Low Reynolds number · Walls

Swimmer Near a Wall: Glide, Turn In or Escape

Why does a microswimmer near a wall glide along it, turn into it or swim away? A wall reflects the flow a swimmer makes, and the reflected flow turns it a little every stroke. In this model a swimmer level with the wall turns toward it, and the turn weakens as the fourth power of height. Started low, it rises and dips in a steady glide; started higher, it turns in and reaches the wall; started pointing away steeply enough, it escapes.

Release a three-sphere swimmer above a wall and follow thousands of strokes: its start height and tilt decide whether it glides, turns in or escapes.

Start height and tilt

Choose where it starts

Start height above the wall

h_0

10\,\mu\mathrm{m}

Start tilt toward the wall

\psi_0

0^{\circ}

Phase lag between the arms

\delta

90^{\circ}

Drag the swimmer up or down in the scene to pick its start height.

Time-lapse

Time-lapse speed

N

800\,\mathrm{cycles/s}

Strokes per second of animation. The arms are drawn slowed down to about one stroke per second so the motion stays readable; at 5 strokes per second a real swimmer would need hours for this path.

How the swimmer is computed

Three spheres of radius 1 µm on arms of mean length 10 µm. Each arm changes by 1 µm, harmonically, with the chosen phase lag, at 5 strokes per second. Every sphere’s velocity follows from the forces on all three through the wall-bounded Stokes flow; the swimmer as a whole exerts no force and no torque.

One stroke is integrated exactly in time (fourth-order Runge–Kutta, 16 steps). Its net change of position and tilt is then used as a rate per stroke and integrated over thousands of strokes in steps of at most 50 strokes.

Model assumptions

The wall is a no-slip plane. The spheres act on the water as point forces, and the wall reflects their flow through the Blake image system; each sphere’s own drag uses the wall correction to first order in radius over height. The model holds only while every sphere stays more than about two radii from the wall, so a run stops there: contact and lubrication forces are not included. The path is a time-lapse in which each step is the exact net motion of whole strokes. Brownian motion is left out.

Height above the wall

7.92\,\mu\mathrm{m}

Tilt toward the wall

7.33^{\circ}

Distance along the wall

29.9\,\mu\mathrm{m}

Strokes so far

1{,}600

Pink is the simulated swimmer. The right chart checks the model against a published far-field formula for a swimmer level with the wall; they agree within 1% at 12 arm lengths and part ways close to the wall, where the formula no longer applies.

More numbers and the raw record

Swimming time at 5 strokes per second $320\,\mathrm{s}$

Lowest sphere centre above the wall $6.65\,\mu\mathrm{m}$

Model turn over far-field formula, at 12 arm lengths $0.994$

The JSON record lists the model version, units, geometry, every start or parameter change, the sampled positions and tilts, and the far-field check values.

Sphere size, stroke and the far-field formula (Eq. 28c) follow Daddi-Moussa-Ider, Lisicki, Hoell and Löwen (2018). The wall tensor is Blake’s (1971), written as in their Eqs. 5–7. Read the 2018 paper

Physics tutorial

A wall turns a swimmer through the flow it makes

BackgroundBacteria and sperm cells collect near glass slides and channel walls. Nothing pulls them there: the water a swimmer pushes cannot slip along a solid surface, so the wall sends part of that flow back at the swimmer.

Why it mattersDaddi-Moussa-Ider, Lisicki, Hoell and Löwen followed the same three-sphere swimmer next to a wall and found three outcomes: trapped at the wall, escaping, or gliding at a steady mean height. This Lab uses their sphere size and stroke, with a simpler far-field description of the wall.

Start with the essentials

Focus question: Why does a microswimmer near a wall glide along it, turn into it or swim away?
One-sentence intuition: Level with the wall, the swimmer turns toward it, and the turn per stroke falls as the fourth power of height. Low down, each dip toward the wall is followed by a rise, so it glides; higher up, it turns in before anything pushes it back; pointed away steeply enough, it outruns the turn.

Core mathematical model

A point force next to a no-slip wall

\mathbf{v}(\mathbf{r})=\mathbf{G}^{\mathrm{W}}(\mathbf{r},\mathbf{r}_0)\cdot\mathbf{f},\qquad \mathbf{G}^{\mathrm{W}}=\mathbf{G}^{\mathrm{O}}(\mathbf{r}-\mathbf{r}_0)-\mathbf{G}^{\mathrm{O}}(\mathbf{r}-\mathbf{r}_0^{*})+\text{doublets at }\mathbf{r}_0^{*},\qquad \mathbf{v}\big|_{z=0}=0

Blake’s image system: the free-space flow of the force, minus a mirror copy below the wall, plus two corrections at the mirror point, makes the water stand still on the wall. The Lab couples the three spheres through this tensor.

Each sphere drags more near the wall

\mu_{\parallel}=\frac{1}{6\pi\eta a}\left(1-\frac{9a}{16h}\right),\qquad \mu_{\perp}=\frac{1}{6\pi\eta a}\left(1-\frac{9a}{8h}\right)

A sphere at height h moves more slowly under a given force, more so toward the wall than along it. These first-order terms stop being accurate within a few radii, which is why a run stops when a sphere comes within two radii.

Far from the wall: the turn falls as the fourth power of height

\frac{d\psi}{dt}\approx-\frac{9aK}{512\,z^{4}}\left(15+\frac{45a}{2}\right),\qquad K=-u_{10}u_{20}\sin\delta

Daddi-Moussa-Ider and co-workers derived this for a swimmer level with the wall, with lengths in units of the mean arm length and time in units of one over the stroke angular frequency. The lower chart compares it with the full model at each height.

Common difficulties

The wall does not attract by force

Typical misconceptionSomething must pull the swimmer onto the wall.

Better mental modelThe swimmer exerts no net force at any time. The wall only changes where the swimmer’s own flow goes, and that changed flow turns the swimmer’s axis; its own swimming then carries it in.

What this model leaves out

Typical misconceptionThe run shows what happens when the swimmer reaches the wall.

Better mental modelThe run stops when a sphere comes within two radii of the wall, because this model has no contact or lubrication forces. In the more detailed calculations of Daddi-Moussa-Ider and co-workers, a trapped swimmer ends up hovering at a fixed height, pointing almost straight at the wall.

Run the experiment

01
Watch the glide
Start with Glide along the wall and follow the height readout for several rises and dips.
What to observe: The height swings between about 6 and 10 µm and returns to the same maximum each time. The track chart shows a steady wave.
02
Start higher
Choose Start higher, turn into the wall, or drag the swimmer up to about 20 µm.
What to observe: It glides level at first, slowly tips toward the wall and does not pull out of the dive. The run stops when the lowest sphere reaches the model limit.
03
Point it away, then stop the loop
Choose Point away and escape. Then set the phase lag to zero.
What to observe: Tilted away by 26°, it climbs past the height where the wall could turn it back. With zero phase lag the arms move back and forth together and the swimmer stays put, wall or no wall.

Swimmer Near a Wall: Glide, Turn In or Escape

A wall turns a swimmer through the flow it makes

Core mathematical model

A point force next to a no-slip wall

Each sphere drags more near the wall

Far from the wall: the turn falls as the fourth power of height

Common difficulties

The wall does not attract by force

What this model leaves out

Run the experiment

Watch the glide

Start higher

Point it away, then stop the loop

Related experiments and articles

Swimming without inertia, from Purcell’s scallop to pairs of swimmers