---
title: "Swimmer Near a Wall: Glide, Turn In or Escape Interactive Lab | Sandbox Physics"
description: "Why does a microswimmer near a wall glide along it, turn into it or swim away?"
canonical_url: "https://sandboxphysics.com/labs/swimmer-near-wall/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/swimmer-near-wall/"
last_modified: "2026-10-29"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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](https://arxiv.org/abs/1801.01162)

Physics tutorial

## A wall turns a swimmer through the flow it makes

**Background** Bacteria 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 matters** Daddi-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 misconception** Something must pull the swimmer onto the wall.

**Better mental model** The 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 misconception** The run shows what happens when the swimmer reaches the wall.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Near a wall, a swimmer’s fate is decided by its height and heading, not by any attraction. The wall reflects the swimmer’s own flow, the reflection turns it, and that turn fades as the fourth power of height.

## Related experiments and articles

Microswimming · Reading path

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

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **A wall slows each sphere and turns the swimmer**](https://sandboxphysics.com/stories/microswimming/walls-change-the-flow/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)