A wall slows each sphere and turns the swimmer

A wall must hold the fluid still at its surface, so every point force acquires an image flow behind the wall, first worked out by Blake. Each sphere moves more slowly near the wall, and more slowly toward it than along it. The spheres also couple through the wall, which can rotate the swimmer. Daddi-Moussa-Ider and colleagues found three outcomes: trapped facing the wall, escaping into open fluid, or gliding along the wall with a slow oscillation.

Sandbox Physics · 2026-10-29

Images hold the wall still

In open fluid, the flow of a point force fades with distance and meets nothing. Near a wall, the same flow would slide fluid along the wall’s surface, which the no-slip condition forbids. Lauga and Powers describe the remedy as a method of images, much as in electrostatics, with the extra difficulty that no-slip constrains every component of the velocity.

Blake found the image system for a point force beside a plane wall. Daddi-Moussa-Ider and colleagues quote it as their equation 5. Start from the free-space flow at the actual separation, subtract the free-space flow from a mirror point behind the wall, then add a force-dipole term and a source-dipole term at that mirror point, weighted by the height of the force above the wall. We took the form shown from their equation 5 and have yet to read Blake’s original paper.

\mathbf{G}(\mathbf{r},\mathbf{r}_{0})=\mathbf{G}^{\mathrm{O}}(\mathbf{s})-\mathbf{G}^{\mathrm{O}}(\mathbf{R})+2z_{0}^{2}\,\mathbf{G}^{\mathrm{D}}(\mathbf{R})-2z_{0}\,\mathbf{G}^{\mathrm{SD}}(\mathbf{R})

Worked example

How much a wall slows one sphere

Evaluate the image flow back at the sphere itself and keep only the leading term in radius over height. A sphere whose centre sits a given height above the wall then moves parallel to the wall with its free mobility reduced by nine sixteenths of the radius-to-height ratio. Toward the wall the reduction is twice as large, nine eighths of that ratio.

At ten radii from the wall, the parallel mobility is 0.944 of its free value and the perpendicular one 0.889. At five radii they are 0.889 and 0.775. At two radii the leading term gives 0.719 and 0.438; that close, higher-order terms matter, and we have not compared these numbers with exact solutions.

The perpendicular reduction is larger because squeezing fluid out of the gap under a sphere is harder than sliding it along the gap. For the swimmer, spheres at different heights now respond differently to the same force. A tilted swimmer has its two ends at different heights, so its spheres move unequally and the wall can rotate the whole body.

\begin{aligned}\frac{\mu_{\parallel}}{\mu_{0}}&\approx1-\frac{9}{16}\,\frac{a}{h},\qquad\frac{\mu_{\perp}}{\mu_{0}}\approx1-\frac{9}{8}\,\frac{a}{h}\\h=5a&:\qquad\frac{\mu_{\parallel}}{\mu_{0}}\approx0.889,\qquad\frac{\mu_{\perp}}{\mu_{0}}\approx0.775\end{aligned}

Three kinds of path beside a wall

Daddi-Moussa-Ider and colleagues drive the two arms harmonically with a quarter-cycle lag, and set the sphere radius and both amplitudes to a tenth of the mean arm length (their equations 13 and 24). Far from the wall the swimmer moves at the speed of their equation 27, which includes the first correction for sphere size. Near the wall they take mobilities from a multipole method that stays accurate close to the surface, and they add a soft repulsion that keeps the spheres off the wall. They state that they checked moderate changes to its parameters.

Three outcomes appear. A swimmer released pointing toward the wall by more than about 0.3 radians is trapped: it curves in, turns until its axis stands perpendicular to the wall, and hovers at a steady height. Released one arm length above the wall, it settles at about 1.12 arm lengths. A swimmer pointing away from the wall by more than about half a radian escapes and carries on in a straight line, as in open fluid. In a bounded region of starting states, about one arm length up and nearly parallel, it glides along the wall while its height and angle oscillate slowly.

The authors note that this oscillation is several orders of magnitude slower than the stroke. Their figures show stroke-averaged positions and angles, which is the right scale for reading these paths.

Two papers on the far field

Far from the wall, how quickly does its influence fade? Zargar, Najafi and Miri expanded in arm length over height, and their 2009 abstract reports a wall-induced quadrupole force that falls as the inverse square of the height. Daddi-Moussa-Ider and colleagues revisit the problem and find that the wall-induced translational velocity falls as the inverse cube of the height and the angular velocity as the inverse fourth power. They state that the earlier far-field decay was reported in error.

A reader can probe the disagreement with symmetry. Lauga and Powers show that a swimmer with a force dipole, a pusher or a puller, feels a wall-induced speed falling as the inverse square of the height. Golestanian and Ajdari show that when the two arms of the three-sphere swimmer have equal mean length, the force dipole cancels and the leading far-field term is a quadrupole, one power of distance weaker. The swimmer studied by Daddi-Moussa-Ider and colleagues is of this symmetric kind, so an inverse-cube leading term is what symmetry suggests. The symmetry check supports their result, and their Section IV supplies the coefficients.

What the near-wall Lab adds

Our calculation: the near-wall Lab uses the swimmer of Daddi-Moussa-Ider and colleagues, with one-micrometre spheres on ten-micrometre arms and a quarter-cycle lag between the arms. It keeps only the far-field part of the wall: Blake’s image flows between the spheres and the leading correction to each sphere’s own mobility. For a swimmer level with the wall, its turn per stroke falls as the fourth power of the height and agrees with their far-field equation 28c within one percent at twelve arm lengths.

Started level at ten micrometres, the model swimmer glides, rising and dipping between about six and ten micrometres. Started at twenty micrometres, it turns in and reaches the model’s near-wall limit after about 4,800 strokes. Tilted 26 degrees away from the wall, it escapes, and at 7.5 micrometres it slides at nearly constant height. The run stops once a sphere comes within two radii of the wall. The hovering state of the paper depends on near-wall mobilities and a soft repulsion, so for that ending the paper remains the reference.

An experiment to think through

Predict, change, explain

Lauga and Powers say the scallop theorem survives beside a wall. Predict how Distance along the wall changes when the two arms move back and forth together, with no phase lag.
Run “Back-and-forth stroke still goes nowhere” and watch Distance along the wall. Then run “Glide along the wall”, “Start higher, turn into the wall” and “Point away and escape”, and compare Height above the wall and Tilt toward the wall.
Explain why the wall turns a swimmer whose stroke traces a loop, while a back-and-forth stroke beside the same wall makes no progress.

Release a swimmer beside a wall

The near-wall Lab keeps the far-field wall terms and stops a run once a sphere comes within two radii of the wall.

Try explaining it yourself

1. A sphere sits four radii above a wall. To leading order, by what factors does the wall reduce its mobility along the wall and toward it?

Read an explained answer

Along the wall, one minus nine sixty-fourths, about 0.86. Toward the wall, one minus nine thirty-seconds, about 0.72.

2. Does a reciprocal stroke make net progress beside a no-slip wall?

Read an explained answer

It stays put. The proof of the scallop theorem uses only the linearity and time independence of the Stokes equations, and both still hold with a wall, so the two halves cancel as they do in open fluid.

3. Why can a wall rotate a three-sphere swimmer whose internal forces all lie along its axis?

Read an explained answer

When the swimmer is tilted, its spheres sit at different heights, and the wall slows each one by a different amount, both directly and through the image flows of its neighbours. Unequal velocities along the body add up to a rotation.

4. A pusher with a force dipole and a symmetric three-sphere swimmer sit at the same large height. How does each one’s wall-induced drift scale with height?

Read an explained answer

The pusher’s drift falls as the inverse square of the height; the symmetric swimmer’s falls as the inverse cube, because its force dipole cancels. Far from the wall, the symmetric swimmer feels the wall much more weakly.

Sources to read alongside

Lauga & Powers, The hydrodynamics of swimming microorganisms, arXiv:0812.2887v1 (2008); Reports on Progress in Physics 72, 096601 (2009)
Preprint version read. Sections 3 and 4 supply the Stokes equations, the point-force flow and the proofs of the scallop theorem; Sections 5 to 7 cover resistive-force theory, flexible filaments, walls and synchronisation.
Golestanian & Ajdari, Three-sphere low Reynolds number swimmer revisited, arXiv:0711.3700v1 (2007); Physical Review E 77, 036308 (2008)
Preprint version 1 read; the journal version was not checked. Equations 1 to 6, 9, 12, 13 and 16, Section IV on the far-field flow and footnote 23 are used.
Daddi-Moussa-Ider, Lisicki, Hoell & Löwen, Swimming trajectories of a three-sphere microswimmer near a wall, arXiv:1801.01162v2 (2018)
Preprint version 2 read; the journal version was not checked. Equations 4 to 7, 13, 24 and 27, the three trajectory types and the far-field discussion are used.
Zargar, Najafi & Miri, Three-sphere low-Reynolds-number swimmer near a wall, Physical Review E 80, 026308 (2009)
Only the abstract was read. It is cited for the earlier far-field claim that a later paper revisits.