Swimming by reciprocal motion at low Reynolds number

Biological microorganisms swim with flagella and cilia that execute nonreciprocal motions for low Reynolds number (Re) propulsion in viscous fluids. This symmetry requirement is a consequence of Purcell’s scallop theorem, which complicates the actuation scheme needed by microswimmers. However, most biomedically important fluids are non-Newtonian where the scallop theorem no longer holds. It should therefore be possible to realize a microswimmer that moves with reciprocal periodic body-shape changes in non-Newtonian fluids. Here we report a symmetric ‘micro-scallop’, a single-hinge microswimmer that can propel in shear thickening and shear thinning (non-Newtonian) fluids by reciprocal motion at low Re. Excellent agreement between our measurements and both numerical and analytical theoretical predictions indicates that the net propulsion is caused by modulation of the fluid viscosity upon varying the shear rate. This reciprocal swimming mechanism opens new possibilities in designing biomedical microdevices that can propel by a simple actuation scheme in non-Newtonian biological fluids.

. Apparent dynamic viscosity of the shear thickening fluid. Power law model (dotted line) is used to fit the viscosity (black squares) in the shear rate range of 1.5~6 s -1 . The change of first normal stress difference N 1 of our shear thickening medium is two orders of magnitude smaller than the Boger fluid used in 8 . Thus, the viscosity change is dominant during the swimming process. The error bars represent standard deviations.
Supplementary Fig. 8. Oscillation test of the shear thickening fluid. The viscous modulus G'' is more than 2 times larger than the elastic modulus G' over the frequency range of 0.1~20 Hz, and the phase angle is 70º ~80º, which both indicate that the viscosity is dominant over elasticity for the shear thickening fluid.
Supplementary Fig. 9. Apparent dynamic viscosity of the shear thinning fluid. Power law model (dotted line) is used to fit the data in the shear rate range of 1~100 s -1 . The error bars represent standard deviations.
Supplementary Fig. 10. Hysteresis of viscosity of the shear thickening fluid. When the shear rate increases from a low viscosity (blue squares), the transition occurs at a higher critical shear rate, while when the shear rate decreases from a high viscosity (red triangles), the transition takes place at a lower shear rate. The error bars represent standard deviations.
Supplementary Fig. 11. The vertices of the computational mesh are concentrated near the surface of the macroscallop to improve resolution of the liquid-solid interface. The color in the lower half shows the velocity field of the fluid.
Supplementary Fig. 12. Approximation of the viscosity for shear thickening fluid for convergence in numerical simulation. Red circles are measured data points in experiment, blue line is the trend line, and the black line is the approximation used in numerical simulation.
Supplementary Fig. 16. Interface-free Micro-scallop swimming test in shear thickening fluid while falling under gravity. (a) Schematic showing the actuation setup for the micro-scallop, which is not drawn to scale. The microscallop is immersed in the fluid, far away from the meniscus and all walls. The propulsion is in the X direction, which is independent from the falling motion caused by gravity in the Y direction. The video is taken by camera from the side. The time interval between dots is 20 s (5 strokes). The swimmer has a propulsion speed of 5.2±1.4 µm/s under asymmetric actuation, which is similar to the results observed when the swimmer is suspended by the air-liquid meniscus. In cases of symmetric actuation or no actuation, the swimmer falls vertically and shows no net displacement in X direction.

Supplementary Note 1. Numerical simulation of the macro-scallop
The numerical simulations of the macro-scallop in non-Newtonian and Newtonian fluids were conducted using the open-source CFD (Computational Fluid Dynamics) package FeatFlow (www.featflow.de). We configured the 3D simulation to use a pseudo 2D setup which means that the thickness of the swimmer and the computational domain are reduced. Thus the number of degrees of freedom in the simulation is significantly reduced as is the computational cost. In the following we will briefly explain the numerical methods used to simulate the macroscallop swimmers in Newtonian and non-Newtonian fluids. In our FeatFlow software the fluid is modelled by the incompressible Navier-Stokes equations which can be formulated as where we denote the constant density by , the shear dependent viscosity by , the unknown velocity and pressure by the pair ⃗ , and the viscous stress tensor by , in which is the rate-of-strain tensor.
This system of equations is discretized using the Finite Element Method (FEM) which is implemented in our The system is then discretized in space using the Galerkin variational formulation of the Navier-Stokes equations. In our FEM framework we use the higher order element pair for the spatial discretization, further aspects of the  Fig. 11).

Supplementary Note 2. Modulation of viscosity as underlying mechanism of propulsion
Let us consider an idealized tethered "scallop" (pump) composed of two infinite plates forming an angle between them. The plates are co-rotating on a common axis with an arbitrary angular velocity . For simplicity we assume that the suspending medium is a Newtonian fluid. It is possible, however, to extend this solution and construct the asymptotic expansion corresponding to weakly non-Newtonian fluid (e.g. shear thinning or thickening) by the method of perturbations 2 , however for our qualitative purposes the leading order solution suffices.
We consider the problem in the polar coordinates The solutions for the streamfunction in the Stokes approximation satisfy . Following 3 and using an ansatz we find the solution for the 'inner' (in between plates) and 'outer' regions, respectively: On the plates at the boundary conditions are satisfied, i.e. The solution in the outer region has an analogous form. The streamlines (isolines of ) are depicted in Supplementary   Fig. 13a-d for four different openings, and . The corresponding vector velocity field is shown for illustration in Supplementary Fig. 13a. Note that the inner and outer solutions in Supplementary Equation (6) The result in Supplementary Equation (7) holds for the outer region with being replaced with . Note that for an infinite scallop is not a function of and depends solely on , i.e., for an arbitrary opening has a constant (but not equal) values along the plates inside and outside.
The corresponding plots of are given in Supplementary Fig. 14a-d  , ̇ where ̇ is the critical shear rate at steady conditions 5 .
It has been found first in 5 , then in 5, 6, 7 that the transition taking place at at ̇ ̇ possesses the properties of the The hysteresis of viscosity explains the asymmetry in propulsion upon exchanging between (fast) closing and opening strokes. Since opening and closing strokes are not identical (at opening the angle between plates changes from 10° up to 295° and at closing it decreases from 295° down to 10°). For (fast) opening, the shear rate decreases from a high viscosity state, and for the (fast) closing cycle the shear rate increases. In agreement with the hysteresis described in 5 the "scallop" propelled by (fast) opening stroke swims better than the one than relies on (fast) closing stroke.
The hysteresis of the shear thickening fluid was measured via a shear rate ramp loop test first from up to and then immediately from down to . The loop was repeated for three times and the average is plotted in Supplementary Fig. 10. In agreement with the previsous works, the transition occurs at a higher critical shear rate when the shear rate increases from a low viscosity (blue squares), and vice versa a lower critical shear rate when decreasing (red triangles).
This hysteresis explains the reason that the average displacement per cycle of the backward stroke is larger than that of the forward stroke, under the same ratio of closing and opening (the absolute value of pre-factor in Fig. 4c is larger than in Fig. 4b). Specifically, the closing and opening strokes of the macro-scallop are not identical: at closing, as the gap between the shells changes from large to small, the shear rate increases and thus the transition of shear thickening occurs at a higher critical shear rate; vice versa at opening, the gap increases, the shear rate decreases, and the critical shear rate is lower. Therefore, in the opening half-cycle, the swimmer exhibits higher average viscosity than that in the closing half-cycle, and consequently results in better propulsion.