Reynolds Number: Why a Bacterium Can't Coast and a Fish Can

The syrup line, as this site uses the term, is the body length at which a swimmer's Reynolds number falls to 1 for a stated fluid, temperature and speed. Change the speed or the fluid and the line moves with it. Only well below it, at Re ≪ 1, can inertia be ignored; near Re = 1, inertia and viscosity are of the same order.

In this site's shorthand, the law code is the same at every size, and below the syrup line in water at 20 °C the local court simply cites viscosity first. Put plainly, when Re is far below 1, the viscous force on a swimmer far exceeds the inertial one. Syrup is a fair picture: by the note that introduces the 1977 reprint (abstract), Purcell's talk "Life at low Reynolds number" was demonstrated with a tall vessel of corn syrup.

Slide the diagram sideways to read its labels, or open it full size.

What does the Reynolds number compare?

Written out, Re = ρUL/μ, where ρ is the fluid's density, U the swimmer's speed, L its length and μ the fluid's viscosity. Lauga and Powers give several readings of it. Among them are inertial terms against viscous terms in the flow, the time a disturbance takes to diffuse away against the time to be carried past the body, and the inertial force on the body, about ρU²L², against the viscous force, about μUL. At low Re, they write, the forces on a body come mainly from viscous drag.

Computed from the international water formulations (IAPWS releases of 2018 and 2008), water at 20 °C has a density of 998.21 kg/m³ and a viscosity of 1.0016 mPa·s. Their ratio, the kinematic viscosity ν = μ/ρ, is 1.0034×10⁻⁶ m²/s, computed for this page, so Re = UL/ν. Where that stickiness comes from is traced on Water Molecule.

Size enters twice. A swimmer that covers k body lengths a second has U = kL, so Re = kL²/ν, and at the same pace a swimmer ten times shorter has a Reynolds number a hundred times smaller. Setting Re = 1 gives L = √(ν/k): 1.0 mm at one body length a second and 0.32 mm at ten, in water at 20 °C, computed for this page.

Illustration

The forces tell the same story. Shrink a swimmer tenfold at the same pace and its inertial force falls ten-thousandfold while its viscous force falls a hundredfold, computed from those scalings. The two meet at Re = 1, where, as Lauga and Powers note after Purcell, both equal μ²/ρ, about 1 nN in water, whatever the swimmer's size.

Slide the diagram sideways to read its labels, or open it full size.

There is no single cut-off for a low Reynolds number. Far below 1, viscous drag dominates; near 1, inertia and viscosity are of the same order; at Re in the thousands, the fish further down this page glide between kicks.

Far below the line: why can't a bacterium coast?

By Lauga and Powers' account, Escherichia coli swims with helical flagella about 20 nm thick, each tracing a helix about 10 µm long; when the motors turn counter-clockwise, the filaments wrap into a bundle that pushes the cell at 25–35 µm/s.

Cultured E. coli measured 1.6–3.9 µm long across 22 growth conditions, with volumes of 1.5–4.4 µm³ (Volkmer and Heinemann 2011). At 25–35 µm/s in water at 20 °C, that gives Re of about 0.00004–0.00014, computed for this page: 7,000 to 25,000 times below 1.

Lauga and Powers estimate how far such a cell glides once it stops pushing. At low Re, their figure is roughly the cell's length times Re times the ratio of its density to the water's: about 0.1 nm for E. coli, by their reckoning.

A sphere model, computed for this page, gives less. Take a sphere of the cell's volume, as dense as the water, slowed by the drag 6πμaU that Lauga and Powers give for a sphere of radius a. Its speed falls by a factor of e within 0.11–0.23 µs, and it coasts 0.003–0.008 nm. A denser cell would coast proportionally farther.

The two figures differ 12- to 36-fold. The scaling estimate drops numerical factors such as the 6π in the drag and uses a length where the sphere model uses a radius; for a sphere, those two choices alone make a factor of 18, computed for this page. Both put the coast at no more than about 0.1 nm, one sixteen-thousandth of the cell's length or less. Nothing carries the cell between strokes; to move, it has to keep pushing.

A non-swimming E. coli still wanders, with a diffusion constant of about 0.1 µm²/s in water at room temperature, by the same review's figure: see Brownian Motion. Bacteria this size are counted in How Many Cells in the Human Body?; How Big Is a Cell? sets them among other cells.

Near the line: paramecia, copepods and the largest single cells

Lisicki and colleagues compiled sizes and swimming speeds for single-celled eukaryotes in 2019 and computed Re for each, taking ν = 10⁻⁶ m²/s for water. Flagellates averaged 31 µm and ciliates 132 µm, with median speeds of 127 and 784 µm/s. For both groups, Re stayed below 1.

From their table, Paramecium caudatum, listed at 225.5 µm and 2,489 µm/s, reaches Re of about 0.56 in water at 20 °C, computed for this page. Its behavior is the subject of Paramecium. The largest single cell in the table, Spirostomum ambiguum, listed at 1,045 µm and 810 µm/s, reaches about 0.84: close to the line, where inertia and viscosity are of the same order.

Copepods cross it. The three planktonic species in Kiørboe and colleagues' 2010 study were 0.3–3.0 mm long (abstract), and adult Temora longicornis swam at 2.1–11.3 mm/s when not mating, in Seuront's 2011 experiments with clean and diesel-tainted seawater at 18 °C. Paired up, those give Re of about 0.6–34 in water at 20 °C, computed for this page. In escape jumps, Kiørboe and colleagues describe power strokes alternating with passive coasting (abstract).

Drifters like these, sorted by size, fill Plankton. By Lisicki and colleagues' account, studies of green algae separate the smallest flagellates from the largest multicellular species by a second ratio, the Péclet number, which weighs transport by the flow against diffusion. The multicellular end of that range on this site is Volvox. The same Reynolds ratio applies to any cell in water, the glass-walled diatoms included, with U as its speed relative to the water.

Slide the diagram sideways to read its labels, or open it full size.

Above the line: how far does a fish glide?

Rummy-nose tetras, Hemigrammus rhodostomus, about 31 mm long on average, swim in bursts: a kick, then a glide during which the speed decays roughly exponentially with a time constant of about 0.80 s (Calovi and colleagues 2018). Their typical speed in active periods was of order 140 mm/s, and they covered about 70 mm between kicks, a little over two body lengths, computed for this page.

At that size and speed, Re is about 4,400–5,500 in water at 25 °C, computed for this page, though the tanks were a little warmer, at 26–28 °C. In a flume study of red-nose tetras, Hemigrammus bleheri, 3.5–4 cm long, Li and colleagues (2021) put Re between 950 and 6,000 and describe the coast as a pause in which the fish glides on its own inertia.

Set side by side, the tetra's gliding speed takes about 0.8 s to fall by a factor of e and the bacterium's takes 0.11–0.23 µs, a ratio of about 3.5 to 7 million, computed for this page. Lauga and Powers sketch the high-Re case: drag grows with the square of the speed, and a body coasts a distance set by its length times its density relative to the water's. In their example, a human swimmer can glide a couple of metres.

Slide the diagram sideways to read its labels, or open it full size.

Does the line move in warm water or in air?

Warm the water to 25 °C and the line slides down a little. With that row of the same water table, 0.8900 mPa·s and 997.05 kg/m³, Re = 1 at 1–10 body lengths per second comes at 0.30–0.94 mm, about 6% shorter than at 20 °C, computed for this page.

Air moves it much further. Farisenkov and colleagues (2022) used a kinematic viscosity of 1.54×10⁻⁵ m²/s for air at 25 °C, about 15 times that of water at 20 °C, which puts the syrup line in air at 25 °C at 1.2–3.9 mm for 1–10 body lengths per second, computed for this page.

Their featherwing beetle, Paratuposa placentis, about 395 µm long and one of the smallest non-parasitic insects, moved forward at about 57 mm/s in four recorded flights, some 140 body lengths a second. Its body reaches Re of about 1.5 in air at 25 °C, computed for this page. The authors put its bristled wings at Re 9 over a wingbeat and up to 20 in power strokes, and describe flight at such sizes as dominated by viscous air friction.

The smallest insects are measured in Smallest Insect, and a bumblebee's wings meet the same ratio in Bumblebee Flight; at the water's surface, the film holds up the water strider.

Slide the diagram sideways to read its labels, or open it full size.

Why can't a cell swim like a scallop?

Below the line the rules of the stroke change as well. At Re close to zero the equations of the flow are linear and do not depend on time. Lauga and Powers draw two consequences: how far a swimmer moves depends only on the sequence of shapes it passes through, not on how fast, and a stroke whose shapes look the same played backward moves it nowhere on average.

That second result is known as Purcell's scallop theorem. By Lauga and Powers' account, Purcell illustrated it with a scallop, which opens and shuts its shell. A hinge that opens slowly and shuts fast still retraces its path when Re is near zero. A real scallop, they add, swims at high Re, where the theorem does not hold.

Cells get round it with strokes that are not reversible. A helix turning one way, like the flagellar bundle of E. coli, is not the same motion played backward, and waves running along a filament push along its whole length, by the same review's account. The review also credits Purcell with a body of three rigid links hinged together and turning out of phase, now called Purcell's swimmer.

Slide the diagram sideways to read its labels, or open it full size.

Somewhere between a paramecium and a tetra, a swimmer stops relying on viscous forces alone and starts handing momentum to the water, as Lauga and Powers describe fish doing. How that switch happens was still drawing much new work when their review appeared in 2009; it reports that as inertia enters, the change can be gradual or sudden depending on the swimmer's symmetry. Cruising copepods, at Re of about 0.6–34 in water at 20 °C, sit in that middle range.

Sources

1. Lauga E, Powers TR. "The hydrodynamics of swimming microorganisms." Reports on Progress in Physics 72(9):096601 (2009). https://doi.org/10.1088/0034-4885/72/9/096601 Read: full text (preprint version).
2. Kiørboe T, Andersen A, Langlois VJ, Jakobsen HH. "Unsteady motion: escape jumps in planktonic copepods, their kinematics and energetics." Journal of the Royal Society Interface 7(52):1591–1602 (2010). https://doi.org/10.1098/rsif.2010.0176 Read: abstract.
3. Purcell EM. "Life at low Reynolds number." American Journal of Physics 45(1):3–11 (1977). https://doi.org/10.1119/1.10903 Read: abstract (the editor's note).
4. International Association for the Properties of Water and Steam. IAPWS R6-95(2018), Revised Release on the IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use (2018). https://iapws.org/technical-guidance/release/IAPWS-95 Read: full text (PDF). Text license: IAPWS: publication in whole or in part allowed with attribution (https://iapws.org/technical-guidance/release/IAPWS-95).
5. International Association for the Properties of Water and Steam. IAPWS R12-08, Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary Water Substance (2008). https://iapws.org/technical-guidance/release/viscosity Read: full text (PDF). Text license: IAPWS: publication in whole or in part allowed with attribution (https://iapws.org/technical-guidance/release/viscosity).
6. Volkmer B, Heinemann M. "Condition-Dependent Cell Volume and Concentration of Escherichia coli to Facilitate Data Conversion for Systems Biology Modeling." PLoS ONE 6(7):e23126 (2011). https://doi.org/10.1371/journal.pone.0023126 Read: full text. Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
7. Lisicki M, Velho Rodrigues MF, Goldstein RE, Lauga E. "Swimming eukaryotic microorganisms exhibit a universal speed distribution." eLife 8:e44907 (2019). https://doi.org/10.7554/eLife.44907 Read: full text, including Appendix 1 (flagellates) and Appendix 2 (ciliates). Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
8. Seuront L. "Hydrocarbon Contamination Decreases Mating Success in a Marine Planktonic Copepod." PLoS ONE 6(10):e26283 (2011). https://doi.org/10.1371/journal.pone.0026283 Read: full text. Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
9. Calovi DS, Litchinko A, Lecheval V, Lopez U, Pérez Escudero A, Chaté H, Sire C, Theraulaz G. "Disentangling and modeling interactions in fish with burst-and-coast swimming reveal distinct alignment and attraction behaviors." PLoS Computational Biology 14(1):e1005933 (2018). https://doi.org/10.1371/journal.pcbi.1005933 Read: full text. Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
10. Li G, Ashraf I, François B, Kolomenskiy D, Lechenault F, Godoy-Diana R, Thiria B. "Burst-and-coast swimmers optimize gait by adapting unique intrinsic cycle." Communications Biology 4:40 (2021). https://doi.org/10.1038/s42003-020-01521-z Read: full text. Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
11. Farisenkov SE, Kolomenskiy D, Petrov PN, Engels T, Lapina NA, Lehmann F-O, Onishi R, Liu H, Polilov AA. "Novel flight style and light wings boost flight performance of tiny beetles." Nature 602(7895):96–100 (2022). https://doi.org/10.1038/s41586-021-04303-7 Read: full text. Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).
12. GBIF Secretariat. GBIF Backbone Taxonomy. Checklist dataset https://doi.org/10.15468/39omei, accessed 2026-10-24; usage keys for the taxa named on this page are listed in swimmers-reynolds.csv. Read: dataset record (species match for the taxa named). Text license: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).

Cite this page

Ants, Cells & Atoms. “Reynolds Number: Why a Bacterium Can't Coast and a Fish Can.” Published 2026-10-25. https://antscellsandatoms.com/scale/the-syrup-line/. Accessed (add the date you read it).

No DOI is issued for entries. Figures made here are CC BY 4.0; see each caption for the license of photographs.

How this entry is kept

Published

25 October 2026

Last substantive revision

none since publication

• How entries are checked
• Corrected entries
• Contact and feedback
• Report an error in this entry

Related entries

Scale & Estimates · Scale estimate · [10⁻⁶ m]

How Many Cells in the Human Body? About Half Are Bacteria

Image: “E. coli Bacteria (7316101966)” by NIAID (2002) · Public domain

"E. coli Bacteria (7316101966)" by NIAID (2002) — https://commons.wikimedia.org/wiki/File:E._coli_Bacteria_(7316101966).jpg — Public domain (work of a United States federal agency (NIAID, part of NIH), public domain in the United States under 17 U.S.C. §105; Commons tag PD-USGov-NIH-NIAID; Flickr source page credits NIAID with no contractor note; dated 2002-11-14 in the file metadata; no copyright term applies in the EU for a US government work released as public domain by its author agency (used as the source's explicit PD mark))

Scale & Estimates · Scale estimate · [10⁻⁷–10⁻⁴ m]

How Big Is a Cell? Most Are a Hair's Width or Far Narrower

Image: “SEM blood cells” by Bruce Wetzel and Harry Schaefer, National Cancer Institute (1982) · Public domain

"SEM blood cells" by Bruce Wetzel and Harry Schaefer, National Cancer Institute (1982) — https://commons.wikimedia.org/wiki/File:SEM_blood_cells.jpg — Public domain (work of United States federal employees at the National Cancer Institute (NIH), public domain in the United States under 17 U.S.C. §105; Commons tag PD-USGov; the NCI Visuals Online source states that all images are in the public domain and may be reproduced without permission, with credit to the listed source or author; dated February 1982; used in the EU on the source's explicit public-domain statement)

Illustration

Scale & Estimates · Scale estimate · [10⁻¹⁰ m]

How Small Is an Atom? For Most Elements, About 0.1–0.5 nm

Image: Illustration: Ants, Cells & Atoms, all rights reserved

Illustration: Ants, Cells & Atoms. Generated schematic illustration — not a photograph, micrograph or specimen record. © Ants, Cells & Atoms, all rights reserved.