Reading Najafi and Golestanian’s three-sphere paper

The four-page paper sets up the swimmer, solves the point-force equations for a four-stroke square, prints the per-stroke steps of one example and offers a velocity formula. The per-stroke steps reproduce to three figures. The printed net step and the formula need care: the net is lower than both its own printed steps and our integration, and Golestanian and Ajdari later describe the formula as a numerical fit.

Sandbox Physics · 2026-10-29

What was read, and how claims are labelled

This reading uses arXiv version 1 of Najafi and Golestanian (2004) and arXiv version 1 of Golestanian and Ajdari (2008). The journal versions were not checked, so equation numbers follow the preprints. Each claim below is marked as something the paper states, something a later paper states, or something our own calculation finds.

Our calculation is the point-force model of Chapter 4, integrated with eight-point Gauss–Legendre quadrature along each edge of the stroke loop. Most of its numbers sit in a benchmark file that the repository command npm run fr23:benchmarks regenerates. The value of 0.156 radii quoted below comes from a separate evaluation of Golestanian and Ajdari’s equation 9.

The setup in equations 1 to 8

Paper: equations 1 and 2 are the Stokes equations, and equation 3 is the no-slip condition on each sphere. Equation 4 writes the sphere velocities as a hydrodynamic tensor acting on the forces, and equation 5 states that the forces add up to zero. Equation 6 gives the tensor in the Oseen form used in Chapter 4, and equations 7 and 8 express its coefficients through sphere radius over separation.

Two conventions differ from this course. The paper calls the middle sphere sphere 1, and it reports the middle sphere’s displacement. Over a full cycle every sphere moves by the same amount, so per-cycle results agree with the centroid used here, while per-stroke numbers refer to the middle sphere.

Worked example

The square stroke and its four steps

Paper: Figure 3 introduces an auxiliary move in which one arm shortens by the stroke while the other stays fixed. Reflection and time reversal map all four strokes of the cycle onto this one move, and equation 9 writes the net displacement as twice the difference between the auxiliary move at the full arm length and at the shortened length. Equation 10 divides by the cycle time, four strokes over the arm speed.

Paper: the in-text example takes arms of ten radii and a stroke of four radii, and prints steps of 1.35 radii backward, 1.44 forward, 1.44 forward and 1.35 backward. Our calculation: minus 1.3531, plus 1.4391, plus 1.4391 and minus 1.3531, which agrees to three figures. The auxiliary move gives minus 1.353 radii at the full arm length and minus 1.439 at the shortened one, so equation 9 yields 0.172 radii per cycle.

Here the reading gets interesting. The paper prints a net of 0.16 radii. Its own printed steps add up to 0.18, and our integration gives 0.172. The text gives no derivation for 0.16. For comparison, our evaluation of Golestanian and Ajdari’s leading-order equation 9 for large deformations, around the same square, gives 0.156 radii, close to the printed value. We record the three numbers side by side and leave the model as it is.

\begin{aligned}\Delta&=2\left[\Delta_{f}(D)-\Delta_{f}(D-\epsilon)\right]=2\left[(-1.353)-(-1.439)\right]R\approx0.172\,R\\V_{s}&=W\,\frac{\Delta}{4\epsilon}=30\ \mu\mathrm{m\,s^{-1}}\times\frac{0.172}{16}\approx0.323\ \mu\mathrm{m\,s^{-1}}\qquad(R=1\ \mu\mathrm{m})\end{aligned}

Worked example

The fitted formula in equation 11

Paper: equation 11 gives the swimming speed as 0.7 times arm speed, times radius over arm length, times the square of stroke over arm length, presented as the leading order of an expansion in small deformation. Later paper: footnote 23 of Golestanian and Ajdari describes it as a relation that numerically fits a large part of the parameter range, small and large deformations alike, without a systematic perturbative calculation.

The scaling shows why this matters. Multiply equation 11 by the cycle time and the distance per cycle grows as the cube of the stroke. The small-loop area law of Chapter 5 makes it grow as the square, because the loop area does. Our calculation makes the gap concrete for arms of ten radii. At a stroke of half a radius, equation 11 underestimates the distance by a factor of 4.7; between three and five radii it comes within about ten percent. Figure 4 of the paper plots the full curve, which we left undigitised.

\Delta_{11}=V_{s}\,\frac{4\epsilon}{W}=2.8\,R\left(\frac{\epsilon}{D}\right)^{3},\qquad\Delta_{K}\approx\frac{7}{12}\,R\left(\frac{\epsilon}{D}\right)^{2}\qquad(\epsilon\ll D)

What later papers added

Later paper: Golestanian and Ajdari (2008) generalise the model to unequal radii and arbitrary strokes. Their equation 12 writes the mean velocity for small deformations as a coefficient times the area-sweeping combination of the two arm displacements, and equation 13 gives the coefficient. Equation 16 covers harmonic strokes with a phase difference, and Section IV shows that the force dipole cancels for a symmetric swimmer. They also derive a stall force and the dissipated power; this course leaves those aside.

Later paper: Daddi-Moussa-Ider and colleagues (arXiv version 2, 2018) include the first correction for finite sphere size in the bulk swimming speed of their equation 27, a factor of one plus five sevenths of radius over arm length. Our calculation, extrapolated to tiny loops, gives 0.7141 for that coefficient against the published 0.7143. This check is independent, since the correction comes from their analysis and the number from our code.

Pooley, Alexander and Yeomans (2007) took the swimmer into pairs, the subject of Chapter 8. Read together, the later papers let you test each step of the 2004 argument against a separate calculation.

An experiment to think through

Predict, change, explain

Before opening the Lab, use equation 9 and the steps above to predict Distance per cycle for ten-micrometre arms, a four-micrometre stroke and one-micrometre spheres.
Run “Four strokes swim forward” and read Distance per cycle, then open More numbers and the raw record and read Average swimming speed. Then lower Stroke size and compare with equation 11.
Explain why the per-stroke steps agree with the paper while the printed net and equation 11 depart from the computed values.

Stroke a three-sphere swimmer

The three-sphere Lab couples three equal spheres through point-force Stokes flow in unbounded water. It leaves out walls, neighbours and thermal noise.

Try explaining it yourself

1. Use equation 10 to compute the average swimming speed for the in-text example, with a radius of one micrometre and an arm speed of thirty micrometres per second.

Read an explained answer

One cycle takes four strokes of four micrometres at thirty micrometres per second, 0.533 seconds. Dividing 0.172 micrometres by that time gives about 0.323 micrometres per second.

2. The printed steps add up to 0.18 radii, the printed net is 0.16, and the integration gives 0.172. How should a reader report this?

Read an explained answer

Quote all three numbers with their origins. The steps are rounded to two decimals, so their sum can be off by up to 0.02 radii, which covers 0.172. The text gives no derivation for 0.16, so it stays an unexplained printed value.

3. At a stroke of half a radius, equation 11 underestimates the distance per cycle by a factor of 4.7. Why does the error grow as the stroke shrinks?

Read an explained answer

Equation 11 makes the distance per cycle grow as the cube of the stroke, while the area law makes it grow as the square. Their ratio scales as one over the stroke, so the fitted formula falls further behind as the stroke gets smaller.

4. Which result in this chapter counts as an independent check of the model, and why?

Read an explained answer

The first-order radius correction. Its coefficient of five sevenths comes from Daddi-Moussa-Ider and colleagues’ separate analysis, and the model reaches 0.7141 by extrapolation. A test that repeated the model’s own formula would add no new evidence.

Sources to read alongside

Najafi & Golestanian, A simplest swimmer at low Reynolds number: three linked spheres, arXiv:cond-mat/0402070v1 (2004); Physical Review E 69, 062901 (2004)
Preprint version 1 read; the journal version was not checked. Equations 1 to 11, Figure 3 and the in-text example below Equation 10 are used. The curve of Figure 4 was not digitised.
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.
Pooley, Alexander & Yeomans, Swimming with a friend at low Reynolds number, arXiv:0705.3612v1 (2007); Physical Review Letters 99, 228103 (2007)
Preprint version 1 read; the journal version was not checked. Equations 1 and 4 to 8, Figure 3 and the trajectories of Figure 4 are described in words; no curve was digitised.
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.