3.1. Angular dynamics

The translational dynamics (2.3) decouple from the rest of the system. Hence, we first investigate the angular dynamics given by the system (2.1)–(2.2) in the $\varOmega _{\parallel } \gg 1$ and $\varOmega _{\perp } = \mathit {O}(1)$ limit. To this end, we introduce the ‘fast’ time scale $T$ via

(3.1)\begin{equation} T = \varOmega_{{\parallel}} t, \end{equation}

and refer to the original time scale $t$ as the ‘slow’ time scale. Proceeding via the standard method of multiple scales (Hinch Reference Hinch1991; Bender & Orszag Reference Bender and Orszag1999), we treat each dependent variable as a function of both the fast- and slow-time scales, converting ordinary differential equations (ODEs) into partial differential equations. The additional degrees of freedom this introduces will be removed later by imposing appropriate periodicity constraints in the fast-time scale.

With the new time scale (3.1), the time derivative becomes

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

which transforms the system (2.1) into

(3.3a)$$\begin{gather} \varOmega_{{\parallel}} \frac{\partial{\theta}}{\partial{T}} + \frac{\partial{\theta}}{\partial{t}} = \varOmega_{{\perp}} \cos \psi + f_1(\theta,\phi), \end{gather}$$

(3.3b)$$\begin{gather}\varOmega_{{\parallel}} \frac{\partial{\psi}}{\partial{T}} + \frac{\partial{\psi}}{\partial{t}} = \varOmega_{{\parallel}} - \varOmega_{{\perp}} \dfrac{\cos \theta \sin \psi}{\sin \theta} + f_2(\theta,\phi), \end{gather}$$

(3.3c)$$\begin{gather}\varOmega_{{\parallel}} \frac{\partial{\phi}}{\partial{T}} + \frac{\partial{\phi}}{\partial{t}} = \varOmega_{{\perp}} \dfrac{\sin \psi}{\sin \theta} + f_3(\phi). \end{gather}$$

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

(3.4)\begin{equation} \zeta(T,t) \sim \zeta_0(T,t) + \dfrac{1}{\varOmega_{{\parallel}}} \zeta_1(T,t) \quad \text{as } \varOmega_{{\parallel}} \to \infty, \quad \text{for } \zeta \in \{\theta, \psi, \phi \}. \end{equation}

Using the asymptotic expansions (3.4) in the transformed governing equations (3.3), we obtain the leading-order (i.e. $\mathit {O}(\varOmega _{\parallel })$) system

(3.5a–c)\begin{equation} \frac{\partial{\theta_0}}{\partial{T}} = 0, \quad \frac{\partial{\psi_0}}{\partial{T}} = 1, \quad \frac{\partial{\phi_0}}{\partial{T}} = 0. \end{equation}

The system (3.5a–c) is straightforward to directly integrate. This will not be the case for the more general problem in § 4, when we have to solve a non-trivial nonlinear problem at leading order. Directly integrating (3.5a–c), we obtain the solutions

(3.6a–c)\begin{equation} \theta_0 = \bar{\vartheta}(t), \quad \psi_0 = T + \bar{\varPsi}(t), \quad \phi_0 = \bar{\varphi}(t). \end{equation}

Since $\psi$ and $\phi$ are angles interpreted modulo $2 {\rm \pi}$, the leading-order solutions (3.6a–c) are $2{\rm \pi}$-periodic in the fast time $T$. Importantly, $\bar {\vartheta }$, $\bar {\varPsi }$ and $\bar {\varphi }$ are as-of-yet undetermined functions of the slow time, directly related to $\theta _0$, $\psi _0$ and $\phi _0$, respectively. Our remaining goal is to determine the governing equations for these slow-time functions. To do this, we must proceed to the next asymptotic order and determine the solvability conditions.

After posing the asymptotic expansions (3.4) and moving the slow-time derivatives to the right-hand side, the $\mathit {O}(1)$ terms in (3.3) are

(3.7a)\begin{gather} \frac{\partial{\theta_1}}{\partial{T}} = \varOmega_{{\perp}} \cos \psi_0 + f_1(\theta_0,\phi_0) - \frac{\mathrm{d} \theta_0}{\mathrm{d} t} = \varOmega_{{\perp}} \cos (T + \bar{\varPsi}) + f_1(\bar{\vartheta},\bar{\varphi}) - \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t}, \end{gather}

(3.7b)\begin{gather} \frac{\partial{\psi_1}}{\partial{T}} =- \varOmega_{{\perp}} \dfrac{\cos \theta_0 \sin \psi_0}{\sin \theta_0} + f_2(\theta_0,\phi_0) - \frac{\partial{\psi_0}}{\partial{t}} \nonumber\\ = - \varOmega_{{\perp}} \dfrac{\cos \bar{\vartheta} \sin (T + \bar{\varPsi})}{\sin \bar{\vartheta}} + f_2(\bar{\vartheta},\bar{\varphi}) - \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t}, \end{gather}

(3.7c)\begin{gather} \frac{\partial{\phi_1}}{\partial{T}} = \varOmega_{{\perp}} \dfrac{\sin \psi_0}{\sin \theta_0} + f_3(\phi_0) - \frac{\mathrm{d} \phi_0}{\mathrm{d} t} = \varOmega_{{\perp}} \dfrac{\sin (T + \bar{\varPsi})}{\sin \bar{\vartheta}} + f_3(\bar{\varphi}) - \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t}, \end{gather}

now accompanied by fast-time periodicity constraints over any $2{\rm \pi}$ period in $T$. These remove the additional degrees of freedom introduced by initially taking $T$ and $t$ to be independent. For the system (3.7), it is straightforward to determine the requisite solvability conditions. This is because the linear operators acting on $\theta _1$, $\psi _1$ and $\phi _1$ (on the left-hand sides) decouple from one another, even though the full problem (2.1) is fully coupled. This will not be the case for the more general problem in § 4, when we have to derive and solve a non-trivial adjoint problem.

Nevertheless, for the sublimit of bacterial spinning, we consider in this section, this decoupling means that we can straightforwardly obtain our desired solvability conditions by integrating (3.7) with respect to $T$ from $0$ to $2 {\rm \pi}$. The integrated fast-time derivatives on the left-hand side vanish due to the imposed periodicity, and the terms multiplied by $\varOmega _{\perp }$ on the right-hand sides also vanish due to their specific trigonometric forms. Therefore, this integration yields the system

(3.8a–c)\begin{equation} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} = f_1(\bar{\vartheta},\bar{\varphi};B), \quad \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = f_2(\bar{\vartheta},\bar{\varphi};B), \quad \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = f_3(\bar{\varphi};B), \end{equation}

which provide the governing equations we seek for the slow-time functions in the leading-order solutions (3.6a–c). Importantly, we see that the slow-time system (3.8a–c) is equivalent to the full system (2.1) without the rotation terms (i.e. $\varOmega _{\parallel } = \varOmega _{\perp } = 0$) using the equivalence $(\theta, \psi, \phi ) \leftrightarrow (\bar {\vartheta }, \bar {\varPsi }, \bar {\varphi })$. One consequence of this is that $\bar {\varPsi }$ decouples from $\bar {\vartheta }$ and $\bar {\varphi }$, since $\psi$ decouples from $\phi$ and $\theta$ in the absence of rotation. This property is maintained in the more general analysis we present in § 4. Moreover, we note that the Bretherton parameter $B$ remains unchanged in the emergent behaviour in this limit. This property is not maintained in § 4.

To dwell on the general point for clarity, in the sublimit of fast bacterial spinning (with $\varOmega _{\parallel } \gg 1$ and $\varOmega _{\perp } = \mathit {O}(1)$) the emergent behaviour is the same as it would be without any fast spinning. While $\psi$ does vary rapidly due to this fast spinning, the leading-order emergent angular dynamics (up to and including time scales of $t = \mathit {O}(1)$) are not otherwise affected by the fast spinning. Importantly, this emergent indifference to fast spinning is not the case for the more general rotation we consider in § 4.

Figure 2 illustrates these observations through numerical solution of the full and the systematically averaged governing equations, displaying the evolution of the swimmer's orientation in several different cases. In this figure we illustrate the changing orientation of the object by plotting a trajectory on the surface of the unit sphere, which $\theta$ and $\phi$ naturally parameterise via standard spherical coordinates. With the rapid evolution of $\psi$ not illustrated, the particle's orientation traces out a closed orbit on the sphere, corresponding to a Jeffery's orbit that depends on the Bretherton parameter $B$ and the initial conditions. The orbits of passive particles, with $\varOmega _{\perp }=\varOmega _{\parallel }=0$, are illustrated in (a), with varying shape parameters and initial conditions ($\theta (0)={\rm \pi} /18$, $\psi (0)=-{\rm \pi} /4$ in all columns, and $\phi (0)={\rm \pi} /2$ in all but the final column, where $\phi (0) = 0$). We see from (a,b) that the emergent angular dynamics is not affected by the fast spinning as long as $\varOmega _{\perp }$ is not significant, and so active spinning particles generate the same orbits as passive particles if the off-axis spinning is not significant. However, we see from (c) that this result breaks down once $\varOmega _{\perp }$ starts to become a little larger; for an active particle with a significant component of off-axis spinning, the evolution of $\phi$ and $\psi$ differs from that of a passive particle.

Figure 2. Rotational dynamics in the bacterial limit. Each plot displays the evolution of the particle orientation on the unit sphere, parameterised with $(\theta,\phi )$ as standard spherical coordinates, with the vertical axis corresponding to $\theta =0$. (a) Standard 3-D Jeffery's orbits of passive particles for three different values of the Bretherton parameter $B = (r^2 - 1)/(r^2 + 1)$. (b) Incorporating active spinning in the bacterial limit, with $\varOmega _{\parallel } \gg 1, \varOmega _{\perp } = \mathit {O}(1)$. The full particle orientation dynamics is shown as a thin blue line, while the thicker green line shows the solution for the emergent, averaged behaviour calculated in § 3. Notably, this thick green line traces the same orbit as the blue curves in (a); as predicted by (3.8a–c), the leading-order dynamics is not affected by spinning in this limit. (c) Further increasing $\varOmega _{\perp }$ beyond the bacterial limit causes a loss in the validity of our predictions thus far. The full evolution of the particle orientation (2.1)–(2.2) is shown by the thin blue line, while the emergent dynamics predicted by (3.8a–c) are shown by the thick green line, in poor agreement with the full dynamics. Accurately capturing these emergent trajectories requires the more general analysis of § 4, the predictions of which are shown as red curves. (a) $\varOmega _{\parallel } = \varOmega_{\bot} = 0$, (b) $\varOmega_{\parallel} = 15, \varOmega_{\bot} = 0.5$ and (c) $\varOmega_{\parallel} = 15, \varOmega_{\bot} = 3$.

In more detail, in figure 2(b) the swimmer spins with $\varOmega _{\parallel }=15$ and $\varOmega _{\perp }=0.5$ and its orientation over time is represented as a thin dark blue line. The trajectory of the leading-order dynamics is shown as a thick green line, and can be seen to be in excellent agreement with the average evolution of the swimmer's orientation; moreover, it is indeed essentially identical to the non-spinning Jeffery orbit from figure 2(a), in line with the conclusions of our asymptotic analysis.

The analysis in this section starts to lose validity when the magnitude of the off-axis rotation $\varOmega _{\perp }$ increases, since this voids the requirement $\varOmega _{\perp } = \mathit {O}(1)$. This can be seen in figure 2(c), where the dynamics predicted by (3.8a–c) start to diverge from the actual emergent behaviour of the swimmer, here with $\varOmega _{\parallel }=15$ and $\varOmega _{\perp }=3$. As before, the blue line shows the full dynamics of the swimmer, and the green line shows the emergent dynamics predicted by the analysis of this section. The latter fails to well approximate the average evolution of the orientation since $\varOmega _{\perp }$ is now significant. In § 4 we generalise our analysis to capture the emergent behaviour outside the bacterial sublimit considered in this section. The predictions of this later analysis are shown as red curves in figure 2(c), which exhibit excellent agreement with the averaged behaviour of the full system.

We have now calculated the emergent behaviour of the angular dynamics in the sublimit of bacterial spinning. We close this problem by considering the emergent behaviour of the translational dynamics.

3.2. Translational dynamics

The governing equations for translation are given in (2.3). Using the transformation (3.2), the translational governing equations (2.3) become

(3.9)\begin{equation} \varOmega_{{\parallel}} \frac{\partial{\boldsymbol{X}}}{\partial{T}} + \frac{\partial{\boldsymbol{X}}}{\partial{t}} = V_1\hat{\boldsymbol{e}}_{1}(\theta, \phi) + V_2\hat{\boldsymbol{e}}_{2}(\theta,\psi, \phi) + V_3\hat{\boldsymbol{e}}_{3}(\theta,\psi, \phi) + Y \boldsymbol{e}_{3}, \end{equation}

with the $\hat {\boldsymbol {e}}_{i}$ denoting the swimmer frame basis vectors, given in terms of the laboratory frame basis vectors $\boldsymbol {e}_{i}$ in (A1).

Following (3.4) and expanding each dependent variable as an asymptotic series in inverse powers of $\varOmega _{\parallel }$, we obtain the $\mathit {O}(\varOmega _{\parallel })$ (leading-order) system

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

Hence, the vector position $\boldsymbol {X}$ is independent of the fast time $T$ at leading order. That is, $\boldsymbol {X}_0 = \boldsymbol {X}_0(t)$. Our remaining goal is to derive the dependence of $\boldsymbol {X}_0$ on the slow time via the calculation of solvability conditions at higher order.

At next order (i.e. ${O}(1)$), the translational governing equations (3.9) are

(3.11)\begin{equation} \frac{\partial{\boldsymbol{X}_1}}{\partial{T}} + \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} = V_1 \hat{\boldsymbol{e}}_{1}(\theta_0,\phi_0) + V_2 \hat{\boldsymbol{e}}_{2}(\theta_0,\psi_0,\phi_0) + V_3 \hat{\boldsymbol{e}}_{3} (\theta_0,\psi_0,\phi_0) + Y_0 \boldsymbol{e}_{3}, \end{equation}

along with fast-time periodicity constraints over any $2 {\rm \pi}$ period in $T$. It is straightforward to obtain the appropriate solvability conditions by integrating (3.11) with respect to the fast time $T$ from $0$ to $2 {\rm \pi}$. In the same way as in § 3.1, the fast-time derivative on the left-hand side vanishes due to the imposition of fast-time periodicity. Substituting the leading-order angular dynamics solutions (3.6a–c) into the explicit forms of $\hat {\boldsymbol {e}}_{i}$ given in (A1), we find that

(3.12a–c) \begin{equation} \langle \hat{\boldsymbol{e}}_{1}(\theta_0,\phi_0) \rangle = \tilde{\boldsymbol{e}}_{1}(\bar{\vartheta},\bar{\varphi}), \quad \langle \hat{\boldsymbol{e}}_{2}(\theta_0,\psi_0,\phi_0) \rangle = \boldsymbol{0}, \quad \langle \hat{\boldsymbol{e}}_{3}(\theta_0,\psi_0,\phi_0) \rangle = \boldsymbol{0}, \end{equation}

using the notation $\left \langle \boldsymbol {\cdot } \right \rangle$, defined as the average of its argument over one fast-time oscillation

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

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

Hence, using the results (3.12a–c), taking the fast-time average of (3.11) yields

(3.14)\begin{equation} \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} = V_1 \tilde{\boldsymbol{e}}_{1}(\bar{\vartheta},\bar{\varphi}) + Y_0 \boldsymbol{e}_{3}. \end{equation}

In particular, comparing the slow-time equation (3.14) to the full translational dynamics of (2.3) highlights that the structure of the translational dynamics is broadly unchanged by the rapid spinning, with basis vectors being replaced by their averages. However, this means that the self-induced velocity components in the original $\hat {\boldsymbol {e}}_{2}(\theta,\psi,\phi )$ and $\hat {\boldsymbol {e}}_{3}(\theta,\psi,\phi )$ directions are cancelled out by the fast spinning of the swimmer, so that only the velocity component in the original $\hat {\boldsymbol {e}}_{1}(\theta,\phi )$ direction contributes to the average velocity alongside the shear flow. Hence, this analysis shows that one can effectively neglect the off-axis components of the self-induced propulsive velocity of such a swimmer, at least at leading order and for $t = \mathit {O}(1)$. Interpreted physically, this result establishes that rapid axial spinning, in the absence of significant off-axis rotation, is sufficient to justify modelling the translation of such a swimmer using only the axial component of propulsive linear velocity, since the other self-velocity contributions cancel out at leading order. We show in § 4.5 that this intuitive result requires modification when off-axis rotation is significant.

We show some examples of the swimming behaviours of actively spinning particles in figure 3 and supplementary movie 1. Each panel displays the evolution of a swimmer's position as a black line. A superposed ribbon illustrates the evolution of the intrinsic orientation angle $\psi$; its shade of grey indicates the value of $\psi \in [-{\rm \pi}, {\rm \pi}]$ according to the colourbar on the right-hand side of the figure.

Figure 3. Translational dynamics of a spheroidal swimmer in the bacterial limit. A dynamic version of this figure is given in supplementary movie 1. In each panel, the black line traces the swimmer position $\boldsymbol {X}$ over time via (2.1)–(2.3), with $\boldsymbol {X} = \boldsymbol {0}$ and $(\theta,\psi,\phi ) = (2{\rm \pi} /5,{\rm \pi} /2,-7{\rm \pi} /15)$ initially. Here, we have fixed $B=0.5$ and $V_1=1$. The shaded ribbons represent the evolution of the swimmer orientation, with its twist and shading encoding the value of $\psi$ according to the colourbar. In panel (a) the body does not have an intrinsic rotation ($\varOmega _{\parallel } = \varOmega _{\perp } = 0$), nor any off-axis propulsive velocities ($V_2 = V_3 = 0$). In all other panels the particle is spinning with $\varOmega _{\parallel } = 15$ (and $\varOmega _{\perp } = 0$) and the thick red line shows the predicted translational dynamics (3.8a–c), (3.14), in excellent agreement with full dynamics. The panels differ in their prescribed off-axis propulsive velocities: (b) $V_2 = V_3 = 0$; (c) $V_2 = 1, V_3 = 0$; (d) $V_2 = V_3 = 1$; (e) $V_2 = 5, V_3 = 0$; (f) $V_2 = V_3 = 5$. Despite the range of prescribed propulsive velocities and correspondingly intricate fast-time-scale trajectories seen here, the emergent average translation is consistent between panels, as predicted by our analysis.

Figure 3(a) showcases the relatively simple dynamics of a particle that is not actively spinning ($\varOmega _{\parallel }=\varOmega _{\perp }=0$) and is only propulsing along its symmetry axis ($V_1 = 1, V_2 = V_3 = 0$), while figures 3(b)–3(f) illustrate the behaviours of spinning particles with various propulsive velocities, having fixed $V_1 = 1$ in each panel and taking $\varOmega _{\parallel }=15$ and $\varOmega _{\perp }=0$ in all but figure 3(a). In each panel, the trajectories predicted by the asymptotic analysis are shown as thick red lines, showcasing excellent agreement with the full dynamics in all but the final case, as expected, wherein the components of propulsive velocity become comparable in magnitude to the fast rate of active rotation.

The near-indistinguishable nature of the red and black lines in panel (b), where $V_2 = V_3 = 0$, highlights that the rapid spinning illustrated by the twisting ribbon need not significantly modify the emergent translational dynamics when there is no off-axis propulsion. In panels (c–e) we allow $(V_2,V_3)\neq\ (0,0)$, which causes far more complex fast-scale dynamics, as can be seen by the cork-screwing of the black line. However, as predicted by our asymptotic analysis, the combination of active spinning and non-trivial self-propulsion does not materially modify the predicted average emergent dynamics (red line). A slight exception to this is shown in figure 3(f), which corresponds to a case where the propulsive velocity is no longer asymptotically smaller than the rate of the rapid rotation. This starts to break the requirement of a separation of scales, giving rise to a discrepancy between the predictions of the asymptotic analysis and the full dynamics.

4.1. Leading-order angular dynamics

For our analysis of the general problem, we start by considering the angular dynamics. The time-derivative transformation (4.4) converts the system (2.1) into

(4.5a)$$\begin{gather} \lambda \varOmega_{{\parallel}} \frac{\partial{\theta}}{\partial{T}} + \frac{\partial{\theta}}{\partial{t}} = \omega \varOmega_{{\parallel}} \cos \psi + f_1(\theta,\phi), \end{gather}$$

(4.5b)$$\begin{gather}\lambda \varOmega_{{\parallel}} \frac{\partial{\psi}}{\partial{T}} + \frac{\partial{\psi}}{\partial{t}} = \varOmega_{{\parallel}}\left(1 - \omega \dfrac{\cos \theta \sin \psi}{\sin \theta}\right) + f_2(\theta,\phi), \end{gather}$$

(4.5c)$$\begin{gather}\lambda \varOmega_{{\parallel}} \frac{\partial{\phi}}{\partial{T}} + \frac{\partial{\phi}}{\partial{t}} = \omega \varOmega_{{\parallel}} \dfrac{\sin \psi}{\sin \theta} + f_3(\phi). \end{gather}$$

We expand each dependent variable as an asymptotic series in inverse powers of $\varOmega _{\parallel }$, following (3.4). Then the leading-order (i.e. $\mathit {O}(\varOmega _{\parallel })$) version of (4.5) is

(4.6a)$$\begin{gather} \lambda \frac{\partial{\theta_0}}{\partial{T}} = \omega\cos \psi_0, \end{gather}$$

(4.6b)$$\begin{gather}\lambda \frac{\partial{\psi_0}}{\partial{T}} = 1 - \omega \dfrac{\cos \theta_0 \sin \psi_0}{\sin \theta_0}, \end{gather}$$

(4.6c)$$\begin{gather}\lambda \frac{\partial{\phi_0}}{\partial{T}} = \omega\dfrac{\sin \psi_0}{\sin \theta_0}. \end{gather}$$

The leading-order system (4.6) is forced solely by the rapid spinning of the swimmer, and constitutes a nonlinear 3-D dynamical system for $\boldsymbol {x}_0 := (\theta _0, \psi _0, \phi _0)^{{\rm T}}$, which is notation we will use later.

A key difference in our analysis of the general problem in this section is that the nonlinear, coupled leading-order system (4.6) is not straightforward to solve. This is in direct contrast to the linear, decoupled leading-order system (3.5a–c) of the previous section. One consequence of this is that the important slow-time variables are not immediately obvious.

Of great help in our task is the fact that we can solve the leading-order system (4.6) in closed form. This is abetted by two key observations. Firstly, (4.6c) decouples from the other two subequations, and hence, we treat it last. Secondly, (4.6a) and (4.6b) can be combined and integrated to admit a first integral, which constitutes one of the three slow-time functions we expect to arise. To see this, we first divide (4.6b) by (4.6a) and multiply through by $\omega \cos \psi _0$ to yield

(4.7)\begin{equation} \omega\cos \psi_0 \frac{\partial{\psi_0}}{\partial{\theta_0}} = 1 - \omega \dfrac{\cos \theta_0 \sin \psi_0}{\sin \theta_0}. \end{equation}

Since the left-hand side of (4.7) can be written as $\omega \partial (\sin \psi _0)/\partial \theta _0$, (4.7) can be rewritten as a first-order linear differential equation for $\sin \psi _0$ as a function of $\theta _0$, i.e. implicitly making the change of variables $(T,t) \mapsto (\theta _0,t)$. Multiplying through by the appropriate integrating factor of $\sin \theta _0$ and grouping everything on the same side, we may rewrite (4.7) as

(4.8)\begin{equation} 0 = \dfrac{\partial}{\partial \theta_0} \left(\omega \sin \theta_0 \sin \psi_0 \right) - \sin \theta_0. \end{equation}

Hence, the required first integral is obtained by directly integrating (4.8) to yield

(4.9)\begin{equation} H(t) = \omega \sin \theta_0 \sin \psi_0 + \cos \theta_0; \end{equation}

thus generating the function of integration $H(t)$, emphasising that $H$ is independent of the fast time $T$. As will become clear once we complete the full analysis, it will turn out to be particularly convenient to redefine $H(t) = \lambda \cos \bar {\vartheta }(t)$, so we formally write

(4.10)\begin{equation} \lambda \cos \bar{\vartheta}(t) =\omega \sin \theta_0 \sin \psi_0 + \cos \theta_0, \end{equation}

where $\bar {\vartheta } = \bar {\vartheta }(t)$ is the first of three slow-time functions (each marked with an overbar) we obtain from our leading-order analysis in this section. We note that our definition of $\bar {\vartheta }$ in (4.10) in the limit $\omega \to 0$ coincides with its former definition in (3.6a). The goal of the next-order analysis in § 4.2 is to derive the governing equations satisfied by $\bar {\vartheta }$ and the two other appropriate slow-time functions we derive in the remainder of this section.

Substituting the fast-time conserved quantity (4.10) into (4.6a), we obtain the following nonlinear first-order differential equation:

(4.11)\begin{equation} \lambda \frac{\partial{\theta_0}}{\partial{T}} = \dfrac{\pm\sqrt{\omega^2 \sin^2 \theta_0 - (\lambda \cos \bar{\vartheta} - \cos \theta_0)^2}}{\sin \theta_0}. \end{equation}

Since (4.11) is an autonomous first-order differential equation for $\theta _0$ as a function of $T$, it admits a solution in the implicit form

(4.12)\begin{equation} \pm \int^{\theta_0} \dfrac{\lambda \sin s \, \mathrm{d} s}{\sqrt{\omega^2 \sin^2 s - (\lambda \cos \bar{\vartheta} - \cos s)^2}} = T + \bar{\varPsi}(t), \end{equation}

where $\bar {\varPsi } = \bar {\varPsi }(t)$ is the second of the three slow-time functions we obtain from our leading-order analysis. In fact, we can evaluate (4.12) explicitly using the substitution $\lambda \cos s = \cos \bar {\vartheta } + u \sin \bar {\vartheta }$ to convert (4.12) into

(4.13)\begin{equation} \mp \int \dfrac{\mathrm{d} u}{\sqrt{\omega^2 - u^2}} = T + \bar{\varPsi}, \end{equation}

omitting the transformed limit of the integral for notational convenience. It is then straightforward to integrate the left-hand side of (4.13) and rearrange to deduce that

(4.14a)\begin{equation} \lambda \cos \theta_0 = \cos \bar{\vartheta} - \omega \sin \bar{\vartheta} \cos (T + \bar{\varPsi}). \end{equation}

We note that the $\mp$ in (4.13) (as well as the choice of $\cos (T +\bar {\varPsi })$ in (4.14a) instead of $\sin (T +\bar {\varPsi })$) can be absorbed into an initial phase shift in $\bar {\varPsi }$. Therefore, we have taken the negative root in moving from (4.13) to (4.14a) for later convenience, without loss of generality. Rearranging (4.14a), we deduce that

(4.14b)\begin{equation} \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{equation}

It will also be helpful later to note that the combination of (4.10) and (4.14a) yields the relationship

(4.14c)\begin{equation} \lambda \sin \theta_0 \sin \psi_0 = \omega \cos \bar{\vartheta} + \sin \bar{\vartheta} \cos (T + \bar{\varPsi}), \end{equation}

and that differentiating (4.14a) with respect to $T$ and invoking (4.6a) yields the relationship

(4.14d)\begin{equation} \sin \theta_0 \cos \psi_0 =-\sin \bar{\vartheta} \sin (T + \bar{\varPsi}). \end{equation}

Given the relationships (4.14), we see that the trigonometric quantities $\cos \theta _0$, $\sin \theta _0 \sin \psi _0$ and $\sin \theta _0 \cos \psi _0$ are fast oscillations in time whose mean, amplitude and phase are slowly varying, coupled quantities.

The final result required from the leading-order system is the solution of (4.6c), and the subsequent appearance of the third (and final) slow-time function. Rewriting the right-hand side of (4.6c) as $\sin \theta _0 \sin \psi _0/\sin ^2 \theta _0$, we can use the expressions (4.14b)–(4.14c) to rewrite (4.6c) as

(4.15)\begin{equation} \frac{\partial{\phi_0}}{\partial{T}} = \dfrac{\omega^2 \cos \bar{\vartheta} + \omega\sin \bar{\vartheta} \cos (T + \bar{\varPsi})}{(\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta})^2 + \omega^2 \sin^2 (T + \bar{\varPsi})}. \end{equation}

Then, observing that

(4.16)\begin{equation} \dfrac{\partial}{\partial T}\left( \dfrac{\omega \sin (T + \bar{\varPsi})}{\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta}} \right) = \dfrac{\omega^2 \cos \bar{\vartheta} + \omega\sin \bar{\vartheta} \cos (T + \bar{\varPsi})}{(\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta})^2}, \end{equation}

integrating (4.15) directly with respect to the variable $\omega \sin (T + \bar {\varPsi }) / (\omega \cos \bar {\vartheta } \cos (T + \bar {\varPsi }) + \sin \bar {\vartheta })$ yields the solution

(4.17)\begin{equation} \tan(\phi_0 - \bar{\varphi}(t)) = \dfrac{\omega \sin (T + \bar{\varPsi})}{\omega \cos \bar{\vartheta} \cos (T + \bar{\varPsi}) + \sin \bar{\vartheta}}, \end{equation}

where $\bar {\varphi } = \bar {\varphi }(t)$ is the final slow-time function we obtain from our leading-order analysis. For later convenience, it is also helpful to expand out the left-hand side of (4.17) and rearrange to obtain the relationship

(4.18) \begin{equation} \tan \phi_0 = \dfrac{\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{equation}

We have now fully solved the nonlinear, coupled leading-order problem (4.6) to obtain the fast-time solutions (4.14) and (4.18), in terms of the three slow-time functions of integration $\bar {\vartheta }(t)$, $\bar {\varPsi }(t)$ and $\bar {\varphi }(t)$. Importantly for the method of multiple scales, the leading-order solutions given in (4.14) and (4.18) are $2{\rm \pi}$-periodic in the fast time $T$ (appropriately interpreting $\psi _0$ and $\phi _0$ modulo $2 {\rm \pi}$). (Despite starting with a nonlinear oscillator system, we are able to use multiple scales instead of the more general method of Kuzmak (Reference Kuzmak1959) because the period of the fast-time oscillation is independent of the slow time.) The slow-time functions $\bar {\vartheta }(t)$, $\bar {\varPsi }(t)$ and $\bar {\varphi }(t)$ are currently undetermined, and the goal of our next-order analysis in § 4.2 is to derive the governing equations for these functions. For generic initial conditions of the full system, one can also derive the appropriate initial conditions for these slow-time functions. We present this derivation in Appendix C, noting that some care is required in order to ensure that the correct branches are taken.

Over the slow time, one can think of $\bar {\vartheta }$ as controlling some emergent amplitude or mean of oscillation, $\bar {\varPsi }$ as controlling some emergent phase of oscillation and $\bar {\varphi }$ as an emergent drift in yawing. Additionally, in a sense to be made precise later, we can interpret each slow-time function as being associated with an underlying variable. Specifically, and in the same way as their equivalent variables in § 3, we can associate $\bar {\vartheta }$ with $\theta$, $\bar {\varPsi }$ with $\psi$ and $\bar {\varphi }$ with $\phi$.

4.2. Solvability conditions from the 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 $\bar {\varphi }(t)$. We note that the procedure to do this is non-standard for our problem, as it involves a non-trivial system of equations.

Specifically, when using the method of multiple scales for standard problems involving first-order ODEs, procedures to obtain governing equations for the slow-time functions often involve solving the next-order system of the general form

(4.19)\begin{equation} L \boldsymbol{x}_1 = \boldsymbol{F}(\boldsymbol{x}_0, \partial \boldsymbol{x}_{0}/\partial t), \end{equation}

where $\boldsymbol {x}_1$ is periodic in $T$ (say $2 {\rm \pi}$-periodic, without loss of generality), $L$ is a differential-algebraic linear operator (generally dependent on $\boldsymbol {x}_0(T,t)$) and $\boldsymbol {F}$ is a function of both $\boldsymbol {x}_0$ and its slow-time derivative. Then one proceeds by visually identifying secular terms in the solution and forcing their coefficients to vanish, generating the appropriate slow-time governing equations (Hinch Reference Hinch1991; Bender & Orszag Reference Bender and Orszag1999). However, this procedure can be challenging for more complicated systems where full solutions of the next-order system are prohibitively difficult to obtain, or where secular terms are difficult to identify.

In such systems it can often be simpler to obtain the requisite governing equations by deriving solvability conditions for the next-order system instead, through the imposition of periodicity on the fast scale. This can be simpler because it circumvents the need to solve an inhomogeneous problem. The appropriate way to derive these solvability conditions is via 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), which follows from the Fredholm alternative theorem applied to a periodic system. To briefly summarise here: a next-order linear system such as (4.19) yields appropriate slow-time evolution equations via its solvability condition

(4.20)\begin{equation} \int_0^{2{\rm \pi}} \boldsymbol{P} \boldsymbol{\cdot} \boldsymbol{F} \, \mathrm{d}T = 0, \end{equation}

where $\boldsymbol {P}$ is the fast-time periodic solution of the homogeneous adjoint problem

(4.21)\begin{equation} L^* \boldsymbol{P} = \boldsymbol{0}, \end{equation}

with fast-time periodicity conditions, where $L^*$ is the differential-algebraic linear adjoint operator. The solvability conditions (4.20) will generate the appropriate slow-time governing equations, reducing our task to explicitly solving the homogeneous 3-D system (4.21) instead of the inhomogeneous 3-D system (4.19). Generally, each linearly independent solution of the adjoint problem (4.21) will contribute one solvability condition.

For the specific problem we consider, the next-order system (4.19) constitutes a 3-D dynamical system for $\boldsymbol {x}_1(T,t) := (\theta _1, \psi _1, \phi _1)^{{\rm T}}$, and is generated from the $\mathit {O}(1)$ terms in (4.5) after posing the asymptotic expansions (3.4). Specifically, the linear operator $L$ and the right-hand side $\boldsymbol {F}$ are

(4.22)$$\begin{gather} L = \begin{pmatrix} \lambda \partial_{T} & \omega \sin \psi_0 & 0\\[5pt] -\omega \dfrac{\sin \psi_0}{\sin^2 \theta_0} & \lambda \partial_{T} + \omega \dfrac{\cos \theta_0 \cos \psi_0}{\sin \theta_0} & 0 \\[10pt] \omega\dfrac{\cos \theta_0 \sin \psi_0}{\sin^2 \theta_0} & -\omega\dfrac{\cos \psi_0}{\sin \theta_0} & \lambda \partial_{T} \end{pmatrix}, \end{gather}$$

(4.23)$$\begin{gather}\boldsymbol{F} =\begin{pmatrix} f_1(\theta_0,\phi_0) - \partial \theta_0 / \partial t \\ f_2(\theta_0,\phi_0) - \partial \psi_0 / \partial t \\ f_3(\phi_0) - \partial \phi_0 / \partial t \end{pmatrix}, \end{gather}$$

where $\partial _{T} \equiv \partial /\partial T$. The homogeneous adjoint operator $L^*$ is the transpose of the matrix operator taking the adjoint of each element, and is therefore defined as

(4.24)\begin{equation} L^* =\begin{pmatrix} -\lambda \partial_{T} & -\omega\dfrac{\sin \psi_0}{\sin^2 \theta_0} & \omega\dfrac{\cos \theta_0 \sin \psi_0}{\sin^2 \theta_0} \\[10pt] \omega \sin \psi_0 & -\lambda \partial_{T} + \omega \dfrac{\cos \theta_0 \cos \psi_0}{\sin \theta_0} & -\omega\dfrac{\cos \psi_0}{\sin \theta_0} \\[10pt] 0 & 0 & -\lambda \partial_{T} \\ \end{pmatrix}. \end{equation}

We emphasise that $L \neq L^*$, i.e. this problem is not self-adjoint.

To generate the slow-time equations we are seeking for the three slow-time functions $\bar {\vartheta }(t)$, $\bar {\varPsi }(t)$ and $\bar {\varphi }(t)$, we must derive the appropriate solvability conditions (4.20) by solving the homogeneous adjoint problem (4.21) with $L^*$ defined in (4.24). Given that the coefficients in (4.24) depend on the fast time $T$, this is a non-trivial task. Nevertheless, we are able to solve (4.21) in Appendix D, where we deduce that the general solution is

(4.25) \begin{align} \boldsymbol{P} = C_1 \begin{pmatrix} \omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 \\ \omega \sin \theta_0 \cos \psi_0 \\ 0 \end{pmatrix} + C_2 \begin{pmatrix} \omega \cos \psi_0 \\ \sin \theta_0 \left( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0\right) \\ 0 \end{pmatrix} + C_3 \begin{pmatrix} 0 \\ \cos \theta_0 \\ 1 \end{pmatrix} \end{align}

for arbitrary constants $C_1$, $C_2$ and $C_3$.

Therefore, we obtain our required solvability conditions by substituting the adjoint solutions (4.25) into the general solvability condition (4.20), with $\boldsymbol {F}$ defined in (4.23), and setting the resulting coefficients of $C_i$ ($i = 1, 2, 3$) to zero. This procedure yields the following three solvability conditions:

(4.26a)\begin{align} &\left\langle \theta_{0 t} \left(\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 \right) + \psi_{0 t} \omega \sin \theta_0 \cos \psi_0 \right\rangle \nonumber\\ &\quad= \left\langle \,f_1 \left(\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 \right) + f_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle, \end{align}

(4.26b)\begin{align} &\left\langle \theta_{0 t} \omega \cos \psi_0 + \psi_{0 t} \sin \theta_0 \left(\sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 \right) \right\rangle \nonumber\\ &\quad= \left\langle \,f_1 \omega \cos \psi_0 + f_2 \sin \theta_0 \left( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 \right) \right\rangle, \end{align}

(4.26c)\begin{align} &\left\langle \psi_{0t} \cos \theta_0 + \phi_{0t} \right\rangle = \left\langle \,f_2 \cos \theta_0 + f_3 \right\rangle.\end{align}

Here we use the subscript $t$ to denote partial differentiation with respect to $t$, and we use the notation $\left \langle \boldsymbol {\cdot } \right \rangle$ to denote the average of its argument over one fast-time oscillation, as defined in (3.13).

Our remaining task is to evaluate the averages in (4.26) in terms of the slow-term functions $\bar {\vartheta }$, $\bar {\varPsi }$ and $\bar {\varphi }$, and this will lead us to the emergent slow-time equations for the system. The left-hand sides of (4.26) are currently written in terms of the original variables $\theta _0(T,t)$, $\psi _0(T,t)$ and $\phi _0(T,t)$. To proceed, we need to write them in terms of the slow-time functions $\bar {\vartheta }(t)$, $\bar {\varPsi }(t)$ and $\bar {\varphi }(t)$ and then take appropriate averages. We carry out this task in Appendix E, which allows us to rewrite (4.26) as

(4.27a)$$\begin{gather} - \lambda \sin \bar{\vartheta} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} = \left\langle \,f_1 \left(\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 \right) + f_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle, \end{gather}$$

(4.27b)$$\begin{gather}\lambda \sin^2 \bar{\vartheta} \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = \left\langle \,f_1 \omega \cos \psi_0 + f_2 \sin \theta_0 \left( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 \right) \right\rangle, \end{gather}$$

(4.27c)$$\begin{gather}\cos \bar{\vartheta} \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} + \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = \left\langle \,f_2 \cos \theta_0 + f_3 \right\rangle. \end{gather}$$

Equations (4.27) represent the appropriate solvability conditions for any $f_i$ acting on a rapidly spinning object in Stokes flow. In the remainder of Part 1, we evaluate these solvability conditions for the $f_i$ defined in (2.2), which represent the hydrodynamic interaction of shear flow with a spheroidal object. In Reference Dalwadi, Moreau, Gaffney, Ishimoto and WalkerPart 2 we consider the interaction of shear flow with general chiral objects with helicoidal symmetry. This requires evaluating a different set of functions on the right-hand side of (4.27), resulting in fundamentally different emergent equations to Part 1.

4.3. Evaluating the right-hand side of the solvability conditions

Our final task is to evaluate the right-hand side of the three solvability conditions in (4.27), using the $f_i$ defined in (2.2). While (4.14) provide helpful expressions for the required trigonometric functions of $\theta _0$ and $\psi _0$, currently our only expression for $\phi _0$ is (4.18), given in terms of $\tan \phi _0$. Given that the $f_i$ defined in (2.2) involve $\sin 2 \phi _0$ and $\cos 2 \phi _0$, it is helpful to derive expressions for these quantities. To do this, we first calculate expressions involving $\sin \phi _0$ and $\cos \phi _0$ from $\tan \phi _0$ defined in (4.18), i.e.

(4.28a)$$\begin{gather} \lambda \sin \theta_0 \sin \phi_0 = \omega \cos \bar{\varphi} \sin \sigma + \sin \bar{\varphi} \left(\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta} \right), \end{gather}$$

(4.28b)$$\begin{gather}\lambda \sin \theta_0 \cos \phi_0 = \cos \bar{\varphi} \left(\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta} \right) - \omega \sin \bar{\varphi} \sin \sigma, \end{gather}$$

using the shorthand notation

(4.29)\begin{equation} \sigma = T + \bar{\varPsi}(t) \end{equation}

for algebraic convenience. The factors of $\lambda \sin \theta _0$ in (4.28) arise from imposing the constraint $\sin ^2 \phi _0 + \cos ^2 \phi _0 = 1$ and using (4.14b) to evaluate $\sin ^2 \theta _0$. While it is not essential, since the sign differences would cancel with one another in our subsequent analysis, for definiteness, we choose the positive square root, essentially exploiting the fact that $\sin \theta _0 > 0$ (since $\theta _0 \in [0, {\rm \pi}]$), and using our choice of $\bar {\varphi }(0)$ up to an additive multiple of ${\rm \pi}$ (see Appendix C). Then, from the expressions (4.28), we may use appropriate double-angle formulae to deduce that

(4.30a)$$\begin{gather} \lambda^2 \sin^2 \theta_0 \cos 2 \phi_0 = \mathcal{C}(\sigma,t) \cos 2\bar{\varphi} - \mathcal{S}(\sigma,t) \sin 2 \bar{\varphi}, \end{gather}$$

(4.30b)$$\begin{gather}\lambda^2 \sin^2 \theta_0 \sin 2 \phi_0 = \mathcal{S}(\sigma,t) \cos 2\bar{\varphi} + \mathcal{C}(\sigma,t) \sin 2 \bar{\varphi}, \end{gather}$$

where we have introduced the functions

(4.30c)$$\begin{gather} \mathcal{C}(\sigma,t) := (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta})^2 - \omega^2 \sin^2\sigma, \end{gather}$$

(4.30d)$$\begin{gather}\mathcal{S}(\sigma,t) := 2 \omega \sin\sigma (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta}). \end{gather}$$

To evaluate the fast-time averages on the right-hand side of (4.27), it is helpful to exploit the parity of various quantities. In particular, we note that $\mathcal {C}$ and $\mathcal {S}$ are even and odd, respectively, around $\sigma = {\rm \pi}$, which follows immediately from their definitions (4.30c)–(4.30d). Since $\sin ^2 \theta _0$ is also even, as follows from (4.14b), this means that $\cos 2 \phi _0$ and $\sin 2 \phi _0$ defined in (4.30a)–(4.30b) are naturally decomposed into their odd and even parts. Additionally, since the expressions in (4.14) tell us that $\cos \theta _0$ and $\sin \theta _0 \sin \psi _0$ are even, and that $\sin \theta _0 \cos \psi _0$ is odd (each around $\sigma = {\rm \pi}$), several terms on the right-hand side of (4.27) will vanish immediately under fast-time averaging. Exploiting this parity, we find that

(4.31a)\begin{align} &\left\langle \,f_1 \left(\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 \right) + f_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle \nonumber\\ & \quad = \dfrac{B}{2 \lambda^2}\sin 2 \bar{\varphi} \left\langle \cos \theta_0 \left[\mathcal{C} \left(1 - \dfrac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0}\right) - \dfrac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} \right] \right\rangle, \end{align}

(4.31b)\begin{align} &\left\langle \,f_1 \omega \cos \psi_0 + f_2 \sin \theta_0 \left( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 \right) \right\rangle \nonumber\\ & \quad = \dfrac{B}{2 \lambda^2}\cos 2 \bar{\varphi} \left\langle \cos \theta_0 \left[\mathcal{C} \left(1 - \dfrac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0}\right) - \dfrac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} \right] \right\rangle, \end{align}

(4.31c)\begin{align} &\left\langle \,f_2 \cos \theta_0 + f_3 \right\rangle = \dfrac{1}{2}\left(1 - \dfrac{B}{\lambda^2} \left\langle \mathcal{C} \right\rangle \cos 2 \bar{\varphi}_0 \right). \end{align}

To evaluate the required average in (4.31a) and (4.31b) (which is the same), we use the relationships (4.14), (4.30c)–(4.30d), and simplify the resulting expressions to deduce that

(4.32a) \begin{align} &\left\langle \cos \theta_0 \left[\mathcal{C} \left(1 - \dfrac{\omega \cos \theta_0 \sin \psi_0}{\sin \theta_0}\right) - \dfrac{\mathcal{S} \omega \cos \psi_0}{\sin \theta_0} \right] \right\rangle \nonumber\\ & \quad = \dfrac{\lambda \sin \bar{\vartheta}}{2} \langle 2 \omega \cos \sigma \cos 2 \bar{\vartheta} + (2 - \omega^2(1 + \cos 2\sigma ) ) \sin \bar{\vartheta} \cos \bar{\vartheta}\rangle \nonumber\\ & \quad = \dfrac{(2 - \omega^2 ) \lambda \sin^2 \bar{\vartheta} \cos \bar{\vartheta}}{2}. \end{align}

Similarly, the required average in (4.31c) can be evaluated using (4.14b) and (4.30c)–(4.30d), to obtain

(4.32b) \begin{equation} \langle \mathcal{C} \rangle = \langle (\omega \cos \bar{\vartheta} \cos \sigma + \sin \bar{\vartheta})^2 - \omega^2 \sin^2\sigma \rangle = \dfrac{2 - \omega^2}{2} \sin^2 \bar{\vartheta}. \end{equation}

Then using the results (4.32) in (4.31), we can write

(4.33a)$$\begin{gather} \left\langle \,f_1 \left(\omega \cos \theta_0 \sin \psi_0 - \sin \theta_0 \right) + f_2 \omega \sin \theta_0 \cos \psi_0 \right\rangle = \dfrac{2 - \omega^2}{4 \lambda} B \sin^2 \bar{\vartheta} \cos \bar{\vartheta} \sin 2 \bar{\varphi}, \end{gather}$$

(4.33b)$$\begin{gather}\left\langle \,f_1 \omega \cos \psi_0 + f_2 \sin \theta_0 \left( \sin \theta_0 - \omega \cos \theta_0 \sin \psi_0 \right) \right\rangle = \dfrac{2 - \omega^2}{4 \lambda} B \sin^2 \bar{\vartheta} \cos \bar{\vartheta} \cos 2 \bar{\varphi}, \end{gather}$$

(4.33c)$$\begin{gather}\left\langle \,f_2 \cos \theta_0 + f_3 \right\rangle = \dfrac{1}{2}\left(1 - \dfrac{2 - \omega^2}{2 \lambda^2}B \sin^2 \bar{\vartheta} \cos 2 \bar{\varphi} \right). \end{gather}$$

Equation (4.33) concludes our evaluation of the right-hand sides of the solvability conditions. The next step is to piece this all together to obtain the emergent angular dynamics.

4.4. The emergent angular dynamics

We now have the requisite information to obtain the governing equations for $\bar {\vartheta }$, $\bar {\varPsi }$ and $\bar {\varphi }$, which are the main results we have been seeking through this analysis. To this end, we substitute the right-hand side results (4.33) into the solvability conditions (4.27), then rearrange to obtain the following nonlinear system:

(4.34a)$$\begin{gather} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} =-\dfrac{B}{4} \dfrac{2 - \omega^2}{1 + \omega^2}\sin \bar{\vartheta} \cos \bar{\vartheta} \sin 2 \bar{\varphi}, \end{gather}$$

(4.34b)$$\begin{gather}\frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = \dfrac{B}{4} \dfrac{2 - \omega^2}{1 + \omega^2} \cos \bar{\vartheta} \cos 2 \bar{\varphi}, \end{gather}$$

(4.34c)$$\begin{gather}\frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = \dfrac{1}{2}\left(1 - \dfrac{2 - \omega^2}{2 (1 + \omega^2)}B \cos 2 \bar{\varphi} \right). \end{gather}$$

Here $\bar {\vartheta }$, $\bar {\varPsi }$ and $\bar {\varphi }$ feed into the full leading-order asymptotic solution through (4.14) and (4.18).

Remarkably, (4.34) can be rewritten in terms of the functions $f_i$, which are defined in (2.2), to yield

(4.35a)\begin{equation} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} = f_1(\bar{\vartheta},\bar{\varphi}; \hat{B}), \quad \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = f_2(\bar{\vartheta},\bar{\varphi}; \hat{B}), \quad \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = f_3(\bar{\varphi}; \hat{B}), \end{equation}

where we define $\hat {B}$ to be the effective Bretherton parameter, given explicitly by

(4.35b)\begin{equation} \hat{B} := \dfrac{2 - \omega^2}{2 (1 + \omega^2)}B. \end{equation}

Writing the system (4.34) in the form (4.35) leads to several key deductions. Firstly, by comparison with the full system (2.1)–(2.2), we see that we can formally identify each slow-time function with an underlying variable: $\bar {\vartheta }$ with $\theta$, $\bar {\varPsi }$ with $\psi$, and $\bar {\varphi }$ with $\phi$. Moreover, the compact relationship described by (4.35) retroactively motivates our notational choice to define the effective fast-time conserved quantity as $\lambda \cos \bar {\vartheta }(t)$ in (4.10) instead of $H(t)$ as in (4.9).

Secondly, the effective Bretherton parameter is given in (4.35b), and plotted with respect to $\omega$ in figure 4(a), from which we note that the sign of the effective Bretherton parameter is different to that of the actual Bretherton parameter if $|\omega | > \sqrt {2}$. Moreover, the effective Bretherton parameter vanishes (i.e. the effective particle behaves as a sphere) if $|\omega | = \sqrt {2}$. More generally, since (4.35) is exactly equivalent to Jeffery's equations, we have found that the rapid spinning of spheroidal swimmers generates Jeffery's orbits with an effective Bretherton parameter that we have calculated analytically. Therefore, the effect of rapid spinning is to modify the effective shape of the spheroidal swimmer, as quantified through the relationship (4.35b).

Figure 4. The effective parameters as a function of $\omega$: (a) $\hat {B}$ relative to $B$ from (4.35b), and (b) $\hat {V}$ relative to $|V| = \sqrt {V_1^2 + V_2^2}$ from (4.42). Of note, reversing the sign of $V_2$ would reflect each line in the $\omega = 0$ axis.

Finally, it is of interest to briefly consider the limiting cases of $\omega \to 0$ and $|\omega | \to \infty$. We have already considered the limit of $\omega \to 0$ (where the rotation axis coincides with the spheroidal symmetry axis) explicitly in § 3. It is reassuring to note that in this limit, our generalised effective Bretherton parameter $\hat {B} \to B$ from (4.35b), and hence, the system (4.35) reduces to

(4.36a–c)\begin{equation} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} = f_1(\bar{\vartheta},\bar{\varphi}; B), \quad \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = f_2(\bar{\vartheta},\bar{\varphi}; B), \quad \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = f_3\left(\bar{\varphi}; B \right), \end{equation}

in agreement. That is, this type of spinning leaves the system essentially unchanged, recapturing the explicit results of § 3.

In the limit of $|\omega | \to \infty$ (where the rotation axis is perpendicular to the spheroidal symmetry axis), our generalised effective Bretherton parameter $\hat {B} \to -B/2$ from (4.35b), and the system (4.35) reduces to

(4.37a–c)\begin{equation} \frac{\mathrm{d} \bar{\vartheta}}{\mathrm{d} t} = f_1\left(\bar{\vartheta},\bar{\varphi}; -\frac{B}{2}\right), \quad \frac{\mathrm{d} \bar{\varPsi}}{\mathrm{d} t} = f_2\left(\bar{\vartheta},\bar{\varphi}; -\frac{B}{2}\right), \quad \frac{\mathrm{d} \bar{\varphi}}{\mathrm{d} t} = f_3\left(\bar{\varphi}; -\frac{B}{2} \right). \end{equation}

That is, the sign of the effective Bretherton parameter changes (so an active oblate spheroid behaves as a passive prolate spheroid, and vice versa) and its magnitude halves (making it behave more like a sphere, in some sense).

We illustrate our asymptotic results in figure 5, showing the rotational dynamics of the swimmer orientation $(\theta,\phi )$ via the same representation on the unit sphere as in figure 2. With figure 5(a) showing the Jeffery's orbits of passive particles for reference, it is clear that rapid spinning with non-zero $\omega$ can significantly impact on the emergent dynamics, with $\omega$ and the Bretherton parameter $B$ varying between panels. Within each column, the initial configuration is fixed and shown as a black dot for reference. The average trajectory predicted by the asymptotic analysis is shown as a red curve in each plot, and the full numerical solution is shown as a blue curve. In each case, the full asymptotic solution obtained from combining the leading-order fast-time solutions (4.14) and (4.18) with the slow-time equations (4.35) successfully captures the leading-order behaviour of the full oscillatory dynamics, even when it is markedly complex, though we omit this comparison in figure 5 to allow the details of the trajectories to be seen cleanly.

Figure 5. Rotational dynamics of a spheroidal swimmer with general axis of rotation. Each plot displays the evolution of the particle orientation $(\theta,\phi )$, with vertical axis corresponding to $\theta =0$. In each column we fix $B$ and vary $\omega$, so each column corresponds to the same object shape. In rows (b–e) the full evolution of the particle orientation (2.1)–(2.2) is shown as a thin blue line, while the emergent dynamics (4.35) are shown as a thick red line. (a) Jeffery's orbits of passive particles, for reference. (b) The bacterial limit, with $\left \lvert \omega \right \rvert$ small. (c–e) Larger values of $\omega$ yield Jeffery's orbits with an effective Bretherton parameter for the emergent dynamics. (d) The critical value $\omega =\sqrt {2}$ leads to $\hat {B}=0$ for each particle. (e) Here $|\omega| > \sqrt{2}$, and so B and $\hat {B}$ have opposite signs. We use $\varOmega _{\parallel } = 15$ and initial conditions $\theta (0) = {\rm \pi}/18$, $\psi (0) = -{\rm \pi} /4$ throughout. We use initial condition $\phi (0) = {\rm \pi}/2$ for the first two columns, and $\phi (0) = 0$ for the final column. The initial condition for each trajectory is shown as a black dot. (a) $\omega = 0$, (b) $\omega = 0.1$, (c) $\omega = 0.5$, (d) $\omega = \sqrt{2}$ and (e) $\omega \rightarrow \infty$.

As predicted by (4.35), the emergent dynamics of the spinning objects are simply Jeffery's orbits in transformed variables with modified Bretherton parameters. That is, a rapidly spinning object behaves as though it is a differently shaped spin-free object. In particular, with $\left \lvert \hat {B} \right \rvert < \left \lvert B \right \rvert$, the emergent Jeffery's orbits are somewhat simpler, in that they more closely resemble the straightforward behaviour exhibited by a sphere. This is particularly evident as $\left \lvert \omega \right \rvert \rightarrow \sqrt {2}$, with the dynamics approaching those of a sphere for all particles irrespective of their shape, as shown in figure 5(d).

Our final task is to use our results for the emergent angular dynamics to calculate the emergent translational dynamics.

4.5. The emergent translational dynamics

In this final subsection for the general problem, we calculate the emergent translational dynamics, for which the full governing equations are given in (2.3). Our aim is to derive emergent (effective) governing equations for the translation, again using the method of multiple scales. Using the time-derivative transformation (4.4), the governing equations (2.3) become

(4.38)\begin{equation} \varOmega_{{\parallel}} \lambda \frac{\partial{\boldsymbol{X}}}{\partial{T}} + \frac{\partial{\boldsymbol{X}}}{\partial{t}} = V_1\hat{\boldsymbol{e}}_{1}(\theta, \phi) + V_2\hat{\boldsymbol{e}}_{2}(\theta,\psi, \phi) + V_3\hat{\boldsymbol{e}}_{3}(\theta,\psi, \phi) + Y \boldsymbol{e}_{3}. \end{equation}

We proceed as before, taking an asymptotic expansion of the dependent variables in inverse powers of $\varOmega _{\parallel }$, as in (3.4). At leading order (i.e. ${O}(\varOmega _{\parallel })$), we obtain

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

Therefore, in the same way as for the bacterial sublimit in § 3, $\boldsymbol {X}_0 = \boldsymbol {X}_0(t)$ and the vector position $\boldsymbol {X}$ is independent of the fast time $T$ at leading order.

At next order (i.e. ${O}(1)$), substituting the leading-order fast-time angular dynamics solutions (4.14) and (4.18) in (4.38), we obtain the system

(4.40)\begin{equation} \lambda \frac{\partial{\boldsymbol{X}_1}}{\partial{T}} + \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} = V_1\hat{\boldsymbol{e}}_{1}(\theta_0, \phi_0) + V_2\hat{\boldsymbol{e}}_{2}(\theta_0,\psi_0, \phi_0) + V_3\hat{\boldsymbol{e}}_{3}(\theta_0,\psi_0, \phi_0) + Y_0 \boldsymbol{e}_{3}, \end{equation}

with a fast-time periodicity constraint on $\boldsymbol {X}_1$. The solvability conditions are straightforward to obtain by integrating (4.40) with respect to the fast time $T$ from $0$ to $2 {\rm \pi}$. As before, the fast-time derivative vanishes due to periodicity. In Appendix F we calculate the fast-time averages of the unit vectors as

(4.41a–c) \begin{gather} \langle \lambda \hat{\boldsymbol{e}}_{1}(\theta_0, \phi_0)\rangle = \tilde{\boldsymbol{e}}_{1}(\bar{\vartheta},\bar{\varphi}), \quad \langle \lambda \hat{\boldsymbol{e}}_{2}(\theta_0,\psi_0, \phi_0) \rangle = \omega \tilde{\boldsymbol{e}}_{1}(\bar{\vartheta},\bar{\varphi}), \quad \langle \hat{\boldsymbol{e}}_{3}(\theta_0,\psi_0, \phi_0) \rangle = \boldsymbol{0}, \end{gather}

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

Hence, substituting (4.41a–c) into the fast-time averaged version of (4.40), the emergent governing equation for translation is

(4.42)\begin{equation} \frac{\mathrm{d} \boldsymbol{X}_0}{\mathrm{d} t} = \hat{V} \tilde{\boldsymbol{e}}_{1}(\bar{\vartheta},\bar{\varphi}) + Y_0 \boldsymbol{e}_{3}, \quad \text{with } \hat{V} = \dfrac{V_1 + \omega V_2}{\lambda} = \dfrac{V_1 + \omega V_2}{\sqrt{1 + \omega^2}}. \end{equation}

Of note, and generalising the case explored in § 3, we now retain a contribution from an off-axis direction of swimming, namely $V_2$. That is, this off-axis swimming component does not average to zero in the general case. The evolution of the effective swimming velocity $\hat {V}$ (normalised by $|V| = \sqrt {V_1^2+V_2^2}$) with respect to $\omega$ is shown in figure 4(b) for different values of off-axis velocity. Perhaps surprisingly, the coefficient of $V_2$ in (4.42) is odd in $\omega$, suggesting that the behaviour of a swimmer with $\omega <0$ can be markedly different to that of one with $\omega >0$, recalling that the sign of $\omega$ is determined solely by the relative directions of spinning via (4.1). Thus, supposing that $V_1=\left \lvert \omega \right \rvert V _2$, a swimmer with both $\varOmega _{\perp }>0$ and $\varOmega _{\parallel }>0$ propels itself at an effective speed of $2\left \lvert V_1 \right \rvert /\lambda$, while swapping either one of the directions of spinning gives an effective swimming speed of zero. The role of $V_2$ is further highlighted in the limit $\left \lvert \omega \right \rvert \to \infty$, in which the contribution of $V_1$ vanishes. Hence, in this limit, which might be considered the ‘opposite’ to the bacterial case, the translational dynamics are only modulated by $V_2$ and the shape-independent advection by the shear.

The conditional significance of $V_2$ is illustrated in figures 6, 7 and supplementary movie 2, which explore the emergent translational dynamics more generally. Figures 6(a) and 6(b) highlight how $V_3$ has no effect on the leading-order translation, with the average trajectories in each panel being qualitatively indistinct, while we see an overall trend that increasing $\omega$ decreases the effective swimming speed when $V_2=0$, as predicted by our analytic result (4.42). In figures 6(c) and 6(d), however, we see that increasing $V_2$ from zero does modify the swimming trajectory, with these modifications increasing in significance as $\left \lvert \omega \right \rvert$ increases, in line with the predictions of (4.42). We emphasise that $\hat {V} \to V_2$ as $\left \lvert \omega \right \rvert \to \infty$, in line with physical intuition, despite the apparent importance of the sign of $\omega$. This is because $\tilde {\boldsymbol {e}}_{1}(\bar {\vartheta },\bar {\varphi })$ depends on $\omega$ and, in fact, tends to exact opposite directions in the limits $\omega \to \pm \infty$ (when the initial conditions derived in Appendix C are taken into account). Figure 7 further emphasises the significant differences that can be observed on the translational dynamics upon switching the sign of $\omega$.

Figure 6. Translational dynamics of a spheroidal swimmer with general axis of rotation. A dynamic version of this figure is given in supplementary movie 2. The solutions of the full dynamical system (2.1)–(2.3) are plotted as ribbons with a black centreline, while the emergent dynamics (4.35), (4.42) are shown as thick red lines that accurately capture the behaviour of the full system. The orthogonal velocities vary for each panel: (a) $(V_2,V_3) = (0,0)$, for which we see that increasing $\omega$ reduces the effective swimming speed; (b) $(V_2,V_3) = (0,0.5)$, for which we see that the average dynamics is unaltered from (a), despite significant modifications in the fast-time variation; (c) $(V_2,V_3) = (0.5,0)$; (d) $(V_2,V_3) = (1.5,0)$. In (c,d) we see that the dynamics is now significantly modified from (a,b) (an effect that increases with $\omega$), but is still captured by the asymptotic solution at leading order. In each panel we use $B = 0.5$, $\varOmega _{\parallel } = 10$, $V_1 = 1$, initial position $\boldsymbol {X} = \boldsymbol {0}$ and initial orientation $(\theta,\psi,\phi ) = (2{\rm \pi} /5,{\rm \pi} /2,-7{\rm \pi} /15)$, simulating over the same time interval.

Figure 7. Translational dynamics of a spheroidal swimmer with general axis of rotation, comparing to negative values of $\omega$. In each case, the solutions of the full dynamical system (2.1)–(2.3) are plotted as ribbons with a black centreline, while the emergent dynamics (4.35), (4.42) are shown as thick coloured lines that accurately capture the behaviour of the full system. The colour of the thick lines indicates the value of the orthogonal velocity $V_2$, ranging in $\{ 0, 0.25, 1/\sqrt {2}, 1, 2\}$, with the associated colours ranging from red to yellow. Each panel corresponds to a different value of $\omega$, as indicated at the top, showing a large range of behaviours for the translational dynamics depending on $\omega$ and $V_2$. In particular, the top row features negative values of $\omega$, for which the effective velocity $\hat {V}$ vanishes when $V_2 = -V_1/\omega$; this makes the associated trajectories near invisible on the figure, because they essentially remain at the origin. In each panel we use $B = 0.5$, $\varOmega _{\parallel } = 10$, $V_1 = 1$, $V_3 = 0$, initial position $\boldsymbol {X} = \boldsymbol {0}$ and initial orientation $(\theta,\psi,\phi ) = (2{\rm \pi} /5,{\rm \pi} /2,-7{\rm \pi} /15)$, simulating over the same time interval.