2.1. Squirmer model

We employ the squirmer model (Lighthill Reference Lighthill1952; Blake Reference Blake1971; Pak & Lauga Reference Pak and Lauga2014) to represent microswimmers, where propulsion is generated by a slip velocity at the swimmer’s surface. This model has been widely used in biological fluid mechanics to elucidate the hydrodynamics of swimming micro-organisms (Pedley Reference Pedley2016; Ishikawa Reference Ishikawa2024). Building on previous work by Datt & Elfring (Reference Datt and Elfring2019), we investigate a squirmer with complete polar and azimuthal squirming modes in viscosity gradients. The general slip velocity at the surface is given by

(2.1) \begin{align} \boldsymbol{u}^{\mathcal{S}}(\theta , \phi ) = \sum _{n = 1}^\infty \left [\frac {2}{n(n + 1)}B_nP^{\prime}_n(\cos \theta )\sin \theta \hat {\boldsymbol{\theta }} + C_nP^{\prime}_n(\cos \theta )\sin \theta \hat {\boldsymbol{\phi }}\right ], \end{align}

where $P^{\prime}_n$ is the derivative of the $n$ th-order Legendre polynomial, $B_n$ and $C_n$ are the coefficients for the $n$ th polar and azimuthal squirming modes, respectively, with $\hat {\boldsymbol{\theta }}$ and $\hat {\boldsymbol{\phi }}$ denoting the unit polar and azimuthal tangent vectors, respectively. The first polar and azimuthal modes, $B_1$ and $C_1$ , determine the steady swimming speed in a uniform Newtonian fluid at zero Reynolds number, given by $U_{N} = (2/3)B_1$ and $\Omega _N = -C_1 / a$ , respectively. The azimuthal squirming modes have been used to model the chirality of microswimmers (Ghose & Adhikari Reference Ghose and Adhikari2014; Pedley Reference Pedley2016; Pedley, Brumley & Goldstein Reference Pedley, Brumley and Goldstein2016). When the polar and azimuthal squirming axes are aligned, squirmers swim in straight paths with rotational velocity $\Omega _N$ . However, the effects of their misalignment have only recently begun to receive attention. In such cases, squirmers follow chiral (spiral) trajectories (Burada et al. Reference Burada, Maity and Jülicher2022; Samatas & Lintuvuori Reference Samatas and Lintuvuori2023). In addition to the rotlet mode $C_1$ , rotlet dipoles $C_2$ , which describe the flow pattern generated by a microswimmer with rotating flagella and a counter-rotating body, play a significant role in propulsion near walls (Fadda et al. Reference Fadda, Molina and Yamamoto2020) and in complex fluids (Binagia et al. Reference Binagia, Phoa, Housiadas and Shaqfeh2020; Nganguia et al. Reference Nganguia, Zheng, Chen, Pak and Zhu2020; Housiadas et al. Reference Housiadas, Binagia and Shaqfeh2021; Kobayashi et al. Reference Kobayashi, Jung, Matsuoka, Nakayama, Molina and Yamamoto2023, Reference Kobayashi, Molina and Yamamoto2024). In this study, we examine how chirality, represented by the full azimuthal squirming modes, influences swimming kinematics in viscosity gradients.

2.2. Fluids with viscosity gradients

The fluid is incompressible and governed by the Stokes equations:

(2.2) \begin{align} \boldsymbol{\nabla }\cdot \boldsymbol{u} = 0,\qquad \boldsymbol{\nabla }\cdot \boldsymbol{\sigma } = \boldsymbol{0}, \end{align}

where $\boldsymbol{u}$ and $\boldsymbol{\sigma }$ represent the velocity and stress fields, respectively. We investigate a Newtonian fluid with viscosity variations influenced by factors such as temperature $T$ (Oppenheimer, Navardi & Stone Reference Oppenheimer, Navardi and Stone2016) and nutrient concentration $C$ (Shoele & Eastham Reference Shoele and Eastham2018), expressed as $\eta = \eta (T, C)$ . Since both temperature $T$ and concentration $C$ are governed by the advection–diffusion equations, they evolve over time $t$ and space $\boldsymbol{r}$ . Consequently, viscosity fields can be written as $\eta = \eta [T(t, \boldsymbol{r}), C(t, \boldsymbol{r})]$ . While swimming motion can influence viscosity variations, we follow the assumptions of Oppenheimer et al. (Reference Oppenheimer, Navardi and Stone2016) and Shoele & Eastham (Reference Shoele and Eastham2018), assuming that advective transport of thermal energy and concentration is negligible compared with diffusion. Specifically, we assume that the Péclet numbers $\textrm{Pe}_T = Ua / \kappa _T \ll 1$ and $\textrm{Pe}_C = Ua/ \kappa _C \ll 1$ , where $U$ is the characteristic velocity, and $\kappa _T$ and $\kappa _C$ represent diffusivities of thermal energy and concentration, respectively. Under these conditions, the flow-induced distortions of the temperature and concentration fields can be ignored, allowing us to express viscosity as $\eta = \eta (\boldsymbol{r})$ . For the micron-scale organisms considered here, we assume that the viscosity field varies only weakly, meaning the characteristic length scale $L$ significantly exceeds the swimmer’s size $a$ $(\varepsilon \equiv a / L \ll 1)$ , as assumed in previous studies (Oppenheimer et al. Reference Oppenheimer, Navardi and Stone2016; Shoele & Eastham Reference Shoele and Eastham2018; Liebchen et al. Reference Liebchen, Monderkamp, ten Hagen and Löwen2018; Datt & Elfring Reference Datt and Elfring2019; Shaik & Elfring Reference Shaik and Elfring2021; Gong et al. Reference Gong, Shaik and Elfring2024a ,Reference Gong, Shaik and Elfring b ). The stress field follows the Newtonian model but incorporates a spatially varying viscosity $\eta (\boldsymbol{r})$ , given by

(2.3) \begin{align} \boldsymbol{\sigma } = -p\boldsymbol{1} + \eta (\boldsymbol{r})\big[\boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^T\big]. \end{align}

Therefore, the governing equation of fluids is given by

(2.4) \begin{align} -\boldsymbol{\nabla }p + \eta (\boldsymbol{r})\boldsymbol{\nabla }^2\boldsymbol{u} + (\boldsymbol{\nabla }\eta )\cdot \big[\boldsymbol{\nabla }\boldsymbol{u} + (\boldsymbol{\nabla }\boldsymbol{u})^T\big] = \boldsymbol{0}, \end{align}

where $p$ is the pressure. We consider a linear viscosity gradient,

(2.5) \begin{align} \boldsymbol{\nabla }\eta = \frac {\eta _0}{L} \boldsymbol{e}_\eta = \varepsilon \frac {\eta _0}{a}\boldsymbol{e}_\eta ,\qquad \left (\varepsilon \equiv \frac {a}{L}\right ), \end{align}

with the gradient directed along $\boldsymbol{e}_\eta$ . We ignore thermal fluctuations, focusing only on the deterministic motion of microswimmers (Liebchen et al. Reference Liebchen, Monderkamp, ten Hagen and Löwen2018; Datt & Elfring Reference Datt and Elfring2019; Shaik & Elfring Reference Shaik and Elfring2021; Gong et al. Reference Gong, Shaik and Elfring2023, Reference Gong, Shaik and Elfring2024a ,Reference Gong, Shaik and Elfring b ).

We employ the integral theorem to derive the swimming speed of a chiral squirmer in viscosity gradients without directly computing the flow fields (Oppenheimer et al. Reference Oppenheimer, Navardi and Stone2016; Elfring Reference Elfring2017; Shoele & Eastham Reference Shoele and Eastham2018; Masoud & Stone Reference Masoud and Stone2019; Datt & Elfring Reference Datt and Elfring2019). The integral theorem is given by

(2.6) \begin{align} -\int _{\mathcal{S}}\boldsymbol{n}\cdot \boldsymbol{\sigma }\cdot \hat {\boldsymbol{u}}\ {\textrm d}S + \int _{\mathcal{S}}\boldsymbol{n}\cdot \hat {\boldsymbol{\sigma }}\cdot \boldsymbol{u}\ {\textrm d}S = 2\int _{\mathcal{V}}[\eta (\boldsymbol{r}) -\eta _0]\boldsymbol{E}:\boldsymbol{\nabla }\hat {\boldsymbol{u}}\ {\textrm d}V, \end{align}

where $\mathcal{V}$ and $\mathcal{S}$ denote the fluid volume and the surface of the particle, respectively, and $\boldsymbol{n}$ is the normal surface vector directed into the fluid. As an auxiliary problem, we consider $\hat {\boldsymbol{u}}$ and $\hat {\boldsymbol{\sigma }}$ , which satisfy the Stokes equation with constant viscosity $\eta _0$ .

3.1. Alignment of the polar and azimuthal squirming axes

We now derive the translational and angular velocities $(\boldsymbol{U}, \boldsymbol{\Omega })$ for squirmers with all polar and azimuthal squirming modes in linear viscosity fields up to $\mathcal{O}(\varepsilon )$ ,

(3.7a) \begin{align} \boldsymbol{U} & = \frac {2}{3}B_1\mathbf{p} - \frac {aB_2}{5}(\boldsymbol{1} - 3\mathbf{pp})\cdot \boldsymbol{\nabla }\left (\frac {\eta }{\eta _0}\right ),\\[-10pt]\nonumber \end{align}

(3.7b) \begin{align} \boldsymbol{\Omega } & =-\frac {C_1}{a}\mathbf{p} -\frac {B_1}{3}\mathbf{p}\times \boldsymbol{\nabla }\left (\frac {\eta }{\eta _0}\right ) + \frac {2}{5}C_2(\boldsymbol{1}-3\mathbf{pp})\cdot \boldsymbol{\nabla }\left (\frac {\eta }{\eta _0}\right ). \end{align}

Datt & Elfring (Reference Datt and Elfring2019) showed that among all polar squirming modes $B_n$ , only the first two modes contribute to the translational (second term in (3.7a )) and rotational velocities (second term in (3.7b )) in the linear gradients. The first polar mode $B_1$ causes the squirmer to rotate towards regions of lower viscosity (negative viscotaxis) when not perfectly aligned with the viscosity gradient (Datt & Elfring Reference Datt and Elfring2019). In this study, we find that among all azimuthal squirming modes, only the second mode $C_2$ contributes to the rotational velocity in the linear viscosity gradients. Therefore, we focus exclusively on the first polar mode $B_1$ and the second azimuthal mode $C_2$ , as the coupling between viscosity gradients and the swimmer’s chirality arises solely from the rotlet dipole mode $C_2$ .

The angular velocity due to the squirmer with $B_1$ and $C_2$ modes in the linear viscosity gradient can be decomposed by

(3.8) \begin{align} \boldsymbol{\Omega } & = \Omega _{\textit{Sp}} (\boldsymbol{e}_\eta \cdot \mathbf{p})\mathbf{p} + \Omega _{{ Pr}}\boldsymbol{e}_\eta + \Omega _{\textit{Da}}\boldsymbol{e}_\eta \times \mathbf{p},\nonumber\\[5pt] \Omega _{\textit{Sp}} & = -\frac {6C_2}{5L},\quad \Omega _{\textit{Pr}} = \frac {2C_2}{5L},\quad \Omega _{\textit{Da}} = \frac {B_1}{3L}. \end{align}

Using (3.8) and $\dot {\mathbf{p}} = \boldsymbol{\Omega }\times \mathbf{p}$ , we derive the orientational dynamics of the chiral squirmer (see figure 2 a), governed by the Landau–Lifshitz–Gilbert (LLG) equation, capturing the essential physics of the alignment behaviour (Gilbert Reference Gilbert2004; Lakshmanan Reference Lakshmanan2011)

(3.9) \begin{align} \dot {\mathbf{p}} = \Omega _{\textit{Pr}}(\boldsymbol{e}_\eta \times \mathbf{p}) + \Omega _{\textit{Da}}(\boldsymbol{e}_\eta \times \mathbf{p})\times \mathbf{p}. \end{align}

The swimmer exhibits damping ( $\Omega _{\textit{Da}}$ ) leading to negative viscotaxis, spinning ( $\Omega _{\textit{Sp}}$ ) around the propulsion axis $\mathbf{p}$ and precession ( $\Omega _{\textit{Pr}}$ ) around the axis of viscosity gradients $\boldsymbol{e}_\eta$ .

Interestingly, the orientation dynamics of achiral squirmers in chiral fluids with odd viscosity is governed by the fixed LLG equation, exhibiting parallel or antiparallel alignments with the axis of odd viscosity (Hosaka et al. Reference Hosaka, Chatzittofi, Golestanian and Vilfan2024). The interplay between a polar vector (orientation) and a pseudovector (axis of odd viscosity) makes parallel and antiparallel alignments equivalent. In spin systems, however, parallel and antiparallel alignments are not equivalent, as both magnetisation (or spin) and the magnetic field are pseudovectors. In viscosity gradients, both the orientation $\mathbf{p}$ and $\boldsymbol{e}_\eta$ are polar vectors, rendering parallel and antiparallel alignments non-equivalent. Consequently, the orientation dynamics can be elegantly described by the LLG equation. Moreover, while experiments on microswimmers in fluids with odd viscosity remain elusive, numerous studies on viscosity gradients (Kaiser & Doetsch Reference Kaiser and Doetsch1975; Petrino & Doetsch Reference Petrino and Doetsch1978; Daniels et al. Reference Daniels, Longland and Gilbart1980; Sherman et al. Reference Sherman, Timkina and Glagolev1982; Takabe et al. Reference Takabe, Tahara, Islam, Affroze, Kudo and Nakamura2017; Stehnach et al. Reference Stehnach, Waisbord, Walkama and Guasto2021; Coppola & Kantsler Reference Coppola and Kantsler2021) suggest our findings are experimentally reproducible.

The intriguing orientation dynamics can be understood by examining the effect of the viscosity gradient on the thrust torque. The thrust torque, generated by the slip velocity at the squirmer’s surface, dominates in regions of higher viscosity. As shown in figures 2(b) and 2(c), when the swimming axis is perpendicular (antiparallel) to the viscosity gradient, the torque is predominantly governed by the traction generated by the right (lower) hemisphere, leading to rotating around $\boldsymbol{e}_\eta$ ( $\mathbf{p}$ ). When not perfectly aligned with the viscosity gradient, chiral squirmers move towards regions of lower viscosity, following spiral paths. Once aligned with the viscosity gradient, the squirmer swims in a straight line with spinning. Chiral squirmers typically transition from chiral (spiral) to achiral trajectories (linear trajectories with spinning) unless perfectly aligned with viscosity gradients.

Now, we describe the swimmer’s rotation using the polar angle $\vartheta$ , measured from the $\boldsymbol{e}_\eta$ -axis, meaning $\vartheta = \cos ^{-1}(\mathbf{p}\cdot \boldsymbol{e}_\eta )$ , and the azimuthal angle $\varphi$ . The swimming axis is $\mathbf{p} = (\sin \vartheta \cos \varphi , \sin \vartheta \sin \varphi , \cos \vartheta )$ , and (3.9) becomes

(3.10) \begin{align} \dot {\vartheta } = \Omega _{\textit{Da}}\sin \vartheta ,\qquad \dot {\varphi } = \Omega _{\textit{Pr}}. \end{align}

The dynamics of $\vartheta$ are characterised by one unstable fixed point at $\vartheta = 0$ (corresponding to positive viscotaxis) and one stable fixed point at $\vartheta = \pi$ (corresponding to negative viscotaxis). Equation (3.10) solves to

(3.11) \begin{align} \tan \frac {\vartheta (t)}{2} = \tan \frac {\vartheta (0)}{2}e^{\Omega _{\textit{Da}}t},\quad \varphi (t) = \Omega _{\textit{Pr}}t + \varphi (0). \end{align}

We can estimate the characteristic time scales as follows. First, the time scale for the transition from spiral to linear trajectories corresponds to the time required for the orientation to become perfectly anti-aligned with the viscosity gradients. This time scale is governed solely by the strength of the source dipole $B_1$ and the viscosity gradients $\varepsilon$ , and is given by $\tau \sim 1 / \Omega _{\textit{Da}}$ . Second, the time scale associated with the spiral trajectory – specifically, the period of the helix – is determined by the strength of the rotlet dipole $C_2$ and the viscosity gradients $\varepsilon$ , and is given by $\tau = 2\pi / (\Omega _{\textit{Pr}}\sin \theta )$ .

We demonstrate negative viscotaxis, accompanied by spiral trajectories, for various chiral parameters $\zeta = C_2 / B_1$ and initial polar angles $\vartheta (0)$ in figure 3. Chirality allows the swimmer to follow a three-dimensional trajectory, in contrast to an achiral swimmer, which moves within a two-dimensional plane (Datt & Elfring Reference Datt and Elfring2019). As shown in figure 3, the specific trajectories vary depending on the swimmer’s chirality, orientation and viscosity gradient strengths. When the swimmer’s orientation is parallel or antiparallel to the axis of viscosity gradients $\boldsymbol{e}_\eta$ , it swims in a straight line. In contrast, all chiral squirmers follow spiral trajectories and exhibit negative viscotaxis unless perfectly aligned with the axis of viscosity gradients. Eventually, the swimmer will align perfectly with $-\boldsymbol{e}_\eta$ and exhibit straight-line motion. Particularly, the angular velocity associated with precession $\Omega _{\textit{Pr}}$ increases with the chiral parameter $\zeta$ , resulting in smaller radii for the spiral trajectories. Additionally, stronger viscosity gradients lead to larger angular velocities associated with both precession and damping, resulting in tighter spiral trajectories. These distinctions in trajectories could potentially be exploited for sorting swimmers (Tamura et al. Reference Tamura, Yanagawa, Sugiura, Takagi, Sumaru, Matsui and Kanamori2014). Furthermore, the swimmer’s paths could be used to measure the fluid’s viscosity gradient strengths.

To evaluate the impact of swimmer’s chirality, represented by the rotlet dipole, we estimate the chiral parameter $\zeta = C_2 / B_1$ for E. coli, which propels itself using a rotating flagellum and a counter-rotating body. Here, $B_1$ can be determined from the Newtonian swimming speed, given by $U_N =(2/3)B_1$ (Lighthill Reference Lighthill1952; Blake Reference Blake1971), where $U_N =$ $8.3$ µm s $^{-1}$ (Patteson et al. Reference Patteson, Gopinath, Goulian and Arratia2015). To compute the chiral parameter $\zeta$ , we estimate $C_2$ based on the motion of E. coli’s helical flagellum, which spins at $100$ Hz with a helical diameter of $0.5$ µm (Lauga Reference Lauga2016). This analysis yields $\zeta \approx 4.0$ (Binagia et al. Reference Binagia, Phoa, Housiadas and Shaqfeh2020), suggesting that chirality plays a crucial role in the dynamics of chiral swimmers in viscosity gradients, as seen in figure 3.

Figure 4.

Orientation of a chiral swimmer (sphere) with a misalignment between source and rotlet dipoles within the body-fixed frame $(\boldsymbol{e}_1,\boldsymbol{e}_2,\boldsymbol{e}_3)$ . The rotlet dipole, oriented along the $\mathbf{d}$ direction, forms an angle $\gamma$ with the swimming axis $\mathbf{p}$ . The body-fixed frame $(\boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3)$ (red axes) is related to the laboratory frame $(\boldsymbol{e}_x, \boldsymbol{e}_y, \boldsymbol{e}_z)$ (black axes) through the Euler angles $(\varphi , \vartheta , \psi )$ . Without loss of generality, we set $\boldsymbol{e}_\eta = \boldsymbol{e}_z$ and $\mathbf{p} = \boldsymbol{e}_3$ .

3.2. Misalignment of the polar and azimuthal squirming axes

Ciliated microorganisms generally exhibit not only translational motion, but also body rotation, resulting in helical trajectories (Crenshaw Reference Crenshaw1989). Pak & Lauga (Reference Pak and Lauga2014) introduced a general squirmer model that includes non-axisymmetric squirming motion, while Maity & Burada (Reference Maity and Burada2019) proposed the chiral squirmer, which explicitly incorporates body rotation. Burada et al. (Reference Burada, Maity and Jülicher2022) and Samatas & Lintuvuori (Reference Samatas and Lintuvuori2023) identified intriguing hydrodynamic interactions in chiral squirmers, charactrised by a deterministic misalignment of source dipoles $B_1$ and rotlets $C_1$ , distinguishing them from achiral squirmers. Next, we employ their chiral squirmer models, which incorporate misaligned polar and azimuthal squirming modes (Burada et al. Reference Burada, Maity and Jülicher2022; Maity & Burada Reference Maity and Burada2019, Reference Maity and Burada2022a ,Reference Maity and Burada b , Reference Maity and Burada2023; Samatas & Lintuvuori Reference Samatas and Lintuvuori2023), to investigate the behaviour of chiral squirmers in viscosity gradients. Specifically, we examine a chiral swimmer with a deterministic misalignment between the source dipole $B_1$ and rotlet dipole $C_2$ . The rotlet dipole is oriented along the $\mathbf{d}$ axis (blue axis in figure 4) in the body frame, forming an angle $\gamma$ with the swimming axis $\mathbf{p}$ . This configuration can resemble E. coli’s misaligned axes of propulsion and rotation of flagella and body in E. coli during its tumbling state (Macnab Reference Macnab1977). While E. coli’s tumbling is a stochastic process, our deterministic chiral squirmer model, defined by the misalignment of polar and azimuthal modes, offers fresh insights into the underlying mechanisms of E. coli’s behaviour in viscosity gradients.

We solve for the velocities in linear viscosity fields up to $\mathcal{O}(\varepsilon )$ ,

(3.12) \begin{align} \boldsymbol{\Omega } = \Omega _{\textit{Sp}}(\boldsymbol{e}_\eta \cdot \mathbf{d})\mathbf{d} + \Omega _{\textit{Pr}}\boldsymbol{e}_\eta + \Omega _{\textit{Da}}\boldsymbol{e}_\eta \times \mathbf{p}. \end{align}

To describe the orientational dynamics of a chiral microswimmer with a misalignment between the source and rotlet dipoles, it is convenient to work in a body-fixed coordinate system with the basis $(\boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3)$ , chosen such that $\boldsymbol{e}_3 \equiv \mathbf{p}$ . The swimmer’s orientation relative to the laboratory frame – defined by the basis $(\boldsymbol{e}_x, \boldsymbol{e}_y, \boldsymbol{e}_z)$ , where $\boldsymbol{e}_z \equiv \boldsymbol{e}_\eta$ – is described using the Euler angles $(\varphi , \vartheta , \psi )$ . Here, $\varphi$ and $\vartheta$ denote the azimuthal and polar angles measured from the $\boldsymbol{e}_z = \boldsymbol{e}_\eta$ axis, and $\psi$ represents the roll angle about the direction $\mathbf{p}$ (see figure 4). As shown in figure 4, the rotation follows the $zxz$ -convention, consisting of a rotation by $\varphi$ about $\boldsymbol{e}_z$ , followed by a rotation by $\vartheta$ about the $x$ axis of the newly obtained frame (indicated by the green arrow in figure 4) and, finally, a rotation by $\psi$ about the body fixed axis $\boldsymbol{e}_3$ . The rotation matrix $\mathbf{R}$ that transforms the laboratory frame basis vectors to the body-fixed basis vectors is given by

(3.13) \begin{align} \mathbf{R} = \left (\!\!\! \begin{array}{ccc} \cos \varphi \cos \psi - \sin \varphi \cos \vartheta \sin \psi &\!\! \sin \varphi \cos \psi + \cos \varphi \cos \vartheta \sin \psi &\!\! \sin \vartheta \sin \psi \\ -\cos \varphi \sin \psi - \sin \varphi \cos \vartheta \cos \psi &\!\! -\sin \varphi \sin \psi + \cos \varphi \cos \vartheta \cos \psi &\!\! \sin \vartheta \cos \psi \\ \sin \varphi \sin \vartheta &\!\! -\cos \varphi \sin \vartheta &\!\! \cos \vartheta \end{array}\!\!\! \right ) \end{align}

The inverse transformation is given by

(3.14) \begin{align} (\boldsymbol{e}_x, \boldsymbol{e}_y, \boldsymbol{e}_z) = \mathbf{R}^T\cdot (\boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3). \end{align}

We can express the axis of viscosity gradients as $\boldsymbol{e}_\eta = \sin \vartheta \sin \psi \boldsymbol{e}_1 + \sin \vartheta \cos \psi \boldsymbol{e}_2 + \cos \vartheta \boldsymbol{e}_3$ . The angular velocity of the swimmer can be expressed in terms of the derivatives of the Euler angles as

(3.15) \begin{align} \boldsymbol{\Omega } = (\dot {\varphi }\sin \vartheta \sin \psi + \dot {\vartheta }\cos \psi )\boldsymbol{e}_1 + (\dot {\varphi } \sin \vartheta \cos \psi - \dot {\vartheta }\sin \psi )\boldsymbol{e}_2 + (\dot {\varphi } \cos \vartheta + \dot {\psi })\boldsymbol{e}_3. \end{align}

By representing $\mathbf{d} = \sin \gamma \boldsymbol{e}_1 + \cos \gamma \boldsymbol{e}_3$ without loss of generality, and substituting into (3.12), we obtain

(3.16) \begin{align} \boldsymbol{\Omega } &= \Omega _{\textit{Sp}}(\sin \gamma \sin \vartheta \sin \psi + \cos \gamma \cos \vartheta )(\sin \gamma \boldsymbol{e}_1 + \cos \gamma \boldsymbol{e}_3) \nonumber\\&\quad + \Omega _{\textit{Pr}}(\sin \vartheta \sin \psi \boldsymbol{e}_1 + \sin \vartheta \cos \psi \boldsymbol{e}_2 + \cos \vartheta \boldsymbol{e}_3) \nonumber\\ &\quad + \Omega _{\textit{Da}}(\sin \vartheta \cos \psi \boldsymbol{e}_1 - \sin \vartheta \sin \psi \boldsymbol{e}_2). \end{align}

Equating the two expressions for the angular velocity ((3.15) and (3.16)), we obtain the following equations:

(3.17a) \begin{align} \dot {\varphi } & = \Omega _{\textit{Sp}}\sin \gamma \sin \psi \left (\sin \gamma \sin \psi + \frac {\cos \gamma }{\tan \vartheta }\right ) + \Omega _{\textit{Pr}},\\[-10pt]\nonumber \end{align}

(3.17b) \begin{align} \dot {\vartheta } & = \Omega _{\textit{Sp}}\sin \gamma \cos \psi (\sin \gamma \sin \vartheta \sin \psi + \cos \gamma \cos \vartheta )+ \Omega _{\textit{Da}}\sin \vartheta ,\\[-10pt]\nonumber \end{align}

(3.17c) \begin{align}\dot {\psi } & = \Omega _{\textit{Sp}}\left (\cos ^2\gamma \cos \vartheta - \sin ^2\gamma \cos \vartheta \sin ^2\psi + \sin \gamma \cos \gamma \frac {\sin ^2\vartheta - \cos ^2\vartheta }{\sin \vartheta }\sin \psi \right ). \end{align}

Since the equations are independent of $\varphi$ , it suffices to examine the phase portraits in the $(\vartheta ,\psi )$ plane (see figure 5 d–f). The system displays a spiral source and sink, with the sink indicating negative viscotaxis. At the steady state, the axis of angular velocity $\boldsymbol{\Omega }$ is nearly antiparallel to the axis of viscosity gradients (see figure 5 a–c). Consequently, the steady-state trajectories of chiral squirmers are spiral, with the radius of these spiral trajectories decreasing with the angle $\gamma$ (see figure 5 g). Swimmers with $\mathbf{d} \ne \mathbf{p}$ consistently exhibit spiral trajectories, unlike swimmers with $\mathbf{d} = \mathbf{p}$ . Furthermore, the unstable point, which corresponds to the positive viscotaxis for swimmers with $\mathbf{d}=\mathbf{p}$ , disappears for swimmers with $\mathbf{d}\ne \mathbf{p}$ . Additionally, increasing the chiral parameters $\zeta$ results in a smaller radius, while increasing the viscosity gradients $\varepsilon$ leads to tighter spiral trajectories. We can estimate the characteristics time scale for the spiral, specifically the period of the helix. Since the axis of angular velocity $\boldsymbol{\Omega }$ is nearly antiparallel to $\mathbf{d}$ , the period can be approximated as $\tau \sim 2\pi / (\Omega \sin \gamma )$ , with $\Omega = |\boldsymbol{\Omega }|$ .

Figure 5.

(a)–(c) Schematic representations of a chiral swimmer’s steady-state orientation for different angles: (a) $\gamma = \pi / 8$ , (b) $\gamma = \pi / 4$ and (c) $\gamma = 3\pi / 8$ , with fixed parameters $\varepsilon = 0.1$ and $\zeta = 5$ . (d)–(f) Phase portraits of the orientational dynamics of a chiral squirmer for various angles: (d) $\gamma = \pi / 8$ , (e) $\gamma = \pi / 4$ and (f) $\gamma = 3\pi / 8$ , with $\varepsilon = 0.1$ and $\zeta = 5$ . Each phase portrait shows one spiral source (white circle) and one spiral sink (black circle), with red lines representing example trajectories. (g) Steady-state trajectories of swimmers for different angles $\gamma$ . All swimmers exhibit negative viscotaxis.