1. Introduction
Swimming by shape changes at low Reynolds number (LRN) is widely used in biology and micro-robotic design. In this flow regime, inertial effects are negligible, and the micro-organisms or micro-robots propel themselves by exploiting the viscous resistance of the fluid. For example, while a scallop that can only open or close its shell can swim in the ocean by accelerating the surrounding water, such a swimming strategy does not work at LRN, which is generalized by the principle: any time-reversible swimming stroke leads to no net translation at LRN Newtonian fluid, known as the
scallop theorem [
1].
It is important to understand how the performance of swimming depends on the geometric patterns of shape deformations for micro-swimmers. In microbiology, to fight the viscous resistance, different microorganisms adopt various propulsion mechanisms and directed locomotion strategies for searching for food and running from predators. For example, individual cells such as bacteria find food by a combination of taxis and kinesis using a flagellated or ciliated mode of swimming [
2,
3,
4,
5]. Recently, it was discovered that Dd cells can occasionally detach from the substrate and stay completely free in suspension for a few minutes before they slowly sink; during the free suspension stage, cells continue to form pseudopods that convert to rear-ward moving bumps, thereby propelling the cell through the surrounding fluid in a totally adhesion- free fashion [
6]. Human neutrophils can swim to a chemoattractant fMLP (formyl-methionylleucyl-phenylalanine) source at a speed similar to that of cells migrating on a glass coverslip under similar conditions [
7]. Most recently and equally striking, Drosophila fat body cells can actively swim to wounds in an adhesion-independent motility mode associated with actomyosin-driven, peristaltic cell shape deformations [
8]. In micro bio-engineering, medical microrobots revolutionize many aspects of medicine in recent years, which make existing therapeutic and diagnostic procedures less invasive [
9]. Different LRN swimming models and micro-robots have been designed since Purcell’s two-hinge model was advanced [
1]. In particular, various linked-sphere types of models have appeared, since their simple geometry permits both analytical and computational results [
10,
11,
12,
13,
14,
15,
16,
17]. These analytical and numerical results have greatly inspired the designs of micro robotic devices, for example, swimmers with 2 and 3 rotatory cylinders have been built to study the hydrodynamic interaction between a wall and an active swimmer [
18,
19]. Other micro-robots inspired by analytical/numerical works include the Quadroar swimmer, which consists of rotating disks and a linear actuator [
20], as well as the Purcell’s two-hinge model [
21].
An important problem in LRN swimming is to find the optimized swimming stroke of the micro-swimmer, either (1) with respect to time, i.e., the stroke in one swimming cycle that moves the cell farthest, or (2) with respect to energy, i.e., among all strokes with designated starting and end points, find the one that consumes the least energy. These are usually called the
time optimal control and the
energy optimal control problems, respectively. Both optimal problems have attracted substantial interest in optimization as well as geometry [
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33]. Moreover, recently techniques from machine learning have been applied to linked-sphere types of model gait design, which allows incorporating environmental influences on the micro-swimmer’s swimming behavior, including noise and a frictional medium [
34].
Here by investigating the optimal strokes of a group of linked-sphere types of LRN modeling swimmers, we study the efficiencies of propelling mechanisms at LRN of different types of shape deformations. We start from three linked-sphere swimmers–Najafi-Golestanian (NG) 3-sphere accordion model (
Figure 1a) [
10,
11,
12],
pushmepullyou (PMPY) 2-sphere model (
Figure 1b) [
15], and the volume-exchange (VE) 3-sphere model (
Figure 1c) [
17]. All three models have only 2 degrees of freedom in their shape deformations, which is a minimal requirement that enables the swimmer to propel itself at LRN, according to the
scallop theorem [
1]. In particular, the shape deformations in the three models can be generalized as body elongation and/or mass transportation, and we investigate the efficiencies of these two shape deformation modes on the swimming performances. Then we generalize our results into a linear chain of spheres. The results are presented as follows: in
Section 2 we present a brief introduction of the LRN swimming problem; in
Section 3 we review the three existing linked-sphere types of swimmers: NG 3-sphere, PMPY 2-sphere and VE 3-sphere models and discuss their optimization problems in
Section 4; finally in
Section 5 we discuss the optimization problem of models consisting of a chain of spheres.
2. Swimming at LRN by Shape Changes-the Exterior Problem
The Navier-Stokes equations for an incompressible fluid of density
, viscosity
, and velocity
are
where
is the external force field. Herein we assume that the swimmer is self-propelled and does not rely on any exterior force, and therefore we require that
. The Reynolds number based on a characteristic length scale
L and speed scale
U is
, and when converted to dimensionless form and the symbols re-defined, the equations read
Here
is the Strouhal number and
is a characteristic frequency of the shape changes. When
the convective momentum term in Equation (
1) can be neglected, but the time variation requires that
. When both terms are neglected, which we assume throughout, the flow is governed by the Stokes equations:
We also consider the propulsion problem in an infinite domain and impose the condition on the velocity field.
In the LRN regime time does not appear explicitly, momentum is assumed to equilibrate instantaneously, and bodies move by exploiting the viscous resistance of the fluid. As a result, time-reversible deformations produce no motion, which is the content of the
scallop theorem [
1]. For a self-propelled swimmer, there is no net force or torque, and therefore movement is a purely geometric process: the net displacement of a swimmer during a stroke is independent of the rate at which the stroke is executed, as long as the Reynolds and Strouhal numbers remain small enough.
Suppose that a swimmer occupies the closed compact domain
, at time
t, and let
denote its prescribed time-dependent boundary. A
swimming stroke is specified by a time-dependent sequence of the boundary
, and it is
cyclic if the initial and final shapes are identical, i.e.,
where
T is the period [
35]. The swimmer’s boundary velocity
relative to fixed coordinates can be written as a part
that defines the intrinsic shape deformations, and a rigid motion
. If
denotes the velocity field in the fluid exterior to
, then a standard LRN self-propulsion problem is:
Given a cyclic shape deformation
that is specified by
, solve the Stokes equations subject to:
In order to treat general shape changes of a swimmer defined by
with boundary
, one must solve the exterior Stokes Equations (
2) for
, with a prescribed velocity
on
and subject to the decay conditions
and
as
. The solution has the representation:
where
is the free-space Green’s function,
is the associated third-rank stress tensor,
is the exterior normal and
is the force on the boundary [
36]. The constraints that the total force and the total torque vanish determine the center-of-mass translational and angular velocities. When
, this is an integral equation for the force distribution on the boundary, the solution of which determines the forces needed to produce the prescribed shape changes.
6. Discussion
A successful and efficient locomotory gait design is important for micro-organisms and micro-robot who live or present in a viscous fluid environment. While bacteria often adopt a flagellated or ciliated mode of swimming strategy, amoeboid cells deform their cell bodies to propel themselves, resisting the viscous resistance from surrounding fluid. Such shape deformations can be generalized into two modes: stretch or elongation of a part of or the whole cell body, and mass transportation along the the cell body which does not greatly change the cell length. Interestingly, in bio-engineering, there are three linked-sphere LRN swimming models (NG 3-sphere, PMPY 2-sphere and VE 3-sphere models) that adopt a uni- or mixed modes of the aforementioned two shape deformation changes. By analyzing the optimal strokes of the three swimmers, we show that PMPY which adopts the mixed control is the most efficient among the three. We also consider models consisting a chain of spheres, and again we find that the PMPY-type of swimmers that use mixed controls are more efficient than NG-type of swimmers which use uni-controls of length change. We also find that generally speaking, the swimming efficiency decreases as the number of spheres increases, implying that more degrees of freedom in shape deformations is not a good strategy in optimal gait design. When the sphere separation distance increases, the efficiencies of NG type of swimmers greatly decrease, while the efficiencies of PMPY type of swimmers decrease in swimmers with many spheres, but are not affected much for swimmers with less spheres.
The findings in our paper can be potentially applied in the design of micro-robots with more complex structures. In addition, the numerical scheme presented here can be applied to more advanced LRN swimming systems. For example, it can be applied for general 2D and 3D swimmers, when the swimmer shapes can be represented by conformal mappings or spherical harmonics [
35,
56,
57,
58].
Moreover, for micro-organisms, their swimming behaviors are also significantly affected by environmental factors in biological media, therefore complex rheology of the surrounding fluid should be considered. A starting point might be bringing the linked-sphere swimmers into viscoelastic medium. We would like to point out that asymptotic analysis results for NG 3-sphere swimmer in linearized viscoelastic fluid have been obtained [
13].