\sqrt{1-\zeta^{2}},\quad
z=c_{f}\tau\zeta,$ (1)

where $r^{2}=x^{2}+y^{2}$, and $c_{f}=\sqrt{a^{2}-b^{2}}$. The surface of the
spheroidal particle is given by the equation

$\displaystyle\frac{z^{2}}{a^{2}}+\frac{r^{2}}{b^{2}}=1,$ (2)

which translates to $r=b\sqrt{1-\zeta^{2}}$ and $z=a\zeta$. Comparing with
(1), the spheroidal particle surface can be simply represented by

$\displaystyle\tau=\tau_{0}=1/e,$ (3)

where $e=c_{f}/a$ is the eccentricity. The basis vectors in the prolate
spheroidal coordinates, represented as
$(\bm{e}_{\tau},\bm{e}_{\zeta},\bm{e}_{\phi})$, are related to the basis
vectors in cylindrical coordinates, denoted as
$(\bm{e}_{r},\bm{e}_{z},\bm{e}_{\phi})$, in the following manner,

$\displaystyle\begin{split}\bm{e}_{\tau}=\frac{c_{f}\tau}{h_{\zeta}}\bm{e}_{r}+\frac{c_{f}\zeta}{h_{\tau}}\bm{e}_{z},\quad\bm{e}_{\zeta}=-\frac{c_{f}\zeta}{h_{\tau}}\bm{e}_{r}+\frac{c_{f}\tau}{h_{\zeta}}\bm{e}_{z}.\end{split}$
(4)

The metric coefficients for the prolate spheroidal coordinates are given by

$\displaystyle
h_{\tau}=\frac{c_{f}\sqrt{\tau^{2}-\zeta^{2}}}{\sqrt{\tau^{2}-1}},\quad
h_{\zeta}=\frac{c_{f}\sqrt{\tau^{2}-\zeta^{2}}}{\sqrt{1-\zeta^{2}}},\quad
h_{\phi}=c_{f}\sqrt{\tau^{2}-1}\sqrt{1-\zeta^{2}}.$ (5)

On the surface of the prolate spheroidal particle, the unit normal vector
pointing outwards is given by $\bm{n}=\bm{e}_{\tau}$, and the unit tangent
vector pointing upwards is given by $\bm{t}=\bm{e}_{\zeta}$ as illustrated in
figure 1.

###  2.2 Governing equations and boundary conditions

We treat the problem within the continuum framework of self-diffusiophoretic
propulsion (Golestanian et al., 2007; Michelin & Lauga, 2014), where the
particle interacts with a solute species of local concentration $C$. Here, we
consider an axisymmetric Janus spheroidal particle with chemically active and
inert compartments, with the polar position $\zeta_{0}$ specifying the active
surface coverage as illustrated in figure 1. On the active portion of the
particle surface ($\tau=\tau_{0},\ \zeta\leq\zeta_{0}$), we assume that the
solute is emitted/absorbed with a fixed-flux characterised by the activity
$A$,

$\displaystyle D\bm{n}\cdot\nabla C=-A,$ (6)

where $D$ is the diffusivity, $A>0$ corresponds to solute emission, and $A<0$
corresponds to the solute absorption. The activity becomes zero ($A=0$) on the
inert portion of the particle surface ($\tau=\tau_{0},\ \zeta>\zeta_{0}$).
Under the assumption of a thin interaction layer (Golestanian et al., 2005,
2007; Michelin & Lauga, 2014; Datt et al., 2017), the effective slip velocity
at the surface of the particle,

$\displaystyle\bm{u}_{s}$ $\displaystyle=M(\bm{I}-\bm{n}\bm{n})\cdot\gradient
C,$ (7)

is proportional to the tangential concentration gradients and the phoretic
mobility $M$ determined by the interaction potential profile (Anderson, 1989;
Michelin & Lauga, 2014). In general, when the interactions are attractive,
$M<0$ and the slip velocity is opposite to the concentration gradients; when
the interactions are repulsive, $M>0$ and the slip velocity is along the
concentration gradients. In this work, we present results for the case where
$M>0$ and $A>0$ without loss of generality. By symmetry and linearity, a
flipping of the sign of $M$ or $A$ only inverts the direction of swimming
velocity in the results presented below.

In the bulk fluid, the solute concentration is governed by an advection-
diffusion equation

$\displaystyle\frac{\partial C}{\partial t}+\bm{u}\cdot\gradient
C=D\gradient^{2}C,$ (8)

where $\bm{u}$ is the velocity of the flow, and the solute concentration in
the far-field is denoted by $C_{\infty}$. In the inertialess regime, the flow
generated by the phoretic slip velocity is governed by the momentum and
continuity equations, respectively, as

$\displaystyle\gradient\cdot\bm{\sigma}$
$\displaystyle=\mathbf{0},\quad\gradient\cdot\bm{u}=0,$ (9)

where $\bm{\sigma}=-p\bm{I}+\bm{T}$, $p$ is the pressure, $\bm{I}$ is the
identity tensor, and $\bm{T}$ is the deviatoric stress tensor. The boundary
condition for the velocity field on the particle surface in the laboratory
frame is given by

$\displaystyle\bm{u}(\tau=\tau_{0})$ $\displaystyle=\bm{u}_{s}(\zeta)+\bm{U},$
(10)

where $\bm{u}_{s}$ is the phoretic slip velocity given by (7), and
$\bm{U}=U\mathbf{e}_{z}$ is the unknown propulsion velocity, which occurs in
the $z$-direction by axisymmetry. The flow decays to zero in the far field,
$\bm{u}(\tau\to\infty)=\mathbf{0}$. The system of equations is closed by
enforcing the force-free condition on the particle,

$\displaystyle\int_{S}\bm{n}\cdot\bm{\sigma}\;{\rm d}S=\mathbf{0},$ (11)

where $S$ denotes the particle surface.

###  2.3 Shear-thinning rheology

To probe the effect of shear-thinning rheology on the self-diffusiophoretic
motion, we consider here the Carreau constitutive model (Bird et al., 1987),
which has been shown effective in capturing the shear-thinning viscosity of
different biological fluids (Vélez-Cordero & Lauga, 2013). In the Carreau
model, the deviatoric stress is given by

$\displaystyle\bm{T}=\left[\mu_{\infty}+(\mu_{0}-\mu_{\infty})\left(1+\frac{1}{2}\lambda^{2}\dot{\bm{\gamma}}:\dot{\bm{\gamma}}\right)^{\frac{n-1}{2}}\right]\dot{\bm{\gamma}};$
(12)

here $\mu_{0}$ and $\mu_{\infty}$ represent, respectively, the viscosities
when the shear rate is zero and infinite, $1/\lambda$ characterizes the
critical shear rate at which the non-Newtonian behaviour becomes significant,
and $\dot{\bm{\gamma}}=\nabla\bm{u}+\left(\nabla\bm{u}\right)^{T}$ is the
strain rate tensor. For low and high shear rates (relative to the critical
shear rate), the fluid tends to behave as a Newtonian fluid with viscosity,
respectively, $\mu_{0}$ and $\mu_{\infty}$. In the intermediate regime, the
fluid displays a power-fluid behaviour, with the index $n<1$ characterising
the degree of shear-thinning.

###  2.4 Non-dimensionalisation

We non-dimensionalise the problem by scaling lengths with $a$, velocities with
$MA/D$, stresses by $\mu_{0}MA/Da$, and the solute concentration with $Aa/D$.
Hereafter we consider only dimensionless quantities and use the same symbols
as their dimensional counterparts for convenience.

We denote the solute concentration relative to the far-field solute
concentration as $c=C-C_{\infty}$, which satisfies the dimensionless
advection-diffusion equation,

$\displaystyle\Pen\left(\frac{\partial c}{\partial t}+\bm{u}\cdot\gradient
c\right)=\gradient^{2}c.$ (13)

Here, the Péclet number $\Pen=MAa/D^{2}$ characterises the relative importance
of advective to diffusive transport of the solute. We assume the diffusivity
is high enough and neglect the alteration in solute distribution caused by the
flow from phoretic effects, the solute concentration becomes harmonic,

$\displaystyle\gradient^{2}c=0.$ (14)

The dimensionless boundary condition on the active portion of the particle
surface ($\tau=\tau_{0},\ \zeta\leq\zeta_{0}$) is given by

$\displaystyle\bm{n}\cdot\gradient c=-1,$ (15)

whereas that on the inert portion ($\tau=\tau_{0},\ \zeta>\zeta_{0}$) is
simply

$\displaystyle\bm{n}\cdot\gradient c=0.$ (16)

The relative solute concentration decays to zero at infinity,

$\displaystyle c(\tau\rightarrow\infty)=0.$ (17)

Given that the solute concentration is decoupled, the governing equations for
the fluid align with those presented in (9). The Carreau constitutive equation
is rendered dimensionless as

$\displaystyle\bm{T}=\dot{\bm{\gamma}}+(1-\beta)\left[-1+\left(1+\frac{1}{2}\mbox{\it
Cu}^{2}\dot{\bm{\gamma}}:\dot{\bm{\gamma}}\right)^{\frac{n-1}{2}}\right]\dot{\bm{\gamma}},$
(18)

where $\beta=\mu_{\infty}/\mu_{0}$, the viscosity ratio, and $\mbox{\it
Cu}=\lambda MA/aD$, the Carreau number, which compares the characteristic
shear rate $MA/aD$ to the critical shear rate $1/\lambda$.

In the laboratory frame, the dimensionless boundary condition for the velocity
field on the particle surface is given by
$\bm{u}(\tau=\tau_{0})=\bm{u}_{s}+U\bm{e}_{z}$, where the slip velocity in
dimensionless form reads

$\displaystyle\bm{u}_{s}=(\bm{I}-\bm{n}\bm{n})\cdot\gradient c,$ (19)

and the flow decays to zero in the far-field,
$\bm{u}(\tau\to\infty)=\mathbf{0}$. In the following calculations, we
determine the unknown propulsion speed $U$ of the spheroidal self-
diffusiophretic particle in a shear-thinning fluid.

##  3 Asymptotic analysis and numerical simulations

###  3.1 Asymptotic analysis

The solute concentration can be obtained by solving the Laplace equation (14)
with boundary conditions (15)–(17) in the prolate spheroidal coordinates. An
analytical solution in form of a series is given by (Popescu et al., 2010)

$\displaystyle
c(\tau,\zeta)=\sum_{n=0}^{\infty}\rho_{n}Q_{n}(\tau)P_{n}(\zeta),$ (20)

where $P_{n}(\zeta)$ and $Q_{n}(\tau)$ are, respectively, the Legendre
functions of the first and the second kinds. As $\tau\geq\tau_{0}>1$, the
Legredre functions of the the second kind vanish when $\tau\to\infty$,
satisfying the far-field boundary condition for the relative concentration,
(17). By substituting the solution (20) into the boundary conditions at the
particle surface, (15)–(16), and employing the orthogonality of the Legendre
functions, the coefficients $\rho_{n}$ in the series solution are determined
as

$\displaystyle\rho_{n}(\tau_{0},\zeta_{0})=-\frac{2n+1}{2}\frac{1}{Q^{\prime}_{n}(\tau_{0})\tau_{0}\sqrt{\tau_{0}^{2}-1}}\int_{-1}^{\zeta_{0}}\sqrt{\tau_{0}^{2}-\zeta^{2}}P_{n}(\zeta)\
\text{d}\zeta.$ (21)

By employing the solution (20) in (19), the resulting phoretic slip velocity
at the particle surface, $\mathbf{u}_{s}(\zeta)=u_{s}\bm{e}_{\zeta}$, is given
by

$\displaystyle
u_{s}(\zeta)=\tau_{0}\sum_{n=0}^{\infty}B_{n}\frac{P^{1}_{n}(\zeta)}{\sqrt{\tau_{0}^{2}-\zeta^{2}}},$
(22)

where the phoretic modes are given by

$\displaystyle B_{n}=-\rho_{n}(\tau_{0},\zeta_{0})Q_{n}(\tau_{0}),$ (23)

and $P_{n}^{1}$ is the associated Legendre function with order 1.

We perform an asymptotic analysis in the weakly non-Newtonian regime where the
deviation of the viscosity ratio from unity, $\epsilon=1-\beta$, is small. We
expand the physical quantities in powers of $\epsilon$ as

$\displaystyle\\{\bm{u},\dot{\bm{\gamma}},\bm{\sigma},p,\bm{T},U\\}=\\{\bm{u}_{0},\dot{\bm{\gamma}}_{0},\bm{\sigma}_{0},p_{0},\bm{T}_{0},U_{0}\\}+\epsilon\\{\bm{u}_{1},\dot{\bm{\gamma}}_{1},\bm{\sigma}_{1},p_{1},\bm{T}_{1},U_{1}\\}+O(\epsilon^{2}).$
(24)

The zeroth-order problem corresponds to the Newtonian problem, where
$\bm{\sigma}_{0}=-p_{0}\bm{I}+\dot{\bm{\gamma}}_{0}$, and
$\dot{\bm{\gamma}}_{0}=\gradient\bm{u}_{0}+(\gradient\bm{u}_{0})^{T}$. For
boundary conditions, we have
$\bm{u}_{0}(\tau=\tau_{0})=U_{0}\bm{e}_{z}+u_{s}\bm{e}_{\zeta}$ on the
particle surface and $\bm{u}_{0}(\tau\to\infty)=\mathbf{0}$ in the far-field,
where $u_{s}$ is given in (22). The flow field $\bm{u}_{0}$ and propulsion
speed $U_{0}$ of this zeroth-order, Newtonian problem were obtained in
previous works (Leshansky et al., 2007; Lauga & Michelin, 2016; Popescu et
al., 2010; Poehnl et al., 2020), which we summarise in Appendix A.

We consider the first-order non-Newtonian correction to the Newtonian problem.
To the order of $\epsilon$, the flow satisfies

$\displaystyle\gradient\cdot\bm{\sigma}_{1}$ $\displaystyle=\mathbf{0},$ (25)
$\displaystyle\gradient\cdot\bm{u}_{1}$ $\displaystyle=0,$ (26)

where $\bm{\sigma}_{1}=-p_{1}\bm{I}+\bm{T}_{1}$ and the stress tensor
$\bm{T}_{1}=\dot{\bm{\gamma}}_{1}+\mathsfbi{A}$, with

$\displaystyle\mathsfbi{A}=\left[-1+\left(1+\frac{1}{2}\mbox{\it
Cu}^{2}\dot{\bm{\gamma}}_{0}:\dot{\bm{\gamma}}_{0}\right)^{\frac{n-1}{2}}\right]\dot{\bm{\gamma}}_{0}.$
(27)

For boundary conditions, we have the first correction to the Newtonian
propulsion velocity, $\bm{u}_{1}(\tau=\tau_{0})=\bm{U}_{1}=U_{1}\bm{e}_{z}$,
on the particle surface, and $\bm{u}_{1}(\tau\to\infty)=\mathbf{0}$ in the
far-field. To obtain the propulsion speed $U_{1}$, we bypass detailed
calculations of the flow via a reciprocal theorem approach (Lauga, 2014). By
considering an auxiliary Stokes flow due to a prolate spheroid of the same
geometry translating at a velocity $\hat{\bm{U}}$, where the velocity
$\hat{\bm{u}}$ and stress $\hat{\bm{\sigma}}$ fields satisfy
$\gradient\cdot\hat{\bm{\sigma}}=\mathbf{0}$ and
$\gradient\cdot\hat{\bm{u}}=0$, one can form the relation

$\displaystyle\hat{\bm{u}}\cdot(\gradient\cdot\bm{\sigma}_{1})=\bm{u}_{1}\cdot(\gradient\cdot\hat{\bm{\sigma}})=0.$
(28)

By integrating the relation over the fluid volume $V$ exterior to the particle
surface $S$ and applying the divergence theorem, one can obtain

$\displaystyle\int_{S}{\bm{n}\cdot\hat{{\bm{\sigma}}}\cdot{\bm{u}}_{1}}\
\text{d}S-\int_{S}{\bm{n}\cdot{\bm{\sigma}}_{1}\cdot{\hat{\bm{u}}}}\
\text{d}S=\int_{V}{\bm{\sigma}_{1}:\nabla\hat{\bm{u}}}\
\text{d}V-\int_{V}{\hat{\bm{\sigma}}:\nabla\bm{u}_{1}}\ \text{d}V.$ (29)

We note that due to the force-free condition at $O(\epsilon)$,
$\int_{S}{\bm{n}\cdot{\bm{\sigma}}_{1}}\ \text{d}S=\mathbf{0}$, the second
integral on the left hand side of (29) is given by
$\int_{S}{\bm{n}\cdot{\bm{\sigma}}_{1}\cdot{\hat{\bm{u}}}}\
\text{d}S=(\int_{S}{\bm{n}\cdot{\bm{\sigma}}_{1}}\
\text{d}S)\cdot\hat{\bm{U}}=0$. Upon substituting the constitutive equations
for $\hat{\bm{\sigma}}$ and $\bm{\sigma}_{1}$ and applying the boundary
condition $\bm{u}_{1}=\bm{U}_{1}$ on $S$, (29) simplifies to

$\displaystyle\hat{\bm{F}}\cdot\bm{U}_{1}=\int_{V}{\mathsfbi{A}:\nabla\hat{\bm{u}}}\
\text{d}V,$ (30)

where $\hat{\bm{F}}=\int_{S}\bm{n}\cdot\hat{\bm{\sigma}}\
\text{d}S=-8\pi\tau_{0}^{-1}\left[(\tau_{0}^{2}+1)\coth^{-1}\tau_{0}-\tau_{0}\right]^{-1}\bm{e}_{z}$
is the drag on the translating prolate spheroid in the auxiliary problem.
Therefore, the first-order correction to the phoretic speed is given in terms
of a volume integral in prolate spheroidal coordinates as

$\displaystyle
U_{1}=-\frac{(\tau_{0}^{2}+1)\coth^{-1}\tau_{0}-\tau_{0}}{4\tau_{0}^{2}}\int_{\tau_{0}}^{\infty}\int_{-1}^{1}\left(\mathsfbi{A}:\nabla\hat{\bm{u}}\right)(\tau^{2}-\zeta^{2})\;\text{d}\zeta\text{d}\tau,$
(31)

which can be evaluated with quadrature.

Figure 2: $(a)$ Swimming speed of a spheroidal Janus particle $U$ in a shear-
thinning fluid relative to its corresponding Newtonian value $U_{0}$ as a
function of the Carreau number for different values of eccentricity $e$ when
the shear-thinning effect is weak ($\beta=0.9$). The asymptotic results in the
small $\epsilon=1-\beta$ limit (lines) agree well with numerical simulations
(symbols). For large eccentricies (e.g. $e=0.9$ and 0.99), the Janus particle
can swim faster in a shear-thinning fluid than in a Newtonain fluid. $(b)$
Numerical results for strong shear-thinning effect ($\beta=0.1$), the
qualitative behaviours remain the same, the speed variations are substantially
larger. In both $(a,b)$, the active coverage of the particle $\zeta_{0}=0$ and
the shear-thinning power law index $n=0.25$.

###  3.2 Numerical simulation

To extend the results beyond the weakly non-Newtonian regime considered in the
asymptotic analysis in §3.1, we develop numerical simulations based on the
finite element method (FEM) using the partial differential equation (PDE)
module of the commercial package COMSOL to perform fully coupled simulations
of the momentum and continuity equations (9) with the Carreau-Yasuda
constitutive equation (18), and the solute transport equation (14). We use an
axisymmetric computational domain with a dimensionless radius of $500$ to
simulate the self-propulsion of the Janus particle in an unbounded fluid. A
sufficiently large domain size is important to guarantee accuracy due to the
slow spatial decay of flows at low Reynolds numbers. The Janus particle is
modelled as a half-spheroid whose major axis coincides with the axis of
symmetry. The simulations are performed in a reference frame that is co-moving
with the particle, and the far-field velocity is equal to the negative
swimming velocity determined by the force-free condition (11). The
computational domain is discretised by approximately $100000-127000$
triangular elements, and the mesh is locally refined near the particle to
properly resolve the spatial variation of the viscosity. Taylor-Hood and
quadratic Lagrange elements are adopted to discretise the flow field
($\bm{u},p$) and the concentration field $c$, respectively. It is important to
note that, theoretically, there exists a discontinuous alteration in surface
activity between the active and inert compartments of the Janus particle.
However, when modelled numerically, this abrupt transition can cause
significant numerical errors, particularly at lower Cu values. To alleviate
the numerical errors, we introduce a minor smoothing transition, dependent on
the mesh size, to the surface activity in the vicinity of the discontinuity.

In addition to comparing with the asymptotic results in this work, we have
validated our numerical implementation against previous results for a
spherical Janus particle in a shear-thinning fluid (Datt et al., 2017) and a
spheroidal Janus particle in a Newtonian fluid (Popescu et al., 2010); see
Appendix B for more details.

##  4 Results and discussion

Figure 3: $(a)$ Swimming speed of a spheroidal Janus particle in a shear-
thinning fluid with different values of eccentricity $e$ and active coverage
$\zeta_{0}$. The dashed line indicates the particles of which the swimming
speed is enhanced by the shear-thinning effect. $(b)$ Relative swimming
efficiency of a spheroidal Janus particle with different values of
eccentricity and active coverage. For all data points, $\mbox{\it Cu}=20000$,
$\beta=0.1$ and $n=0.25$.

###  4.1 Effect of particle eccentricity on self-diffusiophoresis in a shear-
thinning fluid

In a Newtonian fluid, the dependence of the self-diffusiophoretic propulsion
speed on the particle geometry and catalyst coverage was examined in detail by
previous works (Popescu et al., 2010; Poehnl et al., 2020). Here we
investigate how shear-thinning rheology impacts the propulsion speeds ($U$)
relative to their corresponding Newtonian values ($U_{0}$). The special case
of a spherical Janus particle was examined by Datt et al. (2017) and shown to
always swim slower in a shear-thinning fluid than in a Newtonian fluid across
a wide range of Cu. In figure 2$(a)$, we reproduce these results by setting
the eccentricity to be zero ($e=0$, black solid line and black circles): the
spherical Janus particle displays reduced propulsion speed ($U/U_{0}<1$) as Cu
increases from zero, reaching a local minimum when Cu is around
$\mathcal{O}(1)$, before approaching the Newtonian value again when Cu becomes
exceedingly large. We employ the spherical case as a benchmark to probe the
effect of particle geometry by varying the eccentricity from $e=0$ to
$e=0.99$. From spherical to moderately spheroidal particles (e.g., $e=0.6$),
the increased eccentricity does not affect the qualitative features of the
speed dependence on Cu.

However, for more slender spheroidal particles (e.g., $e=0.99$), our results
reveal that a self-diffusiophoretic particle can also swim faster in a shear-
thinning fluid than in a Newtonian fluid (blue dotted lines and blue upward
triangles), which was shown impossible for a spherical particle (black solid
line and black circles) (Datt et al., 2017). These new behaviors are predicted
by both the asymptotic results by the reciprocal theorem (lines) and results
by numerical simulations (symbols) in the weakly shear-thinning regime
($\beta=0.9$), which display excellent agreements as shown in figure 2$(a)$.
We verify that these new features continue to exist beyond the weakly non-
Newtonian regime by considering a small viscosity ratio ($\beta=0.1$) in
figure 2$(b)$, where we observe the same qualitative behaviours but with
greater magnitudes of speed enhancement and reduction at different Cu.

Figure 4: Swimming speed of $(a)$ a spherical $(e=0)$ and $(b)$ a spheroidal
$(e=0.99)$ Janus particle as a function of $\zeta_{0}$ with $\beta=0.1$. Three
fluids are considered: $\mbox{\it Cu}=0$ (blue downward-pointing triangle,
Newtonian fluid), 500 (red upward-pointing triangle), 20000 (black circle).

###  4.2 Effect of active surface coverage on self-diffusiophoresis in a
shear-thinning fluid

We focus in §4.1 on Janus particles with half active surface coverage
($\zeta_{0}=0$), which was shown to maximise the self-diffusiphoretic
propulsion speed of spherical and spheroidal particles in a Newtonian fluid.
Here, we examine whether this feature remains the same or not when the fluid
displays shear-thinning rheology. In figure 3$(a)$, we display the propulsion
speed relative to its Newtonian value as a function of particle eccentricity
and active surface coverage, which varies between $\zeta_{0}=-1$ (no active
surface coverage) and $\zeta_{0}=1$ (full active surface coverage). It is
observed that, regardless of the active surface coverage, the regime of
enhanced propulsion ($U/U_{0}>1$, indicated by the dashed line in figure
3$(a)$) only occurs when the particle eccentricity goes beyond a threshold
value of approximately 0.7. In addition, the enhanced propulsion occurs for a
wider range of active surface coverage with increased particle eccentricity.
For instance, among all the values of active surface coverage examined in
figure 3$(a)$, while enhanced propulsion is observed only in about $15\%$ of
the cases when $e=0.8$, the percentage increase to more than 60% when
$e=0.99$. Another interesting feature is the asymmetry in the occurrence of
enhanced propulsion with respect to the active surface coverage: the regime is
not symmetrically distributed around $\zeta_{0}$ but instead skewed towards
the positive direction of $\zeta$. This observation also suggests that the
specific case of half active coverage ($\zeta_{0}=0$), which was shown to
maximise self-diffusiophertic propulsion in previous works (Popescu et al.,
2010; Poehnl et al., 2020; Datt et al., 2017), may no longer be optimal for
spheroidal particles in shear-thinning fluids.

In addition to propulsion speed, efficiency is another relevant performance
measure of the swimming motion. Recent studies have investigated how the
geometrical shape of active particles influence their efficiency of swimming
in a Newtonian fluid (Guo et al., 2021; Daddi-Moussa-Ider et al., 2021). Here
we adopt the widely used definition of swimming efficiency introduced by
Lighthill (1975) for low-Reynolds-number swimmers, $\eta=\bm{F}\cdot\bm{U}/P$,
to characterize the efficiency of swimming in a shear-thinning fluid.
Lighthill’s efficiency compares the power dissipation of the swimmer,
$P=\int\bm{n}\cdot\bm{\sigma}\cdot\bm{u}\;\mathrm{d}S$, with the power
required to move a particle with identical shape at the same swimming velocity
$\bm{U}$ against the drag force $\bm{F}$. Our results show that, while speed
enhancement occurs only in a specific domain of eccentricity and active
surface coverage (figure 3$(a)$), the swimming efficiency in a shear-thinning
fluid is consistently enhanced, $\eta/\eta_{0}>1$, relative to the
corresponding swimming efficiency in a Newtonian fluid ($\eta_{0}$) in the
entire domain shown in figure 3$(b)$. Taken together, these results reveal
that self-diffusiophoretic propulsion can be enhanced both speed-wise and
efficiency-wise in a shear-thinning fluid relative to the corresponding case
in a Newtonian fluid.

Figure 5: Swimming speed of $(a)$ a pair of complementarily coated spherical
$(e=0)$ and $(b)$ spheroidal $(e=0.99)$ particles in a shear-thinning fluid
with active coverage $\zeta_{0}=\pm 0.5$. The compementarily coated spherical
swimmers are propelled with the same speed, while spheroidal swimmers break
this symmetry. In both $(a,b)$, $\beta=0.1$ and $n=0.25$.

Next, we further examine the asymmetry observed in the enhanced propulsion
speed with respect to the active surface coverage shown in figure 3$(a)$. We
display in figure 4$(a)$ the absolute propulsion speed of a spherical and a
spheroidal particles as a function of active surface coverage at different
values of Cu. In figure 4$(a)$, we observe that the propulsion speed of a
spherical particle is symmetric about the half surface coverage ($\zeta_{0}$),
which maximises the speed in both Newtonian ($\mbox{\it Cu}=0$) and shear-
thinning ($\mbox{\it Cu}>0$) fluids. In contrast, for a spheroidal particle
with $e=0.99$ shown in figure 4$(b)$, while the aforementioned features still
hold in the Newtonian limit ($\mbox{\it Cu}=0$, blue downward triangles), when
the fluid is shear-thinning (e.g., $\mbox{\it Cu}=500$, red upward triangles)
the variation of the propulsion becomes asymmetric about $\zeta_{0}=0$, which
no longer maximises the self-diffusiophoretic propulsion speed. Instead, the
maximum propulsion speed occurs at a positive active surface coverage
($\zeta_{0}>0$) as shown in figure 4$(b)$, depending on parameters measuring
the shear-thinning effect including $\beta$ and Cu. The emergence of this
novel feature requires the combined presence of both non-Newtonian rheology
and non-spherical geometry, which we attempt to better understand via symmetry
arguments presented in the next section.

###  4.3 Symmetry considerations

Figure 6: Schematics illustrating the symmetry considerations for a pair of
complementarily coated particles. The phoretic slip velocity on the surface of
a particle with an active coverage $\zeta_{0}$ is denoted as
$u_{s}(\zeta;\zeta_{0})$, and the corresponding propulsion speed as
$U(\zeta_{0})$. The slip velocity on a fully-coated spherical particle is zero
everywhere due to the isotropy. Consequently, the flow induced by the
particles with active region $\pm\zeta_{0}$ always shows a symmetry, which
leads to identical speeds. As the slip velocity on a fully-coated anisotropic
particle is not zero, the flow and the slip velocity do not have the
reflection symmetry, and the particle speeds are not the same in general.
However, if the flow is Newtonian, the speeds are the same due to the
linearity.

To examine this feature of symmetry breaking across the full range of Cu, we
compare the swimming speed of two complementarily coated particles with
$\zeta_{0}=\pm 0.5$ for the spherical ($e=0$, figure 5$a$) and spheroidal
($e=0.99$, figure 5$b$) cases. For spherical particles, figure 5($a$) shows
that the swimming speed of particle with $\zeta_{0}=-0.5$ (blue triangles) is
identical to that with $\zeta_{0}=0.5$ (black circles) over the entire range
of Cu, despite the latter having a significantly larger active surface
coverage. On the contrary, Figure 5$(b)$ demonstrates that the swimming speed
of two complementarily coated spheroidal particles ($\zeta_{0}=\pm 0.5$) only
approach the same value when Cu is exceedingly small or large, where the fluid
medium becomes effectively Newtonian. At intermediate values of Cu, the
spheroidal particle with $\zeta_{0}=-0.5$ (blue triangles) generally exhibits
a considerably different swimming speed compared with its complementarily
coated counterpart ($\zeta_{0}=0.5$, black circles) as shown in figure 5$(b)$.

One may understand the above feature as a combined result of symmetry breaking
and nonlinear rheology as illustrated in figure 6. We denote the phoretic slip
velocity on the surface of a particle with an active coverage $\zeta_{0}$ as
$u_{s}(\zeta;\zeta_{0})$ and the corresponding propulsion speed as
$U(\zeta_{0})$. We note that in the zero-$Pe$ limit considered here, the
linearity of the Laplace equation allows superposition in the solute
concentration problem. Now, consider a fully coated particle, the phoretic
slip velocity can be decomposed into two complementary cases,
$u_{s}(\zeta;1)=u_{s}(\zeta;-\zeta_{0})-u_{s}(-\zeta;\zeta_{0})$, as shown in
figure 6 for a spherical and a spheroidal particles. For a fully coated
spherical particle, the slip velocity is zero everywhere on the particle
surface due to isotropy, $u_{s}(\zeta;1)=0$. This property leads to the result
$u_{s}(\zeta;-\zeta_{0})=u_{s}(-\zeta;\zeta_{0})$, which means that the
boundary conditions on two complementarily coated spherical particles become
identical upon a reflection about $z=0$. This result is illustrated in figure
7$(a)$ for the slip velocity of two complementarily coated spherical
particles, which consequently, upon a reflection about $z=0$, generate the
same flow field as shown in figure 7$(c)$. The identical propulsion speed of
these complementarily coated particles, $U(\zeta_{0})=U(-\zeta_{0})$, is
therefore a direct result of isotropy for spherical self-diffusiophertic
particles, regardless of whether the fluid is Newtonian or non-Newtonian.

Figure 7: The slip velocity and the flow field around the particles with
active region $\zeta_{0}=\pm 0.5$, in a shear-thinning fluid with $\mbox{\it
Cu}=1$, $\beta=0.1$ and $n=0.25$. The slip velocity and the flow field around
the spherical swimmers has an upside-down symmetry $(a)(c)$, which does not
appear in those around the spheroidal swimmers ($e=0.99$) $(b)(d)$.

When particle eccentricity is introduced, the anisotropy implies that the slip
velocity of a fully coated spheroidal particle does not vanish everywhere on
the surface of the particle, $u_{s}(\zeta;1)\neq 0$. As illustrated in figure
7$(b)$, the boundary conditions on two complementarily coated spheroidal
particles therefore no longer have the reflection symmetry about $z=0$,
$u_{s}(\zeta;-\zeta_{0})\neq u_{s}(-\zeta;\zeta_{0})$, leading to generally
distinct flows surrounding the spheroidal particles as shown in figure 7$(d)$.
Consequently, unlike spherical particles, one may expect two complementarily
coated spheroidal particles to have distinct propulsion speeds in general.
This conclusion is largely true as shown in figure 5$(b)$, except for the
special case when the fluid is Newtonian. For a Newtonian fluid, the linearity
of the governing equations allows the superposition of the solutions
associated with the pair of particles with complementary coatings to form the
solution of a particle with full coating (figure 6$a$), leading to the result,
$U(1)=U(\zeta_{0})-U(-\zeta_{0})$. Since $U(1)=0$ for a fully coated particle,
we obtain the conclusion $U(\zeta_{0})=U(-\zeta_{0})$ for a Newtonian fluid,
which holds for both spherical and spheroidal particles, despite the absence
of reflection symmetry in their slip velocities. When the fluid is non-
Newtonian, the superposition described above no longer holds, allowing the
propulsion speed of two complementarily coated spheroidal particles to be
different.

To summarise, isotropy in spherical geometry alone guarantees that two
complementarily coated particles propel with identical speeds, regardless of
whether the fluid is Newtonian or not. In parallel, in a Newtonian fluid the
linearity of the problem alone guarantees the same, regardless of whether the
particle is spherical or not. Hence, to propel two complementarily coated
particles with different speeds, both istropy and linearity need to be broken.
The emergence of different speeds for a pair of complementarily coated
spheroidal particles in a shear-thinning fluid reported here, therefore,
serves as a specific example illustrating this general feature.

##  5 Concluding remarks

Shear-thinning viscosity is a non-Newtonian behavior that active particles
often encounter in biological fluids. The investigation into how this
ubiquitous non-Newtonian rheology impacts the propulsion speed of active
particles has garnered considerable recent interest. In particular, previous
studies have demonstrated how shear-thinning rheology slows down spherical
active particles (Datt et al., 2015, 2017). A more recent investigation (van
Gogh et al., 2022) has suggested that, by tuning the geometrical shape of a
squirmer, it is possible for a spheroidal squirmer to swim faster in a shear-
thinning fluid than in a Newtonian fluid. In this work, we have extended the
analysis by van Gogh et al. (2022) on the spheroidal squirmer model to self-
diffusiophoretic particles, a major physico-chemical propulsion mechanism of
synthetic active particles. Unlike the squirmer model, where the velocity
distribution on the particle surface is prescribed, the effective slip
velocity of a self-diffusiophoretic particle is determined by the solute
concentration gradient and the phoretic mobility. Using asymptotic analysis to
probe the weakly non-Newtonian behavior, we have demonstrated that shear-
thinning viscosity can indeed enhance self-diffusiophoretic propulsion of
spheroidal particles with a large particle eccentricity in a specific regime
of Carreau number. This result is in stark contrast with spherical self-
diffusiophoretic particles, which always swim slower in a shear-thinning fluid
(Datt et al., 2017). We have also used numerical simulations to verify that
the new features uncovered by the asymptotic analysis continue to hold beyond
the weakly non-Newtonian regime.

We have also systematically characterised the dependence of the self-
diffusiophoretic propulsion speed on the particle’s active surface coverage in
a shear-thinning fluid. Previous studies showed that a pair of complementarily
coated spherical or spheroidal particles always propel at the same speed in a
Newtonian fluid. When the fluid becomes shear-thinning, the same propulsion
speed still occurs when the complementarily coated particles are spherical in
shape. However, we have found distinct propulsion speeds for two
complementarily coated spheroidal particles in a shear-thinning fluid. We have
also presented symmetry arguments to better understand how this new feature
emerges as a combined effect of anisotropy associated with the spheroidal
geometry and nonlinearity associated with the non-Newtonian rheology. Such
symmetry-breaking might hint at using anisotropic active particles as a tool
for probing microrheology of complex fluids.

We remark on several limitations of the current work and discuss potential
directions for further investigations. First, we have neglected the effect of
solute advection by considering the zero $Pe$ limit. It remains unclear how
the flow modifications due to shear-thinning rheology influence solute
advection and thereby the phoretic propulsion. In particular, the symmetry
considerations presented in Section 4.3, which require the linearity of the
Laplace equation, would no longer hold for finite $Pe$. It would therefore be
interesting to probe how the nonlinearity associated with solute advection
affects the symmetry breaking observed for the propulsion of complementarily
coated particles. Second, we have followed previous work (Datt et al., 2017)
to focus on the non-Newtonian effect in the bulk fluid in this work,
neglecting the influence of fluid rheology on the surface slip of a self-
diffusiophretic particle, which was shown to modify the propulsion speed of a
spherical Janus particle in intriguing manners (Choudhary et al., 2020). In
particular, in a weakly shear-thinning fluid, the effect due to the modified
slip velocity could dominate the retardation due to the bulk non-Newtonian
stress, leading to the speed enhancement of a Janus spherical particle. An
investigation is currently underway to extend the analysis beyond the weakly
non-Newtonian regime and examine the effect of particle geometry in this more
complex physical scenario, where the non-Newtonian effects on both slip and
mobility of self-diffusiophoretic particles are taken into account. We also
call for future efforts in developing a comprehensive physical understanding
of the findings reported in this work. Last, we focus on the effect of shear-
thinning viscosity here while complex biological fluids also display other
non-Newtonian fluid behaviors including viscoelasticity. Future work
accounting for the viscoelastic stress and its combined effects with shear-
thinning rheology will shed light on how the geometric shape of self-
diffusiophretic particles should be tuned to maximize their propulsion in
biological fluids.

Funding. We acknowledge the supports from the National Science Foundation
under grant numbers 1931292 and 2323046 (to O.S.P.), the National Natural
Science Foundation of China under grant number 12372258 (to Y.M.), and the
Fundamental Research Funds for the Central Universities, Peking University (to
Y.M.). L.Z. acknowledges the partial support from the Singapore Ministry of
Education Academic Research Fund Tier 2 (MOE-T2EP50122-0015). The computation
of the work was performed on resources of the National Supercomputing Centre,
Singapore (https://www.nscc.sg).

Declaration of interests. The authors report no conflict of interest.

##  Appendix A Solution to the zeroth-order (Newtonian) problem

Here we summarise the solution to the zeroth-order problem, which corresponds
to the self-diffusiophertic motion of a spheroidal particle in a Newtonian
fluid considered in previous works (Leshansky et al., 2007; Lauga & Michelin,
2016; Popescu et al., 2010; Poehnl et al., 2020). In particular, we follow the
approach by Poehnl et al. (2020) here to determine the unknown velocity field
$\bm{u}_{0}$ and propulsion velocity $\bm{U}_{0}$.

The unknown propulsion velocity $\bm{U}_{0}$ can be obtained using the Lorentz
reciprocal theorem (Stone & Samuel, 1996; Popescu et al., 2010; Poehnl et al.,
2020), by considering an auxiliary Stokes flow problem
$(\hat{\bm{u}},\hat{\bm{\sigma}})$ of a translating prolate spheroidal
particle of the same geometry along its major axis. Via the reciprocal theorem
(Popescu et al., 2010; Poehnl et al., 2020), an integral relation is obtained
as

$\displaystyle\hat{\bm{F}}\cdot\bm{U}_{0}=-\int_{S}\bm{u}_{s}\cdot(\bm{n}\cdot\hat{\bm{\sigma}})\;dS,$
(32)

which relates the force on the translating particle in the auxiliary problem
$\hat{\bm{F}}$ with the unknown propulsion velocity $\bm{U}_{0}$ via a surface
integral involving the surface velocity $\bm{u}_{s}$. By using the known
solution to the auxiliary problem (Happel & Brenner, 2012) and simplifying the
surface integral in the prolate spheroidal coordinates, the propulsion
velocity $\bm{U}_{0}=U_{0}\mathbf{e}_{z}$ is obtained in terms of the integral
(Popescu et al., 2010; Poehnl et al., 2020)

$\displaystyle
U_{0}=-\frac{\tau_{0}}{2}\int_{-1}^{1}u_{s}\frac{\sqrt{1-\zeta^{2}}}{\sqrt{\tau_{0}^{2}-\zeta^{2}}}\;d\zeta\cdot$
(33)

Upon substituting the slip velocity given by (22), one obtain the final
expression of the propulsion speed for $N$ phoretic modes as (Poehnl et al.,
2020)

$U_{0}=\frac{\tau_{0}^{2}}{2}\sum_{n=1,\ \text{odd
$n$}}^{N}B_{n}\int_{-1}^{1}\frac{P_{1}^{1}(\zeta)P_{n}^{1}(\zeta)}{\tau_{0}^{2}-\zeta^{2}}\
d\zeta\cdot$ (34)

To determine the velocity field, we consider a streamfunction $\psi_{0}$ for
the axisymmetric flow in the co-moving frame

$\displaystyle\bm{u}_{0}-\bm{U}_{0}=\frac{1}{h_{\zeta}h_{\phi}}\frac{\partial\psi_{0}}{\partial\zeta}\bm{e}_{\tau}-\frac{1}{h_{\tau}h_{\phi}}\frac{\partial\psi_{0}}{\partial\tau}\bm{e}_{\zeta},$
(35)

where the far-field corresponds to a uniform flow given by $-\bm{U}_{0}$. A
general solution of the streamfunction can be expanded in terms of products of
the Gegenbauer functions in the prolate spheroidal coordinates. For a bounded
solution satisfying the far-field and force-free condition, the streamfunction
takes the form (Poehnl et al., 2020)

$\displaystyle\psi_{0}(\tau,\zeta)=\sum_{n=2}^{\infty}g_{n}(\tau)G_{n}(\zeta),$
(36)

with $g_{2}(\tau)=C_{4}H_{4}(\tau)+D_{2}H_{2}(\tau)-2c^{2}U_{0}G_{2}(\tau)$,
$g_{3}(\tau)=C_{3}+C_{5}H_{5}(\tau)+D_{3}H_{3}(\tau)$, and
$g_{n}(\tau)=C_{n+2}H_{n+2}(\tau)+C_{n}H_{n-2}(\tau)+D_{n}H_{n}(\tau)$ for
$n\geq 4$, where $G_{n}$ and $H_{n}$ are the Gegenbauer polynomials of the
first and the second kinds, respectively. The coefficients $C_{n}$ and $D_{n}$
are determined by the tangential slip velocity and the zero normal velocity on
the particle surface. If we consider a slip velocity (22) expansion of only
$N$ modes and apply the boundary conditions to the stream function expansion
(36) with only $N$ terms ranging from $n=2$ to $n=N+1$, the following system
of equations is obtained

$\displaystyle g_{n}(\tau_{0})$ $\displaystyle=0,\quad\text{for}\ 2\leq n\leq
N+1,$ (37)
$\displaystyle\partialderivative{g_{n}}{\tau}\Big{|}_{\tau=\tau_{0}}$
$\displaystyle=\tau_{0}c^{2}n(n-1)B_{n-1},\quad\text{for}\ 2\leq n\leq N+1.$
(38)

These equations can be separated into a system with coefficients that only
have even indices and a system with coefficients that only have only odd
indices, as all the indices of $C_{n}$ and $D_{n}$ in $g_{n}$ are always the
same parity (Poehnl et al., 2020). The odd system of equations always has one
more unknown than equation; a solve-able system of equations is obtained by
setting

$C_{N+3-m}=0,\ \text{where}\ m=\begin{cases}1\ \text{, when $N$ is odd}\\\ 0\
\text{, when $N$ is even}\end{cases},$ (39)

for $N\geq 2$. Upon obtaining the phoretic modes $B_{n}$ using (21) and (23)
and using the result given by (34), the system of equations (37)–(38) is
solved for the coefficients $C_{n}$ and $D_{n}$ for the zeroth-order velocity
field. Interested readers are referred to previous works for further details
(Popescu et al., 2010; Poehnl et al., 2020).

##  Appendix B Validation of numerical simulations

In this appendix, we include results on the validation of our numerical
approach against previously reported findings. First, we follow the numerical
implementation described in Section 3.2 to simulate the self-propulsion of a
spherical Janus particle in a shear-thinning fluid and compare the numerical
results with the asymptotic solution obtained by Datt et al. (2017) in the
weakly nonlinear limit ($\beta=0.99$). As shown in figure 8$(a)$, the results
display satisfactory agreements for a wide range of Cu for different active
surface coverage, $\zeta_{0}$. Second, we assess the capability of our
numerical approach in handling non-spherical geometries by simulating the
self-propulsion of spheroidal Janus particles with different eccentricities in
a Newtonian fluid and compare the numerical results with the solution obtained
by Popescu et al. (2010). As shown in figure 8$(b)$, the results again agree
satisfactorily, validating our numerical implementation.

Figure 8: Validation of numerical approach against previously reported
results. $(a)$ Comparison of the numerical results (symbols) on the swimming
velocity of a spherical Janus particle as a function of $Cu$ for different
active surface coverage $\zeta_{0}$ with the asymptotic solution (lines) of
the scaled first-order swimming velocity ($U_{1}/U_{0}$) obtained by Datt et
al. (2017) in a weakly shear-thinning fluid ($\beta=0.99$). $(b)$ Comparison
of the numerical results (symbols) on the swimming velocity $U$ of a
spheroidal Janus particle as a function of $\zeta_{0}$ for different
eccentricities $e$ with the solution (lines) obtained by Popescu et al. (2010)
in a Newtonian fluid. Note that the Janus particles simulated here are coated
on the bottom to maintain consistency, and correspondingly, we set $A=-1$ and
$M=1$ in both $(a,b)$.

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