Squirmers with arbitrary shape and slip: modeling, simulation, and optimization

TL;DR

This work develops a Helmholtz-based framework to describe tangential slip on arbitrarily shaped squirmers with spherical topology, enabling a boundary-harmonic expansion of slip and a tractable forward problem via a six-dimensional linear system to obtain $(\mathbf{U},\boldsymbol{\Omega})$. It proves that time-independent slip produces a circular-helix trajectory with axis along $\boldsymbol{\Omega}$ and provides explicit expressions for the translational and rotational velocities in the prolate-spheroid case, highlighting how aspect ratio shapes these motions. The authors then tackle the inverse problem of minimizing power loss under a prescribed net-motion direction and, separately, a global optimization over motion direction, revealing that symmetry planes strongly influence whether optimal motion is straight or helical. The results offer design principles for energy-efficient, chirality-enabled artificial microswimmers and lay groundwork for future work on confinement, boundaries, and collective dynamics, with broader applicability to non-axisymmetric swimmer geometries.

Abstract

We consider arbitrary-shaped microswimmers of spherical topology and propose a framework for expressing their slip velocity in terms of tangential basis functions defined on the boundary of the swimmer using the Helmholtz decomposition. Given a time-independent slip velocity profile, we show that the trajectory followed by the microswimmer is a circular helix. We derive analytical expressions for the translational and rotational velocities of a prolate spheroid swimmer in terms of its Helmholtz decomposition modes and explore the effect of aspect ratio on these rigid body velocities. Then, for a given arbitrary swimmer shape of spherical topology, we investigate which slip profile minimizes the total power loss. A partial minimization is performed in which the direction of net motion of the swimmer is prescribed, followed by a global optimization procedure in which the best net motion direction is determined. The optimization results suggest that the competition between linear and rotational optimal motion is linked to symmetries in the shape of the microswimmer.

Figures (14)

• Figure 1: Summary of first-order modal behavior. Slip velocity profiles and resultant rigid body velocities when all modes are set to zero except (a) $a_1^0 = 1$ (left), $a_1^1 = 1 = -a_1^{-1}$ (center), and $a_1^1 = \mathrm{i} = a_1^{-1}$ (right) and (b) $b_1^0 = 1$ (left), $b_1^1 = 1 = -b_1^{-1}$ (center), and $b_1^1 = \mathrm{i} = b_1^{-1}$ (right). Colour bar represents the magnitude of the normalised slip velocity.
• Figure 2: Shape dependence of rigid body velocities due to each mode. (a) Plots of $U_x$, $U_y$, and $U_z$ versus reduced volume $\nu$ when all modes are set to zero except $a_1^1 = 1 = -a_1^{-1}$ (top), $a_1^1 = \mathrm{i} = a_1^{-1}$ (middle), and $a_1^0 = 1$ (bottom), respectively. (b) Plots of $\varOmega_x$, $\varOmega_y$, and $\varOmega_z$ versus reduced volume $\nu$ when all modes are set to zero except $b_1^1 = 1 = -b_1^{-1}$ (top), $b_1^1 = \mathrm{i} = b_1^{-1}$ (middle), and $b_1^0 = 1$ (bottom). Spheroids have constant surface area $4 \pi$. As $\nu$ decreases, $U_z$, $\varOmega_x,$$\varOmega_y$, $\varOmega_z$ all decrease in magnitude, while $U_x$ and $U_y$ increase in magnitude.
• Figure 3: Plots of swimmer motion as well as the translational velocity $\hbox{\boldmath $U$}$ and its components $\hbox{\boldmath $V$}$ and $\hbox{\boldmath $W$}$ for (a) helical motion with $\hbox{\boldmath $\varOmega$} \neq \hbox{\boldmath $0$}$ and $\hbox{\boldmath $V$} \neq \hbox{\boldmath $0$}$ and (b) straight line motion with $\hbox{\boldmath $\varOmega$} \times \hbox{\boldmath $U$} = \hbox{\boldmath $0$}$ and $\hbox{\boldmath $V$} = \hbox{\boldmath $0$}$.
• Figure 4: Partial slip optimization of the axisymmetric stomatocyte (with $\lambda = 0.4$) when $\hbox{\boldmath $W$} = \hbox{\boldmath $e$}_z$ points along its axis of symmetry. (a) Optimal slip velocity field plotted together with the direction of the net motion $\hbox{\boldmath $W$} = \hbox{\boldmath $U$}$ (red arrow). The angular velocity is computed to be $\hbox{\boldmath $\varOmega$} = \hbox{\boldmath $0$}$, resulting in rotation-free, straight line motion along $\hbox{\boldmath $W$}$. The colour bar represents $|\hbox{\boldmath $v$}^\textrm{S}|$. (b) Plot of $|\hbox{\boldmath $v$}^\textrm{S}|$ as a function of normalised arc length of the generating curve $s$. The arbitrary shape optimizer results (red circles) agree with the axisymmetric optimizer results of guo2021optimal (blue dotted line).
• Figure 5: Partial optimization when $\hbox{\boldmath $W$}$ lies within a symmetry plane. (a) Optimal slip velocity field for a non-axisymmetric triangular antiprism, when $\hbox{\boldmath $W$} = (0,0.7069,0.7073)$ (red arrow) lies within its $x=0$ symmetry plane. The translational velocity is $\hbox{\boldmath $U$} = \hbox{\boldmath $W$}$ and the angular velocity is $\hbox{\boldmath $\varOmega$} = \hbox{\boldmath $0$}$, resulting in rotation-free, straight line motion. Colour bar represents $|\hbox{\boldmath $v$}^\textrm{S}|$. (b) Resulting straight line motion in the direction of $\hbox{\boldmath $W$}$. Swimmer has been enlarged for illustrative purposes. (c) Body frame flow field streamlines in the $y = 0$ plane. The optimization resulted in a normalised minimum power loss of $\widehat{\mathcal{P}}(\hbox{\boldmath $W$})/(12 \pi R_0) = 1.1267$.