Undulatory Propulsion at Milliscale on Water Surface

Abstract The oscillatory pitch motion at the leading edge of a millimeter‐scale flexible sheet on the water surface can generate undulatory locomotion for swimming, similar to a honeybee vibrating its wings for propulsion. The influence of various parameters on such swimming strategy remains unexplored. This study uses magnetic milliswimmers to probe the propulsion mechanics and impact of different parameters. It is found that this undulatory propulsion is driven by capillary forces and added mass effects related to undulatory waves of the milliswimmers, along with radiation stress stemming from capillary waves at the interface. Modifying the parameters such as actuation frequency, pitch amplitude, bending stiffness, and hydrofoil length alters the body waveform, thus, affecting the propulsion speed and energy efficiency. Although undulatory motion is not a prerequisite for water surface propulsion, optimizing body stiffness to achieve a proper undulatory waveform is crucial for efficient swimming, balancing energy consumption, and speed. The study also reveals that the induced water flow is confined near the water surface, and the flow structures evolve with varying factors. These discoveries advance the understanding of undulatory water surface propulsion and have implications for the optimal design of small‐scale swimming soft robots in the future.


Text S1: Calculation of the buoyant force
The heaviest part of the swimmer is the magnetic head, characterized by the volume of  ×  × ℎ = 5 × 2 × 1.5.It has been observed that the failure of the floating typically results from the sinking of the head.So we will primarily focus on analyzing the floating/sinking state of it.Let's assume the head totally submerges under the water surface with the meniscus pinned at four edges (Fig. S4a).In the static state, the total floating force can be estimated as: where   is the water density,  is the gravitational constant,  is the surface tension coefficient, and Θ = 109.4° is the contact angle (Fig. S3).The head is composed of a small cubic magnet with a density of around 7.6 mg/mm 3 and side length of 1 mm, and a polymer shell with a density of around 0.97mg/mm 3 .The floating force and the weight of the head in relation to the scaling factor is depicted in Fig. S4b.It shows that the head can scale up to around 2.3 times the current dimension, which corresponds to 257mg, without sinking at the static state.

Movie S1 (separate file).
Characterization of flow field on water surface.The swimmer has the body bending stiffness of 1.83 × 10 -10 Nm 2 and the body length of 10 mm.The pitch angle amplitude is 26° and the actuation frequency is 50 Hz.

Movie S2 (separate file).
Influence of changing pitch angle amplitude (  ) and frequency () on body's waveform and propulsion speed.

Movie S3 (separate file).
Influence of changing body's bending stiffness ( ) and frequency () on body's waveform and propulsion speed.

Movie S4 (separate file).
Influence of changing body's length () and frequency () on body's waveform and propulsion speed.

Fig. S1 .
Fig. S1.Characterization of the swimmer with stiff body.(A)The body of the swimmer was made of Mylar film with a thickness of 0.05 mm, resulting in a bending stiffness of around 1.56 × 10 -7 Nm 2 , which is orders of magnitude larger than the swimmers reported in the main text.We didn't observe a wavy shape even at a very large pitch angle amplitude.Scale bar: 2 mm.(B) The average propulsion speed of the swimmer.(C) The forces arising from the swimming.Note the vertical axis has a logarithmic scale.The force induced by wave is orders of magnitudes larger than the force due to added mass and the capillary force.

Fig. S3
Fig. S3 Contact angle measurement.(A) Sessile drop experiments were conducted on substrates made of Ecoflex 0010 and PDMS.Three liquids with known disperse and polar components of the surface tension were used.(B) The measurement results were linearly fitted.

Fig. S4
Fig. S4 Calculation of the buoyant force.(A) The force analysis of the floating object.The buoyant force is composed of the buoyancy, as per Archimedes' principle, and the normal component of the surface tension force.(B) The variation of the weight of the swimmer's head and maximum floating force the water surface can provide.The calculation assumes the swimmer's head is at the static condition.