1. Introduction
Following Lighthill and Blake’s seminal works on ciliary propulsion of spherical squirmers in a Newtonian fluid (Lighthill Reference Lighthill1952; Blake Reference Blake1971), Keller & Wu (Reference Keller and Wu1977) developed theory for the ciliary propulsion of spheroidal squirmers. The original model only included the first swimming mode, and was later extended to include higher modes of the tangential surface velocity (Theers et al. Reference Theers, Westphal, Gompper and Winkler2016; Pohnl, Popescu & Uspal Reference Pohnl, Popescu and Uspal2020). Nevertheless, similar to the case of spherical squirmers, the translational velocity of the spheroidal squirmers
$\boldsymbol{U} = U\boldsymbol{e}_z$
, where
\begin{equation} U = B_1\tau _0 \big [\tau _0 - (\tau _0^2-1)\coth ^{-1}\tau _0\big ], \end{equation}
depends on the first polar squirming mode
$B_1$
. Here, the surface parameter
$\tau _0$
is related to the spheroidal squirmer’s eccentricity
$e$
through
$\tau _0 = 1/e$
. In the limit
$\tau _0\rightarrow \infty$
, (1.1) recovers the classical result first obtained by Lighthill (Reference Lighthill1952) for spherical squirmers. That is, for tangentially actuated spherical squirmers,
$U_0=2B_1/3$
. The translational speed also means that spheroidal, more elongated squirmers propel faster compared with spherical squirmers (as illustrated in figure 1
a). Equation (1.1) has proven useful, and even critical, in several studies that analytically investigate the influence of shape (through changes in the eccentricity) on the swimming dynamics of spheroidal squirmers in various environments including porous media (Demir et al. Reference Demir, van Gogh, Palaniappan and Nganguia2024) and shear-thinning fluids (van Gogh et al. Reference van Gogh, Demir, Palaniappan and Pak2022).
In addition to translational motion, many microorganisms also undergo rotation. The rotation could be of the entire organism, such as with Volvox (Pedley, Brumley & Goldstein Reference Pedley, Brumley and Goldstein2016), or an extension of the organism’s body, such as a flagellum (Purcell Reference Purcell1997). Pak & Lauga (Reference Pak and Lauga2014) derived swimming kinematics for generalised squirming motion. Besides recovering Lighthill’s results for translational motion, they provided analytical expressions for the swirling (or rotational) motion (flow field and velocity) of ciliated microorganisms. In particular, they derived the angular velocity to be
\begin{equation} \boldsymbol{\varOmega }_0 = -\frac {C_{01}}{a^3}\boldsymbol{e}_z, \end{equation}
where
$C_{01}$
is the first azimuthal squirming mode, and
$a$
is the squirmer’s radius. Two years later, Pedley et al. (Reference Pedley, Brumley and Goldstein2016) specifically extended Lighthill and Blake’s squirmer model to include axisymmetric swirl velocities.
The angular velocity does not affect the squirming motion in purely viscous fluids (Pak & Lauga Reference Pak and Lauga2014). However, recent studies have reported enhancement in the propulsion speed
$U_0$
of spherical squirmers in complex fluids as a result of swirling motion. Specifically, the swirling motion positively influenced the translational speed of spherical squirmers in viscoelastic fluids (Binagia et al. Reference Binagia, Phoa, Housiadas and Shaqfeh2020; Housiadas, Binagia & Shaqfeh Reference Housiadas, Binagia and Shaqfeh2021) and shear-thinning fluids (Nganguia et al. Reference Nganguia, Zheng, Chen, Pak and Zhu2020). In the latter study, the effect of the swirling motion depended on whether the analysis was carried out relative to purely viscous fluids, or exclusively within shear-thinning fluids. A swirling spherical squirmer in shear-thinning fluid did not propel any faster compared with a non-swirling squirmer in a purely viscous fluid. However, when both swimmers are placed in a shear-thinning fluid, swirling squirmers have a distinctive advantage over rotation-free, translating squirmers. Beyond the effects of swirling motion on translational velocity, the angular velocity can be employed to distinguish spherical squirmers in shear-thinning fluids. Nganguia et al. (Reference Nganguia, Zheng, Chen, Pak and Zhu2020) reported that squirmers have distinct rotational direction: positive (negative) rotation for pullers (pushers), and zero rotation for neutral squirmers.
While the swirling motion of spherical squirmers has garnered increasing interest in recent years, the kinematics and energetics of swirling spheroidal squirmers have not been explored nearly as extensively. Indeed, although the rotation of spheroidal rigid bodies has been studied to great extent since the 1910s (Jeffery Reference Jeffery1915, Reference Jeffery1922; Chwang & Wu Reference Chwang and Wu1974), the rotational motion of squirmers has only recently begun to pick the interest of researchers in the field. The available studies have focused on the dynamics and velocity field of the swimmers in purely viscous (Fortune et al. Reference Fortune, Worley, Sendova-Franks, Franks, Leptos, Lauga and Goldstein2021; Xia et al. Reference Xia, Yu, Zhang, Lin and Ouyang2025b ) and non-Newtonian fluids (Xia et al. Reference Xia, Yu, Lin and Ishikawa2025a ).
Fortune et al. (Reference Fortune, Worley, Sendova-Franks, Franks, Leptos, Lauga and Goldstein2021) used a spheroidal squirmer with axisymmetric swirl as a model for circular mills formed by the marine worm Symsagittifera roscoffensis (Franks et al. Reference Franks, Worley, Grant, Gorman, Vizard, Plackett, Doran, Gamble, Stumpe and Sendova-Franks2016). Specifically, they developed a theoretical framework to describe the flow field generated by the mills. In their study, the dynamics takes place in a purely viscous fluid where the first azimuthal squirming mode does not contribute to the velocity field. Still in purely viscous fluids, Xia et al. (Reference Xia, Yu, Zhang, Lin and Ouyang2025b ) considered the torques experienced by spheroidal squirmers in the absence of azimuthal squirming modes. Here, the torques resulted from the proximity with a wall rather than a self-rotating motion. Later, the authors numerically investigated the dynamics of swirling ellipsoidal squirmers in viscoplastic fluids (Xia et al. Reference Xia, Yu, Lin and Ishikawa2025a ) and reported significant benefit of the swirling motion on the overall performance of the squirmers. Their analysis did not consider the first azimuthal squirming mode, likely because, as in the case of spherical squirmers, it does not contribute to the flow field. Moreover, the authors only investigated swirling neutral squirmers. This choice may have been guided by the finding that neutral squirmers do not rotate in shear-thinning fluids (Nganguia et al. Reference Nganguia, Zheng, Chen, Pak and Zhu2020). Nevertheless, the authors’ approach did not permit the determination of the angular velocity.
To our knowledge, a result analogous to (1.2) for spheroidal squirmers has yet to be reported in the literature. To address this gap in knowledge, our paper is organised as follows. We formulate the problem in § 2, then derive and discuss results for the velocity field, rotational velocity and power dissipation for a purely rotating spheroidal squirmer in § 3. We conclude with a few remarks on the implication of our work in § 4.
2. Mathematical formulation
2.1. Geometrical set-up and swirling spheroidal squirmer model
Given the prolate spheroidal bodies of many ciliates, we employ the prolate spheroidal squirmer for its biological relevance. The prolate spheroidal coordinate system is given by
$(\tau , \eta , \phi )$
, where
$\tau \geqslant 1$
,
$-1\leqslant \eta \leqslant 1$
and
$0\leqslant \phi \leqslant 2\pi$
. The position vector
\begin{equation} \boldsymbol{x} = \frac {c\tau \sqrt {\tau ^2-1}}{\sqrt {\tau ^2-\eta ^2}}\boldsymbol{e}_\tau + \frac {c\eta \sqrt {1-\eta ^2}}{\sqrt {\tau ^2-\eta ^2}}\boldsymbol{e}_\eta , \end{equation}
where
$c=\sqrt {b_z^2-b_x^2}$
is the semi-focal length of a prolate spheroid with semi-major and semi-minor axes
$b_z$
and
$b_x$
, respectively. The unit vectors are related to those of the Cartesian coordinates as
\begin{equation} \boldsymbol{e}_\tau = \frac {\tau \sqrt {1-\eta ^2}}{\sqrt {\tau ^2-\eta ^2}}\boldsymbol{e}_r + \frac {\eta \sqrt {\tau ^2-1}}{\sqrt {\tau ^2-\eta ^2}}\boldsymbol{e}_z, \quad \boldsymbol{e}_\eta = -\frac {\eta \sqrt {\tau ^2-1}}{\sqrt {\tau ^2-\eta ^2}}\boldsymbol{e}_r + \frac {\tau \sqrt {1-\eta ^2}}{\sqrt {\tau ^2-\eta ^2}}\boldsymbol{e}_z, \end{equation}
where
$\boldsymbol{e}_r=\cos \phi \boldsymbol{e}_x+\sin \phi \boldsymbol{e}_y$
.
Several swimming mechanisms have been proposed to describe the dynamics of ciliated microorganisms (Lighthill Reference Lighthill1952; Ishikawa et al. Reference Ishikawa, Pedley, Drescher and Goldstein2020; Rodrigues, Lisicki & Lauga Reference Rodrigues, Lisicki and Lauga2021). In this work, we consider the motion of these organisms through a prescribed, constant surface velocity model. While the rotational velocity does not contribute to the overall propulsion of squirmers in purely viscous fluids (Pak & Lauga Reference Pak and Lauga2014), it is still associated with the first azimuthal squirming mode. Here, we extend the swirling spheroidal squirmer proposed by Xia et al. (Reference Xia, Yu, Lin and Ishikawa2025a ) by retaining the first azimuthal mode. Thus,
\begin{equation} \boldsymbol{u}_{sq} = -\tau _0\frac {\sqrt {1-\eta ^2}}{\sqrt {\tau _0^2-\eta ^2}}\left (C_1+3C_2\eta \right )\boldsymbol{e}_\phi , \end{equation}
where
$\tau =\tau _0$
is a parameter denoting the surface of the spheroidal squirmer. Physically, the second azimuthal mode (
$C_2$
) describes a rotating flagellum and counter-rotating body. Here, we only keep the first two azimuthal modes due to their physical significance (
$C_1$
is associated with rotation while
$C_2$
represents rotating flagellum of some microorganisms). For completeness, we present results for the general case of higher azimuthal modes in Appendix A.
2.2. Governing equations
We consider a spheroidal squirmer experiencing pure rotation through an axisymmetric flow in an unbounded purely viscous fluid. Given the size of the micro-swimmer, inertial effects can be omitted and the flow field is governed by the incompressible Stokes equation
\begin{align} - \boldsymbol{\nabla }\!p + \mu {\nabla} ^2\boldsymbol{u} &= \boldsymbol{0}, \end{align}
\begin{align} \boldsymbol{\nabla } \boldsymbol{\cdot }\boldsymbol{u} &= 0, \end{align}
where
$p$
is the pressure,
$\mu$
is the viscosity and
$\boldsymbol{u}$
is the fluid velocity. Since the focus of this paper is on the purely rotational motion
$\boldsymbol{u} = \left \langle 0,0,u_\phi \right \rangle$
of spheroidal squirmers, the governing equation can be expressed in spheroidal coordinates as
$\mu ({\nabla} ^2\boldsymbol{u})_\phi = 0$
, or (Fortune et al. Reference Fortune, Worley, Sendova-Franks, Franks, Leptos, Lauga and Goldstein2021)
\begin{equation} \big(\tau ^2-1\big)\frac {\partial ^2u_\phi }{\partial \tau ^2} + 2\tau \frac {\partial u_\phi }{\partial \tau } - \frac {u_\phi }{\tau ^2-1} + \big(1-\eta ^2\big)\frac {\partial ^2u_\phi }{\partial \eta ^2} - 2\eta \frac {\partial u_\phi }{\partial \eta } - \frac {u_\phi }{1-\eta ^2} = 0. \end{equation}
In the far field, the velocity in the laboratory frame is given by
\begin{equation} \boldsymbol{u}(\tau \rightarrow \infty , \eta ) = \boldsymbol{0}, \end{equation}
whereas the boundary condition on the squirmer’s surface reads
\begin{equation} \boldsymbol{u}(\tau =\tau _0, \eta ) = \boldsymbol{\varOmega }\times \boldsymbol{x} + \boldsymbol{u}_{sq}, \end{equation}
with
$\boldsymbol{u}_{sq}$
given by (2.3). The unknown angular velocity
$\boldsymbol{\varOmega }$
is obtained by enforcing the torque-free condition
\begin{equation} \int _S\boldsymbol{x}\times (\boldsymbol{\sigma } \boldsymbol{\cdot }\boldsymbol{n})\,\textrm {d}S = \boldsymbol{0}, \end{equation}
where
$S$
denotes the surface of the squirmer, the stress
$\boldsymbol{\sigma } = -p \unicode{x1D644} + \mu \dot {\boldsymbol{\gamma }}$
,
$\dot {\boldsymbol{\gamma }} = \boldsymbol{\nabla }\boldsymbol{u} + ( \boldsymbol{\nabla }\boldsymbol{u})^T$
is the strain tensor and
$\boldsymbol{n} = \boldsymbol{e}_\tau$
is the unit normal vector on the squirmer surface. Note that, in the case of a translating spheroidal squirmer, the swimming velocity
$\boldsymbol{U}$
(1.1) is derived by enforcing the force-free condition
$\int _S\boldsymbol{\sigma } \boldsymbol{\cdot }\boldsymbol{n}\,\textrm {d}S = \boldsymbol{0}$
.
Assuming a squirmer with constant semi-major axis length, we non-dimensionalise lengths using the semi-major axis length
$b_z$
, velocities using
$b_zC_1$
and pressure using
$\mu C_1$
. In this case, the dimensionless governing equations become
\begin{equation} - \boldsymbol{\nabla }\!p + {\nabla} ^2\boldsymbol{u} = \boldsymbol{0}, \quad \boldsymbol{\nabla } \boldsymbol{\cdot }\boldsymbol{u} = 0 , \end{equation}
while the squirming surface velocity
\begin{equation} \boldsymbol{u}_{sq} = -\tau _0\frac {\sqrt {1-\eta ^2}}{\sqrt {\tau _0^2-\eta ^2}}\left (1+3\chi _2\eta \right )\boldsymbol{e}_\phi , \end{equation}
where
$\chi _2 = C_2/(b_zC_1)$
. The focal length
$c = \sqrt {1-(b_x/b_z)^2}$
. Note that, with this choice of scaling, the equivalent radius for a spherical squirmer is
$a=b_z$
and (1.2) becomes
$\boldsymbol{\varOmega }_0=\boldsymbol{e}_z$
(after letting
$C_{01}=-C_1$
). Moreover, in this work the semi-major axis
$b_z$
is held fixed, and the focal length can be expressed as
$c=1/\tau _0$
.
3. Swirling motion
As done in our previous work (Nganguia & Pak Reference Nganguia and Pak2018; Demir et al. Reference Demir, van Gogh, Palaniappan and Nganguia2024), we exploit the linearity of the Stokes equation and decompose the swirling motion into two separate problems: the pumping problem with velocity
$\boldsymbol{u}_P$
and the rotational problem with velocity
$\boldsymbol{u}_R$
. In the pumping problem, the squirmer is held fixed (by an external force) and not allowed to rotate. A net flow is generated as a result of the prescribed surface velocity on the squirmer. In the rotational problem, the squirmer is allowed to swirl freely with angular velocity
$\boldsymbol{\varOmega }$
. The overall flow field around a swirling squirmer is then the superposition of the solutions of these two problems:
$\boldsymbol{u} = \boldsymbol{u}_P + \boldsymbol{u}_R$
.
3.1. Pumping problem
In this section, the squirmer acts as a pump according to the prescribed surface velocity. The solution of (2.5) (vanishing at infinity) is given by
\begin{equation} u_\phi = \sum ^\infty _{n=1} A_n Q^1_n(\cosh \xi ) P^1_n(\cos \zeta ), \end{equation}
where
$P^1_n$
and
$Q^1_n$
are the associated Legendre functions of the first and second kind, respectively. Letting
$\cosh \xi =\tau$
,
$\cos \zeta =\eta$
,
$\sin \zeta = \sqrt {1-\eta ^2}$
, (3.1) can be re-written as
\begin{equation} u_\phi (\tau ,\eta ) = \sum ^\infty _{n=1} A_n Q^1_n(\tau ) P^1_n(\eta ). \end{equation}
We use the boundary condition (2.7) and the orthogonality of the Legendre functions to obtain the constants
$A_n$
. This yields
\begin{equation} A_m = - \frac {\left (m+\dfrac {1}{2}\right )}{m(m+1) Q^1_m(\tau _0)} \tau _0\int ^1_{-1} \left [\frac {\sqrt {1-\eta ^2}}{\sqrt {\tau _0^2-\eta ^2}}\left (1+3\chi _2\eta \right ) \right ] P^1_m(\eta )\,\textrm {d}\eta .\end{equation}
Specifically, for a two-mode squirmer, the coefficients
\begin{align} A_1 &= \frac {3 \tau _0}{4 Q^1_1(\tau _0)} \left [\sqrt {\tau _0^2-1}-\big(\tau _0^2-2\big)\csc ^{-1}\left (\tau _0\right ) \right ]\! , \end{align}
\begin{align} A_2 &= \frac {45 \tau _0}{12 Q^1_2(\tau _0)} \left [ \frac {\sqrt {\tau _0^2-1}\left(3\tau _0^2-2\right)+\tau _0^2\left(4-3\tau _0^2\right)\csc ^{-1}\left (\tau _0\right )}{4} \right ] \chi _2, \end{align}
and the velocity field
\begin{equation} \boldsymbol{u}_P = \left [-A_1 Q^1_1(\tau ) \sqrt {1-\eta ^2}-3A_2Q^1_2(\tau )\eta \sqrt {1-\eta ^2}\right ]\boldsymbol{e}_\phi . \end{equation}
It can be shown that the total torque along the swimming direction is
$\boldsymbol{M}_P = M_P\boldsymbol{e}_z$
, where
\begin{equation} M_P = \displaystyle 2\pi c^3(\tau _0^2-1) \int ^1_{-1} \left [ \sigma _{\phi \tau } \right ] \Big |_{\tau =\tau _0} \sqrt {(\tau _0^2-\eta ^2)(1-\eta ^2)} \,\textrm {d}\eta , \end{equation}
and the stress in the
$\boldsymbol{e}_\phi -\boldsymbol{e}_\tau$
plane is given by
\begin{equation} \sigma _{\phi \tau } = \frac {\sqrt {\tau ^2-1}}{c\sqrt {\tau ^2-\eta ^2}}\frac {\partial u_\phi }{\partial \tau } - \frac {\tau }{c\sqrt {(\tau ^2-1)(\tau ^2-\eta ^2)}}u_\phi . \end{equation}
Carrying out the integral yields the total torque due to the pumping problem
\begin{equation} M_P = \frac {16}{3}\pi c^2 A_1, \end{equation}
where
$A_1$
is given in (3.4a
).
3.2. Rotational problem
We now allow the squirmer to swirl by superposing the solution of the pumping problem with that of the rotation of a rigid spheroidal particle rotating with velocity
$\boldsymbol{\varOmega }$
. In the latter problem, no surface velocity is prescribed. The purely rotational problem was solved by Jeffery (Reference Jeffery1915, Reference Jeffery1922) and Chwang & Wu (Reference Chwang and Wu1974). The velocity field, given in the polar cylindrical coordinates
$(r,\phi ,z)$
, is
\begin{equation} u_\phi = \frac {\beta _0}{c^2r}\left [ (c-z)R_1 + (c+z)R_2 + r^2\ln \frac {R_1-(z+c)}{R_2-(z-c)} \right ]\!, \end{equation}
where
\begin{align} & \beta _0\left [\frac {2e}{1-e^2} - \ln \left (\frac {1+e}{1-e}\right ) \right ] = \omega c^2, \end{align}
\begin{align} &\qquad R_1 = \sqrt {(z+c)^2+r^2}, \end{align}
\begin{align} &\qquad R_2 = \sqrt {(z-c)^2+r^2}, \end{align}
where
$e$
is the eccentricity of the squirmer, and
$\omega$
is the angular speed. The total torque exerted by the fluid on the prolate spheroid is given by
$\boldsymbol{M}_R = -M_R\boldsymbol{e}_z$
, where
\begin{equation} M_R = \frac {32\pi }{3} c \beta _0 .\end{equation}
Noting that the eccentricity
$e=1/\tau _0$
and using the identity
$\coth ^{-1}(x)=\ln [(x+1)/(x-1)]/2$
, the total torque can be expressed in terms of the rotational speed
$\omega$
from (3.10a
) as
\begin{equation} M_R = \frac {16\pi }{3} \frac {\omega c^3 (\tau _0^2-1)}{\left [ \tau _0-(\tau _0^2-1)\coth ^{-1}\left (\tau _0\right ) \right ]}. \end{equation}
Note that, in the spherical limit (
$\tau _0\rightarrow \infty$
) and for a fixed angular speed, the torque
$M_R=8\pi \omega$
, as expected.
The unknown rotational speed
$\omega$
is obtained by applying the torque-free condition on the squirmer
\begin{equation} \boldsymbol{M}_{\!P}+\boldsymbol{M}_R = \boldsymbol{0}. \end{equation}
After carrying out the calculation, (3.13) yields
\begin{equation} \omega = \frac {3 \tau _0 \left [\sqrt { \tau _0^2-1}-\left ( \tau _0^2-2\right ) \csc ^{-1}( \tau _0)\right ]}{4 c \sqrt { \tau _0^2-1}}. \end{equation}
In the spherical limit
$\tau _0\rightarrow \infty$
,
$\omega =1$
, which recovers (1.2) (in dimensionless form and after scaling
$C_{01}=-C_1$
). Figure 1(b) shows the rotational speed as a function of the squirmer’s eccentricity. The result suggests that more elongated swimmers have faster rotation rate. This monotonic behaviour aligns with the translational speed, as illustrated in figure 1(a). Thus a translating and swirling spheroidal squirmer in a Newtonian fluid propels and rotates faster compared with its spherical counterpart.
We remark that the angular velocity (3.14) is independent of the second azimuthal squirming mode,
$\chi _2$
. In general, only the first squirming mode contributes to the angular velocity. This is consistent with the angular velocity of a spherical squirmer (Pak & Lauga Reference Pak and Lauga2014). To illustrate this point, consider the contribution of the third azimuthal squirming mode. The surface velocity (2.10) now includes the term
$-\tau _0(\tau _0^2-\eta ^2)^{-({1}/{2})}(1-\eta ^2)^{({1}/{2})}\chi _3[16-20(1-\eta ^2)]/16 \boldsymbol{e}_\phi$
. In the spherical limit, this term recovers the third azimuthal squirming mode
$\chi _3[\sin \theta +5\sin (3\theta )]/16$
. Then, using (3.3) yields a non-zero term associated with the contribution of the third azimuthal mode to the velocity field. However, the surface stress integrates to zero. This confirms that the total torque
$M_P$
only depends on the velocity field contribution associated with the first azimuthal mode. Thus, the angular velocity
$\boldsymbol{\varOmega }$
is only a function of the first azimuthal mode. This finding contrasts with the translational velocity
$\boldsymbol{U}$
that was shown to be affected by all the higher, odd polar modes
$B_{2n+1}$
(Pohnl et al. Reference Pohnl, Popescu and Uspal2020). However, for spherical swimmers, note that while the third polar squirming mode does not contribute to the translational velocity in Newtonian fluids (Lighthill Reference Lighthill1952; Nganguia & Palaniappan Reference Nganguia and Palaniappan2024), it does induce propulsion in non-Newtonian fluids (Pietrzyk et al. Reference Pietrzyk, Nganguia, Datt, Zhu, Elfring and Pak2019; Housiadas Reference Housiadas2021).
Scaled translational speed
$U/U_0$
(a), rotational speed
$\omega /\omega _0$
(b) and power dissipation
$\mathcal{P}/\mathcal{P}_0$
(c) as a function of the squirmer’s eccentricity
$e$
. In (a), the curve is obtained using (1.1) (Keller & Wu Reference Keller and Wu1977) with
$B_1=1$
. In (b), the symbols are obtained from boundary integral simulations whereas the solid curve is calculated using (3.14). In (c), the curve is obtained using (3.22) with
$\chi _2=1$
. Inset of (c) shows the total power dissipation
$P=\mathcal{P}_B+\mathcal{P}_C$
and the power resulting from the polar squirming modes
$\mathcal{P}_B$
plotted as a function of the eccentricity. Here,
$\mathcal{P}_C = \mathcal{P}$
. The dimensionless quantities in the spherical limit are given by
$U_0=2/3$
,
$\omega _0=-1$
and
$\mathcal{P}_0=96\pi /5$
.
3.3. Velocity field
The velocity field
$\boldsymbol{u} = \boldsymbol{u}_R + \boldsymbol{u}_P$
. After substituting the position vector in Cartesian coordinates
\begin{equation} \boldsymbol{x} = c\sqrt {(\tau ^2-1)(1-\eta ^2)}\left ( \cos \phi \boldsymbol{e}_x + \sin \phi \boldsymbol{e}_y \right ) + c\tau \eta \boldsymbol{e}_z, \end{equation}
in (3.9), evaluating the resulting expression at
$\tau =\tau _0$
and simplifying, the rotational velocity field obtained by Chwang & Wu (Reference Chwang and Wu1974) can be expressed in prolate spheroidal coordinates as
\begin{equation} \boldsymbol{u}_R = \frac {3 }{4}\tau _0 \sqrt {1-\eta ^2} \left [\sqrt {\tau _0^2-1}-\big (\tau _0^2-2\big ) \csc ^{-1}(\tau _0)\right ] \boldsymbol{e}_\phi .\end{equation}
On the other end, the pumping velocity can be expressed in the form
$\boldsymbol{u}_P = \boldsymbol{u}_{C_1} + \boldsymbol{u}_{C_2}$
, where
\begin{equation} \boldsymbol{u}_{C_1} = -\frac {3}{4} \tau _0 \sqrt {1-\eta ^2} \left [\sqrt {\tau _0^2-1}-\big (\tau _0^2-2\big ) \csc ^{-1}(\tau _0)\right ] \boldsymbol{e}_\phi , \end{equation}
and
\begin{equation} \boldsymbol{u}_{C_2} = \frac {45}{16} \tau _0 \eta \sqrt {1-\eta ^2} \Big (\big (2-3 \tau _0^2\big ) \sqrt {\tau _0^2-1}+\big (3 \tau _0^2-4\big ) \tau _0^2 \csc ^{-1}(\tau _0)\Big ) \chi _2 \boldsymbol{e}_\phi . \end{equation}
With
$\eta =\cos \theta$
and lengths scaled by the radius of a spherical squirmer, both (3.16) and (3.17) converge to
$\boldsymbol{u}_R= \sin \theta$
and
$\boldsymbol{u}_{C_1} = -\sin \theta$
in the spherical limit
$\tau _0\rightarrow \infty$
. In the same limit,
$\boldsymbol{u}_{C_2} = -3\chi _2\sin (2\theta )/2$
. This result is consistent with Pak & Lauga (Reference Pak and Lauga2014) for spherical squirmers. Figure 2 shows the rotational velocity field on the surface of the squirmer that is associated with the second azimuthal mode with
$\chi _2=1$
for various values of the squirmer’s eccentricity (ranging from
$\tau _0=7.1$
(spherical limit) to
$\tau _0=1.05$
(needle-like limit)).
Rotational velocity field
$\boldsymbol{v}_{C_2}$
with
$\chi _2=1$
. The surface parameter is (a)
$\tau _0=7.1$
, (b)
$\tau _0=2.1$
, (c)
$\tau _0=1.1$
and (d)
$\tau _0=1.05$
.
3.4. Power dissipation
The power dissipation
$\mathcal{P}=-\int _S\boldsymbol{\sigma }\boldsymbol{\cdot }\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}\,\textrm {d}S$
. For a spherical squirmer, Pak & Lauga (Reference Pak and Lauga2014) and Pedley et al. (Reference Pedley, Brumley and Goldstein2016) extended the analysis by Lighthill (Reference Lighthill1952) to include the rate of work for the swirling motion in terms of the swimming azimuthal modes
$C_n$
. In dimensionless form, the power dissipation is given by
\begin{equation} \mathcal{P}_{0} = \sum ^\infty _{n=2} \frac {4n(n+1)(n+2)\pi }{2n+1} \chi _{0n}^2, \end{equation}
where
$\chi _{0n} = C_{0n}/aC_{01}$
. In the case of a two-mode squirmer (
$n=2$
), (3.19) simplifies to
$96\pi \chi _{02}^2/5$
. We previously derived the power dissipation of a two-mode prolate spheroidal squirmer in the absence of swirl (Demir et al. Reference Demir, van Gogh, Palaniappan and Nganguia2024). Focusing on the azimuthal component in prolate spheroidal coordinates, the power dissipation is given by
\begin{equation} \mathcal{P} = -2\pi c^2\sqrt {\tau _0^2-1} \int ^1_{-1} \big ( u_\phi \sigma _{\tau \phi } \big ) \Big |_{\tau =\tau _0}\sqrt {\tau _0^2-\eta ^2}\,\textrm {d}\eta . \end{equation}
After carrying out the integral and simplifying, the power dissipation reads
\begin{align} \mathcal{P} &= -\frac {24 \pi c A_2^2 }{5 \left (\tau _0^2-1\right )} \left [-3 \tau _0^2+3 \big (\tau _0^2-1\big ) \tau _0 \coth ^{-1}(\tau _0)+2\right ] \nonumber \\ & \quad \times \left [-3 \tau _0^3+3 \big (\tau _0^2-1\big )^2 \coth ^{-1}(\tau _0)+5 \tau _0\right ]\!. \end{align}
Then, substituting the coefficient
$A_2$
(3.4b) yields
\begin{align} \mathcal{P} &= \frac {135 \pi c }{d_2} \tau _0^2 \Big [\big (2-3 \tau _0^2\big ) \sqrt {\tau _0^2-1}+\big (3 \tau _0^2-4\big ) \tau _0^2 \csc ^{-1}(\tau _0)\Big ]^2 \nonumber \\ & \quad \times \Bigg [\tau _0 \big (9 \tau _0^4-21 \tau _0^2+10\big )+3 \big (\tau _0^2-1\big ) \coth ^{-1}(\tau _0) \nonumber \\ & \quad \times \Big (\!-6 \tau _0^4+10 \tau _0^2+3 \big (\tau _0^2-1\big )^2 \tau _0 \coth ^{-1}(\tau _0)-2\Big )\Bigg ] \chi _2^2, \end{align}
where
\begin{equation} d_2 = 8 \left [ 6 \tau _0^2-6 \big (\tau _0^2-1\big ) \tau _0 \coth ^{-1}(\tau _0)-4\right ]^2\! .\end{equation}
In the spherical limit (
$\tau _0\rightarrow \infty$
), (3.22) yields
$\mathcal{P}\rightarrow \mathcal{P}_{0}$
. Figure 1(c) shows the power dissipation generated by the second azimuthal mode as a function of the squirmer’s eccentricity. The power dissipation is scaled by the value of the energy expended by a spherical squirmer. Unlike the monotonically decreasing power dissipation generated from the tangential squirming modes (Demir et al. Reference Demir, van Gogh, Palaniappan and Nganguia2024), here, swirling spheroidal squirmers expend more energy compared with spherical squirmers. Of interest is the non-monotonic behaviour of the power, which reaches a maximum
$\mathcal{P}/\mathcal{P}_0\approx 1.14$
at high eccentricity (
$e\approx 0.95$
). In the limit of needle-like shape (
$\tau _0\rightarrow 1$
), the power dissipation
$\mathcal{P} = 135\pi ^3\chi _2^2/64$
. It is worth noting that this intriguing behaviour does not alter the conclusion drawn in our previous work (Demir et al. Reference Demir, van Gogh, Palaniappan and Nganguia2024); the total power dissipation
$P = P_B + P_C$
of a spheroidal squirmer remains lower compared with that of a spherical squirmer (see inset of figure 1
c).
4. Concluding remarks
In this paper, we derived the rotational velocity (3.14) and power dissipation (3.22) for a swirling spheroidal squirmer in a viscous fluid. Our results show that a spheroidal squirmer always rotates faster than a spherical squirmer (figure 1 b). Moreover, the contribution of the swirling motion to the power dissipation revealed a non-monotonic behaviour (figure 1 c), reaching a maximum near the needle-like limit of the eccentricity range.
We make a critical observation for self-propelling, swirling spheroidal squirmers. After close inspection of the total torque
$M_R$
, (3.12) can be cast in the form
$ M_R = 16\pi c^3 \alpha \tau _0(\tau _0^2-1) \omega /(3U^*)$
, where
$\alpha =B_1/b_zC_1$
and
$U^*=\alpha \tau _0 [ \tau _0-(\tau _0^2-1){} \coth ^{-1} (\tau _0 ) ]$
. This shows that the total torque due to the rotational motion is inversely proportional to the translational swimming speed. In fact, we can write
$M_R = \tilde {P}_{B_1}/U^*$
, where
$\tilde {P}_{B_1}$
approximates the mechanical (swimming) power associated with the first polar squirming modes (
$B_1$
). The exact expression for the power dissipation associated with the first polar mode is given by
$ P_{B_1} = -4\pi c(\tau _0^2-1) [ \tau _0-(\tau _0^2+1)\coth ^{-1}(\tau _0) ]$
(Demir et al. Reference Demir, van Gogh, Palaniappan and Nganguia2024). In the spherical limit, both expressions recover
$16\pi /3$
(Lighthill Reference Lighthill1952).
Total viscous power dissipation can be broken down into two parts: external and internal dissipation (Daddi-Moussa-Ider, Goldstein & Vilfan Reference Daddi-Moussa-Ider, Goldstein and Vilfan2023). The former results from hydrodynamic interaction between the swimmer and the surrounding fluid, while the latter depends on the specific swirling mechanism. For ciliates, internal dissipation is responsible for over 90 % of the total power dissipation (Keller & Wu Reference Keller and Wu1977; Ito, Omori & Ishikawa Reference Ito, Omori and Ishikawa2019). Thus, considering different swirling mechanisms such as the one proposed by Ishikawa et al. (Reference Ishikawa, Pedley, Drescher and Goldstein2020) will yield additional insight into the behaviour reported in figure 1(c). We are currently extending the model of Ishikawa et al. (Reference Ishikawa, Pedley, Drescher and Goldstein2020) to non-spherical squirmers.
Ciliated microorganisms, including Paramecium, Tetrahymena, etc. move along a helical path. This dynamics is in part due to (i) the rotation of their body on their own major axis or (ii) the rotation of individual cilia (Bullington Reference Bullington1925; Omoto & Kung Reference Omoto and Kung1980; Marumo, Yamagishi & Yajima Reference Marumo, Yamagishi and Yajima2021). The biological implications of our findings are twofold. First, similar to (1.1) for the swimming speed (which has been validated against experimental measurements (Rodrigues et al. Reference Rodrigues, Lisicki and Lauga2021)), (3.14) provides an expression to assess the rate of rotation experienced by ciliated microorganisms. Second, our study also proposes a means to estimate the energy expenditure of swirling spheroidal swimmers.
Our results complement and complete the work by Keller & Wu (Reference Keller and Wu1977) by analytically solving the fluid dynamics problem of a rotating spheroidal squirmer in a viscous fluid. Just as the translational velocity has become the benchmark for studying the effect of shape on the dynamics of micro-swimmers in complex fluids, our solution now provides a reference for the rotational velocity of swimmers. For instance, Nganguia et al. (Reference Nganguia, Zheng, Chen, Pak and Zhu2020) showed that spherical pushers and pullers experience opposite, non-zero rotational velocity in shear-thinning fluids. In another study, Binagia et al. (Reference Binagia, Phoa, Housiadas and Shaqfeh2020) and Housiadas et al. (Reference Housiadas, Binagia and Shaqfeh2021) reported a marked increase in the propulsion speed of spherical squirmers in viscoelastic fluids. In simpler viscous fluids, asymmetric azimuthal modes resulted in the propulsion of microswimmers along a helical path (Burada, Maity & Julicher Reference Burada, Maity and Julicher2022). To what extent these conclusions would change when accounting for non-spherical shapes remains unknown. For spheroidal swimmers, we now have an exact analytical expression for the rotational velocity, enabling us to investigate the effect of swirling motion on spheroidal squirmers’ kinematics and energetics in complex fluid environments.
Acknowledgements
H.N. thanks Y.-N. Young of the New Jersey Institute of Technology, USA, and H. Shum of the University of Waterloo, Canada, for insightful discussions, and for providing the numerical simulation used to validate the results.
Funding
H. N. gratefully acknowledges funding support from the National Science Foundation grant no. 2211633.
Declaration of interests
The authors report no conflicts of interest.
Appendix A. Coefficients of the flow field in the pumping problem with generalised surface velocity and power dissipation
We aim to derive an expression for the coefficients of the flow field given a generalised surface velocity that accounts for all the azimuthal modes. Note that the surface velocities for polar and azimuthal squirming modes have the same dependence on
$\eta$
. In dimensionless form, they are given by the unique series representation
\begin{equation} \boldsymbol{u}_{sq} = \tau _0 \sum ^\infty _{n=1}\frac {P_n^1(\eta )}{\sqrt {\tau _0^2-\eta ^2}}\chi _n. \end{equation}
Applying the boundary condition (2.7) with
$\boldsymbol{\varOmega } = \boldsymbol{0}$
(the pumping problem) yields
\begin{equation} \sum ^\infty _{n=1} A_n Q^1_n(\tau _0) P^1_n(\eta ) = \tau _0 \sum ^\infty _{n=1}\frac {P_n^1(\eta )}{\sqrt {\tau _0^2-\eta ^2}}\chi _n .\end{equation}
Multiplying both sides by
$P_m^1(\eta )$
, integrating with respect to
$-1\leqslant \eta \leqslant 1$
and using the the orthogonality property of the associated Legendre polynomials
$P_m^1$
yields
\begin{equation} A_m = - \frac {\left (m+\frac {1}{2}\right )}{m(m+1) Q^1_m(\tau _0)} \tau _0 \left ( \sum ^\infty _{n=1} \int ^1_{-1} \frac {P^1_n(\eta ) P^1_m(\eta )}{\sqrt {\tau _0^2-\eta ^2}} \,\textrm {d}\eta \right ) \chi _n .\end{equation}
The product of two associated Legendre polynomials can be expressed as the sum over a single associated Legendre polynomial (Dong & Lemus Reference Dong and Lemus2002; Pawlak & Stachowiak Reference Pawlak and Stachowiak2021). The product
$P^1_n(\eta ) P^1_m(\eta )$
becomes
\begin{equation} P^1_n(\eta ) P^1_m(\eta ) = -\sqrt {nm(n+1)(m+1)} \sum _k G P_k(\eta ), \end{equation}
where
$|n-m|\leqslant k\leqslant n+m$
,
$k+n+m$
is even, and
\begin{equation} G = (2k+1) \left (\begin{array}{ccc} n & m & k\\0 & 0 & 0 \end{array}\right ) \left (\begin{array}{ccc} n & m & k\\-1 & 1 & 0 \end{array}\right ) \end{equation}
is expressed in terms of the
$3-j$
symbols
$\left(\begin{array}{ccc} s_1 & s_2 & s_3\\t_1 & t_2 & t_3 \end{array}\right)$
(Dennis & Walker Reference Dennis and Walker1971; Dong & Lemus Reference Dong and Lemus2002). Note that the
$3-j$
symbol is zero when
$k+n+m$
is odd. Moreover,
\begin{equation} P_k(\eta ) = \frac {1}{2^k} \sum ^{k/2}_{l=0} (-1)^l \frac {(2k-2l)!}{l!(k-l)!(k-2l)!} \eta ^{k-2l}, \end{equation}
and
\begin{equation} \frac {1}{\sqrt {\tau _0^2-\eta ^2}} = \sum ^\infty _{j=0}\frac {(2j)!}{4^j(j!)^2}\frac {\eta ^{2j}}{\tau _0^{2j+1}} .\end{equation}
Using (A4), (A6) and (A7), the integral
\begin{eqnarray} \int ^1_{-1} \!\frac {P^1_n(\eta ) P^1_m(\eta )}{\sqrt {\tau _0^2-\eta ^2}} \,\textrm {d}\eta =& & -\sqrt {nm(n+1)(m+1)} \sum ^\infty _{j=0} \sum _k \sum ^{k/2}_{l=0} \frac {(-1)^l(2j)!(2k-2l)!}{2^{2j+k-1}(j!)^2 l! (k-l)! (k-2l)!} \nonumber \\ & & \times \frac {G \tau _0^{-(2j+1)}}{k-2l+2j+1}. \end{eqnarray}
Substituting (A8) into (A3) yields the coefficients of the velocity field associated with each azimuthal mode
$m$
\begin{eqnarray} A_m & &= \frac {\left (m+\frac {1}{2}\right )}{m(m+1) Q^1_m(\tau _0)} \tau _0 \Bigg ( \sum ^\infty _{n=1} \sum _k \sum ^{k/2}_{l=0} \frac {(-1)^l \sqrt {nm(n+1)(m+1)}(2k-2l)! }{2^{k-1} l!(k-l)! (k-2l)!} G \nonumber \\ & &\quad \times \sum ^\infty _{j=0} \frac {(2j)!}{4^{j}(j!)^2 } \frac {1}{(k-2l+2j+1) \tau _0^{2j+1}} \Bigg ) \chi _n. \end{eqnarray}
Note that, for
$k=2l$
, the summation over
$j$
is the series representation of the inverse cosecant function. Moreover, while the coefficients
$A_m$
are generally non-zero, the contribution of the higher azimuthal modes to the angular velocity vanishes after applying the torque-free condition.
A general expression for the power dissipation for all azimuthal modes is given by
\begin{align} \mathcal{P} &= -2\pi c \int ^1_{-1} \Bigg [ \big(\tau _0^2-1\big) \sum ^\infty _{n=2}a_n P_n^1(\eta ) \times \sum ^\infty _{n=2}b_n P_n^1(\eta ) - \tau _0 \sum ^\infty _{n=2}a_n P_n^1(\eta )\nonumber\\&\quad \times \sum ^\infty _{n=2}a_n P_n^1(\eta ) \Bigg ]\,\textrm {d}\eta , \end{align}
where
$a_n = A_n Q_n^{1}(\tau _0)$
and
$b_n=A_n Q_n^{1'}(\tau _0)$
. Since the associated Legendre polynomial
\begin{equation} P_n^m(x) = \frac {1}{2^n n!} \big (1-x^2\big )^{m/2} \dfrac {\,\textrm {d}^{n+m}}{\,\textrm {d}x^{n+m}} \big (x^2-1\big )^n , \end{equation}
setting
$m=1$
, the power dissipation can be re-written as
\begin{align} \mathcal{P} &= -2\pi c \int ^1_{-1} \Bigg [ \big(\tau _0^2-1\big) \Bigg ( \sum ^\infty _{n=2}a_n \frac {1}{2^n n!} \dfrac {\,\textrm {d}^{n+1}}{\,\textrm {d}\eta ^{n+1}} \big (\eta ^2-1\big )^n \Bigg )\nonumber\\&\quad\times \Bigg ( \sum ^\infty _{n=2}b_n \frac {1}{2^n n!} \dfrac {\,\textrm {d}^{n+1}}{\,\textrm {d}\eta ^{n+1}} \big (\eta ^2-1\big )^n \Bigg ) \nonumber \\ &\quad - \tau _0 \Bigg ( \sum ^\infty _{n=2}a_n \frac {1}{2^n n!} \dfrac {\,\textrm {d}^{n+1}}{\,\textrm {d}\eta ^{n+1}} \big (\eta ^2-1\big )^n \Bigg )^2 \Bigg ] (1-\eta ^2) \,\textrm {d}\eta. \end{align}
When
$n=2$
, (A12) recovers (3.22), the contribution of the second azimuthal mode to the power dissipation. In this case, (A12) reduces to
\begin{eqnarray} \mathcal{P} &=& -\frac {24\pi c A_2^2}{5} Q^1_2(\tau _0) \Big [ (\tau _0^2-1) Q^{1'}_2(\tau _0) - \tau _0 Q^1_2(\tau _0) \Big ] ,\end{eqnarray}
where the coefficient
$A_2$
is given in (3.4b), and the associated Legendre function of the second kind
$ Q_2^1 = \sqrt {\tau _0^2-1} [ (3\tau _0^2-2)/(\tau _0^2-1) - 3\tau _0 \coth ^{-1}(\tau _0) ]$
.
After substituting
$A_2$
and
$Q_2^1$
and simplifying, we recover the contribution of the second azimuthal mode (3.22).
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