Thrust and Hydrodynamic Efficiency of the Bundled Flagella

Abstract

The motility mechanism of prokaryotic organisms has inspired many untethered microswimmers that could potentially perform minimally invasive medical procedures in stagnant fluid regions inside the human body. Some of these microswimmers are inspired by bacteria with single or multiple helical flagella to propel efficiently and fast. For multiple flagella configurations, the direct measurement of thrust and hydrodynamic propulsion efficiency has been challenging due to the ambiguous mechanical coupling between the flow field and mechanical power input. To address this challenge and to compare alternative micropropulsion designs, a methodology based on volumetric velocity field acquisition is developed to acquire the key propulsive performance parameters from scaled-up swimmer prototypes. A digital particle image velocimetry (PIV) analysis protocol was implemented and experiments were conducted with the aid of computational fluid dynamics (CFD). First, this methodology was validated using a rotating single-flagellum similitude model. In addition to the standard PIV error assessment, validation studies included 2D vs. 3D PIV, axial vs. lateral PIV and simultaneously acquired direct thrust force measurement comparisons. Compatible with typical micropropulsion flow regimes, experiments were conducted both for very low and higher Reynolds (Re) number regimes (up to a Re number = 0.01) than that are reported in the literature. Finally, multiple flagella bundling configurations at 0°, 90° and 180° helical phase-shift angles were studied using scaled-up multiple concentric flagella thrust elements. Thrust generation was found to be maximal for the in-phase (0°) bundling configuration but with ~50% lower hydrodynamic efficiency than the single flagellum. The proposed measurement protocol and static thrust test-bench can be used for bio-inspired microscale propulsion methods, where direct thrust and efficiency measurement are required.

Keywords: bacteria locomotion; computational fluid dynamics; flagellar propulsion; microswimmer; particle image velocimetry.

Figure A1

Computational domain for the CFD simulations of the flagella propulsion (a) comprised of inner rotating reference frame (red rectangle) and the stationary frame (black rectangle). Helical flow structures around the counter-rotating flagella mapped on the uniform pressure field plotted at the center plane of the flagella (b).

Figure 1

The graphic of the experimental set-up. Only the axial laser sheet configuration is displayed while the lateral 3D particle image velocimetry (PIV) data is acquired for validation and verification purposes using the mirror. PTU: Pulse timing unit.

Figure 2

All flagellar configurations tested in the present study. D1: Baseline single-flagellum model. D2: Overlapping bundle state of two nearby flagella. D3: Two flagella with 90-degree relative phase shift. D4: Two flagella with 180-degree relative phase shift. λ =10 mm, A = 3.175 mm.

Figure 3

Volumetric PIV data acquisition and analysis methodology. (a) Structured C-grid generated for radial (2C/3C) PIV data interpolation. (b) 20 PIV slices and (c) interpolated volumetric data. (d) Sketch of the control volume that spans one pitch of flagella segment for thrust and dissipation calculation and corresponding control surfaces.

Figure 4

Comparison of radial (V_x) and axial (V_y) velocity components measured and computed with PIV measurements and computational fluid dynamics (CFD) simulations, respectively for D1 (single helix) configuration.

Figure 5

The axial velocity component, V_y in the center plane acquired using PIV measurements (top row) and CFD simulations (bottom row). Compared to the single flagellum, a slight increase is observed for the in-phase (Φ = 0) double flagella configuration. In comparison, the two out-of-phase (Φ = 90 and Φ = 180) double flagella configurations demonstrated a decrease in V_y affecting the thrust production.

Figure 6

CFD simulations for radial velocity V_x plotted in the center-plane, increases from single flagellum (a) to in-phase double flagella configuration (b) and decreases for the out-of-phase double flagella configuration (c).

Figure 7

CFD simulations for tangential velocity V_z distribution along the center axis of the three major flagella configurations. It shows both inclusion of additional flagellum and phase angle between the flagella increase the flow momentum that is convected in the radial direction. Radial momentum decayed approximately 3, 3^1/2 and 4 helix diameters away from the single flagellum (a), in-phase (b) and out-of-phase (c) double flagella configurations, respectively.

Figure 8

Pressure contours are plotted at the center-plane along the flagella axis for (a) single flagellum, (b) in-phase, (c) out-of-phase double flagella configurations inside less viscous oil (350 cSt) and (d) out-of-phase double flagella inside high viscous oil (30,000 cSt). Pressure drop across the flagella increased slightly for the out of-phase configuration and notably for the high viscous case compared to the single flagellum.

Figure 9

Thrust force comparisons chart: Big chart compares the thrust force for D1 (single) and D2 (overlapping) configurations rotating in 30,000 cSt silicone oil, and the inset chart depicts the thrust force for D1 configuration rotating in 350 cSt silicone oil with the same axes unit as the big chart. Thrust is computed and measured using PIV, load cell measurements and the RFT. Error bars correspond to simultaneous thrust measurements with 20 to 1 confidence level.

Figure 10

PIV thrust force results of the two bundled flagella for four different configurations in 30,000 cSt silicone oil. Error bars correspond to simultaneous thrust measurements with 20 to 1 confidence level.

Figure 11

Axial velocity (a), energy dissipation (b) and hydrodynamic efficiency (c) results for different two-flagellum configurations in 30,000 cSt silicone oil.