Magnetic-field-induced propulsion of jellyfish-inspired soft robotic swimmers

The multifaceted appearance of soft robots in the form of swimmers, catheters, surgical devices, and drug-carrier vehicles in biomedical and microfluidic applications is ubiquitous today. Jellyfish-inspired soft robotic swimmers (jellyfishbots) have been fabricated and experimentally characterized by several researchers that reported their swimming kinematics and multimodal locomotion. However, the underlying physical mechanisms that govern magnetic-field-induced propulsion are not yet fully understood. Here, we use a robust and efficient computational framework to study the jellyfishbot swimming kinematics and the induced flow field dynamics through numerical simulation. We consider a two-dimensional model jellyfishbot that has flexible lappets, which are symmetric about the jellyfishbot center. These lappets exhibit flexural deformation when subjected to external magnetic fields to displace the surrounding fluid, thereby generating the thrust required for propulsion. We perform a parametric sweep to explore the jellyfishbot kinematic performance for different system parameters—structural, fluidic, and magnetic. In jellyfishbots, the soft magnetic composite elastomeric lappets exhibit temporal and spatial asymmetries when subjected to unsteady external magnetic fields. The average speed is observed to be dependent on both these asymmetries, quantified by the glide magnitude and the net area swept by the lappet tips per swimming cycle, respectively. We observe that a judicious choice of the applied magnetic field and remnant magnetization profile in the jellyfishbot lappets enhances both these asymmetries. Furthermore, the dependence of the jellyfishbot swimming speed upon the net area swept (spatial asymmetry) is twice as high as the dependence of speed on the glide ratio (temporal asymmetry). Finally, functional relationships between the swimming speed and different kinematic parameters and nondimensional numbers are developed. Our results provide guidelines for the design of improved jellyfish-inspired magnetic soft robotic swimmers.

Figure 1

Left: Solid and fluid domains, jellyfishbot—magnetic, solid, and fluid properties. Center: Note that the initial, undeformed configuration (without the magnetic field) is a flat plate. Right: Schematic representation of the jellyfishbot trajectory, boundary conditions, and the streamlines in the fluid. The external oscillating magnetic field is a uniform unsteady field in the $y$ direction with no spatial gradient, shown by the two-sided blue arrow.

Figure 2

(a) Variation of jellyfishbot remnant magnetization $M_r$ with respect to the curvilinear axis $s$; the black arrows indicate the direction of $M_r$ at specific segments along the jellyfishbot length. (b) Variation of ${\mathit{B}}_0$ with time $t$ indicating different phases of jellyfishbot swimming kinematics.

Figure 3

Convergence study on the reference case shown in Figs. 1, 2, 4, and 5 to ensure the numerical framework is stable: the system variable of interest (here, the normalized jellyfishbot swimming speed $C$) asymptotically attains a constant value with an increase in the number of beam elements (i.e., decreasing mesh size), and increase in the number of time steps (i.e., decreasing time step).

Figure 4

Chronological sequence of jellyfishbot configurations and surrounding vortex field through the streamlines and magnitude of the vortex field; ($·,·$) represents the applied magnetic field and the corresponding cycle time, respectively ($t, {\mathit{B}}_0$). For an animation of the swimmer, see movie S1 in the Supplemental Material [65].

Figure 5

Chronological sequence of jellyfishbot configurations and surrounding velocity field through the streamlines and magnitude of the velocity field; ($·,·$) represents the applied magnetic field and the corresponding cycle time, respectively ($t, {\mathit{B}}_0$). For an animation of the swimmer, see movie S2 in the Supplemental Material [65].

Figure 6

(a) Steady-state kinematics of the reference jellyfishbot swimmer of length $L=2.0\mathrm{mm}$ for ten cycles. (b) Jellyfishbot displacement showing the three individual phases for one cycle (time period is 0.2 s).

Figure 7

Variation of jellyfishbot average speed $c$ with respect to lateral and vertical proximity using a vertical proximity of 0.5 mm for the lateral variation, and a lateral proximity of 1.5 mm for the vertical variation.

Figure 8

Schematic representation of the area swept by the jellyfishbot lappets per swimming cycle: blue lines show the lappet trajectory.

Figure 9

Variation of $C$, NAS, and GR with respect to (a) $E$, (b) $\rho$, and (c) $h$.

Figure 10

Variation of $C$, NAS, and GR with respect to (a) $\mu$ and (b) ${\rho }_f$.

Figure 11

Schematic representation of the jellyfishbot undergoing ephemeral closure of flexible lappets, or self-locking, when the fluid viscosity is too low (a) or the magnetic field is too high (b).

Figure 12

Variation of $C$, NAS, and GR with respect to (a) ${\mathit{B}}_0$, (b) $M_{r,t}$, (c) $M_{r,n}$, and (d) $f_m$.

Figure 13

Variation of (a) $M_{r,n}$ and (b) $M_{r,t}$ along the curvilinear axis $s$ of the swimmer for different values of the phase.

Figure 14

Variation of $C$, NAS, and GR with respect to phase.

Figure 15

Variation of the remnant magnetization $M_r$ with respect to the curvilinear axis $s$ for different values of phase, and the corresponding lappet trajectories: (a) $\mathrm{phase}=1.875$ (beam mode 1), (b) $\mathrm{phase}=3$, (c) $\mathrm{phase}=4.694$ (beam mode 2), and (d) $\mathrm{phase}=7.855$ (beam mode 3).

Figure 16

Variation of $C$, NAS, and GR with respect to the phase for different modes (i.e., lappet deformation).

Figure 17

Schematic representation of the lappet trajectories (for one swimming cycle) and the corresponding NAS for different values of $M_{r,t}$: (a) 0.37 kA/m, (b) 15.37 kA/m, (c) 45.37 kA/m, and (d) 60.37 kA/m.

Figure 18

Schematic representation of the lappet trajectories (for one swimming cycle) and the corresponding NAS for different values of $M_{r,n}$: (a) 0.86 kA/m, (b) 10.86 kA/m, (c) 20.86 kA/m, and (d) 30.86 kA/m.

Figure 19

Optimized jellyfishbot swimming speed for a wide range of viscous fluids.

Figure 20

(a) Correlation of $C$ with a power-law combination of $I_n, M_n$, and $R_n$. (b) Correlation of $C$ with a linear combination of GR and NAS.