Systematic parameterizations of minimal models of microswimming

Simple models are used throughout the physical sciences as a means of developing intuition, capturing phenomenology, and qualitatively reproducing observations. In studies of microswimming, simple force-dipole models are commonplace, arising generically as the leading-order, far-field descriptions of a range of complex biological and artificial swimmers. Though many of these swimmers are associated with intricate, time varying flow fields and changing shapes, we often turn to models with constant, averaged parameters for intuition, basic understanding, and back-of-the-envelope prediction. In this brief study, via an elementary multitimescale analysis, we examine whether the standard use of a priori-averaged parameters in minimal microswimmer models is justified, asking if their behavioural predictions qualitatively align with those of models that incorporate rapid temporal variation through simple extensions. In doing so, we find that widespread, seemingly innocuous choices of parameters can give rise to qualitatively incorrect conclusions from simple models, with the potential to alter our intuition for swimming on the microscale. Further, we highlight and exemplify how a straightforward asymptotic analysis of the non-autonomous models can result in effective, systematic parametrizations of minimal models of microswimming.

Figure 1

Geometry and parametrization of a minimal model of a swimmer above a no-slip boundary. The swimmer is parameterized by the angle $\theta$ between the vector dipole strength $\mathit{p}$ and the separation $h$ of its centroid from the boundary. The velocity due to self-propulsion is assumed to be along the direction of $\mathit{p}$, shared with swimmer-fixed director $\mathit{d}$.

Figure 2

Steady states and stability of angular evolution for the autonomous system of Eq. (10) are shown as dynamics on a circle, for $\alpha >0$ fixed and for three values of $\beta$. Swimming parallel to the boundary corresponds to states with $x=\pi /2, 3\pi /2$, while $x=0, \pi$ corresponds to swimming aligned with the normal to the boundary. (a) With $\beta <-2$, the system evolves to a steady state with $x=n\pi , n\in \mathbb{Z}$. (b) For $-2<\beta <-1, x=n\pi /2$ are stable for all $n\in \mathbb{Z}$, with unstable configurations present between these attractors. (c) For $\beta >-1$, the system evolves to a steady state with $x=\pi /2+n\pi$, for $n\in \mathbb{Z}$. Stable states are shown as solid points, while unstable points are shown hollow. Stabilities for $\alpha <0$ are obtained by reversing the illustrated dynamics.

Figure 3

Angular evolution of a swimmer at fixed separation from a no-slip boundary. The prediction of the a priori-averaged model can be seen to not align with the prediction of the systematically averaged model of Eq. (8) or the dynamics of the full model of Eq. (2), with qualitatively distinct steady configurations from the same initial condition. Rapid oscillations in the full dynamics are not visible at the resolution of this plot. Here, we have fixed $h=1, \omega =100$, and taken $p\left(T\right)=-6sin\left(T\right)+1$ and $B\left(T\right)=sin\left(T\right)/2$. Schematics of the long-time configurations are shown inset.

Figure 4

Geometry and parametrization of interacting dipole swimmers undergoing mirror-symmetric motion, equivalent to the motion of one dipole in the presence of a free-slip boundary. The orientation of the swimmers is described by an angle $\theta$ between the vector dipole strength $\mathit{p}$ and the boundary normal, with the separation between each swimmer and the plane of mirror symmetry being denoted by $h$. As in the previous section, the swimming velocity of each particle is assumed to align with the vector dipole strength.

Figure 5

Steady states and stability of angular evolution for the autonomous system of Eq. (18) are shown as dynamics on a circle, for $\alpha >0$ fixed and for various values of $\beta$. Swimming parallel to one another corresponds to states with $x=\pi /2, 3\pi /2$, while $x=0, \pi$ corresponds to swimming toward or away from the other swimmer. (a) For $\beta <-1, x=n\pi /2$ are stable for all $n\in \mathbb{Z}$, with unstable configurations present between these attractors. (b) For $\beta >-1$, the system evolves to a steady state with $x=\pi /2+n\pi$, for $n\in \mathbb{Z}$. Stable states are shown as solid points, while unstable points are shown hollow. Stabilities for $\alpha <0$ are obtained by reversing the illustrated dynamics.

Figure 6

Angular evolution of a swimmer at fixed separation from a free-slip boundary. The prediction of the a priori-averaged model can be seen to not align with the prediction of the systematically averaged model of Eq. (15) or the dynamics of the full model of Eq. (14), with qualitatively distinct steady configurations from the same initial condition. Here, we have fixed $h=1, \omega =10$, and taken $p\left(T\right)=6sin\left(T\right)-1$ and $B\left(T\right)=sin\left(T\right)/2$. Schematics of the long-time configurations are shown inset.

Figure 7

Geometry of fixed-position dipole rollers. Each roller is pinned in place at its center, but is free to rotate, driven by the dipolar flow generated by the other. Their orientation is captured by their directors ${\mathit{d}}_1$ and ${\mathit{d}}_2$, parameterized by ${\theta }_1$ and ${\theta }_2$, respectively. The axis of each roller's dipolar flow field is assumed to be directed along their orientation vector.

Figure 8

Fixed points of the two-roller system. Up to symmetry and $\pi$-periodicity, the identified fixed points correspond to four distinct configurations whose stability depend on $\alpha$ and $\beta$ nontrivially. Here, we report the stability for $\alpha >0$, corresponding to a swimmer with $\overline{p}>0$ in Eq. (29), which we might refer to as pusher-type particle. The dynamics of a puller-type particle, with $\alpha <0$, correspond to reversing the direction of motion from $\alpha >0$, swapping the stability of nonsaddle fixed points. (a) Parallel alignment along the direction of separation is stable if $\beta <-1$ and unstable for $\beta >-1$. (b) Orthogonal alignment, with one particle pointing directly toward the other, is stable only if $\beta >0$. (c) Parallel alignment that is perpendicular to the relative displacement is only stable if $\beta <-1/2$. (d) Steady parallel alignment that is neither parallel nor perpendicular to the relative displacement is unstable when it exists, which is for $\beta <-1/2$ and $\beta >1/3$.

Figure 9

Diverse phase portraits of the autonomous system of Eq. (30). Varying $\beta$, we illustrate the qualitatively distinct portraits of the autonomous dynamics, with stable and unstable fixed points shown as solid and empty circles, respectively. Moving from (a) to (b), we identify the transition of (0,0) from a stable node to a saddle point via the coalescence of two saddles on the $x=-y$ manifold; following this transition, the parallel alignment of Fig. 8 is the globally attracting state. Moving from (b) to (c) and from (h) to (g) transitions stable/unstable nodes at $(0,\pi /2)$ and $(\pi /2,0)$ to spirals of the same stability. Significant qualitative changes occur through bifurcations of the saddles at (0,0) and $(\pi /2,\pi /2)$, each into a node and two saddles, visible between panels (c) and (d) and panels (f) and (g). The blue trajectories in each panel, which have a common initial condition, highlight the qualitative change in behavior that can result from changing $\beta$, potentially altering the dynamics from the almost-heteroclinic orbits of panel (d) to the distinct stable attractors of panels (c) and (f), for instance. Here, we have fixed $\alpha >0$, noting that $\alpha <0$ corresponds purely to a reversal of the dynamics.

Figure 10

Comparing the results of a priori and systematic averaging. With fixed initial conditions of $(\pi /8,0)$, numerical solutions of (a) the a priori-averaged system of Eq. (31), (b) the systematically averaged dynamics of Eq. (29), and the full system of Eq. (22) are shown as solid curves. (a) With $p\left(T\right)=-10sinT+1$ and $B\left(T\right)=(sinT)/4+1/2$, the a priori-averaged dynamics approach the $(0,\pi /2)$ steady state, following the dynamics illustrated in Fig. 9, as $\beta =\overline{B}=1/2$. (b) The leading-order, systematically derived dynamics evolve to a distinct steady state, with ${\theta }_{1,0}=-{\theta }_{2,0}=\pi /2$, following the dynamics shown in Fig. 9, as $\beta =\overline{pB}/\overline{p}=-3/4$. The corresponding steady state configurations of the rollers are illustrated in the lower right corner of each panel, highlighting the qualitatively distinct behaviours predicted by the models. Here, we have taken $\omega =200$.