2.1. Flow kinematics and general governing equations

With the time scales defined, we can proceed to consider the background flow in detail and derive the governing equations for the rotational and translation dynamics of the swimmer. We have $ {\boldsymbol{x}} = x {\boldsymbol{e}}_1+y{\boldsymbol{e}}_2+z{\boldsymbol{e}}_3$ as the non-dimensional position of a general point with respect to the laboratory-fixed basis, $\{{\boldsymbol{e}}_1, {\boldsymbol{e}}_2, {\boldsymbol{e}}_3\}$ ; analogously, the swimmer-fixed frame has a basis given by $\{\hat {\boldsymbol{e}}_{1}, \hat {\boldsymbol{e}}_{2}, \hat {\boldsymbol{e}}_{3}\}$ with its origin ${\boldsymbol{x}}_c$ at the centroid of the swimmer. As further detailed below, the swimmer is taken to possess a body-fixed symmetry axis throughout its deformation cycle, which we take to be aligned in the body-fixed direction $\hat {\boldsymbol{e}}_{1}$ . Noting the assumptions of flow planarity and sufficient swimmer symmetry to ensure planar motility (detailed below) we can, with suitable initial conditions implicitly assumed, take the swimmer axis of symmetry to lie in the plane of the flow which, without loss, is the $xy$ -plane of the inertial frame. Note that this entails that $\hat {\boldsymbol{e}}_{1}$ also lies in this plane. Hence, we have the simple relations between the basis vectors

(2.4) \begin{equation} \hat {\boldsymbol{e}}_{1}=\cos \theta {\boldsymbol{e}}_1 +\sin \theta {\boldsymbol{e}}_2, \quad \hat {\boldsymbol{e}}_{2}=-\sin \theta {\boldsymbol{e}}_1 +\cos \theta {\boldsymbol{e}}_2,\quad \hat {\boldsymbol{e}}_{3}={\boldsymbol{e}}_3, \end{equation}

with $\theta$ as depicted in figure 1. In practice, this alignment also requires stability of planar swimming, which we do not explore in this work.

Neglecting the influence of the swimmer on the flow (so that no-slip conditions on the swimmer surface are not imposed), we denote the linear, planar, non-dimensional background flow field as $\boldsymbol{u}^*({\boldsymbol{x}},T)$ with rate of strain and angular velocity given by (Bachelor Reference Bachelor1967)

(2.5) \begin{equation} \unicode{x1D640}^{\,*} (T) = \frac {1}{2} \left [\boldsymbol{\nabla } \boldsymbol{u}^* + (\boldsymbol{\nabla } \boldsymbol{u}^*)^\top \right ] , \quad \boldsymbol{\varOmega }^* (T) = \frac {1}{2} \boldsymbol{\nabla } \wedge \boldsymbol{u}^*, \end{equation}

respectively, where $\boldsymbol{\cdot }^\top$ denotes the transpose. As above, the prospect of rapid background flow oscillation is indicated by the fast time variable dependence of the rate of strain and angular velocity while, by the linearity of the flow, we have $\boldsymbol{\varOmega }^*$ and $\unicode{x1D640}^{\,*}$ are independent of spatial location. As the flow is planar we have the further simplifications

(2.6) \begin{equation} \boldsymbol{\varOmega }^* (T) = \varOmega ^*(T)\hat {\boldsymbol{e}}_{3} = \varOmega ^*(T) {\boldsymbol{e}}_3, \quad \varOmega ^* (T) = \frac {1}{2} {\boldsymbol{e}}_3\boldsymbol{\cdot }\boldsymbol{\nabla } \wedge \boldsymbol{u}^* . \end{equation}

Further, we have that the rate of strain tensor, $ \unicode{x1D640}^{\,*}$ , can be written with respect to the laboratory basis as

(2.7) \begin{equation} \unicode{x1D640}^{\,*} = \begin{bmatrix} E^*_{11}(T) & E^*_{12}(T) & 0\\[4pt] E^*_{12}(T) & -E^*_{11}(T) & 0\\[4pt] 0 & 0 & 0 \end{bmatrix}, \end{equation}

noting that symmetry and flow incompressibility entail that only two degrees of freedom remain. In the same laboratory reference frame, the background flow takes the explicit form

(2.8) \begin{align} \boldsymbol{u}^*({\boldsymbol{x}},T) = &\, \boldsymbol{u}^*_{\textit{tr}}(T)+\underbrace {(-y{\boldsymbol{e}}_1 + x{\boldsymbol{e}}_2)\varOmega ^*(T)}_{\text{pure rotation}} \nonumber \\ &+ \underbrace {(x{\boldsymbol{e}}_1 -y{\boldsymbol{e}}_2)E^*_{11}(T) + (y{\boldsymbol{e}}_1 + x{\boldsymbol{e}}_2)E^*_{12}(T)}_{\text{pure strain}} \nonumber \\ = &\, \boldsymbol{u}^*_{\textit{tr}}(T)+ \boldsymbol{\varOmega }^*(T)\wedge {\boldsymbol{x}} + \unicode{x1D640}^{\,*}(T){\boldsymbol{x}} , \end{align}

where $\boldsymbol{u}^*_{\textit{tr}}(T)$ is a translational flow that has no spatial dependence. It is useful to note that velocity field satisfies the identity

(2.9) \begin{equation} \boldsymbol{u}^*({\boldsymbol{x}},T) = \boldsymbol{u}^*_c + \boldsymbol{\varOmega }^*(T)\wedge ({\boldsymbol{x}}-{\boldsymbol{x}}_c) + \unicode{x1D640}^{\,*}(T)({\boldsymbol{x}}-{\boldsymbol{x}}_c) \end{equation}

by linearity, where $\boldsymbol{u}^*_c = \boldsymbol{u}^*({\boldsymbol{x}}_c,T)$ is the background flow at the centroid ${\boldsymbol{x}}_c$ of the swimmer.

With the background flow specified, we show in Appendix A that the planar motion of the swimmer is governed by

(2.10) \begin{align} \frac {{\textrm{d}}{\boldsymbol{x}}_c}{{\textrm{d}}t} &= \boldsymbol{u}^*_c + \left [V + \omega U(T)\right ] \hat {\boldsymbol{e}}_{1} -\,\tilde {\!\unicode{x1D65C}}\unicode{x1D640}^{\,*} , \end{align}

(2.11) \begin{align} \dot \theta {\boldsymbol{e}}_3 =\dot \theta \hat {\boldsymbol{e}}_{3} &= \left [\varOmega ^*(T)+\varOmega (T)+ \omega \varOmega _f(T)\right ]\hat {\boldsymbol{e}}_{3}- \tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*}. \end{align}

Here, $\omega U(T)$ , $V$ and $\omega \varOmega _f(T) + \varOmega (T)$ are the oscillatory swimming speed, average progressive swimming speed and rate of rotation of the swimmer in the absence of any background flow, each of which are assumed to be given. Note that we have decomposed the latter into a fast, zero-mean ${\textit{O}} (\omega )$ component $\varOmega _f(T)$ and an ${\textit{O}} (1 )$ component $\varOmega (T)$ for later convenience. As described in detail in Appendix A, we assume $V \sim {\textit{O}} (1 )$ and $\omega U(T)\sim {\textit{O}} (\omega )$ , with the average of $U(T)$ evaluating to zero over a period of the fast oscillation, $T= 2\pi$ . This is commensurate with the notion of swimmer inefficiency: swimmer variations that occur on the fast scale do not generate net motion on the fast scale. Later, during our asymptotic analysis of the translational equation of (2.10), we will see that this setting necessitates an additional constraint on $U(T)$ and $\varOmega _f(T)$ , in that a particular nonlinear combination of these functions must have an average that is ${\textit{O}} (1/\omega )$ (see the discussion that follows (3.22) for more details). Notably, relaxing these assumptions of inefficiency will lead to swimmers that, at leading order, do not meaningfully interact with the flow.

Note that setting $\boldsymbol{u}^*_c$ , $\unicode{x1D640}^{\,*}$ and $\varOmega ^*$ to zero gives the equations of motion in the absence of flow. The terms $\,\tilde {\!\unicode{x1D65C}}$ and $\tilde {\unicode{x1D65D}}$ are rank three tensors that capture how the rate of strain of the background influences the swimmer dynamics. The assumption of planarity requires $\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*} \parallel \hat {\boldsymbol{e}}_{3}$ , which we impose by constraining the swimmer shape throughout its gait cycle to have sufficient symmetry. However, we remark that this framework retains validity even when $\,\tilde {\!\unicode{x1D65C}}\unicode{x1D640}^{\,*}$ has a component in the $\hat {\boldsymbol{e}}_{3}={\boldsymbol{e}}_{3}$ direction, so we do not disallow this prospect a priori. Here and throughout, all angular velocities are constrained to the $\hat {\boldsymbol{e}}_{3}={\boldsymbol{e}}_{3}$ direction by the assumption of planarity.

Below, we simplify these equations of motion by enforcing geometrical symmetries of the swimmer. The generality of the resulting derivations and equations naturally requires a relatively large number of parameters and variables. Hence, we summarise these in tables 1 and 2 for reference, together with the parameters and variables already introduced.

Table 1. A list of parameters and variables used in the formulation of the governing equations, including the description of the background flow and the swimmer symmetries. All are non-dimensional except for the first row of scales used to non-dimensionalise the system. Note that the variable $\boldsymbol{u}^*_{\textit{tr}}$ is overloaded and relative to either the 2-D flow plane or three-dimensional (3-D) more generally according to context, with the 3-D expression including the constant $z$ -contribution to the background flow. Parameters and variables introduced in the appendices that do not appear in the main text are not listed.

Table 2. A list of parameters and variables describing the multiscale simplifications and aspects of the explicit solutions to the governing equations for special cases. They are all non-dimensional. An overline of any variable refers to taking a temporal average over a period of the fast time scale, as defined by (3.7). Note that the variables $\overline {{\boldsymbol{x}}}_0$ and $\boldsymbol{\varLambda }$ are overloaded and relative to either the 2-D flow plane, or 3-D more generally, according to context, with 3-D expressions, respectively, including the $z$ -contribution to the leading-order swimmer centroid position and a trivial zero-padding in the third dimension when $\boldsymbol{\varLambda }$ acts on a 3-D vector. Parameters and variables introduced in the appendices that do not appear in the main text are not listed in this table.

2.2. Simplified governing equations

In order to be consistent with our assumption of planarity, we must restrict ourselves to particular classes of swimmer geometry. In full generality, this is necessarily technical and requires significant notation. A reader seeking a concrete example might consider the swimmer to be a body of revolution with fore–aft symmetry, though we remark that much more general geometries are admissible within the present framework. Below, we elaborate on the details of some additional cases, though these can be safely skipped if one is willing to accept the presence of the time-dependent geometrical parameters $B(T)$ , $\lambda _5(T)$ , $\eta _2(T)$ , $\eta _3(T)$ and $\eta _4(T)$ in the explicit and simplified governing equations,

(2.12) \begin{align} \dfrac {{\textrm{d}}{{\boldsymbol{x}}_c}}{{\textrm{d}}{t}} &= \boldsymbol{u}^*_c + \left [V + \omega U(T) \right ]\hat {\boldsymbol{e}}_{1} - \eta _2 (T)[\hat {{\boldsymbol{e}}}^\top _1 \unicode{x1D640}^{\,*} (T)\hat {{\boldsymbol{e}}}_2 ] \hat {{\boldsymbol{e}}}_3 \nonumber \\ & +\, \eta _3(T)\boldsymbol{B}_0(T,\sin 2\theta ,\cos 2\theta )\hat {{\boldsymbol{e}}}_1 -\eta _4 (T)[\hat {{\boldsymbol{e}}}^\top _2 \unicode{x1D640}^{\,*} (T)\hat {{\boldsymbol{e}}}_2 ] \hat {{\boldsymbol{e}}}_2, \end{align}

(2.13) \begin{align} \dfrac {{\textrm{d}}{\theta }}{{\textrm{d}}{t}} & = \varOmega ^*(T)+ \omega \varOmega _f(T)+\varOmega (T) +[\lambda _5(T)E^*_{12}(T)-B(T)E^*_{11}(T)]\sin 2\theta \nonumber \\ & +\, [\lambda _5(T)E^*_{11}(T)+B(T)E^*_{12}(T)]\cos 2\theta . \end{align}

Here, $B(T)$ is the Bretherton parameter and, using $c\equiv \cos 2\theta$ and $s\equiv \sin 2\theta$ , we have that $\boldsymbol{B}_0(T,\sin 2\theta ,\cos 2\theta )$ is given by

(2.14) \begin{align} \boldsymbol{B}_0(T,\sin 2\theta ,\cos 2\theta ) = \unicode{x1D640}^{\,*}- [E^*_{11}(T)\cos 2\theta +E^*_{12}(T)\sin 2\theta ]({\boldsymbol{e}}_1{\boldsymbol{e}}_1^\top +{\boldsymbol{e}}_2{\boldsymbol{e}}_2^\top ) \nonumber \\ = \begin{bmatrix} E^*_{11}(T)(1-c) - E^*_{12}(T)s & E^*_{12}(T) & 0 \\ E^*_{12}(T) & -[E^*_{11}(T)(1+c) + E^*_{12}(T)s] & 0 \\ 0 & 0 & 0 \end{bmatrix}, \end{align}

where the final expression is with respect to the laboratory basis. In the remainder of this section, we explicitly describe the construction of these governing equations from (2.10); we continue with an analysis of these equations in § 3.

2.3. Simplification of the governing equations and detailed geometrical constraints

In order to ensure that the swimmer only rotates in the plane of the flow, we immediately restrict consideration to swimmers whose shape throughout the gait cycle possesses a rotational symmetry of degree $n\geqslant 3$ . That is, the swimmers possess a body fixed axis throughout the gait cycle such that there is a shape invariance to rotations around this axis of angle $2\pi /n$ (Ishimoto Reference Ishimoto2020b ). In Shoenflies notation, such a body with $n$ -fold rotational symmetry is denoted by $C_n$ . In turn, the body symmetry enforces the constraints on the entries of the third-rank tensors, $\,\tilde {\!\unicode{x1D65C}}$ and $\tilde {\unicode{x1D65D}}$ , yielding another type of shape classification based on the symmetry of these tensors. This symmetry is a hydrodynamic symmetry and we refer the interested reader to the detailed definitions of Ishimoto (Reference Ishimoto2020a , Reference Ishimotob , Reference Ishimoto2023) for further elaboration on this rich topic. With this $C_n$ ( $n\geqslant 3$ ) body symmetry, Ishimoto (Reference Ishimoto2020b ) considers the structure of $-\,\tilde {\!\unicode{x1D65C}} \unicode{x1D640}^{\,*}$ and $-\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*}$ and shows that a $C_n$ ( $n\geqslant 3$ ) body has helicoidal symmetry of degree three, for which the body dynamics are explicitly written down in the same form. By taking $n\rightarrow \infty$ , it is known that a simple helix approximately follows the same dynamical equations (Ishimoto Reference Ishimoto2020a ). However, the helicoidal symmetry alone is not sufficient for our purposes. Thus, we closely follow Ishimoto (Reference Ishimoto2020b ) to determine $-\,\tilde {\!\unicode{x1D65C}} \unicode{x1D640}^{\,*}$ and $-\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*}$ , noting that additional simplifications will arise here from both the planar nature of the flow and the restriction of swimmer shapes to those for which $\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*} \parallel \hat {\boldsymbol{e}}_{3}$ . Before doing so, it is convenient to introduce the notation

(2.15) \begin{equation} \hat {E}^*_{ij} (T,\sin 2\theta ,\cos 2\theta ):= \hat {{\boldsymbol{e}}}^\top _i \unicode{x1D640}^{\,*} (T)\hat {{\boldsymbol{e}}}_j = \hat {E}^*_{ji}(T,\sin 2\theta ,\cos 2\theta ) \end{equation}

for $i,j\in \{1,2,3\}$ . Explicitly, for $j\in \{1,2,3\}$ we have

(2.16a) \begin{align} \hat {E}^*_{11}&= - \hat {E}^*_{22} = E^*_{11}(T) \cos 2\theta +E^*_{12}(T)\sin 2\theta , \end{align}

(2.16b) \begin{align} \hat {E}^*_{12} &= E^*_{12} (T)\cos 2\theta -E^*_{11}(T)\sin 2\theta , \end{align}

(2.16c) \begin{align} \hat {E}^*_{3j}&=0, \end{align}

each functions of $T$ , $\sin 2\theta$ and $\cos 2\theta$ . This corresponds to the rate of strain tensor expressed in the $\{\hat {\boldsymbol{e}}_{1},\hat {\boldsymbol{e}}_{2},\hat {\boldsymbol{e}}_{3}\}$ basis. Later, we will make us of these definitions of $\hat {E}^*_{ij}$ as functions of $\sin 2\theta$ and $\cos 2\theta$ , wherein it will be appropriate to substitute $\theta$ for another angular variable.

With our assumptions and simplifications, we can now decompose $-\,\tilde {\!\unicode{x1D65C}} \unicode{x1D640}^{\,*}$ and $-\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*}$ via

(2.17a) \begin{align} -\,\tilde {\!\unicode{x1D65C}} \unicode{x1D640}^{\,*} &= \eta _2(T)\boldsymbol{d}_2(T,\theta )+\eta _3(T)\boldsymbol{d}_3(T,\theta )+\eta _4(T)\boldsymbol{d}_4(T,\theta ), \end{align}

(2.17b) \begin{align} -\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*} &= \lambda _2(T)\boldsymbol{d}_2(T,\theta )+\lambda _5(T)\boldsymbol{d}_5(T,\theta ) \end{align}

with

(2.18a) \begin{align} \boldsymbol{d}_2(T,\theta) &= -\hat {E}_{12}^*\hat {\boldsymbol{e}}_{3}, \quad \boldsymbol{d}_3(T,\theta ) = (\unicode{x1D640}^{\,*}-\hat {E}^*_{11} \unicode{x1D644})\hat {\boldsymbol{e}}_{1} , \end{align}

(2.18b) \begin{align} \boldsymbol{d}_4(T,\theta) &= -\hat {E}_{22}^*\hat {\boldsymbol{e}}_{2}, \quad \boldsymbol{d}_5(T,\theta ) = -\hat {E}_{22}^*\hat {\boldsymbol{e}}_{3}, \end{align}

where $\lambda _2(T)\equiv -B(T)$ , $\lambda _5(T)$ , $\eta _2(T)$ , $\eta _3(T)$ and $\eta _4(T)$ are shape-dependent parameters.

Notably, symmetries have to be imposed on the swimmer in order to ensure that there are no contributions to $ -\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*}$ from $\boldsymbol{d}_3(T,\theta )$ and $\boldsymbol{d}_4(T,\theta )$ , so that $\tilde {\unicode{x1D65D}} \unicode{x1D640}^{\,*} \parallel \hat {\boldsymbol{e}}_{3}$ . For the swimmer shapes we consider throughout, this also ensures that $ -\,\tilde {\!\unicode{x1D65C}} \unicode{x1D640}^{\,*}$ has no contribution from $\boldsymbol{d}_5(T,\theta )$ (Ishimoto Reference Ishimoto2020b ). Thus, we do not consider the full range of shapes for which (2.17) is valid, but instead require additional restrictions. Particular examples of swimmer shapes with sufficient symmetry to be admissible within the present framework, together with the related restrictions on (2.17), are presented in detail in Appendix B for the interested reader. Finally, we note that for the simple canonical example of a body of revolution with fore–aft symmetry, there is extensive simplification, with $\lambda _5=\eta _2=\eta _3=\eta _4=0.$ Such a swimmer does not have the asymmetry needed to generate rotation in the absence of a flow, so that its angular velocity in the absence of flow, $\omega \varOmega _f(T)+\varOmega (T),$ is also zero.

3.1. Multiple scales for the angular dynamics

Defining

(3.1a) \begin{align} a_{\dagger }(T)&=\varOmega ^*(T)+\varOmega (T), \end{align}

(3.1b) \begin{align} b_{\dagger }(T)&=B(T)E^*_{11}(T)-\lambda _5(T)E^*_{12}(T), \end{align}

(3.1c) \begin{align} c_{\dagger }(T)&=-B(T)E^*_{12}(T)-\lambda _5(T)E^*_{11}(T), \\[9pt] \nonumber \end{align}

for notational convenience, the angular evolution equation becomes

(3.2) \begin{equation} \dfrac {{\textrm{d}}{\theta }}{{\textrm{d}}{t}} = \omega \varOmega _f(T) + a_{\dagger }(T) - b_{\dagger }(T)\sin {2\theta } -c_{\dagger }(T)\cos {2\theta } , \end{equation}

which is decoupled from the equations for translational motion and thus may be treated in isolation.

To study this angular dynamics, we use the method of multiple time scales, exploiting $\omega \gg 1$ . The slow time scale, $t$ , is associated with the flow, and the fast time scale, $T=\omega t$ , is associated with the swimmer deformation and treadmilling. Hence, the total time derivative decomposes via

(3.3) \begin{equation} \dfrac {{\textrm{d}}{}}{{\textrm{d}}{t}} = \dfrac {\partial {}}{\partial {t}} + \omega \dfrac {\partial {}}{\partial {T}}. \end{equation}

With a zero subscript denoting the leading order, we expand $\theta$ via

(3.4) \begin{equation} \theta = \theta _0(t,T) + \frac {1}{\omega } \theta _1(t,T) + \ldots , \end{equation}

with $\theta _1(t,T)$ inheriting the $2\pi$ -periodicity of the fast time dynamics, as is standard in the multiple time scales method. Thus, at ${\textit{O}} (\omega )$ and ${\textit{O}} (1 )$ we have, respectively,

(3.5a) \begin{align} \theta _{0T} &= \varOmega _f(T), \end{align}

(3.5b) \begin{align} \theta _{1T} &= -\theta _{0t}+ a_{\dagger }(T) - b_{\dagger }(T)\sin {2\theta _0} -c_{\dagger }(T)\cos {2\theta _0}. \end{align}

This gives

(3.6) \begin{equation} \theta _0 (t,T)= \varPsi (T) + \tilde {\theta }_0(t), \quad \varPsi (T) = \int _0^T \varOmega _f(S) \mathop {}\!{\textrm{d}}{S}, \end{equation}

where $\tilde {\theta }_{0}(t)$ is an undetermined function of $t$ alone. We denote fast time scale averages of functions that are $2\pi$ -periodic in the fast variable $T$ using a bar, that is

(3.7) \begin{equation} \overline {Q} := \frac 1 {2\pi } \int _{T_0}^{T_0+2\pi } Q(t,T) \mathop {}\!{\textrm{d}}{T} = \frac 1 {2\pi } \int _{0}^{2\pi } Q(t,T) \mathop {}\!{\textrm{d}}{T}, \end{equation}

where the $T_0$ dependence drops due to the periodicity in the fast variable. Noting that $\varPsi (T)$ is $2\pi$ -periodic, as $\varOmega _f(T)$ is $2\pi$ -periodic with zero mean, averaging (3.6) gives

(3.8) \begin{equation} \overline {\theta }_0 (t)= \overline {\varPsi } + \tilde {\theta }_{0}(t), \end{equation}

with $\bar {\varPsi }$ a constant, so that ${\textrm{d}}\tilde {\theta }_0/{\textrm{d}}t = {\textrm{d}}\overline {\theta }_0/{\textrm{d}}t$ .

Given (3.6), we expand

(3.9) \begin{align} b_{\dagger }(T)\sin {2\theta _0} +c_{\dagger }(T)\cos {2\theta _0} &= b_{\dagger }(T)\sin (2\tilde {\theta }_0 + 2\varPsi ) +c_{\dagger }(T)\cos (2\tilde {\theta }_0 + 2\varPsi ) \nonumber \\ &= b_{\ddagger }(T)\sin (2\tilde {\theta }_0) +c_{\ddagger }(T)\cos (2\tilde {\theta }_0), \end{align}

where we define

(3.10a) \begin{align} b_{\ddagger }(T) &= B(T)\hat {E}^*_{11}(T,\sin (2\varPsi ),\cos (2\varPsi ))-\lambda _5(T)\hat {E}^*_{12}(T,\sin (2\varPsi ),\cos (2\varPsi )) \end{align}

(3.10b) \begin{align} c_{\ddagger }(T) &= -B(T)\hat {E}^*_{12}(T,\sin (2\varPsi ),\cos (2\varPsi ))-\lambda _5(T)\hat {E}^*_{11}(T,\sin (2\varPsi ),\cos (2\varPsi )) \end{align}

using the definitions of (2.16). Hence, the impact of the fast angular dynamics is that the contributions of the rate of strain tensor are taken only after a rotating the basis by an angle $\varPsi (T)$ .

As the only periodic homogeneous solution of (3.5b ) for $\theta _1$ is the constant solution, (3.9) and the Fredholm alternative theorem give

(3.11) \begin{equation} \int _0^{2\pi } \left \{-\theta _{0t}+\chi (T) \right \} \mathop {}\!{\textrm{d}}{T}=0, \end{equation}

defining

(3.12) \begin{equation} \chi (T):= a_{\dagger }(T) - b_{\ddagger }(T)\sin {2\tilde {\theta }_0} -c_{\ddagger }(T)\cos {2\tilde {\theta }_0} \end{equation}

for later convenience. Thus, the leading-order dynamics is governed by the simpler differential equation

(3.13) \begin{equation} \dfrac {{\textrm{d}}{\tilde {\theta }_0}}{{\textrm{d}}{t}} = a - b \sin 2\tilde {\theta }_0 - c \cos 2\tilde {\theta }_0 \, , \quad a=\overline {a}_{\dagger }, \, b=\overline {b}_{\ddagger } , \, c= \overline {c}_{\ddagger }. \end{equation}

Note that we have defined $a,b,c$ as the averages of $a_{\dagger },b_{\ddagger },c_{\ddagger }$ , respectively, for ease of notation in what follows. These quantities will be key in exploring the emergent behaviour in § 4 to § 6. With the initial condition that

(3.14) \begin{equation} \tilde {\theta }_0(t=0)=\theta _0(t=0)=\theta _{00} \end{equation}

and the definitions

(3.15) \begin{equation} p=(b^2+c^2-a^2)^{1/2}, \quad q=(a^2-b^2-c^2)^{1/2}, \end{equation}

one can readily determine

(3.16) \begin{equation} \tan \tilde {\theta }_0 = \left \{ \begin{array}{lr} \dfrac {p\tan \theta _{00}+[a-c-b\tan \theta _{00}]\tanh (pt)}{p+[b-(a+c)\tan \theta _{00}]\tanh (pt)} , & a^2 \lt b^2 + c^2, \\ & \\ \dfrac {\tan \theta _{00} + [a-c-b\tan \theta _{00}]t}{1+[b-(a+c)\tan \theta _{00}]t} , & \,\,\,\,\,\,\,\,a =(b^2 + c^2)^{1/2} , \\ & \\ \dfrac {q\tan \theta _{00} + [a-c-b\tan \theta _{00}]\tan (qt)}{q+[b-(a+c)\tan \theta _{00}]\tan (qt)} , & a^2 \gt b^2 + c^2 , \end{array} \right . \end{equation}

where the appropriate branches of $\arctan$ are chosen so that $\tilde {\theta }_0$ is continuous. Note that, by (3.8), this also gives the angular evolution of $\overline {\theta }_0(t)$ up to a constant shift.

3.2. Multiple scales for the translational dynamics

We proceed to consider the translational dynamics by applying the method of multiple scales to (2.12) using the expansions

(3.17a) \begin{align} {\boldsymbol{x}}_c &= {\boldsymbol{x}}_{0}+\frac {1}{\omega } {\boldsymbol{x}}_{1} +\ldots , \end{align}

(3.17b) \begin{align} \theta &= \theta _0(t,T) + \frac {1}{\omega } \theta _1(t,T) + \ldots = \tilde {\theta }_0(t) + \varPsi (T)+ \frac {1}{\omega } \theta _1(t,T) + \ldots . \end{align}

Before doing so, it will prove convenient to expand the swimmer basis vectors of (2.4) in powers of $\omega$ using (3.17b ). We write

(3.18a) \begin{align} \hat {\boldsymbol{e}}_{1}(\theta ) &= \cos {\varPsi }\tilde {{\boldsymbol{e}}}_{10} + \sin {\varPsi }\tilde {{\boldsymbol{e}}}_{20} + \frac {1}{\omega }\theta _1\left [-\sin {\varPsi }\tilde {{\boldsymbol{e}}}_{10} + \cos {\varPsi }\tilde {{\boldsymbol{e}}}_{20} \right ] + {\textit{O}}\left (\frac {1}{\omega ^2}\right ), \end{align}

(3.18b) \begin{align} \hat {\boldsymbol{e}}_{2}(\theta ) &= -\sin {\varPsi }\tilde {{\boldsymbol{e}}}_{10} + \cos {\varPsi }\tilde {{\boldsymbol{e}}}_{20} + {\textit{O}}\left (\frac {1}{\omega }\right ), \\[9pt] \nonumber \end{align}

defining

(3.19) \begin{equation} \tilde {{\boldsymbol{e}}}_{10}(t) = \cos {\tilde {\theta }_0}{\boldsymbol{e}}_1 + \sin {\tilde {\theta }_0}{\boldsymbol{e}}_2, \quad \tilde {{\boldsymbol{e}}}_{20}(t) = -\sin {\tilde {\theta }_0}{\boldsymbol{e}}_1 + \cos {\tilde {\theta }_0}{\boldsymbol{e}}_2. \end{equation}

We also write the translational equation in a more concise form by defining the matrix operator

(3.20) \begin{equation} \boldsymbol{\varLambda }(T) := \begin{bmatrix} E^*_{11}(T) & E^*_{12}(T) -\varOmega ^*(T)& 0\\ E^*_{12}(T) +\varOmega ^*(T)& -E^*_{11}(T) & 0\\ 0 & 0 & 0 \end{bmatrix}. \end{equation}

This allows us to write the background flow explicitly as an affine map of position, with

(3.21) \begin{equation} \boldsymbol{u}^*_c = \boldsymbol{u}^*_{\textit{tr}}(T) +\boldsymbol{\varLambda }(T){\boldsymbol{x}}_c, \end{equation}

so that the governing equation for the translational motion can be written as

(3.22) \begin{equation} \dfrac {{\textrm{d}}{{\boldsymbol{x}}_c}}{{\textrm{d}}{t}} = \boldsymbol{u}^*_{\textit{tr}}(T) + \boldsymbol{\varLambda }(T){\boldsymbol{x}}_c + \left [V + \omega U(T) \right ]\hat {e}_{1}(\theta ) -\,\tilde {\!\unicode{x1D65C}}(\theta ,T) \unicode{x1D640}^{\,*}(T). \end{equation}

Later, we will make use of the explicit expression derived for $-\,\tilde {\!\unicode{x1D65C}} \unicode{x1D640}^{\,*}$ in (2.17), but we leave it implicit for now.

We are now in a position to discuss a further constraint imposed by our assumption of swimmer inefficiency. Whilst we have already assumed that $\overline {U}=\overline {\varOmega _f}=0$ , for consistency we must also impose that the product of $\omega U(T)$ and $\hat {\boldsymbol{e}}_{1}(\theta )$ in (3.22) does not generate terms that scale with $\omega$ ; otherwise, this would represent a fast effective swimming speed. In particular, noting the expansion of $\hat {\boldsymbol{e}}_{1}(\theta )$ in (3.18), we additionally impose that

(3.23) \begin{equation} \overline {U\cos {\varPsi }},\ \overline {U\sin {\varPsi }} = {\textit{O}}\left (\frac {1}{\omega }\right ), \end{equation}

so that there is no average swimming at leading order in (3.22). This motivates us to define the $2\pi$ -periodic functions of the fast time scale

(3.24a) \begin{align} U_c(T)&:= U(T)\cos {\varPsi (T)} - \overline {U\cos {\varPsi }}, \end{align}

(3.24b) \begin{align} U_s(T)&:= U(T)\sin {\varPsi (T)} - \overline {U\sin {\varPsi }}\,, \end{align}

noting that $\overline {U_c}=\overline {U_s} = 0$ . For later convenience, we write $V_{Uc}:=\omega \overline {U\cos {\varPsi }}$ and $V_{Us}:=\omega \overline {U\sin {\varPsi }}$ , both ${\textit{O}} (1 )$ by assumption. These represent the average swimming speeds that arise due to the combination of angular oscillations and variations in linear swimming speed. With our notion of swimmer inefficiency now made precise, we write the translational governing equation of (3.22) as

(3.25) \begin{align} \dfrac {{\textrm{d}}{{\boldsymbol{x}}_c}}{{\textrm{d}}{t}} &= \boldsymbol{u}^*_{\textit{tr}}(T) + \boldsymbol{\varLambda }(T){\boldsymbol{x}}_c + \left [V\cos {\varPsi (T)} + V_{Uc}\right ]\tilde {{\boldsymbol{e}}}_{10} + \left [V\sin {\varPsi (T)} + V_{Us}\right ]\tilde {{\boldsymbol{e}}}_{20}\nonumber \\ &\quad +\, \boldsymbol{n} \theta _1 + \omega \left [U_c(T)\tilde {{\boldsymbol{e}}}_{10} + U_s(T)\tilde {{\boldsymbol{e}}}_{20}\right ] - \,\tilde {\!\unicode{x1D65C}}(\theta ,T) \unicode{x1D640}^{\,*}(T) + {\textit{O}}\left (\frac {1}{\omega }\right ), \end{align}

writing $\boldsymbol{n} = -U_s(T)\tilde {{\boldsymbol{e}}}_{10} + U_c(T)\tilde {{\boldsymbol{e}}}_{20}$ . Here, we have made use of (3.18) to make the dependence on the asymptotic approximation to $\theta$ explicit in all but $\,\tilde {\!\unicode{x1D65C}}$ .

With this expanded governing equation, we now apply the method of multiple scales as in the case of the angular dynamics. Transforming the time derivatives as in (3.3) and inserting the expansion of (3.17), we have the leading-order balance

(3.26) \begin{equation} \dfrac {\partial {{\boldsymbol{x}}_0}}{\partial {T}} = U_c(T)\tilde {{\boldsymbol{e}}}_{10} + U_s(T)\tilde {{\boldsymbol{e}}}_{20} \end{equation}

at ${\textit{O}} (\omega )$ . This has solution

(3.27) \begin{equation} {\boldsymbol{x}}_0 = I_{U c}(T)\tilde {{\boldsymbol{e}}}_{10}(t) + I_{U s}(T)\tilde {{\boldsymbol{e}}}_{20}(t) + \overline {{\boldsymbol{x}}}_0(t), \end{equation}

defining the $2\pi$ -periodic functions

(3.28a) \begin{align} I_{U c} &:= \int _0^T U_c(s)\mathop {}\!{\textrm{d}}{s} - C_c, \end{align}

(3.28b) \begin{align} I_{U s} &:= \int _0^T U_s(s)\mathop {}\!{\textrm{d}}{s} - C_s, \end{align}

where the constants $C_c$ and $C_s$ are chosen such that $\overline {I_{U c}}=\overline {I_{U s}}=0$ . Note that the periodicity of $I_{U c}$ and $I_{U s}$ follows from $\overline {U_c}=\overline {U_s}=0$ , in turn a consequence of swimmer inefficiency and the definitions of (3.24). These functions capture the oscillatory translational motion that arises from interactions between angular oscillations and the rapidly varying speed $U(T)$ .

At next order, we have the more complex balance

(3.29) \begin{align} \dfrac {\partial {{\boldsymbol{x}}_1}}{\partial {T}} + \dfrac {\partial {{\boldsymbol{x}}_0}}{\partial {t}} &= \boldsymbol{u}^*_{\textit{tr}}(T) + \boldsymbol{\varLambda }(T){\boldsymbol{x}}_0 + \left [V\cos {\varPsi (T)} + V_{Uc}\right ]\tilde {{\boldsymbol{e}}}_{10} + \left [V\sin {\varPsi (T)} + V_{Us}\right ]\tilde {{\boldsymbol{e}}}_{20} \nonumber \\ &\quad +\, \boldsymbol{n} \theta _1 - \,\tilde {\!\unicode{x1D65C}}(\tilde {\theta }_0 + \varPsi (T),T) \unicode{x1D640}^{\,*}(T). \end{align}

To make progress via the Fredholm alternative, we consider the linear operator

(3.30) \begin{equation} \mathcal{L}:= \begin{bmatrix} \dfrac {\partial {}}{\partial {T}} & 0 & 0 & -\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{e}}_1\\ 0 & \dfrac {\partial {}}{\partial {T}} & 0 & -\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{e}}_2\\ 0 & 0 & \dfrac {\partial {}}{\partial {T}} & 0 \\ 0 & 0 & 0 & \dfrac {\partial {}}{\partial {T}} \end{bmatrix} \end{equation}

and the system

(3.31) \begin{equation} \mathcal{L}\begin{bmatrix} {\boldsymbol{x}}_1\\\theta _1 \end{bmatrix} = \boldsymbol{\mathcal{G}}, \end{equation}

with the forcing $\boldsymbol{\mathcal{G}}$ given by concatenating the remaining terms of (3.29) and the integrand of (3.11), the latter being the forcing of the analogous angular evolution equation. We seek periodic solutions of the homogeneous adjoint problem, $\mathcal{L}^*\boldsymbol{s}=0$ , where

(3.32) \begin{equation} \mathcal{L}^* = -\begin{bmatrix} \dfrac {\partial {}}{\partial {T}} & 0 & 0 & 0\\ 0 & \dfrac {\partial {}}{\partial {T}} & 0 & 0\\ 0 & 0 & \dfrac {\partial {}}{\partial {T}} & 0\\ \boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{e}}_1 & \boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{e}}_2 & 0 & \dfrac {\partial {}}{\partial {T}} \end{bmatrix} \end{equation}

and we compute $-\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{e}}_1=U_s(T)\cos {\tilde {\theta }_0} + U_c(T)\sin {\tilde {\theta }_0}$ and $-\boldsymbol{n}\boldsymbol{\cdot }{\boldsymbol{e}}_2 = U_s(T)\sin {\tilde {\theta }_0} - U_c(T)\cos {\tilde {\theta }_0}$ . Note that, in order for solutions of the original forced system to exist, we require that $\boldsymbol{\mathcal{G}}$ is orthogonal to the nullspace of $\mathcal{L}^*$ , writing $\left \langle \boldsymbol{\cdot }, \boldsymbol{\cdot }\right \rangle$ for the appropriate inner product.

Solving the adjoint problem leads to

(3.33) \begin{equation} \boldsymbol{s} = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ c_1\left [I_{U s}(T)\cos {\tilde {\theta }_0} + I_{U c}(T)\sin {\tilde {\theta }_0}\right ] + c_2\left [I_{U s}(T)\sin {\tilde {\theta }_0} - I_{U c}(T)\cos {\tilde {\theta }_0}\right ] + c_4 \end{bmatrix}, \end{equation}

written in terms of arbitrary constants $c_1$ , $c_2$ , $c_3$ and $c_4$ . Abusing notation to write ${\boldsymbol{e}}_4$ for the angular direction here, we have the independent solutions

(3.34a) \begin{align} \boldsymbol{s}_1 &= {\boldsymbol{e}}_1 + \left [I_{U s}(T)\cos {\tilde {\theta }_0} + I_{U c}(T)\sin {\tilde {\theta }_0}\right ]{\boldsymbol{e}}_4, \end{align}

(3.34b) \begin{align} \boldsymbol{s}_2 &= {\boldsymbol{e}}_2 + \left [I_{U s}(T)\sin {\tilde {\theta }_0} - I_{U c}(T)\cos {\tilde {\theta }_0}\right ]{\boldsymbol{e}}_4, \end{align}

(3.34c) \begin{align} \boldsymbol{s}_3 &= {\boldsymbol{e}}_3, \end{align}

(3.34d) \begin{align} \boldsymbol{s}_4 &= {\boldsymbol{e}}_4. \end{align}

These $\boldsymbol{s}_i$ each generate a solvability condition for the system, which will yield governing equations for the components of the motion. Notice that $\left \langle \boldsymbol{s}_4, \boldsymbol{\mathcal{G}} \right \rangle =0$ collapses onto the condition found in our analysis of the angular evolution in (3.11), so that these approaches are consistent.

All the out-of-plane motion is generated by $\left \langle \boldsymbol{s}_3, \boldsymbol{\mathcal{G}} \right \rangle =0$ , which leads to the evolution equation

(3.35) \begin{equation} \dfrac {{\textrm{d}}{\overline {z}_0}}{{\textrm{d}}{t}} = \overline {\boldsymbol{u}^*_{\textit{tr}}}\boldsymbol{\cdot }{\boldsymbol{e}}_3 - \overline {\,\tilde {\!\unicode{x1D65C}}(\tilde {\theta }_0 + \varPsi (T),T) \unicode{x1D640}^{\,*}(T)}\boldsymbol{\cdot }{\boldsymbol{e}}_3. \end{equation}

This corresponds to constant translation out of the plane and, as we will see that there is no dependence on $\overline {z}_0$ in the other governing equations, these dynamics decouple. Thus, without loss of generality, we overload our notation and neglect ${\boldsymbol{e}}_3$ components of all quantities in what follows.

With this simplification and overloaded notation, the solvability conditions $\left \langle \boldsymbol{s}_1, \boldsymbol{\mathcal{G}} \right \rangle =\left \langle \boldsymbol{s}_2, \boldsymbol{\mathcal{G}} \right \rangle =0$ generate the equations of in-plane motion, which we write as

(3.36) \begin{align} \dfrac {{\textrm{d}}{\overline {{\boldsymbol{x}}}_0}}{{\textrm{d}}{t}} &= \overline {\boldsymbol{u}^*_{\textit{tr}}} + \overline {\boldsymbol{\varLambda }}\overline {{\boldsymbol{x}}}_0 + \overline {I_{U c}\boldsymbol{\varLambda }}\tilde {{\boldsymbol{e}}}_{10} + \overline {I_{U s}\boldsymbol{\varLambda }}\tilde {{\boldsymbol{e}}}_{20} + \left [V\overline {\cos {\varPsi }} + V_{Uc} + \overline {\chi I_{U s}}\right ]\tilde {{\boldsymbol{e}}}_{10} \nonumber \\ &\quad +\, \left [V\overline {\sin {\varPsi }} + V_{Us} - \overline {\chi I_{U c}}\right ]\tilde {{\boldsymbol{e}}}_{20} - \overline {\,\tilde {\!\unicode{x1D65C}}(\tilde {\theta }_0 + \varPsi ,T) \unicode{x1D640}^{\,*}}, \end{align}

recalling the definition of $\chi (T)$ from (3.12). It remains to compute the average of the shape-flow interaction terms captured by

(3.37) \begin{align} \overline {\,\tilde {\!\unicode{x1D65C}}(\tilde {\theta }_0 + \varPsi ,T) \unicode{x1D640}^{\,*}}. \end{align}

This can be done in a straightforward manner using the expressions and definitions of (2.12) and (3.18), though the calculation is somewhat lengthy and broadly uninformative. For the purpose of our analysis, it suffices to summarise these expressions simply in terms of a sum of $\tilde {{\boldsymbol{e}}}_{10}$ and $\tilde {{\boldsymbol{e}}}_{20}$ , weighted by functions that are in linear in $\cos {2\tilde {\theta }_0}$ , $\sin {2\tilde {\theta }_0}$ , with coefficients formed of nonlinear averages of shape parameters ( $\eta _3$ and $\eta _4$ ), background flow strain rates $E^*_{ij}$ , and the fast oscillations of the swimmer orientation through $\varPsi$ . We write the average contribution as

(3.38) \begin{equation} -\overline {\,\tilde {\!\unicode{x1D65C}}(\tilde {\theta }_0 + \varPsi ,T) \unicode{x1D640}^{\,*}} = F_1(\cos {2\tilde {\theta }_0},\sin {2\tilde {\theta }_0})\tilde {{\boldsymbol{e}}}_{10} + F_2(\cos {2\tilde {\theta }_0},\sin {2\tilde {\theta }_0})\tilde {{\boldsymbol{e}}}_{20} \end{equation}

for implicitly defined linear functions $F_1$ and $F_2$ . These, along with other terms in the effective governing equations, simplify extensively under the assumption of no fast angular oscillations ( $\varOmega _f\equiv 0$ ), as explored in detail in Appendix C.

With this, the governing equation looks to be sufficiently complex as to be unwieldy. However, it turns out that there are further simplifications to be made through seemingly serendipitous cancellation. For instance, gross cancellation occurs in the following summation:

(3.39) \begin{equation} \overline {I_{U c}\boldsymbol{\varLambda }}\tilde {{\boldsymbol{e}}}_{10} + \overline {I_{U s}\boldsymbol{\varLambda }}\tilde {{\boldsymbol{e}}}_{20} + \overline {\chi I_{U s}}\tilde {{\boldsymbol{e}}}_{10} - \overline {\chi I_{U c}}\tilde {{\boldsymbol{e}}}_{20} = \left [\overline {\varOmega I_{U s}} + G_1\right ]\tilde {{\boldsymbol{e}}}_{10} + \left [-\overline {\varOmega I_{U c}} + G_2\right ]\tilde {{\boldsymbol{e}}}_{20}, \end{equation}

with all terms involving $\varOmega ^*$ vanishing. The unspecified functions $G_1$ and $G_2$ are linear in $\cos {2\tilde {\theta }_0}$ and $\sin {2\tilde {\theta }_0}$ , analogous to $F_1$ and $F_2$ . Writing $\unicode{x1D63C}=\overline {\boldsymbol{\varLambda }}$ for notational convenience in what follows, this entails that the governing equation can be written in the significantly simplified form

(3.40) \begin{align} \dfrac {{\textrm{d}}{\overline {{\boldsymbol{x}}}_0}}{{\textrm{d}}{t}} & = \overline {\boldsymbol{u}^*_{\textit{tr}}} + \unicode{x1D63C}\overline {{\boldsymbol{x}}}_0 + \left [V\overline {\cos {\varPsi }} + V_{Uc} + \overline {\varOmega I_{U s}} + H_1(2\tilde {\theta }_0)\right ]\tilde {{\boldsymbol{e}}}_{10}(\tilde {\theta }_0) \nonumber \\ &\quad +\, \left [V\overline {\sin {\varPsi }} + V_{Us} - \overline {\varOmega I_{U c}} + H_2(2\tilde {\theta }_0)\right ]\tilde {{\boldsymbol{e}}}_{20}(\tilde {\theta }_0), \end{align}

with $H_1 = F_1 + G_1$ and $H_2 = F_2 + G_2$ . To emphasise the structure of these equations of motion, we have explicitly included any dependence on the angular dynamics. In particular, all other quantities (save for $\overline {{\boldsymbol{x}}}_0$ ) are constants – effective quantities that have arisen from the systematic multiscale analysis. Thus, the motion consists of constant translation by the background flow, linear interaction with the background flow via $\unicode{x1D63C}=\overline {\boldsymbol{\varLambda }}$ , and an effective self-propelled swimming that is modulated by the average angular dynamics in a nonlinear fashion. In what follows, we set $\overline {\boldsymbol{u}^*_{\textit{tr}}}=\boldsymbol{0}$ by choice of reference frame without loss of generality and, thus, the solutions for the equations of motion below are relative to the mean translational component of the background flow.

Remarkably, further progress can be made now that the evolution of $\overline {{\boldsymbol{x}}}_0$ is written in this form. To proceed, we recall that

(3.41) \begin{equation} \unicode{x1D63C} = \overline {\boldsymbol{\varLambda }} = \begin{bmatrix} \overline {E^*_{11}} & \overline {E^*_{12}} - \overline {\varOmega ^*} \\ \overline {E^*_{12}} + \overline {\varOmega ^*} & -\overline {E^*_{11}} \end{bmatrix}. \end{equation}

We note that its eigenvalues are given by

(3.42) \begin{equation} \pm \big(\overline {E^*_{11}}^2+\overline {E^*_{12}}^2-\overline {\varOmega ^*}^2\big)^{1/2} \end{equation}

and that

(3.43) \begin{equation} \unicode{x1D63C}^2 = \big(\overline {E^*_{11}}^2+\overline {E^*_{12}}^2-\overline {\varOmega ^*}^2\big) \boldsymbol{I}, \end{equation}

as is most readily deduced from the Cayley–Hamilton theorem. Hence, with the definitions

(3.44) \begin{equation} \nu := \big(\overline {E^*_{11}}^2+\overline {E^*_{12}}^2-\overline {\varOmega ^*}^2\big)^{1/2}, \quad \mu := \big(\overline {\varOmega ^*}^2-\overline {E^*_{11}}^2-\overline {E^*_{12}}^2\big)^{1/2}, \end{equation}

we can compute $\exp [\unicode{x1D63C}t]$ to give

(3.45) \begin{equation} \unicode{x1D646}(t):=\exp [\unicode{x1D63C} t]= \begin{cases} \cosh (\nu t) \boldsymbol{I} + \dfrac {1}{\nu }\sinh (\nu t) \unicode{x1D63C}, & \overline {E^*_{11}}^2+\overline {E^*_{12}}^2\gt \overline {\varOmega ^*}^2, \\ \unicode{x1D644} + \unicode{x1D63C} t, & \overline {E^*_{11}}^2+\overline {E^*_{12}}^2=\overline {\varOmega ^*}^2, \\ \cos (\mu t) \unicode{x1D644} + \dfrac {1}{\mu } \sin (\mu t) \unicode{x1D63C}, & \overline {E^*_{11}}^2+\overline {E^*_{12}}^2\lt \overline {\varOmega ^*}^2. \end{cases} \end{equation}

The effective rates $\nu$ and $\mu$ capture the extent to which the rate of strain and the angular velocity of the background flow dominate one another, respectively. Note that at most one of $\nu$ and $\mu$ is real and non-zero for a given background flow.

Thus, solving (3.40) in terms of $\unicode{x1D646}$ and its convolution reveals

(3.46) \begin{align} \overline {{\boldsymbol{x}}}_{0} (t) =\, & \unicode{x1D646}(t) \overline {{\boldsymbol{x}}}_{0}(t=0)\nonumber \\ & + \int _0^t \unicode{x1D646} (t-s) \boldsymbol{C}( \sin 2 \tilde {\theta }_{0} (s),\cos 2 \tilde {\theta }_{0}(s), \sin \tilde {\theta }_{0} (s),\cos \tilde {\theta }_{0}(s))\mathop {}\!{\textrm{d}}{s}, \end{align}

where we define

(3.47) \begin{align} \boldsymbol{C} & := \left [V\overline {\cos {\varPsi }} + V_{Uc} + \overline {\varOmega I_{U s}} + H_1(2\tilde {\theta }_0)\right ]\tilde {{\boldsymbol{e}}}_{10}(\tilde {\theta }_0) \nonumber\\ &\quad +\, \left [V\overline {\sin {\varPsi }} + V_{Us} - \overline {\varOmega I_{U c}} + H_2(2\tilde {\theta }_0)\right ]\tilde {{\boldsymbol{e}}}_{20}(\tilde {\theta }_0), \end{align}

with $H_1$ and $H_2$ linear in $\cos {2\tilde {\theta }_0}$ and $\sin {2\tilde {\theta }_0}$ .

5.1. Exponential temporal dynamics in the plane

If the average strain rate of the flow dominates the average rotation rate of the flow, such that

(5.1) \begin{equation} \overline {E^*_{11}}^2+\overline {E^*_{12}}^2 \gt \overline {\varOmega ^*}^2, \end{equation}

then the matrix exponential of (3.45) entails that the swimmer will drift away from its starting point at an exponential rate, irrespective of its angular dynamics and with the possible exception of edge cases. Such edge cases can occur when $\unicode{x1D646}(t) \overline {{\boldsymbol{x}}}_{0}(t=0)$ is precisely balanced by

(5.2) \begin{equation} \int _0^t \unicode{x1D646} (t-s) \boldsymbol{C}( \sin 2 \tilde {\theta }_{0} (s),\cos 2 \tilde {\theta }_{0}(s), \sin \tilde {\theta }_{0} (s),\cos \tilde {\theta }_{0}(s))\mathop {}\!{\textrm{d}}{s}. \end{equation}

However, (5.2) is independent of the initial location of the swimmer. Thus, mathematical precision is required in the initial conditions for such an edge case, which would not be realisable in practice.

5.2. Linear temporal dynamics in the plane

We proceed to consider the degenerate case

(5.3) \begin{equation} \overline {E^*_{11}}^2+\overline {E^*_{12}}^2 = \overline {\varOmega ^*}^2, \end{equation}

where $\unicode{x1D63C}^2=\boldsymbol{\mathsf{0}}$ from (3.43). This splits into two further subcases: $\unicode{x1D63C}=\boldsymbol{\mathsf{0}}$ or $\unicode{x1D63C}\neq \boldsymbol{\mathsf{0}}$ .

If $\unicode{x1D63C}=\boldsymbol{\mathsf{0}}$ , the flow is either trivial or it is oscillatory with zero mean, so that $\overline {E^*_{11}}=\overline {E^*_{12}}= \overline {\varOmega ^*}=0$ . If the flow is trivial, the equations for translation collapse to

(5.4) \begin{align} \dfrac {{\textrm{d}}{\overline {{\boldsymbol{x}}}_0}}{{\textrm{d}}{t}} &= (V \overline {\cos \varPsi }+V_{Uc}) \tilde {{\boldsymbol{e}}}_{10}(\tilde {\theta }_0)+ (V \overline {\sin \varPsi }+V_{Us})\tilde {{\boldsymbol{e}}}_{20}(\tilde {\theta }_0) \nonumber \\[3pt] &\quad +\, \overline {\varOmega I_{U s}}\tilde {{\boldsymbol{e}}}_{10}(\tilde {\theta }_0) - \overline {\varOmega I_{U c}}\tilde {{\boldsymbol{e}}}_{20}(\tilde {\theta }_0), \end{align}

and the swimmer progresses, in general, on a curved trajectory.

For example, even when the swimmer is not rotating on a fast time scale, so that

(5.5) \begin{align} \varOmega _f &= 0, \quad \varPsi = \sin \varPsi = I_{U s} = V_{Us} = 0, \quad V_{Uc}=\omega \overline {U\cos {\varPsi }}=\omega \overline {U}=0 \nonumber \\[3pt] \tilde {\theta }_0 &= \overline {\theta }_0 , \quad \tilde {{\boldsymbol{e}}}_{10}(\tilde {\theta }_0) = \hat {\boldsymbol{e}}_{10}(\overline {\theta }_0), \quad \tilde {{\boldsymbol{e}}}_{20}(\tilde {\theta }_0)= \hat {\boldsymbol{e}}_{20}(\overline {\theta }_0), \end{align}

the equation of motion reduces to

(5.6) \begin{equation} \dfrac {{\textrm{d}}{\overline {{\boldsymbol{x}}}_0}}{{\textrm{d}}{t}} = V \hat {\boldsymbol{e}}_{10}(\overline {\theta }_0) - \overline {\varOmega I_{U c}}\hat {\boldsymbol{e}}_{20}(\overline {\theta }_0) \end{equation}

and the swimming corresponds to a curved trajectory with radial velocity $V$ and a velocity of $-\overline {\varOmega I_{U c}}$ in the $\overline {\theta }_0$ direction. Moreover, the swimmer trajectory is further modulated by the fast rotation of the swimmer $\varOmega _f(T)$ through $\varPsi$ , $V_{Us}$ , $V_{Uc}$ , $I_{U s}$ , $I_{U c}$ , albeit in a much more complex manner than suggested by naive averaging, with $\overline {\varOmega }_f=0$ insufficient to simplify further.

If instead we have $\unicode{x1D63C} = \boldsymbol{\mathsf{0}}$ via a non-trivial mean-zero oscillatory flow, we have $\unicode{x1D646} = \unicode{x1D644}$ and

(5.7) \begin{equation} \overline {{\boldsymbol{x}}}_0(t)= \overline {{\boldsymbol{x}}}_0(t=0)+\int _0^t \boldsymbol{C}(\sin 2\tilde {\theta }_{0}(s),\cos 2\tilde {\theta }_{0}(s), \sin \tilde {\theta }_{0} (s),\cos \tilde {\theta }_{0}(s))\mathop {}\!{\textrm{d}}{s}, \end{equation}

with $\boldsymbol{C}$ as in (3.47). Suppose that we are in one of the regimes explored in § 4 in which the orientation of the swimmer asymptotes to a fixed value. Then, the integrand of (5.7) becomes constant for large values of the integration variable and, thus, the swimmer will drift to infinity linearly in time (neglecting possible edge cases of perfect cancellation between terms). Notably, this linear drift can happen even if there is only reciprocal swimming, i.e. $V =\overline {\varOmega I_{U s}}=\overline {\varOmega I_{U c}}=0$ . This can be seen explicitly in the example setting of fore–aft symmetric swimmers that are a body of revolution, with the equations reducing to those of (C11c ) and where $\tilde {\theta }_0=\overline {\theta }_0$ . In this case, contributions to swimming arise from terms such as $\overline {I_{U c}E^*_{11}}\hat {\boldsymbol{e}}_{10}(\overline {\theta }_0)$ , which need not be zero even for reciprocal swimmers. Thus, seemingly non-progressive swimming can generate a drift to spatial infinity through interactions with purely oscillatory, mean-zero flows. In other words, this provides an explicit mechanism by which a swimmer might circumvent Purcell’s scallop theorem.

Suppose instead that we are in a regime in which the swimmer tumbles endlessly in the zero-mean oscillatory flow. Note that these flows necessarily have $\overline {\varOmega ^*}=0$ , so that swimmer tumbling requires $\overline {\varOmega }^2\gt 0$ in this setting (consider (4.6) with $\overline {\varOmega ^*}=0$ ), thus excluding swimmers with fore–aft symmetry. Proceeding with a swimmer that is tumbling, we can evaluate the contribution of the term associated with $V \overline {\cos \varPsi }$ in the inertial frame ${\boldsymbol{e}}_1$ direction in (5.7) over a single period of rotation. For one such tumble, which we recall has period $2\pi /q$ , starting from some $t=t_s$ , this contribution is $V \overline {\cos \varPsi }$ multiplied by

(5.8) \begin{eqnarray} && \int _{t_s}^{t_s+2\pi /q} \cos \tilde {\theta }_0 {\textrm{d}}t = \int _{\tilde {\theta }_0(t_s)}^{\tilde {\theta }_0(t_s)+2\pi } \displaystyle\frac {\cos \tilde {\theta }_0}{a-b\sin 2\tilde {\theta }_0-c\cos 2\tilde {\theta }_0} \mathop {}\!{\textrm{d}}{\tilde {\theta }_0} \nonumber\\[6pt] &=& \displaystyle\frac {1}{a}\int _{-\pi }^{\pi } \displaystyle\frac {\cos \tilde {\theta }_0}{1-R\cos (2\tilde {\theta }_0-2\xi )} \mathop {}\!{\textrm{d}}{\tilde {\theta }_0}, \text{ with } R = \displaystyle\frac {(b^2+c^2)^{1/2}}a \lt 1, \end{eqnarray}

where $\xi$ is a phase shift. Here, we have recalled the governing equation of (3.13) to change variables to $\tilde {\theta }_0$ in the integrals. Using the periodicity of the cosines, this can be written as a linear combination of the following integrals:

(5.9) \begin{equation} \frac {1}{a} \int _{-\pi }^{ \pi } \frac {\cos \tilde {\theta }_0}{1 -R\cos (2\tilde {\theta }_0)} \mathop {}\!{\textrm{d}}{\tilde {\theta }_0}=\frac {2}{a} \int _{0}^{\pi } \frac {\cos \tilde {\theta }_0}{1-R\cos (2\tilde {\theta }_0)} \mathop {}\!{\textrm{d}}{\tilde {\theta }_0}, \quad \frac {1}{a}\int _{-\pi }^{ \pi } \frac {\sin \tilde {\theta }_0}{1-R\cos (2\tilde {\theta }_0)} \mathop {}\!{\textrm{d}}{\tilde {\theta }_0}. \end{equation}

Both of these integrals are zero, by the odd parity in reflection about $\theta _0=\pi /2$ for the first integral and about $\theta _0=0$ for the second. Similarly, all other terms contributing to the translational motility in (5.7) can be written as a linear combination of integrals over a temporal period, with integrands

(5.10) \begin{equation} \sin \theta _0, \quad \sin 2\theta _0\cos \theta _0, \quad \cos 2\theta _0\cos \theta _0, \quad \sin 2\theta _0\sin \theta _0,\quad \cos 2\theta _0\sin \theta _0 , \end{equation}

which all integrate to zero using the same arguments as above. Hence, we may conclude that a tumbling swimmer in a purely oscillatory linear planar flow does not drift indefinitely.

Now we consider the final subcase of linear temporal dynamics in the plane, assuming that $\unicode{x1D63C}\neq \boldsymbol{\mathsf{0}}$ and $\unicode{x1D63C}^2 = \boldsymbol{\mathsf{0}}$ . We consider the Jordan normal form for $\unicode{x1D63C}$ to within an overall scaling, though it is also useful to explicitly demonstrate that the transformation required in this particular case is a rotation. First, let $\boldsymbol{e}_A$ denote the zero-eigenvalue unit eigenvector of $\unicode{x1D63C}$ (unique up to sign) and, thus, $\unicode{x1D63C}\boldsymbol{e}_A= \boldsymbol{0}$ . Additionally, let $\boldsymbol{e}_A^\perp$ denote the unit vector perpendicular to $\boldsymbol{e}_A$ so that $\{\boldsymbol{e}_A,\boldsymbol{e}_A^\perp \}$ is a right-handed orthonormal basis. We have $\unicode{x1D63C}\boldsymbol{e}_A^\perp \neq \boldsymbol{0}$ , otherwise $\unicode{x1D63C}=\boldsymbol{\mathsf{0}}$ . Then, with $\alpha _1$ and $\alpha _2$ defined by

(5.11) \begin{equation} \unicode{x1D63C}\boldsymbol{e}_A^\perp = \alpha _1\boldsymbol{e}_A+\alpha _2 \boldsymbol{e}_A^\perp \end{equation}

we have $\boldsymbol{0} = \unicode{x1D63C}^2\boldsymbol{e}_A^\perp = \alpha _2 \unicode{x1D63C}\boldsymbol{e}_A^\perp$ and, hence, $\alpha _2=0$ and $\alpha _1\neq 0$ . Thus,

(5.12) \begin{equation} \unicode{x1D63C} \left [\left . \boldsymbol{e}_A \right | \boldsymbol{e}_A^\perp \right ] = \left [\left . \boldsymbol{0}\right | \alpha _1 \boldsymbol{e}_A \right ] = \left [\left . \boldsymbol{e}_A \right | \boldsymbol{e}_A^\perp \right ] \begin{bmatrix} 0 & \alpha _1 \\ 0 & 0 \end{bmatrix}. \end{equation}

Noting that $ [\left . \boldsymbol{e}_A \right | \boldsymbol{e}_A^\perp ]$ is an orthogonal matrix, using its transpose to left-multiply both sides shows that a rotation of the axes can be found to transform $\unicode{x1D63C}$ to a matrix that is zero except for the upper right-hand off-diagonal entry (this is the Jordan normal form of $\unicode{x1D63C}$ to within scaling). Hence, the flows we are considering here are, on averaging, those of pure shear with $\overline {\boldsymbol{u}^*} =2\overline {E^*_{12}} y\boldsymbol{e}_1$ for a suitable choice of orthonormal basis, and we have $\overline {E^*_{12}}\neq 0$ as $\unicode{x1D63C}\neq \boldsymbol{{0}}$ .

For swimmers that asymptote to a fixed angle, the presence of pure shear results in infinite drift in the ${\boldsymbol{e}}_1$ direction, parallel to the flow, that increases quadratically in time in general. This arises from the fact that the dominant term in the ${\boldsymbol{e}}_1$ direction for $t\gg t_*\gg 1$ (sufficiently large) is approximately

(5.13) \begin{align} &\left [ \int _{t_*}^{t} \unicode{x1D644} +\unicode{x1D63C}(t-s) \mathop {}\!{\textrm{d}}{s}\right ]\left .\boldsymbol{C}( \sin 2 \tilde {\theta }_{0} ,\cos 2 \tilde {\theta }_{0}, \sin \tilde {\theta }_{0} ,\cos \tilde {\theta }_{0})\right \rvert _{\tilde {\theta }_{0}=\tilde {\theta }_0(\infty )} \nonumber \\ &= \left [ \int _{t_*}^{t} \begin{bmatrix} 1 & 2\overline {E^*_{12}} (t-s) \\ 0 & 1 \end{bmatrix}\mathop {}\!{\textrm{d}}{s}\right ] \left .\boldsymbol{C}( \sin 2 \tilde {\theta }_{0} ,\cos 2 \tilde {\theta }_{0}, \sin \tilde {\theta }_{0} ,\cos \tilde {\theta }_{0})\right |_{\tilde {\theta }_{0}=\tilde {\theta }_0(\infty )} \nonumber \\ &\sim \begin{bmatrix} {\textrm{ord}}\left (t^2\right ) \\ {\textrm{ord}}\left (t\right ) \end{bmatrix}. \end{align}

Finally, we note that a tumbling swimmer’s trajectory will drift in the ${\boldsymbol{e}}_1$ direction in all but edge cases due to the

(5.14) \begin{equation} \unicode{x1D646}(t) \overline {{\boldsymbol{x}}}_{0}(t=0)= (\unicode{x1D644} + \unicode{x1D63C} t) \overline {{\boldsymbol{x}}}_{0}(t=0) \end{equation}

term. However, it will not drift in the ${\boldsymbol{e}}_2$ direction. In particular, given the swimmer is tumbling and noting ${\boldsymbol{e}}_2 ^\top \boldsymbol{A}=\boldsymbol{0}$ , we have that its location in the ${\boldsymbol{e}}_2$ direction is given by

(5.15a) \begin{align} {\boldsymbol{e}}_2^\top \overline {{\boldsymbol{x}}}_0(t) &= {\boldsymbol{e}}_2^\top \overline {{\boldsymbol{x}}}_0(t=0)\nonumber \\ &\quad + {\boldsymbol{e}}_2^\top \int _0^t(\unicode{x1D644} +\unicode{x1D63C}(t-s) )\boldsymbol{C}(\sin 2\tilde {\theta }_{0}(s),\cos 2\tilde {\theta }_{0}(s), \sin \tilde {\theta }_{0} (s),\cos \tilde {\theta }_{0}(s))\mathop {}\!{\textrm{d}}{s}, \end{align}

(5.15b) \begin{align} &= {\boldsymbol{e}}_2^\top \overline {{\boldsymbol{x}}}_0(t=0)+\int _0^t {\boldsymbol{e}}_2^\top \boldsymbol{C}(\sin 2\tilde {\theta }_{0}(s),\cos 2\tilde {\theta }_{0}(s), \sin \tilde {\theta }_{0} (s),\cos \tilde {\theta }_{0}(s))\mathop {}\!{\textrm{d}}{s}. \end{align}

The integral above is the second component of those occurring in (5.7) for a tumbling swimmer, as considered in (5.8) and (5.9), and, thus, by an inheritance of this analysis we can deduce that there is no drift in the ${\boldsymbol{e}}_2$ direction for a tumbling swimmer in a flow that is pure shear and non-trivial after time averaging.

5.3. Oscillatory temporal dynamics in the plane

The final class of translational dynamics to consider occurs if the average strain rate is dominated by the rotation rate, such that

(5.16) \begin{equation} \overline {E^*_{11}}^2+\overline {E^*_{12}}^2 \lt \overline {\varOmega ^*}^2. \end{equation}

Neglecting edge cases, we first consider swimming in which $a^2 \lt b^2+c^2$ , so that there is no tumbling and, hence, the swimmer tends to a fixed angle for large time (see § 4). Noting from (3.45) that $\unicode{x1D646}(t)$ is periodic on the slow time scale with period $2\pi /\mu$ , where

(5.17) \begin{equation} \mu = (\overline {\varOmega ^*}^2-\overline {E^*_{11}}^2-\overline {E^*_{12}}^2)^{1/2}, \end{equation}

we have for $t_*$ sufficiently large

(5.18) \begin{align} &\overline {{\boldsymbol{x}}}_0(t_*+2\pi /\mu )- \overline {{\boldsymbol{x}}}_0(t_*) \nonumber \\ & \approx \left [ \int _{t_*}^{t_*+2\pi /\mu }\unicode{x1D646}(t_*-s)\mathop {}\!{\textrm{d}}{s}\right ] \left .\boldsymbol{C}( \sin 2 \tilde {\theta }_{0} ,\cos 2 \tilde {\theta }_{0}, \sin \tilde {\theta }_{0} ,\cos \tilde {\theta }_{0})\right |_{\tilde {\theta }_0=\tilde {\theta }_0(\infty )}=0. \end{align}

The error in approximating $\theta _0$ by its asymptote is exponentially small for large time (see (3.16)) and, hence, these errors do not accumulate. Thus, the trajectory is asymptotically periodic and bounded for large time. Furthermore, even with net motion of the swimmer, so that at least one of $V$ , $\overline {\varOmega I_{U c}}$ or $\overline {\varOmega I_{U s}}$ is not zero, the contribution of the progressive swimming to the motion for $t\gt t_*$ (with $t_*$ sufficiently large) is zero. For example, when $V \gt 0$ , the contribution to the motion for $t\gt t_*$ that scales with $V \overline {\cos \varPsi }$ is (to within exponentially small errors) given by

(5.19) \begin{align} & V \overline {\cos \varPsi } \left [\int _{0}^t \unicode{x1D646} (t -s) \mathop {}\!{\textrm{d}}{s} -\int ^{t_*}_0 \unicode{x1D646} (t_*-s) \mathop {}\!{\textrm{d}}{s} \right ] \hat {\boldsymbol{e}}_{10}(\infty ) \nonumber \\ &= V \overline {\cos \varPsi } \left ( \frac {\sin (\mu t_*)-\sin (\mu t)}\mu + \frac {\cos (\mu t)-\cos (\mu t_*)}{\mu ^2}\unicode{x1D63C} \right ) \hat {\boldsymbol{e}}_{10}(\infty ), \end{align}

with $\unicode{x1D646}$ as in (3.45). In particular, this contribution is oscillatory as $t$ increases. Directly analogous and oscillatory results also hold for all terms involving

(5.20) \begin{align} V ,\,V_{Uc},\,V_{Us}, \,\overline {\varOmega I_{U c}}, \,\overline {\varOmega I_{U s}}. \end{align}

In turn, this explicitly demonstrates that the progressive movement of the swimmer has been converted to an oscillatory movement by the background flow.

In contrast, if the tumbling condition holds, that is $a^2 \gt b^2 + c^2$ , the dynamics will involve the convolution of oscillations at the tumbling frequency and at the frequency associated with the flow

(5.21) \begin{equation} \mu = (\overline {\varOmega ^*}^2-\overline {E^*_{11}}^2-\overline {E^*_{12}}^2)^{1/2}. \end{equation}

Whether such dynamics induces oscillations or unbounded dynamics in the trajectory at long time is contingent on whether or not there is resonance between the different oscillatory contributions, though oscillations may be typically expected as resonance requires parameter fine-tuning. Though general results in this case are less forthcoming, we can examine special cases of (3.16) and (3.46) to yield additional insights, as we pursue below.

6.1. Rotational flow

A particularly simple case is that of rotational flow with non-zero mean, for which $E^*_{11}=E^*_{12}=0$ and $\overline {\varOmega ^*} \neq 0$ . In this regime, the tumbling condition of § 4 holds, with $a= \overline {\varOmega ^*}\neq 0$ and $b=c=0$ , so that (3.16) reduces to

(6.3) \begin{equation} \overline {\theta }_0 = \theta _{00}+\overline {\varOmega ^*}t, \end{equation}

where $\theta _{00}=\overline {\theta }_0(t=0)$ . Further, with $\overline {x}_0={\boldsymbol{e}}_1\boldsymbol{\cdot }\overline {{\boldsymbol{x}}}_0$ , $\overline {y}_0={\boldsymbol{e}}_2\boldsymbol{\cdot }\overline {{\boldsymbol{x}}}_0$ , we have

(6.4) \begin{equation} \dfrac {{\textrm{d}}{\overline {x}_{0}}}{{\textrm{d}}{t}} = - \overline {\varOmega ^*}\overline {y}_0 +V \cos \overline {\theta }_0, \quad \dfrac {{\textrm{d}}{\overline {y}_{0}}}{{\textrm{d}}{t}} = \overline {\varOmega ^*}\overline {x}_0 +V \sin \overline {\theta }_0. \end{equation}

Hence, if there is no intrinsic net swimming ( $V =0$ ), the overall motion is that of simple harmonic oscillations, with

(6.5) \begin{equation} \dfrac {{\textrm{d}}{^2\overline {x}_0}}{{\textrm{d}}{t^2}} + \overline {\varOmega ^*}^2 \overline {x}_0 = 0 \end{equation}

and, thus, there is no net motion on the long time scale for non-progressive swimmers with this level of symmetry in purely rotational flow.

For $V \neq 0$ , we instead have the dynamics of a forced oscillator, with

(6.6) \begin{equation} \dfrac {{\textrm{d}}{^2\overline {x}_0}}{{\textrm{d}}{t^2}} + \overline {\varOmega ^*}^2 \overline {x}_{0} = -2V\overline {\varOmega ^*} \sin \left (\theta _{00}+\overline {\varOmega ^*}t\right ), \quad \overline {y}_{0} = \frac {1}{\overline {\varOmega ^*}}\left [V \cos \overline {\theta }_0 - \dfrac {{\textrm{d}}{\overline {x}_{0}}}{{\textrm{d}}{t}}\right ], \end{equation}

which generates resonance with no parameter fine-tuning beyond that needed to force a purely rotational flow. Thus, for purely rotational flow and even slightly progressive swimmers, we conclude that swimmer motility generates a resonant oscillation, so that the swimmer distance from the origin scales linearly with time. Recovering additional generality by reinstating the shape parameters $\lambda _5$ , $\eta _2$ , $\eta _3$ and $\eta _4$ does not change these observations, as they enter the equations of motion through the rate of strain tensor, which here is zero.

However, if we relax the constraint of fore–aft symmetry and allow the swimmer to generate a slow time scale rotation $\varOmega \neq 0$ , then $a = \overline {\varOmega } +\overline {\varOmega ^*} \neq \overline {\varOmega ^*}$ . In turn, (6.3) no longer holds and the corresponding forcing in (6.6) is modified to a term scaling with $\sin (\theta _{00} + at )$ . Thus, the swimmer oscillates and resonance does not occur, demonstrating that an element of fine tuning (i.e. perfect fore–aft symmetry) is indeed required for resonance.

6.2. Irrotational flow

For an irrotational flow with non-trivial mean, we have $\varOmega ^*=0,$ $\overline {E^*_{11}}^2 +\overline {E^*_{12}}^2\gt 0$ . Recalling $\varOmega =0$ by fore–aft symmetry, we also have $a=0$ and, by (3.16), $\overline {\theta }_0$ tends to a constant asymptote for large time. From (5.1) et seq., we have exponential dynamics in the plane and, recognising the possible exception of edge cases, the swimmer drifts off to spatial infinity. This long-time trajectory can be examined in more detail without further loss of generality. As $\varOmega ^*=0$ , the matrix $\unicode{x1D63C}$ of (6.2) is symmetric and, hence, diagonalisable. Thus, with a suitable choice of laboratory basis, the equations of motion in the plane of the flow in the long-time limit are given approximately by

(6.7) \begin{equation} \dfrac {{\textrm{d}}{\overline {x}_{0}}}{{\textrm{d}}{t}} = \left (\overline {E^*_{11}}^2 +\overline {E^*_{12}}^2\right )^{1/2} ( \overline {x}_{0}+\alpha ), \quad \dfrac {{\textrm{d}}{\overline {y}_{0}}}{{\textrm{d}}{t}}= -\left (\overline {E^*_{11}}^2 +\overline {E^*_{12}}^2\right )^{1/2} ( \overline {y}_{0}+\beta ), \end{equation}

where

(6.8) \begin{align} \alpha \left (\overline {E^*_{11}}^2 +\overline {E^*_{12}}^2\right )^{1/2}, \quad -\beta \left (\overline {E^*_{11}}^2 +\overline {E^*_{12}}^2\right )^{1/2} \end{align}

are the first and second components of the long-time limits of the other terms in (6.2). Noting the opposing signs of the eigenvalues in (6.7), this explicitly demonstrates an exponential drift to infinity in one direction accompanied by an exponential decay towards a constant in the orthogonal direction. Further, we have

(6.9) \begin{equation} \dfrac {{\textrm{d}}{\overline {y}_{0}}}{{\textrm{d}}{\overline {x}_{0}}} = -\frac { \overline {y}_{0}+\beta }{ \overline {x}_{0}+\alpha } \end{equation}

and, hence,

(6.10) \begin{equation} \overline {y}_0+\beta =\frac {M}{\overline {x}_{0}+\alpha }, \end{equation}

where $M$ is a constant of integration. Thus, independent of any further details, a swimmer in such an irrotational flow moves along hyperbolae in the plane once initial transients have decayed. Furthermore, noting the change of basis to bring the equation of motion into the form of (6.7), the hyperbola asymptote is parallel to an eigenvector of the averaged rate of strain tensor, $\overline {\unicode{x1D640}^{\,*}}$ . We showcase an example of asymptoting towards a hyperbola in figure 2, and note that the long-time direction of motion of this swimmer is given by an eigenvector of $\overline {\unicode{x1D640}^{\,*}}$ , in this case $[(1+\sqrt {5})/2,1]^\top$ .

Figure 2. Exponential translational dynamics of a swimmer in irrotational oscillatory flow. (a) The temporal evolution of swimmer position and orientation, highlighting long-time exponential growth of $x$ and $y$ and a constant-average orientation $\theta$ . (b) The path of the swimmer from $(x,y) = (0,0)$ is shown in grey, which exhibits large oscillations around an eventually hyperbolic trajectory. This hyperbola asymptotes to a line with gradient $2/(1+\sqrt {5})$ , parallel to an eigenvector of the averaged rate of strain tensor $\overline {\unicode{x1D640}^{\,*}}$ . The leading-order approximations to the average evolution are shown as black dashed lines, evidencing excellent agreement with the full numerical solutions that expectedly lessens during the exponential translational motion. Here, we have taken $\omega =100$ and set all swimmer and flow parameters to be zero apart from $U(T)=0.1\cos T$ , $E^*_{11}(T)=0.2+\sin T$ and $E^*_{12}(T)=0.4+\sin T$ .

Notably, these broad conclusions apply for both progressive and non-progressive swimmers. They also apply if we relax many of our symmetry constraints (with the exception of fore–aft reflection invariance), with these results holding for non-zero $\eta _2$ , $\eta _3$ , $\eta _4$ and $\lambda _5$ . The breaking of fore–aft symmetry, however, allows for self-induced rotation that has the potential to invalidate these conclusions, as we plausibly obtain $a\gt 0$ and lose the asymptoting behaviour of the swimmer orientation.

6.3. Motility in stationary shear

The classical example of a stationary shear flow can be recovered by setting $E^*_{11}=0$ and $\varOmega ^*=-E^*_{12}\neq 0$ and constant, without loss of generality. In this case, (3.16) and (3.46) entail that $\bar {y}_0$ and $\bar {\theta }_0$ decouple from $\overline {x}_0$ , with motion in the latter direction including an inexorable drift in all but edge cases. Hence, we focus on the dynamics of $\bar {y}_0$ and $\bar {\theta }_0$ , which here are governed by the reduced system

(6.11a) \begin{align} \dfrac {{\textrm{d}}{\bar {y}_0}}{{\textrm{d}}{t}} &= -E^*_{12}\overline {I_{U c}{B}}\cos {\bar {\theta }_0}\cos {2\bar {\theta }_0} +V \sin {\bar {\theta }_0}, \end{align}

(6.11b) \begin{align} \dfrac {{\textrm{d}}{\bar {\theta }_0}}{{\textrm{d}}{t}} &= -E^*_{12}\left (1 - \overline {B}\cos {2\bar {\theta }_0}\right ), \end{align}

recalling that $\overline {I_{U c}}=0$ .

We first consider the case with $|{\overline {B}}|\lt 1$ . Equation (6.11b ) immediately implies that $\bar {\theta }_0$ is periodic, which can also be seen in the general formalism of (3.13) and (3.16) by setting $a=-E^*_{12}$ , $b=0$ , $c= -E^*_{12} \overline {B}$ and $q=(a^2-b^2-c^2)^{1/2}=|{E^*_{12}}|(1-\overline {B}^2)^{1/2}\gt 0$ . This dynamics corresponds precisely to that of planar Jeffery orbits (Jeffery Reference Jeffery1922), as generalised by Bretherton (Reference Bretherton1962) and identified in planar shape-changing swimmers by Gaffney et al. (Reference Gaffney, Dalwadi, Moreau, Ishimoto and Walker2022a ).

We also have that $\bar {y}_0$ is periodic, as can be deduced by observing that the system is conservative, with

(6.12) \begin{equation} \dfrac {{\textrm{d}}{H}}{{\textrm{d}}{t}}= 0, \quad H:= \bar {y}_0 - \int _0^{\bar {\theta }_0} \frac {1}{1-\overline {B}\cos {2\psi }}\left (\overline {I_{U c} B} \cos \psi \cos {2\psi } -\frac V {E^*_{12}}\sin {\psi }\right )\mathop {}\!{\textrm{d}}{\psi }. \end{equation}

Thus, as $H$ is constant, we have that $\bar {y}_0$ is an integral of a smooth, bounded integrand with periodic limits, up to an additive constant. Hence, $\bar {y}_0$ is bounded and periodic for all time. It inherits the period of $\bar {\theta }_0$ , which here is given by

(6.13) \begin{equation} \frac {2\pi }{q} = \frac {2\pi }{(a^2-c^2)^{1/2}} = \frac {2\pi }{|{E^*_{12}}|(1-\overline {B}^2)^{1/2}} \end{equation}

in units of the slow time scale.

Figure 3. Systematically averaged dynamics of a reciprocal swimmer in stationary shear flow, a scenario schematically illustrated in figure 1 with a depiction of $\theta$ , $y$ . In the above, note that $\overline {\theta _0},\overline {y}_0$ are the fast time scale averages of the leading-order approximations, that is $\theta _0$ , $y_0$ , to the variables $\theta$ , $y$ . In the above, we illustrate the semiperiodic phase space that corresponds to a reciprocal swimmer in stationary shear flow, shown both as dynamics on a cylinder and in the plane. We showcase three qualitatively distinct regimes: in (a) and (d), we have $(\overline {I_{U c} B}, \overline {B}) = (0,0.5)$ , leading to no motion at leading order; in (b) and (e), we have $(\overline {I_{U c} B}, \overline {B}) = (1,0.5)$ and long-time periodic motion; in (c) and (f), we have $(\overline {I_{U c} B}, \overline {B}) = (1,1.5)$ and progression. In (f), the dashed lines correspond to stable states of the angular dynamics.

Now suppose that $|{\overline {B}}|\geqslant 1$ , a case that requires extreme shape elongation (Bretherton Reference Bretherton1962). This gives rise to fundamentally different dynamics, with the swimmer no longer tumbling. Instead, its angle asymptotes to a constant that, in turn, induces a drift to infinity along the ${\boldsymbol{e}}_2$ direction (perpendicular to the flow direction) for large time. This is even true for reciprocal swimmers if $\overline {I_{U c} B}\neq 0$ . In other words, it is possible for a highly elongated reciprocal swimmer with fore–aft symmetry to self-propel indefinitely across pathlines in a stationary shear flow via the interaction between shear flow and the swimmer deformation.

The range of possible leading-order dynamics for a reciprocal swimmer ( $V =0$ ) in various flows is illustrated in figure 3, with figure 3(b,e) showcasing periodicity on a long time scale Jeffery orbit. Non-trivial motility is highlighted in figure 3(c,f), in which the reciprocal swimmer approaches a steady state of the angular dynamics and achieves net propulsion across pathlines of the flow.

6.4. Motility in oscillatory shear

A natural generalisation of stationary shear flow is oscillatory shear flow. For such a flow, we have $E^*_{11}=0$ and $\varOmega ^*(T)=-E^*_{12}(T)\neq 0$ , without loss of generality. The equations of motion simplify to

(6.14a) \begin{align} \dfrac {{\textrm{d}}{\overline {x}_0}}{{\textrm{d}}{t}} &= 2\overline {E^*_{12}} \overline {y}_0 + \overline {E^*_{12} I_{U c} }\sin {\overline {\theta }_0} + \overline {E^*_{12} I_{U c} B}\sin {\overline {\theta }_0}\cos {2\overline {\theta }_0}+V \cos {\overline {\theta }_0}, \end{align}

(6.14b) \begin{align} \dfrac {{\textrm{d}}{\bar {y}_0}}{{\textrm{d}}{t}} &= \overline {E^*_{12}I_{U c}}\cos {\bar {\theta }_0} - \overline {E^*_{12}I_{U c}{B}}\cos {\bar {\theta }_0}\cos {2\bar {\theta }_0} +V \sin {\overline {\theta }_0}, \end{align}

(6.14c) \begin{align} \dfrac {{\textrm{d}}{\bar {\theta }_0}}{{\textrm{d}}{t}} &= - \overline {E^*_{12}} \left (1- \frac {\overline {E^*_{12}{B}}}{\overline {E^*_{12}}} \cos {2\bar {\theta }_0}\right ), \end{align}

noting the appearance of an effective Bretherton parameter of $\overline {E^*_{12}{B}}/\overline {E^*_{12}}$ in (6.14c ). Importantly, this quantity need not have magnitude less than unity, even if $|{B(T)}|\lt 1$ for all $T$ . Here, one could replace $E^*_{12}$ with the usual simple shear rate $\gamma$ via the relation $\gamma (T):= -2E^*_{12}(T)$ , corresponding to $\boldsymbol{u}^* = -\gamma (T) y{\boldsymbol{e}}_1$ , though we do not do so here.

Below, we only consider cases where the effective Bretherton parameter has a magnitude greater than unity, noting that this does not necessitate the geometrical constraint of severe elongation, though it does mean the angular dynamics is asymptoting. In this case, the angular dynamics evolves to an asymptotically constant angle at large time. This holds even if the average shear rate is zero (i.e. $\overline {E^*_{12}} =0$ ) but $\overline {E^*_{12}{B}}\neq 0$ , in which case we instead have

(6.15) \begin{equation} \dfrac {{\textrm{d}}{\bar {\theta }_0}}{{\textrm{d}}{t}} = \overline {E^*_{12}{B}}\cos {2\bar {\theta }_0} \end{equation}

and the angular dynamics evolves to an asymptotic state with $\cos {2\bar {\theta }_0}=0$ .

For the moment, we assume that $\overline {E^*_{12}}\neq 0$ and define $\mathcal{C} = \overline {E^*_{12}}/\overline {E^*_{12} B} \in [-1,1]$ as the reciprocal of the effective Bretherton parameter, the long-time asymptote for $\cos {2\bar {\theta }_0}$ . We also assume that the long time asymptote is a stable equilibrium and, thus, exclude edge cases of an initial condition at an unstable equilibrium for $\bar {\theta }_0$ . We then have that the long-time asymptote for $\bar {\theta }_0$ is either in the interval $\bar {\theta }_0\in (0,\pi /2)$ or the interval $\bar {\theta }_0\in (\pi ,3\pi /2)$ and, thus, the signs of $\sin \bar {\theta }_0$ and $\cos \bar {\theta }_0$ are the same for the long-time asymptote. With this, we have

(6.16a) \begin{align} \mathcal{A} &:= \left . (\overline {E^*_{12} I_{U c} }\sin {\overline {\theta }_0} + \overline {E^*_{12} I_{U c} B}\sin {\overline {\theta }_0}\cos {2\overline {\theta }_0}+V \cos {\overline {\theta }_0})\right \rvert _{\cos 2\bar {\theta }_0=\mathcal{C}} \end{align}

(6.16b) \begin{align} &= \pm \left [\frac {\sqrt {1-\mathcal{C}}}{\sqrt {2}} \left ( \overline {E^*_{12} I_{U c} }+\overline {E^*_{12} I_{U c} B}\mathcal{C} \right ) + \frac { V \sqrt {1+\mathcal{C}}}{\sqrt {2}} \right ], \end{align}

(6.16c) \begin{align} \mathcal{B} &:= \left . (\overline {E^*_{12}I_{U c}}\cos {\bar {\theta }_0} - \overline {E^*_{12}I_{U c}{B}}\cos {\bar {\theta }_0}\cos {2\bar {\theta }_0}+V \sin {\overline {\theta }_0})\right \rvert _{\cos 2\bar {\theta }_0=\mathcal{C}} \end{align}

(6.16d) \begin{align} &= \pm \left [ \frac { \sqrt {1+\mathcal{C}}}{\sqrt {2}}\left (\overline {E^*_{12}I_{U c}}-\overline {E^*_{12}I_{U c}{B}} \mathcal{C}\right ) +\frac {V \sqrt {1-\mathcal{C}}}{\sqrt {2}}\right ]. \end{align}

In turn, for large time, we have

(6.17) \begin{equation} \dfrac {{\textrm{d}}{\overline {y}_0}}{{\textrm{d}}{\overline {x}_0}} \approx \frac {\mathcal{B}}{\mathcal{A}+2\overline {E^*_{12}} \overline {y}_0 }. \end{equation}

Neglecting the asymptotically small errors in this approximation, one may integrate to give

(6.18) \begin{equation} \mathcal{B} \overline {x}_0 = \mathcal{K}+ \mathcal{A} \overline {y}_0+\overline {E^*_{12}} \overline {y}_0^2, \end{equation}

where $\mathcal{K}$ is a constant. Thus, excluding possible edge cases, the long-time swimmer trajectories are parabolic, regardless of the details of the flow and the swimming. One such edge case is $\mathcal{B}=0$ , which gives the limiting case of a line of constant $\overline {y}_0$ , for instance.

Finally, for the case where there is no net shear, we have $ \overline {E^*_{12}}=0$ and assume that $\overline {E^*_{12}B} \neq 0$ . Here, the trajectory is also linear, even though naive averaging would predict that the reciprocal swimmer has no net motion. In particular, we have $\mathcal{C}=0$ and $\mathcal{A}= \mathcal{B}$ , regardless of whether $V =0$ or $V \neq 0$ . Then we have

(6.19) \begin{equation} \dfrac {{\textrm{d}}{\overline {y}_0}}{{\textrm{d}}{\overline {x}_0}} \approx 1, \end{equation}

so that the trajectories are simply straight lines of unit gradient, independent of the details of the flow, swimmer and initial conditions, at least once the angular dynamics is asymptoting. Example such trajectories are shown for reciprocal swimmers in figure 4(b), with the associated angular dynamics illustrated in figure 4(c). This highlights two explicit examples of reciprocal swimmers in a zero-mean oscillating shear swimming across pathlines, breaking Purcell’s scallop theorem. Notably, extensive swimmer elongation is not required to be in this dynamical regime.

Figure 4. Behaviours of reciprocal swimmers ( $V=0$ ) in unsteady shear flow with zero average shear. (a) An illustration of the oscillatory flow field. (b) Swimmer trajectories within the oscillatory shear, with the associated angular dynamics and parameters shown in (c). The fast time scale oscillations are clearly visible, as well as the emergent long time behaviours, which can include net motion across pathlines despite reciprocal swimming. The numerical solution of the effective governing equations of (6.14) is shown for two parameter sets as black dashed curves, demonstrating excellent agreement with the average behaviour of the full system. In all cases, the orientation $\theta$ asymptotes to a root of $\cos 2\overline {\theta }_0$ , here $\pi /4$ , and the gradient of the swimmer trajectory asymptotes to unity, as predicted by the asymptotic analysis. Note that only two curves are visible in (c), as two parameter sets have identical averaged angular dynamics. These are universal predictions for swimmers with sufficient symmetry (such as maintaining a shape that is always a body of revolution with fore–aft symmetry).