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Hydrodynamic interactions between a sedimenting squirmer and a planar wall

Published online by Cambridge University Press:  09 May 2025

Henry Shum Open the ORCID record for Henry Shum [Opens in a new window] ,
D. Palaniappan  and
Yuan-Nan Young Open the ORCID record for Yuan-Nan Young [Opens in a new window]
Henry Shum
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
D. Palaniappan
Affiliation:
Department of Mathematics and Statistics, Texas A&M University – Corpus Christi, TX 78412, USA
Yuan-Nan Young*
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
*
Corresponding author: Yuan-Nan Young, [email protected]
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Abstract

The hydrodynamic interactions between a sedimenting microswimmer and a solid wall have ubiquitous biological and technological applications. A plethora of gravity-induced swimming dynamics near a planar no-slip wall provide a platform for designing artificial microswimmers that can generate directed propulsion through their translation–rotation coupling near a wall. In this work, we provide exact solutions for a squirmer (a model swimmer of spherical shape with a prescribed slip velocity) facing either towards or away from a planar wall perpendicular to gravity. These exact solutions are used to validate a numerical code based on the boundary integral method with an adaptive mesh for distances from the wall down to 0.1 % of the squirmer radius. This boundary integral code is then used to investigate the rich gravity-induced dynamics near a wall, mapping out the detailed bifurcation structures of the swimming dynamics in terms of orientation and distance to the wall. Simulation results show that a squirmer may traverse the wall, move to a fixed point at a given height with a fixed orientation in a monotonic way or in an oscillatory fashion, or oscillate in a limit cycle in the presence of wall repulsion.

JFM classification

Complex Fluids: Active matter Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Supplementary material: File

Shum et al. supplementary material movie 1

Three-dimensional rendering of a simulation of a squirmer near a no-slip wall at $(z=0)$ with parameters $(B_1 = 1, \alpha = 5, \beta = -2.25, C^\mathrm{rep} = 10^4)$. The green line segment indicates the orientation of the squirmer, and the trailing red curve traces the trajectory of the center of the squirmer.
Download Shum et al. supplementary material movie 1(File)
File 1.7 MB
Supplementary material: File

Shum et al. supplementary material movie 2

Three-dimensional rendering of a simulation of a squirmer near a no-slip wall at $(z=0)$ with parameters $(B_1 = 1, \alpha = 5, \beta = -10, C^\mathrm{rep} = 10^4)$. The green line segment indicates the orientation of the squirmer, and the trailing red curve traces the trajectory of the center of the squirmer. This movie corresponds to the parameters and initial conditions used for the lower inset of figure 11.
Download Shum et al. supplementary material movie 2(File)
File 311.6 KB