Hydrodynamic interactions between a sedimenting squirmer and a planar wall

TL;DR

The study addresses how gravity modifies the hydrodynamics of a spherical squirmer near a planar wall by deriving an exact axisymmetric solution for a squirmer oriented perpendicular to the wall and validating a boundary-integral solver. It then maps gravity-induced dynamics in the parameter plane defined by $\alpha=V/V_g$ and $\beta=B_2/B_1$, revealing regimes of escape, pinned equilibrium, wall sliding, and oscillatory bouncing, with bifurcations between stable spirals, stable nodes, and limit cycles. Far-field and near-field expressions for the squirmer speed are developed to connect analytic limits with numerical results. These insights inform the design of wall-following and obstacle-avoiding microswimmers and open avenues for exploring more complex fluids and porous-media environments.

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 provides 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 transverse along 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.

Figures (13)

• Figure 1: Schematic of a spherical squirmer of radius $R$ at a height $h$ above a no-slip planar boundary with gravity pointing toward the wall.
• Figure 2: Comparison of the series expression \ref{['eq:Squirmer_Speed_U']} with series truncated to 5000 terms, boundary integral simulation solutions, the near-field formula \ref{['eq:Cooley_NearField']}, and the far-field formula \ref{['eq:U_g']} for the vertical speed $v=U^\mathrm{grav}/V_g$ of a passive sphere ($B_1 = B_2 = 0$) sedimenting under gravity near a no-slip wall without repulsion. Parts (a) and (b) display the same quantities focusing over different ranges of separation from the wall. (c) Errors with respect to the series solution.
• Figure 3: Comparison of the series expression \ref{['eq:Squirmer_Speed_U']} with series truncated to 5000 terms, boundary integral simulation solutions, the near-field formula \ref{['eq:our_NearField']}, and the far-field formula \ref{['eq:U_g_B1_B2']} for the vertical speed of a neutral squirmer ($B_1 = 1, B_2 = 0, V_g=0$) perpendicular to and near a no-slip wall without repulsion. Parts (a) and (b) display the same quantities focusing over different ranges of separation from the wall. (c) Errors with respect to the series solution.
• Figure 4: Comparison of the series expression \ref{['eq:Squirmer_Speed_U']} with series truncated to 5000 terms, boundary integral simulation solutions, the near-field formula \ref{['eq:our_NearField']}, and the far-field formula \ref{['eq:U_g_B1_B2']} for the vertical speed of a contractile squirmer ($B_1 = 0, B_2 = 1, V_g=0$) perpendicular to and near a no-slip wall without repulsion. Parts (a) and (b) display the same quantities focusing over different ranges of separation from the wall. (c) Errors with respect to the series solution.
• Figure 5: Long-time behaviors of a squirmer under gravity next to a flat wall, with the squirmer initially pointing nearly vertically downwards ($\theta(t=0)=-0.99\pi/2$) at a starting height $h/R=10$. The squirmer is not bound to the wall in the "Escape" region. For squirmers bound to the wall under gravity, they settle to either a steady state at a fixed height with a steady tilt angle sliding along the wall (red--blue color bar), or they oscillate in the "Oscillations" region where the squirmer-wall distance oscillates with amplitudes in height indicated by the right (green--yellow) color bar. Negative values of $\alpha$ signify that gravity acts vertically away from the wall. The inset shows the detailed distribution of swimming dynamics for $0<\alpha<0.15$ and $-10<\beta<-2$.