2. Governing equations

Our physical set-up in Part 2 is similar to that in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, but now with a more complex swimmer geometry. That is, we now consider a general helicoidal swimmer, as discussed in § 1, in the presence of a far-field shear flow. This will result in additional hydrodynamic effects due to object chirality and other asymmetries. We scale time with inverse shear rate, and space with a characteristic swimmer length, working in dimensionless quantities henceforth. Specifically, we consider the motion of a rigid, self-propelled helicoidal object in a shear flow, which has swimming velocity $\boldsymbol {V}$ and angular velocity $\boldsymbol {\varOmega }$ in a quiescent fluid. As before, these propulsion and rotation vectors are fixed in direction and magnitude in a swimmer-fixed basis, but the orientation of this swimmer basis will vary rapidly in the laboratory frame through its dependence on $\boldsymbol {\varOmega }$.

We define the swimmer-fixed axis of helicoidal symmetry by $\hat {\boldsymbol {e}}_{1}$. Therefore, we may take $\hat {\boldsymbol {e}}_{2}$ such that the self-generated angular velocity $\boldsymbol {\varOmega }$ is in a plane spanned by $\hat {\boldsymbol {e}}_{1}$ and $\hat {\boldsymbol {e}}_{2}$, where $\boldsymbol {\varOmega }$ makes an angle $\alpha$ with $\hat {\boldsymbol {e}}_{1}$. Therefore, we may write $\boldsymbol {\varOmega } = \varOmega _{\parallel }\hat {\boldsymbol {e}}_{1} + \varOmega _{\perp }\hat {\boldsymbol {e}}_{2}$, with $\varOmega _{\parallel }$ and $\varOmega _{\perp }$ being the constant components of angular velocity that are parallel and perpendicular, respectively, to the axis of helicoidal symmetry. This generates the relationship $\tan \alpha = \varOmega _{\perp }/\varOmega _{\parallel }$. We then define $\hat {\boldsymbol {e}}_{3} = \hat {\boldsymbol {e}}_{1} \times \hat {\boldsymbol {e}}_{2}$. In this swimmer-fixed basis, we write the self-generated propulsion $\boldsymbol {V} = V_1\hat {\boldsymbol {e}}_{1} + V_2\hat {\boldsymbol {e}}_{2} + V_3\hat {\boldsymbol {e}}_{3}$. The position of the particle is given by $\boldsymbol {X} = X \boldsymbol {e}_{1} + Y \boldsymbol {e}_{2} + Z \boldsymbol {e}_{3}$ with respect to the orthonormal basis $\{\boldsymbol {e}_{1},\boldsymbol {e}_{2},\boldsymbol {e}_{3}\}$ of the laboratory frame. These vectors are illustrated in figure 1.

Figure 1. A schematic of the notation and the physical set-up that we consider in Part 2. We investigate the dynamics of a chiral, helicoidal swimmer with axis of symmetry $\hat {\boldsymbol {e}}_{1}$. The swimmer has self-generated translational and rotational velocities $\boldsymbol {V} = V_1\hat {\boldsymbol {e}}_{1} + V_2\hat {\boldsymbol {e}}_{2} + V_3\hat {\boldsymbol {e}}_{3}$ and $\boldsymbol {\varOmega } = \varOmega _{\parallel }\hat {\boldsymbol {e}}_{1} + \varOmega _{\perp }\hat {\boldsymbol {e}}_{2}$, respectively, and these interact with a background shear flow $\boldsymbol {u} = y \boldsymbol {e}_{3}$.

Finally, we impose the far-field flow. Specifically, we are interested in the motion of the particle in the presence of a far-field shear Stokes flow with velocity field $\boldsymbol {u}(x,y,z) = y \boldsymbol {e}_{3}$, with coordinates $x,y,z$ in the laboratory frame. The flow interacts with the particle; we derive the resulting governing equations of motion for the particle in Appendix A, and present the resulting equations below. The dynamics for the orientation of the swimmer frame is given in terms of the Euler angles $(\theta, \psi, \phi )$, also defined formally in Appendix A.

The rotational dynamics is given by

(2.1a)$$\begin{gather} \frac{\mathrm{d} \theta}{\mathrm{d} t} = \varOmega_{{\perp}} \cos \psi + h_1(\theta,\phi; B,C), \end{gather}$$

(2.1b)$$\begin{gather}\frac{\mathrm{d} \psi}{\mathrm{d} t} = \varOmega_{{\parallel}} - \varOmega_{{\perp}}\,\frac{\cos \theta \sin \psi}{\sin \theta} + h_2(\theta,\phi; B,C,D), \end{gather}$$

(2.1c)$$\begin{gather}\frac{\mathrm{d} \phi}{\mathrm{d} t} = \varOmega_{{\perp}}\,\frac{\sin \psi}{\sin \theta} + h_3(\theta,\phi; B,C), \end{gather}$$

where the functions $h_i = f_i + g_i$ ($i = 1, 2, 3$) capture the effects of the Stokes flow interacting with the swimmer. The $f_i$ encode the rotational effects of the achiral aspects of the swimmer (and are the same as in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1). These functions are

(2.2a)$$\begin{gather} f_1(\theta,\phi; B) :={-}\frac{ B}{2} \cos \theta \sin \theta \sin 2 \phi, \end{gather}$$

(2.2b)$$\begin{gather}f_2(\theta,\phi; B) := \frac{B}{2} \cos \theta \cos 2 \phi, \end{gather}$$

(2.2c)$$\begin{gather}f_3(\phi; B) := \tfrac{1}{2} \left(1 - B \cos 2 \phi\right), \end{gather}$$

where $B$ is the shape-capturing Bretherton parameter (Bretherton Reference Bretherton1962), which typically satisfies $\left \lvert B \right \rvert <1$ for all but the most elongated of bodies (Bretherton Reference Bretherton1962; Singh, Koch & Stroock Reference Singh, Koch and Stroock2013).

The $g_i$ encode the rotational effects of the chiral aspects of the swimmer, and were therefore not present in the spheroidal analysis of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1. These chiral functions are

(2.3a)$$\begin{gather} g_1(\theta,\phi; C) :={-}\frac{C}{2} \sin \theta \cos 2 \phi, \end{gather}$$

(2.3b)$$\begin{gather}g_2(\theta,\phi; C, D) :={-} \frac{C}{2} \cos^2 \theta \sin 2 \phi - \frac{ D}{2} \sin^2 \theta \sin 2 \phi, \end{gather}$$

(2.3c)$$\begin{gather}g_3(\theta,\phi;C) := \frac{C}{2} \cos \theta \sin 2\phi. \end{gather}$$

Here, $C$ and $D$ are chirality parameters, where $C$ is sometimes referred to as the Ishimoto parameter (Ohmura et al. Reference Ohmura, Nishigami, Taniguchi, Nonaka, Ishikawa and Ichikawa2021). They represent rotational drift due to moments of chirality along the axis of helicoidal symmetry, as summarised in table 1. If the particle is spheroidal, then $C = D = 0$ and the governing equations for the rotational dynamics reduce to those in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1. For brevity, when referring to $f_i$ and $g_i$, we will often suppress the explicit parameter dependence on $B$, $C$ and $D$ unless specifically relevant. The typical ranges of $C$ and $D$ are not well explored in the literature for different swimmers, with the notable exception of experimental measurements for bacterial swimmers, giving $C \approx 0.01$ (Jing et al. Reference Jing, Zöttl, Clément and Lindner2020; Ronteix et al. Reference Ronteix, Josserand, Lety-Stefanka, Baroud and Amselem2022; Zöttl et al. Reference Zöttl, Tesser, Matsunaga, Laurent, Roure and Lindner2022). Given this, for reference we approximate plausible ranges of these parameters for a simple bacterial model using resistive force theory in Appendix B, which suggest $|C| \approx 0.01$ and $|D| \approx 0.5$. Since $\psi$ decouples from the system (2.1)–(2.3) for passive swimmers (i.e. for $\varOmega _{\parallel } = \varOmega _{\perp } = 0$), and $C$ appears to be small, one might assume that the effects of chirality are unimportant to Jeffery's orbits. We will show that this is not the case in general for the active swimmers that we consider. Therefore, we retain both $C$ and $D$ in our analysis, and we will see that this is important to capture comprehensively the nature of the emergent dynamics.

Table 1. Summary of the six parameters that characterise objects with hydrodynamic helicoidal symmetry.

We now consider the governing equations for $\boldsymbol {X}(t)$, the position of the swimmer in the laboratory frame. While the equivalent equations in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1 were fairly intuitive and straightforward to state, this was due to the intrinsic symmetry of spheroidal particles, which removed several of the more general contributions to translation. Since we now consider a more general class of objects, the translational dynamics in Part 2 feature additional contributions. We derive the resulting governing equations of motion in Appendix A, which are

(2.4)\begin{equation} \frac{\mathrm{d} \boldsymbol{X}}{\mathrm{d} t} = \boldsymbol{V} + Y \boldsymbol{e}_{3} - \beta \left ( \hat{\boldsymbol{e}}_{2} \hat{\boldsymbol{e}}_{3}^{\rm T} - \hat{\boldsymbol{e}}_{3} \hat{\boldsymbol{e}}_{2}^{\rm T} \right ) \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + \gamma \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + (\delta - \gamma) \left(\hat{\boldsymbol{e}}_{1}^{\rm T}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}\right) \hat{\boldsymbol{e}}_{1} . \end{equation}

We emphasise that $\boldsymbol {V}$ and $\hat {\boldsymbol {e}}_{i}$ depend on the orientation of the object through the Euler angles $(\theta, \psi, \phi )$, which evolve via (2.1)–(2.3). The additional terms in (2.4) not present in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1 involve the rate of strain tensor $\boldsymbol {E}^*$, and three additional degrees of freedom encoded through the shape parameters $\beta$, $\gamma$ and $\delta$.

These shape parameters can be interpreted as measures of translational drift induced by the coupling between the shear-induced strain and asymmetries in the object shape. As summarised in table 1, $\beta$ represents a measure of drift due to chirality of the object, and $\gamma$, $\delta$ represent measures of drift due to fore–aft asymmetry of the object. These additional translational terms arise in a similar manner to the additional terms (2.3) in the rotational dynamics. If the particle is spheroidal, then $\beta = \gamma = \delta = 0$ and the governing equations for the translational dynamics reduce to those in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1. In Appendix B, we estimate typical ranges of the shape parameters $\beta$, $\gamma$ and $\delta$ using resistive force theory for a simple model bacterium swimmer.

The full dynamics comprising (2.1)–(2.4) governs the motion of any hydrodynamically helicoidal object in shear flow, by definition. That is, as discussed above, helicoidal objects are defined as objects that follow this dynamics, rather than by any necessary geometric properties. However, as we discuss below, there are important sufficient geometric conditions that give rise to hydrodynamic helicoidicity. In the hydrodynamic sense, the behaviour of helicoidal objects in shear flow is therefore fully characterised by the six parameters $B,C,D,\beta,\gamma,\delta$, summarised in table 1. This general class of shapes contains several subclasses of hydrodynamic symmetries, discussed extensively in Ishimoto (Reference Ishimoto2020a). These subclasses include shapes that possess additional geometric symmetries, and are characterised mathematically by particular combinations of the six shape parameters vanishing.

We illustrate an example of a general hydrodynamically helicoidal body in figure 2(a), recalling that this includes (but is not limited to) objects possessing $n$-fold rotational symmetry along an axis for some integer $n\geqslant 4$. In particular, this allows the object to be chiral and to be free of any fore–aft symmetry constraints.

Figure 2. Examples of the class of shapes considered in Part 2. (a) The bodies that we investigate possess ‘helicoidal’ symmetry, which allows us to characterise their dynamics in shear flow with six parameters, $B,C,D,\beta,\gamma,\delta$. (b) We distinguish the specific subcases that we discuss in the main text: homochirality, heterochirality, achirality and Jeffery bodies, with this last class of objects following the simpler dynamics investigated in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1. Each category of shape is illustrated with an example particle possessing additional geometrical symmetries. (c) For comparison, we provide an example of a shape that does not possess helicoidal symmetry and therefore is not captured by our analysis.

In figure 2(b), we illustrate four main hydrodynamic symmetry subcases of interest, giving examples of geometric symmetries that generate the specific subcases. An object that is geometrically symmetric with respect to a rotation of ${\rm \pi}$ around an axis perpendicular to the helicoidal symmetry axis has $C = D = \gamma = \delta = 0$; we describe an object satisfying these parameter constraints as possessing homochiral hydrodynamic symmetry, following the terminology employed in Ishimoto (Reference Ishimoto2020a). A homochiral object does not experience any chirality-induced rotational drift. That is, from the governing equations (2.1)–(2.4), the effects of chirality in the dynamics of homochiral objects will manifest only through the drift velocity terms in the translational dynamics (2.4). Thus the rotational dynamics will remain as classic Jeffery orbits, while the translational dynamics will differ.

An object that is geometrically symmetric with respect to reflection in a plane normal to the axis of helicoidal symmetry has $\beta = \gamma =\delta = 0$; we describe an object satisfying these parameter constraints as possessing heterochiral hydrodynamic symmetry. A heterochiral object does not experience any chirality-induced translational drift. In particular, such an object always satisfies $\gamma = \delta = 0$. In contrast to homochiral objects, the effect of chirality in the dynamics of heterochiral objects will appear in the rotational drift terms in the rotational dynamics (2.1)–(2.3), resulting in chiral Jeffery orbits. These, in turn, will also influence the translational dynamics (2.4), which is coupled to the evolution of the object orientation. Given the geometric symmetries that generate homochiral or heterochiral objects, we describe an object in either subclass as possessing hydrodynamic fore–aft symmetry.

An object that is geometrically symmetric with respect to continuous rotation around the helicoidal axis (i.e. a body of revolution) has $C = D = \beta = 0$; we describe an object satisfying these parameter constraints as possessing achiral hydrodynamic symmetry. Similar to the homochiral case, an achiral object does not experience any chirality-induced rotational drift. However, an achiral object will experience a different translational drift to a homochiral object in general.

Finally, any object with at least two of the geometric symmetries described above (e.g. a spheroid) has $C = D = \beta = \gamma = \delta = 0$; as noted in the Introduction, we describe an object satisfying these parameter constraints as a Jeffery body. We considered the simpler dynamics of these highly symmetric objects in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1.

More generally, in this study we investigate the emergent dynamics of the nonlinear, autonomous dynamical system defined by (2.1)–(2.4) for general helicoidal objects in shear flow. In the same manner as in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, we consider the regime where the swimmer rotation rate is much larger than the external shear rate. This means that $|\boldsymbol {\varOmega }| = |\varOmega _{\parallel } \hat {\boldsymbol {e}}_{1} + \varOmega _{\perp } \hat {\boldsymbol {e}}_{2}|$ is large. Taking $\varOmega _{\parallel } > 0$ without loss of generality, we consider the distinguished limit where $\varOmega _{\perp } = \mathit {O}(\varOmega _{\parallel })$ with $\varOmega _{\parallel } \gg 1$ (which will give the same information as taking $\left \lvert \boldsymbol {\varOmega } \right \rvert \gg 1$ with $\alpha = \mathit {O}(1)$), and all other parameters are of $\mathit {O}(1)$. This asymptotic limit is distinguished in the sense that it is a general case from which the subcases $|\varOmega _{\perp }| \gg \varOmega _{\parallel }$ and $\varOmega _{\parallel } \gg |\varOmega _{\perp }|$ can be distilled as regular asymptotic sublimits of the results we derive.

3. Setting up the multiple scales analysis

We analyse the system (2.1)–(2.4) in the limit of rapid spinning. We animate the full dynamics of this system for various scenarios in supplementary movies 1–5 available at https://doi.org/10.1017/jfm.2023.924. We consider the distinguished limit $\varOmega _{\perp } = \mathit {O}(\varOmega _{\parallel })$ with $\varOmega _{\parallel } \gg 1$ (treating $\varOmega _{\parallel } > 0$ without loss of generality). Given this, it is helpful to introduce the notation $\omega = \mathit {O}(1)$ such that

(3.1)\begin{equation} \varOmega_{{\perp}} = \omega \varOmega_{{\parallel}}, \end{equation}

and to formally consider the single asymptotic limit $\varOmega _{\parallel } \gg 1$.

Our approach is similar to that in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1; we analyse the system (2.1)–(2.4) using the method of multiple scales in the limit of large $\varOmega _{\parallel }$, with the goal of deriving effective equations that govern the emergent behaviour. Moreover, we will see that the leading-order system is equivalent to that of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, so we are able to exploit the multiscale framework that we derived therein. The set-up for the multiple scales analysis is therefore equivalent to that in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, and we repeat it here for convenience. We reintroduce $T$, the fast time scale, via

(3.2)\begin{equation} T = \big(\varOmega_{{\perp}}^2 +\varOmega_{{\parallel}}^2\big)^{1/2} t = \lambda \varOmega_{{\parallel}} t, \end{equation}

where we use

(3.3)\begin{equation} \lambda := \sqrt{1 +\omega^2} \end{equation}

for notational convenience, and we refer to the original time scale $t$ as the slow time scale. Treating the fast and slow time scales as independent and using (3.2), the time derivative becomes

(3.4)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} t} \mapsto \lambda \varOmega_{{\parallel}}\,\frac{\partial }{\partial T} + \frac{\partial }{\partial t}. \end{equation}

Under the time derivative mapping (3.4), the rotational dynamics system (2.1) is transformed to

(3.5a)$$\begin{gather} \varOmega_{{\parallel}} \lambda\,\frac{\partial \theta}{\partial T} + \frac{\partial \theta}{\partial t} = \varOmega_{{\parallel}} \omega \cos \psi + h_1(\theta,\phi), \end{gather}$$

(3.5b)$$\begin{gather}\varOmega_{{\parallel}} \lambda\,\frac{\partial \psi}{\partial T} + \frac{\partial \psi}{\partial t} = \varOmega_{{\parallel}}\left(1 - \omega\,\dfrac{\cos \theta \sin \psi}{\sin \theta}\right) + h_2(\theta,\phi), \end{gather}$$

(3.5c)$$\begin{gather}\varOmega_{{\parallel}} \lambda\,\frac{\partial \phi}{\partial T} + \frac{\partial \phi}{\partial t} = \varOmega_{{\parallel}} \omega\,\dfrac{\sin \psi}{\sin \theta} + h_3(\theta,\phi), \end{gather}$$

and the translational dynamics system (2.4) is transformed to

(3.6)\begin{align} \lambda \varOmega_{{\parallel}}\,\frac{\partial \boldsymbol{X}}{\partial T} + \frac{\partial \boldsymbol{X}}{\partial t} &= \boldsymbol{V} + Y \boldsymbol{e}_{3} - \beta \left ( \hat{\boldsymbol{e}}_{2} \hat{\boldsymbol{e}}_{3}^{\rm T} - \hat{\boldsymbol{e}}_{3} \hat{\boldsymbol{e}}_{2}^{\rm T} \right ) \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + (\delta - \gamma) \left(\hat{\boldsymbol{e}}_{1}^{\rm T}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}\right) \hat{\boldsymbol{e}}_{1} + \gamma \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}. \end{align}

We expand each dependent variable as an asymptotic series in inverse powers of $\varOmega _{\parallel }$, writing

(3.7)\begin{equation} \xi(T,t) \sim \xi_0(T,t) + \frac{1}{\varOmega_{{\parallel}}}\,\xi_1(T,t) \quad \text{as } \varOmega_{{\parallel}} \to \infty, \quad \text{for } \xi \in \{\phi, \theta, \psi, X, Y, Z \}. \end{equation}

Since the leading-order (fast) terms in (3.5) and (3.6) are $\mathit {O}(\varOmega _{\parallel })$, but the new chiral and asymmetric terms are all of $\mathit {O}(1)$, these new terms do not appear in the leading-order analysis. This means that the leading-order analysis and the adjoint solution used to derive the solvability conditions at next order are equivalent to those in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, and we can therefore use directly the equivalent results therein. Consequently, we are fairly brief with the leading-order analysis and the derivation of the solvability conditions in the full analysis below, directing the interested reader to Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1.

4. Deriving the emergent angular dynamics

4.1. Leading-order analysis

Using the asymptotic expansions (3.7) in the transformed governing equations (3.5), we obtain the leading-order (i.e. $\mathit {O}(\varOmega _{\parallel })$) system

(4.1a)$$\begin{gather} \lambda\,\frac{\partial \theta_0}{\partial T} = \omega\cos \psi_0, \end{gather}$$

(4.1b)$$\begin{gather}\lambda\,\frac{\partial \psi_0}{\partial T} = 1 - \omega\,\frac{\cos \theta_0 \sin \psi_0}{\sin \theta_0}, \end{gather}$$

(4.1c)$$\begin{gather}\lambda\,\frac{\partial \phi_0}{\partial T} = \omega\,\frac{\sin \psi_0}{\sin \theta_0}. \end{gather}$$

We show in § 4.1 of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1 that the solution to the nonlinear system (4.1) is

(4.2a)$$\begin{gather} \lambda \cos \theta_0 = \cos \bar{\vartheta} - \omega \sin \bar{\vartheta} \cos (T + \bar{\varPsi}), \end{gather}$$

(4.2b)$$\begin{gather}\lambda \sin \theta_0 \sin \psi_0 = \omega \cos \bar{\vartheta} + \sin \bar{\vartheta} \cos (T + \bar{\varPsi}), \end{gather}$$

(4.2c)$$\begin{gather}\tan \phi_0 = \frac{\omega \cos \bar{\varphi} \sin (T + \bar{\varPsi}) + \sin \bar{\varphi} (\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta})}{\cos \bar{\varphi} (\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta}) - \omega \sin \bar{\varphi} \sin (T + \bar{\varPsi})}, \end{gather}$$

where $\bar {\vartheta } = \bar {\vartheta }(t)$, $\bar {\varPsi } = \bar {\varPsi }(t)$ and $\bar {\varphi } = \bar {\varphi }(t)$ are the three slow-time functions of integration that remain undetermined from our leading-order analysis. The goal of the next-order analysis in § 4.2 is to derive the governing equations satisfied by $\bar {\vartheta }$, $\bar {\varPsi }$ and $\bar {\varphi }$. As in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, one can think of $\bar {\vartheta }$ as controlling some emergent amplitude of oscillation, $\bar {\varPsi }$ as controlling some emergent phase of oscillation, and $\bar {\varphi }$ as the emergent drift in yawing. We will also show later that $\bar {\vartheta }$ can be associated with $\theta$, $\bar {\varPsi }$ with $\psi$, and $\bar {\varphi }$ with $\phi$.

Before proceeding, it will be helpful later to note the additional relationships

(4.2d)$$\begin{gather} \sin \theta_0 \cos \psi_0 ={-}\sin \bar{\vartheta} \sin (T + \bar{\varPsi}), \end{gather}$$

(4.2e)$$\begin{gather}\lambda^2 \sin^2 \theta_0 = (\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta})^2 + \omega^2\sin^2 (T + \bar{\varPsi}), \end{gather}$$

where the former follows from differentiating (4.2a) with respect to $T$ and imposing (4.1a), and the latter follows from rearranging (4.2a).

4.2. Next-order system

Our remaining goal is to determine the governing equations satisfied by the slow-time functions $\bar {\vartheta }(t)$, $\bar {\varPsi }(t)$ and $\bar {\varphi }(t)$. To do this, we must determine the solvability conditions required for the first-order correction (i.e. $\mathit {O}(1)$) terms in (3.5) after posing the asymptotic expansions (3.7). These $\mathit {O}(1)$ terms are

(4.3a)$$\begin{gather} \lambda\,\frac{\partial \theta_1}{\partial T} + \omega \psi_1 \sin \psi_0 = h_1(\theta_0,\phi_0) - \frac{\partial \theta_0}{\partial t}, \end{gather}$$

(4.3b)$$\begin{gather}\lambda\,\frac{\partial \psi_1}{\partial T} - \omega \theta_1\,\frac{\sin \psi_0}{\sin^2 \theta_0} + \omega \psi_1\, \frac{\cos \theta_0 \cos \psi_0}{\sin \theta_0} = h_2(\theta_0,\phi_0) - \frac{\partial \psi_0}{\partial t}, \end{gather}$$

(4.3c)$$\begin{gather}\lambda\,\frac{\partial \phi_1}{\partial T} + \omega \theta_1\,\frac{\cos \theta_0 \sin \psi_0}{\sin^2 \theta_0} - \omega \psi_1\,\frac{\cos \psi_0}{\sin \theta_0} = h_3(\theta_0,\phi_0) - \frac{\partial \phi_0}{\partial t}, \end{gather}$$

along with $2 {\rm \pi}$-periodicity in $T$. The system (4.3) constitutes a non-autonomous linear coupled three-dimensional system for $(\theta _1, \psi _1, \phi _1)$ with an inhomogeneous forcing in terms of the leading-order solution.

To derive the required solvability conditions, we use the method of multiple scales for systems (see, for example, pp. 127–128 of Dalwadi (Reference Dalwadi2014) or p. 22 of Dalwadi et al. (Reference Dalwadi, Chapman, Oliver and Waters2018)). As detailed in § 4.2 of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, this entails taking the dot product of the vector solution to the homogeneous adjoint version of (4.3) with the vector right-hand side of (4.3), and averaging over one fast-time oscillation. We calculate the adjoint solution in § 4.3 and Appendix D of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1; using this to apply the procedure outlined above yields the three solvability conditions

(4.4a)\begin{align} - \lambda \sin \bar{\vartheta}\,\frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} &= \frac{\lambda \hat{B}}{2} \sin^2 \bar{\vartheta} \cos \bar{\vartheta} \sin 2 \bar{\varphi} \nonumber\\ &\quad + \left\langle g_1 (\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 ) + g_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle, \end{align}

(4.4b)\begin{align} \lambda \sin^2 \bar{\vartheta}\,\frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} &= \frac{\lambda \hat{B}}{2} \sin^2 \bar{\vartheta} \cos \bar{\vartheta} \cos 2 \bar{\varphi} \nonumber\\ &\quad + \left\langle g_1 \omega \cos \psi_0 + g_2 \sin \theta_0\, ( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 ) \right\rangle, \end{align}

(4.4c)\begin{align} \cos \bar{\vartheta}\,\frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} + \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} &= \frac{1}{2}(1 - \hat{B} \sin^2 \bar{\vartheta} \cos 2 \bar{\varphi}) + \left\langle g_2 \cos \theta_0 + g_3 \right\rangle, \end{align}

where we have used the results from §§ 4.2 and 4.3 of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1 to evaluate all the non-chiral terms (i.e. all terms not involving $g_i$), including the use of the effective Bretherton parameter that we derived in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1:

(4.5)\begin{equation} \hat{B} := \frac{(2 - \omega^2)B}{2 (1 + \omega^2 )}. \end{equation}

Additionally, we use the notation $\left \langle\, \cdot\, \right \rangle$ to denote the average of its argument over one fast-time oscillation, explicitly defining

(4.6)\begin{equation} \left\langle y \right\rangle = \frac{1}{2 {\rm \pi}}\int_0^{2{\rm \pi}} y \, \mathrm{d}T. \end{equation}

Our remaining task is to evaluate the outstanding averages in the three solvability conditions (4.4), each of which involves the chiral contributions $g_i$ defined in (2.3). We have explicit representations of the terms involving the trigonometric functions of $\theta _0$ and $\psi _0$ through the leading-order solutions (4.2). The terms involving $\sin 2 \phi _0$ and $\cos 2 \phi _0$, which arise from the $g_i$ defined in (2.2), require additional calculation. To derive expressions for these double-angle quantities, we first note

(4.7a)$$\begin{gather} \lambda \sin \theta_0 \sin \phi_0 = \omega \cos \bar{\varphi} \sin \sigma + \sin \bar{\varphi} (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta}), \end{gather}$$

(4.7b)$$\begin{gather}\lambda \sin \theta_0 \cos \phi_0 = \cos \bar{\varphi} (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta}) - \omega \sin \bar{\varphi} \sin \sigma, \end{gather}$$

using the shorthand $\sigma = T + \bar {\varPsi }$. The expressions (4.7) are calculated via the identities $\tan \phi _0 = (\lambda \sin \theta _0 \sin \phi _0)/(\lambda \sin \theta _0 \cos \phi _0)$, and $\lambda ^2 \sin ^2 \theta _0 = (\lambda \sin \theta _0 \sin \phi _0)^2 + (\lambda \sin \theta _0 \cos \phi _0)^2$, the left-hand sides of which are defined in (4.2c) and (4.2e). Then, from the expressions of (4.7), appropriate double-angle formulae imply that

(4.8a)$$\begin{gather} \lambda^2 \sin^2 \theta_0 \cos 2 \phi_0 = \mathcal{C}(\sigma,t) \cos 2\bar{\varphi} - \mathcal{S}(\sigma,t) \sin 2 \bar{\varphi}, \end{gather}$$

(4.8b)$$\begin{gather}\lambda^2 \sin^2 \theta_0 \sin 2 \phi_0 = \mathcal{S}(\sigma,t) \cos 2\bar{\varphi} + \mathcal{C}(\sigma,t) \sin 2 \bar{\varphi}, \end{gather}$$

(4.8c)$$\begin{gather}\mathcal{C}(\sigma,t) := (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta})^2 - \omega^2 \sin^2\sigma, \end{gather}$$

(4.8d)$$\begin{gather}\mathcal{S}(\sigma,t) := 2 \omega \sin\sigma (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta} ). \end{gather}$$

We can now simplify the remaining fast-time averages in the right-hand side of (4.4). We start by exploiting the parity of various expressions. Specifically, we use the evenness of $\cos \theta _0$, $\sin \theta _0 \sin \psi _0$, $\sin ^2 \theta _0$ and $\mathcal {C}$ around $\sigma = {\rm \pi}$ (from (4.2a), (4.2b), (4.2e) and (4.8c), respectively), and the oddness of $\sin \theta _0 \cos \psi _0$ and $\mathcal {S}$ around $\sigma = {\rm \pi}$ (from (4.2d) and (4.8d), respectively). This allows us to write the fast-time averages in the right-hand side of (4.4) as

(4.9a)\begin{align} & \left\langle g_1 (\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 ) + g_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle \nonumber\\ &\quad ={-}\frac{C}{2 \lambda^2} \cos 2 \bar{\varphi} \left\langle \mathcal{C} \left(\frac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0} - 1\right)+ \frac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} \right\rangle \nonumber\\ &\qquad +\frac{C - D}{2 \lambda^2} \cos 2 \bar{\varphi} \left\langle \mathcal{S} \omega \sin \theta_0 \cos \psi_0 \right\rangle, \end{align}

(4.9b)\begin{align} & \left\langle g_1\omega \cos \psi_0 + g_2 \sin \theta_0 ( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 ) \right\rangle \nonumber\\ &\quad = \frac{C}{2 \lambda^2} \sin 2 \bar{\varphi} \left\langle \mathcal{C} \left(\frac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0} - 1 \right) + \frac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} \right\rangle \nonumber\\ &\qquad + \frac{D - C}{2 \lambda^2} \sin 2 \bar{\varphi} \langle \mathcal{C} ( \omega \cos \theta_0 \sin \theta_0 \sin \psi_0 - \sin^2 \theta_0 ) \rangle, \end{align}

(4.9c)\begin{align} &\left\langle g_2 \cos \theta_0 + g_3 \right\rangle= \frac{C - D}{2 \lambda^2} \sin 2 \bar{\varphi} \left\langle \mathcal{C} \cos \theta_0 \right\rangle. \end{align}

Using the leading-order solutions (4.2) with the definitions of $\mathcal {C}$ and $\mathcal {S}$ in (4.8c)–(4.8d), we may write the terms within the averages on the right-hand sides of (4.9) explicitly as

(4.10a)$$\begin{gather} \mathcal{C} \left( \frac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0} - 1\right)+ \frac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} ={-}(1 + \omega^2 )\sin \bar{\vartheta}\, ( \sin \bar{\vartheta} + \omega \cos \bar{\vartheta} \cos \sigma ), \end{gather}$$

(4.10b)$$\begin{gather}\mathcal{S} \omega \sin \theta_0 \cos \psi_0 ={-}2 \omega^2 \sin \bar{\vartheta} \sin^2 \sigma\,( \sin \bar{\vartheta} + \omega \cos \bar{\vartheta} \cos \sigma ), \end{gather}$$

(4.10c)\begin{gather} \mathcal{C} (\omega \cos \theta_0 \sin \theta_0 \sin \psi_0 - \sin^2 \theta_0 ) \nonumber\\ =\sin \bar{\vartheta}\,( \sin \bar{\vartheta} + \omega \cos \bar{\vartheta} \cos \sigma ) (\omega^2 \sin^2 \sigma - ( \sin \bar{\vartheta} + \omega \cos \bar{\vartheta} \cos \sigma )^2 ), \end{gather}

(4.10d)\begin{gather} \mathcal{C} \cos \theta_0 = \frac{\cos \bar{\vartheta} - \omega \sin \bar{\vartheta} \cos \sigma}{\lambda} ((\sin \bar{\vartheta} + \omega \cos \bar{\vartheta} \cos \sigma )^2 - \omega^2 \sin^2\sigma ). \end{gather}

We can now calculate explicitly the averages of the right-hand sides of (4.10) over one fast-time oscillation, to deduce that

(4.11a)$$\begin{gather} \left\langle \mathcal{C} \left( \frac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0} - 1\right)+ \frac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} \right\rangle ={-}(1 + \omega^2)\sin^2 \bar{\vartheta}, \end{gather}$$

(4.11b)$$\begin{gather}\left\langle \mathcal{S} \omega \sin \theta_0 \cos \psi_0 \right\rangle ={-}\omega^2 \sin^2 \bar{\vartheta}, \end{gather}$$

(4.11c)$$\begin{gather}\langle \mathcal{C} (\omega \cos \theta_0 \sin \theta_0 \sin \psi_0 - \sin^2 \theta_0 ) \rangle ={-}\frac{\sin^2 \bar{\vartheta}}{2} (2 \omega^2 \cos^2 \bar{\vartheta} + (2 - \omega^2) \sin^2 \bar{\vartheta} ), \end{gather}$$

(4.11d)$$\begin{gather}\left\langle \mathcal{C} \cos \theta_0 \right\rangle = \frac{2 - 3 \omega^2}{2 \lambda} \cos \bar{\vartheta} \sin^2 \bar{\vartheta}. \end{gather}$$

Substituting (4.11) into (4.9), we deduce the following expressions for the averages of the chiral terms:

(4.12a)\begin{gather} \left\langle g_1 (\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 ) + g_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle = \frac{C + \omega^2 D}{2 \lambda^2} \cos 2 \bar{\varphi} \sin^2 \bar{\vartheta}, \end{gather}

(4.12b)\begin{gather} \left\langle g_1\omega \cos \psi_0 + g_2 \sin \theta_0\,( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 ) \right\rangle \nonumber\\ ={-}\frac{1}{2 \lambda^2} \sin 2 \bar{\varphi} \sin^2 \bar{\vartheta} \left( (C + \omega^2 D ) \cos^2 \bar{\vartheta} + \frac{3 \omega^2 C + (2 - \omega^2)D}{2} \sin^2 \bar{\vartheta} \right), \end{gather}

(4.12c)\begin{gather} \left\langle g_2 \cos \theta_0 + g_3 \right\rangle = \frac{(C - D) (2 - 3\omega^2)}{4 \lambda^2} \sin 2 \bar{\varphi} \cos \bar{\vartheta} \sin^2 \bar{\vartheta}. \end{gather}

Finally, to obtain the slow-time governing equations for $\bar {\vartheta }$, $\bar {\varPsi }$ and $\bar {\varphi }$ that we have been seeking, we substitute the explicit averages (4.12) into the solvability conditions (4.4), and rearrange to obtain the reduced system

(4.13a)$$\begin{gather} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} ={-}\frac{\hat{B}}{2} \sin \bar{\vartheta} \cos \bar{\vartheta} \sin 2 \bar{\varphi} - \frac{\hat{C}}{2} \sin \bar{\vartheta} \cos 2 \bar{\varphi}, \end{gather}$$

(4.13b)$$\begin{gather}\frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = \frac{\hat{B}}{2} \cos \bar{\vartheta} \cos 2 \bar{\varphi} - \frac{\hat{C}}{2} \cos^2 \bar{\vartheta} \sin 2 \bar{\varphi} - \frac{\hat{D}}{2} \sin^2 \bar{\vartheta} \sin 2 \bar{\varphi}, \end{gather}$$

(4.13c)$$\begin{gather}\frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = \frac{1}{2}(1 - \hat{B} \cos 2 \bar{\varphi} ) + \frac{\hat{C}}{2} \cos \bar{\vartheta} \sin 2 \bar{\varphi}, \end{gather}$$

where we define the effective chiral coefficients

(4.14a,b)\begin{equation} \hat{C} := \frac{C + \omega^2 D}{(1 + \omega^2)^{3/2}},\quad \hat{D} := \frac{3 \omega^2 C + (2 - \omega^2) D}{2 (1 + \omega^2)^{3/2}}. \end{equation}

We illustrate these effective parameters in terms of $\omega$ in figure 3.

Figure 3. The effective parameters $\hat {B}, \hat {C}, \hat {D}, \hat {\beta }, \hat {\gamma }, \hat {\delta }$ as functions of $\omega$, normalised by their intrinsic equivalents. (a) $\hat {B}$ and $\hat {\beta }$ are functions only of $\omega$, and exhibit the same dependence on $\omega$ following normalisation. (b,c) The remaining effective parameters are functions of three parameters. All are coupled to $\omega$; the orientational shape parameters are also coupled to $C$ and $D$, while the translational shape parameters are also coupled to $\gamma$ and $\delta$ instead. We show selected curves for different parameter values. Several of the effective coefficients display non-trivial zeros as functions of $\omega$. This suggests that specific activity-induced spinning can effectively eliminate certain parameters, and hence the associated physical interactions of an object with the flow.

4.3. Summary

By comparison with the original angular dynamical system, defined in (2.1)–(2.3), we see that the emergent dynamics governed by (4.13) can be rewritten in terms of the combined achiral and chiral functions $h_i = f_i + g_i$ as

(4.15a–c)\begin{equation} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} = h_1(\bar{\vartheta},\bar{\varphi}; \hat{B}, \hat{C}), \quad \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = h_2(\bar{\vartheta},\bar{\varphi};\hat{B}, \hat{C}, \hat{D}), \quad \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = h_3(\bar{\vartheta},\bar{\varphi};\hat{B}, \hat{C}), \end{equation}

where the effective Bretherton parameter $\hat {B}$ is defined in (4.5), and the effective chiral coefficients $\hat {C}$ and $\hat {D}$ are defined in (4.14a,b).

Therefore, similar to Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, the emergent dynamics for rapidly spinning chiral particles are governed by a system that has the same functional form as the original dynamical system without rapid spinning, but with modified coefficients (4.14a,b) that account for the effect of the spinning. As before, we can identify each slow-time function with an underlying variable: $\bar {\vartheta }$ with $\theta$, $\bar {\varPsi }$ with $\psi$, and $\phi$ with $\bar {\varphi }$. Since the slow terms in the original dynamical system represent the generalised Jeffery's equations for chiral particles, we can say that rapidly spinning chiral particles behave as particles with an effective chirality, as quantified through the effective coefficients (4.14a,b).

We explore the effect of rotation on the orientational dynamics in figure 4 and supplementary movies 1–4. In figure 4, we illustrate trajectories in the $(\phi,\theta )$-plane and set $D = 0$ for simplicity. In figures 4(a–c), we fix the Bretherton parameter $B = 0.7$ and vary the chirality parameter $C$ in order to highlight the qualitative changes that chirality can induce. In figure 4(a), we set $C = 0$ and present standard Jeffery orbits for homochiral particles for the purpose of comparison, which are periodic as $\left \lvert B \right \rvert < 1$. Since this sublimit is a regular limit of the achiral analysis of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, the trajectories shown in these plots are identical to those explored in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1. In figure 4(b), we increase the chirality parameter to $C = 0.7$, illustrating the trajectories of chiral objects. Here, the chirality breaks the periodicity of the slow-time generalised Jeffery trajectories for smaller values of $\omega$, instead inducing a drift towards the pole $\theta =0$. However, this periodicity-breaking effect appears to weaken for larger values of $\omega$, when the effective chirality $\hat {C}$ of the object is reduced following (4.14a,b). In figure 4(c), we show trajectories for a strongly chiral object, increasing the chirality parameter to $C = 1.5$. Here, the chirality induced periodicity-breaking effect is stronger, with the notable appearance of attractive and repulsive points away from the poles at $\theta = 0,{\rm \pi}$, and persists for larger values of $\omega$ before eventually leading to approximately periodic trajectories as $\omega$ increases further.

Figure 4. Exploring the orientational dynamics in the $(\phi,\theta )$-plane for various values of $B$, $C$ and $\omega$, with sample, rapidly oscillating full dynamics shown in blue for $\omega \neq 0$, and the corresponding averaged dynamics shown in red: (a) $(B,C) = (0.7,0)$, (b) $(B,C) = (0.7,0.7)$, (c) $(B,C) = (0.7,1.5)$, (d) $(B,C) = (0,0.7)$, and (e) $(B,C) = (0,1.5)$. We use $D=0$ and $(\theta,\phi ) = ({\rm \pi} /2,-{\rm \pi} )$ at initial time throughout. For the blue lines, we also set $\varOmega _{\parallel } = 10$ and $\varOmega _{\perp } = 10 \omega$. Dynamic versions of the full dynamics of the highlighted trajectories in (b–e) are given in supplementary movies 1–4.

In figures 4(d,e), we consider the effects of chirality on an object with vanishing Bretherton parameter, setting $B = 0$. In figure 4(d), we take $C = 0.7$, observing periodic trajectories whose behaviour is significantly more oscillatory in the $\theta$ variable than in the classical Jeffery orbits of figure 4(a). Further, $\theta = {\rm \pi}/2$ is no longer a steady solution, which can also be seen by considering directly the contribution of the chiral function $g_1$ of (2.3a) in the governing equation (2.1a). As $\omega$ increases, we see a general reduction in these oscillations towards those of a sphere (with $B=C=0$), as predicted by our explicit result for the effective chirality $\hat {C}$ in (4.14a,b). In figure 4(e), we consider a strongly chiral object by taking $C = 1.5$. In this case, the strongly chiral effects induce periodic orbits that, curiously, do not encircle the pole for smaller values of $\omega$, instead orbiting around non-trivial fixed points in the $(\phi,\theta )$-plane. However, as $\omega$ increases and decreases $\hat {C}$, these orbits collapse, and the trajectories begin to approach those seen in figure 4(d) for smaller values of $\omega$, as expected. The existence of periodic orbits that do not encircle the pole for larger values of $C$ is due to the pole becoming a repulsive fixed point when $B^2+C^2 > 1$ in the case of a passive object (Ishimoto Reference Ishimoto2020a,Reference Ishimotob), with non-trivial attractors emerging as a result of the bifurcation. In figure 5, we provide a visual characterisation of the qualitative behaviour of the solution space for the orientational dynamics in terms of the effective parameters $\hat {B}$ and $\hat {C}$.

Figure 5. Schematic showing the qualitative nature of the orientational dynamics within the parameter space $(B,C)$, for different values of $\omega$. The darker regions within each ellipse indicate that trajectories drift towards a pole ($\theta = 0$ for the yellow region, $\theta = {\rm \pi}$ for the blue region). Outside the ellipses, the pole solutions become repulsive points and non-trivial attractors exist. The lighter regions external to each ellipse indicate that these non-trivial attractors are in the northern (yellow) and southern (blue) hemispheres, respectively. The thicker red lines (solid and dashed) on the axes indicate periodic trajectories. Dashed lines indicate the existence of orbits that are not centred around one of the poles at $\theta = 0, {\rm \pi}$. In the critical cases $\omega = \sqrt {2}$ and $\omega \to \infty$, all trajectories are orbits, so there exist only red regions. There is a distinction between orbits centred around a pole (darker red) and not centred around a pole (lighter red). Finally, the influence of the third shape parameter $D$ is shown: (a) $D=0$, (b) $D=0.3$, and (c) $D=-0.3$.

Given these observations, it is of interest to note the limiting cases $\omega \to 0$ and $|\omega | \to \infty$. In the limit $\omega \to 0$, the effective chiral parameters remain the same, i.e. $\hat {C} \to C$ and $\hat {D} \to D$. That is, when spinning is rapid only around the axis of helicoidal symmetry, the effective shape of the chiral swimmer is unchanged; the rapid rotation does not impact significantly the emergent angular dynamics. On the other hand, in the limit $|\omega | \to \infty$, the effective chiral parameters vanish, i.e. $\hat {C} \to 0$ and $\hat {D} \to 0$. That is, when rapid spinning only around an axis perpendicular to the axis of helicoidal symmetry, the rapid rotation causes the chiral swimmer to lose the effect of its chirality and for its orientation to evolve as though it were an achiral particle. This is because the coefficients $C$ and $D$ can be thought of as moments of chirality along the axis of helicoidal symmetry, and rapid rotation around an axis perpendicular to this will ‘spread out’ the chirality on average, reducing the effective moment to zero.

Additionally, we see that a chiral particle with $C = 0$ but $D \neq 0$ (or $D = 0$ but $C \neq 0$) can result in $\hat {C} \neq 0$ and $\hat {D} \neq 0$. That is, in certain cases with chiral particles, rapid spinning can generate effective terms that were not present in the original equations. Moreover, rapid spinning can either enhance or diminish the effects of chirality, depending on the specific values of $C$ and $D$, and the relative rotation ratio $\omega$.

A helpful way to interpret the effective chirality parameters $\hat {C}$ and $\hat {D}$, defined in (4.14a,b), is in terms of their relative sizes with respect to $\sqrt {C^2 + D^2}$, which can be considered a measure of the overall chirality of the object. To study this, it is helpful to introduce the parameter $\zeta$, defined as the principal argument of the complex number

(4.16)\begin{equation} \exp(i \zeta) = \frac{C + {\rm i} D}{|C + {\rm i} D|}. \end{equation}

Therefore, the introduction of $\zeta$ collapses the two-dimensional parameter space $(C, D)$ onto a single parameter via the complex unit circle. Then, utilising the relationship $\tan \alpha = \omega$, where $\alpha$ is the angle between the rotational and helicoidal axes, we can rewrite (4.14a,b) as

(4.17a)$$\begin{gather} \frac{\hat{C}}{|C + {\rm i} D|} = \cos \alpha (\cos^2 \alpha \cos \zeta + \sin^2 \alpha \sin \zeta ), \end{gather}$$

(4.17b)$$\begin{gather}\frac{\hat{D}}{|C + {\rm i} D|} = \frac{\cos \alpha}{2} (3\sin^2 \alpha \cos \zeta + (2\cos^2 \alpha- \sin^2 \alpha) \sin \zeta ), \end{gather}$$

which means that we can illustrate the left-hand sides of (4.17) in terms of just two parameters: $\alpha$ and $\zeta$ (see figures 6a,b). Through explicit calculation, it can also be shown that

(4.18)\begin{equation} \hat{C}^2 + \hat{D}^2 \leqslant C^2 + D^2, \end{equation}

which is illustrated in figure 6(c). Interpreting $\sqrt {C^2 + D^2}$ as a measure of the overall chirality of the object, we can deduce that rotation never increases the overall effective chirality. In fact, in general, rotation reduces the overall chirality, only leaving the overall chirality unchanged for $\alpha = 0$. While this reduction is a general property for the overall chirality, it is notable that (4.14a,b) implies that rotation can cause specific individual chirality parameters to increase. That is, rotation can cause $|\hat {C}| > |C|$ or $|\hat {D}| > |D|$, but the constraint (4.18) means that these cannot occur at the same time. Since $C$ and $D$ represent different aspects of chirality, we can interpret this as rotation allowing different aspects of chirality to be overemphasised or underemphasised, even though rotation reduces the overall chirality of the object.

Figure 6. Representations of the scaled effective chiral coefficients: (a) $\hat {C}/ \sqrt {C^2 + D^2} \in [-1, 1]$, (b) $\hat {D}/ \sqrt {C^2 + D^2} \in [-1, 1]$, (c) $\sqrt {(\hat {C}^2 + \hat {D}^2)/(C^2 + D^2)} \in [0, 1]$. We define these quantities in (4.14a,b) and (4.17). Notably, the magnitude of each quantity is bounded above by 1, so we may conclude that the effect of rapid rotation is to reduce the effective overall chirality of an object.

5. Deriving the emergent translational dynamics

Using the asymptotic expansions (3.7) in the transformed governing equations (3.6), we obtain the trivial leading-order (i.e. $\mathit {O}(\varOmega _{\parallel })$) system

(5.1)\begin{equation} \frac{\partial \boldsymbol{X}_0}{\partial T} = \boldsymbol{0}, \end{equation}

which tells us that $\boldsymbol {X}_0 = \boldsymbol {X}_0(t)$.

At next order (i.e. $\mathit {O}(1)$), we obtain the system

(5.2)\begin{align} \lambda\,\frac{\partial \boldsymbol{X}_1}{\partial T} + \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} &= \boldsymbol{V} + Y_0 \boldsymbol{e}_{3} - \beta \left( \hat{\boldsymbol{e}}_{2} \hat{\boldsymbol{e}}_{3}^{\rm T} - \hat{\boldsymbol{e}}_{3} \hat{\boldsymbol{e}}_{2}^{\rm T} \right) \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + \gamma \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + (\delta - \gamma) \left(\hat{\boldsymbol{e}}_{1}^{\rm T}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}\right) \hat{\boldsymbol{e}}_{1}, \end{align}

with $2 {\rm \pi}$-periodicity in $T$, recalling that $\boldsymbol {V} = V_1\hat {\boldsymbol {e}}_{1} + V_2\hat {\boldsymbol {e}}_{2} + V_3\hat {\boldsymbol {e}}_{3}$. The solvability condition that will give our emergent dynamics is obtained simply by averaging (5.2) over $T \in (0, 2{\rm \pi} )$. Performing this averaging and imposing periodicity in $T$, (5.2) becomes

(5.3)\begin{equation} \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} = \left\langle\boldsymbol{V} + Y_0 \boldsymbol{e}_{3} - \beta \left( \hat{\boldsymbol{e}}_{2} \hat{\boldsymbol{e}}_{3}^{\rm T} - \hat{\boldsymbol{e}}_{3} \hat{\boldsymbol{e}}_{2}^{\rm T} \right) \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + \gamma \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} + (\delta - \gamma) \left(\hat{\boldsymbol{e}}_{1}^{\rm T}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}\right) \hat{\boldsymbol{e}}_{1} \right\rangle. \end{equation}

Some care needs to be taken in evaluating the right-hand side of (5.3), since the swimmer-frame basis vectors $\hat {\boldsymbol {e}}_{i}$ are dependent on $T$ through their dependence on the Euler angles, with the explicit dependence given in (A1). Importantly, since the leading-order analysis is the same between Parts 1 and 2, and the first two terms on the right-hand side of (5.3) are present in Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1, we can use our results of § 4.5 of Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 1 to state immediately that

(5.4)\begin{equation} \left \langle \boldsymbol{V} + Y_0 \boldsymbol{e}_{3} \right \rangle = \hat{V} \tilde{\boldsymbol{e}}_{1}(\bar{\vartheta},\bar{\varphi}) + Y_0 \boldsymbol{e}_{3}, \end{equation}

where

(5.5)\begin{equation} \hat{V} := \frac{V_1 + \omega V_2}{\sqrt{1 + \omega^2}}, \end{equation}

and $\tilde {\boldsymbol {e}}_{1}(\bar {\vartheta },\bar {\varphi })$ can be considered equivalent to the (hatted) basis vector $\hat {\boldsymbol {e}}_{1}$ in (A1), but with argument $(\theta,\phi )$ replaced by $(\bar {\vartheta },\bar {\varphi })$.

To calculate the remaining averages on the right-hand side of (5.3), we start by writing them in terms of the laboratory basis, using the swimmer-to-laboratory transformation (A1) and the definition of $\boldsymbol {E}^*$ (A5a,b). This yields

(5.6a) $$\begin{gather} \big( \hat{\boldsymbol{e}}_{2} \hat{\boldsymbol{e}}_{3}^{\rm T} - \hat{\boldsymbol{e}}_{3} \hat{\boldsymbol{e}}_{2}^{\rm T} \big) \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} = \tfrac{1}{2}\big(\big[s_{\theta}^2 c_{2\phi}\big] \boldsymbol{e}_{1} + [ c_{\theta} s_{\theta} s_{\phi}] \boldsymbol{e}_{2} + [ c_{\theta} s_{\theta} c_{\phi}] \boldsymbol{e}_{3}\big), \end{gather}$$

(5.6b)$$\begin{gather}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} ={-}\tfrac{1}{2} ( s_{\theta} c_{\phi} \boldsymbol{e}_{2} - s_{\theta} s_{\phi} \boldsymbol{e}_{3}), \end{gather}$$

(5.6c) $$\begin{gather} \big(\hat{\boldsymbol{e}}_{1}^{{\rm T}}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}\big)\hat{\boldsymbol{e}}_{1} ={-}\tfrac{1}{2} \big(\big[c_{\theta} s_{\theta}^2 s_{2 \phi}\big] \boldsymbol{e}_{1} + \big[s_{\theta}^3 s_{2 \phi} s_{\phi} \big] \boldsymbol{e}_{2} - \big[s_{\theta}^3 s_{2 \phi} c_{\phi}\big] \boldsymbol{e}_{3}\big), \end{gather}$$

where we have used shorthand notation with $c_{\theta }$, $s_{\theta }$, $c_{\phi }$, $s_{\phi }$ denoting $\cos \theta _0$, $\sin \theta _0$, $\cos \phi _0$, $\sin \phi _0$, and so on. We can then calculate the averages of (5.6) using the expressions (4.2) and (4.7)–(4.8) that we derived previously, to deduce that

(5.7a) \begin{gather} \big\langle \big( \hat{\boldsymbol{e}}_{2} \hat{\boldsymbol{e}}_{3}^{\rm T} - \hat{\boldsymbol{e}}_{3} \hat{\boldsymbol{e}}_{2}^{\rm T} \big) \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} \big\rangle = \frac{2 - \omega^2}{4 \lambda^2} \big( \big[s_{\bar{\vartheta}}^2 c_{2\bar{\varphi}}\big] \boldsymbol{e}_{1} + [ c_{\bar{\vartheta}} s_{\bar{\vartheta}} s_{\bar{\varphi}} ] \boldsymbol{e}_{2} + [ c_{\bar{\vartheta}} s_{\bar{\vartheta}} c_{\bar{\varphi}} ] \boldsymbol{e}_{3}\big) \nonumber\\ = \frac{2 - \omega^2}{2 \lambda^2} \big( \tilde{\boldsymbol{e}}_{2} \tilde{\boldsymbol{e}}_{3}^{\rm T} - \tilde{\boldsymbol{e}}_{3} \tilde{\boldsymbol{e}}_{2}^{\rm T}\big) \boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1}, \end{gather}

(5.7b)\begin{gather} \langle \boldsymbol{E}^* \hat{\boldsymbol{e}}_{1} \rangle ={-}\frac{1}{2 \lambda} ( s_{\bar{\vartheta}} c_{\bar{\varphi}} \boldsymbol{e}_{2} - s_{\bar{\vartheta}} s_{\bar{\varphi}} \boldsymbol{e}_{3}) = \frac{\boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1}}{\lambda}, \end{gather}

(5.7c) \begin{gather} \big\langle \big(\hat{\boldsymbol{e}}_{1}^{{\rm T}}\boldsymbol{E}^* \hat{\boldsymbol{e}}_{1}\big)\hat{\boldsymbol{e}}_{1} \big\rangle ={-}\frac{1}{4 \lambda^3} \big[( 2 - 3 \omega^2)\big\{ \big[c_{\bar{\vartheta}} s_{\bar{\vartheta}}^2 s_{2 \bar{\varphi}}\big] \boldsymbol{e}_{1} + \big[s_{\bar{\vartheta}}^3 s_{2 \bar{\varphi}} s_{\bar{\varphi}}\big] \boldsymbol{e}_{2}\big.\big. \nonumber\\ - \,\big.\big. \big[s_{\bar{\vartheta}}^3 s_{2 \bar{\varphi}} c_{\bar{\varphi}}\big] \boldsymbol{e}_{3}\big\} + 2\omega^2 ( s_{\bar{\vartheta}} c_{\bar{\varphi}} \boldsymbol{e}_{2} - s_{\bar{\vartheta}} s_{\bar{\varphi}} \boldsymbol{e}_{3})\big] \nonumber\\ = \frac{2 - 3\omega^2}{2 \lambda^3} \big(\tilde{\boldsymbol{e}}_{1}^{{\rm T}}\boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1}\big)\tilde{\boldsymbol{e}}_{1} + \frac{\omega^2}{\lambda^3} \boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1}, \end{gather}

where $\tilde {\boldsymbol {e}}_{i} = \tilde {\boldsymbol {e}}_{i}(\bar {\vartheta },\bar {\varphi })$ can be considered equivalent to their $\hat {\boldsymbol {e}}_{i}$ (hatted) versions in (5.6) with arguments $(\theta _0,\phi _0)$ replaced by $(\bar {\vartheta },\bar {\varphi })$.

Finally, substituting (5.4) and (5.7) into (5.3), we obtain our effective equation for the emergent translational dynamics:

(5.8)\begin{equation} \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} = \hat{V} \tilde{\boldsymbol{e}}_{1} + Y_0 \boldsymbol{e}_{3} - \hat{\beta} \left(\tilde{\boldsymbol{e}}_{2} \tilde{\boldsymbol{e}}_{3}^{\rm T} - \tilde{\boldsymbol{e}}_{3} \tilde{\boldsymbol{e}}_{2}^{\rm T} \right) \boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1} + \hat{\gamma} \boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1} + (\hat{\delta} - \hat{\gamma}) \big(\tilde{\boldsymbol{e}}_{1}^{\rm T}\boldsymbol{E}^* \tilde{\boldsymbol{e}}_{1}\big) \tilde{\boldsymbol{e}}_{1}, \end{equation}

emphasising that $\tilde {\boldsymbol {e}}_{i}$ are functions of the slow-time variables $\bar {\vartheta }$ and $\bar {\varphi }$, and that we have defined the effective coefficients

(5.9a–c)\begin{equation} \hat{\beta} = \frac{(2 - \omega^2) \beta}{2(1 + \omega^2)},\quad \hat{\gamma} = \frac{\gamma + \omega^2 \delta}{(1 + \omega^2)^{3/2}},\quad \hat{\delta} = \frac{3 \omega^2 \gamma + (2 - \omega^2) \delta}{2 (1 + \omega^2)^{3/2}}, \end{equation}

and we illustrate these effective coefficients as functions of $\omega$ in figure 3. Therefore, we see that the effective translational equation (5.8) has the same functional form as the original equation (2.4), but with dependence on the fast-varying Euler angles switched to dependence on the slow-time functions that we derived in § 4, and modified coefficients (5.9a–c) that account systematically for the effect of the fast spinning. Therefore, we can say that rapidly spinning chiral particles are translated as particles with an effective chiral shape, as quantified through the effective shape coefficients defined in (5.9a–c). The excellent agreement between the complex full translational dynamics and the emergent dynamics predicted by (5.8) is illustrated on an example in figure 7, and we explore further the effect of varying the intrinsic shape parameters in figure 8.

Figure 7. Illustration of the agreement between the full spinning translational dynamics and the emergent system that we derive. (a) The predictions of the emergent dynamics are shown as a red curve, while the full dynamics is shown as a black line with attached ribbon, coloured according to the spin angle $\psi$ of the object. Differences between the dynamics are barely visible at the resolution of this plot. (b) A portion of the trajectory in (a), showing the small (expected) discrepancy between the full and emergent solutions. Here, we have taken $\varOmega _{\parallel } = \varOmega _{\perp } = 100$, $B=0.8$, $C=-0.3$, $D=-0.5$, $\beta =0.01$, $\gamma = 0.3$, $\delta = -3$, $V_1 = 1$ and $V_2 = V_3 = 0.5$.

Figure 8. Exploring the influence of the effective geometric parameters $\hat {\beta }$, $\hat {\gamma }$, $\hat {\delta }$ and $\hat {C}$ on the emergent translational dynamics. In each panel, we vary each effective parameter independently from $(\hat {\beta }, \hat {\gamma }, \hat {\delta }, \hat {C}) = (0,0,0,0)$, highlighting the distinct role that each parameter plays in determining the emergent translational dynamics. In each column, we show three-dimensional trajectories and traces of laboratory-frame coordinates over time. Throughout, we use initial conditions $\boldsymbol {X} = 0$ and $(\theta,\phi,\psi ) = ({\rm \pi} /3,{\rm \pi} /6,2{\rm \pi} /3)$.

Finally, we consider the limiting cases $\omega \to 0$ and $|\omega | \to \infty$. In the limit $\omega \to 0$, the effective coefficients are unchanged (i.e. $\hat {\beta } \to \beta$, $\hat {\gamma } \to \gamma$, $\hat {\delta } \to \delta$). That is, when the axis of rapid spinning coincides with the axis of helicoidal symmetry, the effective shape of the chiral swimmer is unchanged; the rapid rotation does not impact significantly the emergent translational dynamics. In contrast, in the limit $|\omega | \to \infty$ the effective coefficients are changed, with $\hat {\beta } \to -\beta /2$ and $\hat {\gamma }, \hat {\delta } \to 0$. We recall that the results of § 4 state that the effective chirality coefficients also vanish in the same limit (i.e. $\hat {C}, \hat {D} \to 0$ as $|\omega | \to \infty$), and that passive homochiral objects satisfy $C = D = \gamma = \delta = 0$ (see figure 2). Therefore, we may conclude that when rapid spinning occurs around an axis perpendicular to the axis of helicoidal symmetry, a general active helicoidal swimmer will behave as though it is a passive homochiral swimmer. This can be interpreted intuitively by noting that a rapidly rotating swimmer with rotation axis perpendicular to its helicoidal axis can be thought of as exhibiting a geometric rotational symmetry of ${\rm \pi}$ around its rotation axis.