Abstract
A liquid carries one length, $\ell_c=\sqrt{\gamma/\rho g}$. This paper asks whose size that length is——the answer is nobody’s. No new mathematical theorem and no new law is claimed.
First, water gives $\mathbf{2.7273}$ mm. Mercury $1.9116$ mm, ethanol $1.6977$ mm, liquid helium $0.2625$ mm (Section 2).
Second, this is the core of the paper. Mercury has $\mathbf{6.6621}$ times the surface tension of water and a capillary length only $\mathbf{0.7009}$ times as long——its density is $13.5611$ times greater, and only the square root of the ratio survives (Section 2).
Third, the length is a boundary. The Bond number $\mathrm{Bo}=(L/\ell_c)^2$ is exactly $1$ there (Section 3).
Fourth, capillarity is not what raises sap. In a $20 \mu$m vessel it lifts water only $\mathbf{0.74}$ m (Section 4).
Fifth, in an alveolus it is $\gamma$ that must move. Bare water gives $1456$ Pa, surfactant $\mathbf{500}$ Pa (Section 5).
Sixth, the separator is whether $\gamma$ enters the formula. The break-up threshold $2\pi R$ does not contain it (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the Laplace pressure, Jurin’s law, the Bond number and the Rayleigh--Plateau instability are all standard. We do not build fluid mechanics——all we use is one length and one square root. We do not treat the contact angle——we compute with $\cos\theta=1$ (complete wetting). Real contact angles, hysteresis and roughness are not treated. We do not treat dynamics——we say whether a column breaks, never how fast. We do not build plant physiology——whether the cohesion--tension theory is right is out of scope. We give only the number showing capillarity does not suffice. We do not build lung physiology——neither surfactant composition nor real alveolar shape is treated. Only the Laplace estimate. We claim no accuracy for representative values——$\gamma=0.0728$ N/m and $\rho=998$ kg/m$^3$ are values for showing orders and move with temperature. Relation to earlier papers: Paper 258 showed that $4\pi$ appears only when the source is a point, using capillaries as its example of a cylindrical source——this paper stays in the same capillary setting but asks about a length rather than a solid angle. 258 treats potential, this paper treats an interface, and they share not one quantity. Paper 291 showed one sea holding three lengths——the $\ell_c$ here is not a fourth length but a length that is the boundary between two forces. Paper 306 showed that the number of base units is a promise——$\ell_c$ is not a promise: it comes out of two material quantities. Paper 140 showed that symmetry fixes ratios and dynamics fixes the scale——$\ell_c$ is on the scale side. What is added is computing $\ell_c$ for five liquids, showing that surface tension and capillary length reverse their order for mercury, checking that the Bond number equals $(L/\ell_c)^2$, counting that capillarity reaches only $0.74$ m against a tree’s height, and pointing out that the break-up threshold $2\pi R$ does not contain $\gamma$.
1. Notation and the Question
Setting: Build the length at which gravity balances surface tension.
$$\ell_c=\sqrt{\frac{\gamma}{\rho g}}$$with $\gamma$ the surface tension, $\rho$ the density, $g=9.80665$ m/s$^2$.
Observation 1.1 Counting dimensions, $[\gamma]=\mathrm{N/m}=\mathrm{kg/s^2}$ and $[\rho g]=\mathrm{kg/(m^2s^2)}$. The ratio is $\mathrm{m^2}$, and a square root makes it a length——there is only one way to combine them.
Question: Whose size is this length?
2. Five Liquids
Check 2.1:
| Liquid | $\gamma$ [N/m] | $\rho$ [kg/m$^3$] | $\ell_c$ [mm] |
|---|---|---|---|
| Water ($20$ °C) | $0.0728$ | $998$ | $\mathbf{2.7273}$ |
| Mercury | $0.4850$ | $13534$ | $\mathbf{1.9116}$ |
| Glycerol | $0.0634$ | $1261$ | $2.2643$ |
| Ethanol | $0.0223$ | $789$ | $1.6977$ |
| Liquid helium $4$ | $0.000098$ | $145$ | $0.2625$ |
Observation 2.2 This is the core of the paper. Mercury’s surface tension is $\mathbf{6.6621}$ times that of water——and yet its capillary length is $\mathbf{0.7009}$ times as long, i.e. shorter.
Observation 2.3 The reason is the density. Mercury is $13.5611$ times denser than water, a larger rise than the surface tension’s——and since $\ell_c$ takes only the square root of the ratio, $\sqrt{6.6621/13.5611}=0.7009$.
Observation 2.4 The five values sit between $0.2625$ and $2.7273$ mm. $\gamma$ spans a factor of $4950$, from $0.000098$ to $0.4850$, while $\ell_c$ spans only $10.4$——the square root acts twice and flattens the spread.
Observation 2.5 Paper 292 wrote that only the logarithm adds. What happens here is the other side of that——on a logarithmic scale, $3.69$ decades of $\gamma$ shrink to $1.02$ decades of $\ell_c$.
3. The Length Is a Boundary
Setting: The Bond number is $\mathrm{Bo}=\rho g L^2/\gamma$.
Check 3.1: Computed for water.
| $L$ [mm] | $\mathrm{Bo}$ | Which wins |
|---|---|---|
| $0.1000$ | $0.001344$ | surface tension |
| $1.0000$ | $0.134437$ | surface tension |
| $\mathbf{2.7273}$ | $\mathbf{0.999967}$ | balance |
| $10.0000$ | $13.443732$ | gravity |
| $100.0000$ | $1344.373173$ | gravity |
Observation 3.2 $\mathrm{Bo}=(L/\ell_c)^2$. The third row is $1$ not by accident but because $L=\ell_c$ was substituted——the Bond number and the capillary length are two ways of writing one thing.
Observation 3.3 So $\ell_c$ is nobody’s size. It is fixed by the liquid alone and never looks at the object placed in it——what it looks at is whom that object’s size is compared with.
Observation 3.4 This is the scale side in the sense of Paper 140. Symmetry fixes the ratios and dynamics fixes the scale——$\ell_c$ is a scale built from two dynamical quantities, $\gamma$ and $\rho g$, and never from geometry.
4. Capillarity Is Not What Raises Sap
Setting: Jurin’s law: in a tube of radius $r$, $h=2\gamma\cos\theta/(\rho g r)$. Take $\cos\theta=1$.
Check 4.1:
| $r$ [$\mu$m] | $h$ [m] |
|---|---|
| $10$ | $1.4877$ |
| $20$ | $\mathbf{0.7438}$ |
| $100$ | $0.1488$ |
| $1000$ | $0.0149$ |
Observation 4.2 Real xylem vessels have radii of $5$–$50 \mu$m. There capillarity lifts $\mathbf{0.74}$ m——tall trees reach $100$ m.
Check 4.3: Inverting: $h=100$ m demands $r=\mathbf{0.1488} \mu$m.
Observation 4.4 That is $134$ times narrower than a real vessel. No such vessel exists——capillarity does not account for it.
Observation 4.5 A point to write honestly——this paper claims nothing about what does raise it. The cohesion--tension theory (transpiration at the leaves pulls the column) is the standard answer, but whether it is right is out of scope——all that is said is the one number: capillarity is not enough.
Observation 4.6 $h$ grows as $1/r$, while $\ell_c$ contains no $r$ at all. The same surface tension yields both a quantity that sees $r$ and one that does not——the same shape as Paper 258’s finding that the source’s form fixes the solid angle.
5. In an Alveolus, What Must Move Is $\gamma$
Setting: The Laplace pressure of a sphere is $\Delta P=2\gamma/R$. Take an alveolar radius $R=100 \mu$m.
Check 5.1:
| State | $\gamma$ [N/m] | $\Delta P$ [Pa] |
|---|---|---|
| Bare water | $0.0728$ | $1456.00$ |
| With surfactant | $0.0250$ | $\mathbf{500.00}$ |
| At its lowest | $0.0010$ | $20.00$ |
Check 5.2: Holding $\gamma$ at bare water and moving only $R$.
| $R$ [$\mu$m] | $\Delta P$ [Pa] |
|---|---|
| $50$ | $2912.00$ |
| $100$ | $1456.00$ |
| $200$ | $728.00$ |
Observation 5.3 At constant $\gamma$ the smaller one is at higher pressure. If they are connected, the small alveolus empties into the large one and collapses——constant $\gamma$ is unstable.
Observation 5.4 Surfactant lowers $\gamma$ as the alveolus shrinks. As $R$ falls so does $\gamma$, so $2\gamma/R$ does not run away——what removes the instability is not the value but the fact that $\gamma$ is not a constant.
Observation 5.5 $\ell_c$ does not appear here. An alveolus at $100 \mu$m is $27$ times smaller than $\ell_c=2.7273$ mm——it lies on the side where gravity does not act, so $\rho g$ drops out.
6. Separator: Whether $\gamma$ Enters
Setting: Rayleigh--Plateau: a liquid column breaks when the wavelength exceeds $2\pi R$.
Check 6.1:
| $R$ [$\mu$m] | Break-up wavelength $2\pi R$ [$\mu$m] |
|---|---|
| $1000$ | $6283.1853$ |
| $100$ | $628.3185$ |
| $10$ | $62.8319$ |
Observation 6.2 Separator——whether $\gamma$ enters the formula. The threshold $2\pi R$ does not contain $\gamma$——surface tension does not decide whether it breaks, only how fast.
Check 6.3: Three quantities side by side.
| Quantity | What enters | What it decides |
|---|---|---|
| $\ell_c=\sqrt{\gamma/\rho g}$ | $\gamma$ and $\rho g$ | which force wins |
| $h=2\gamma/(\rho g r)$ | $\gamma$, $\rho g$, $r$ | how high it climbs |
| $\lambda>2\pi R$ | $R$ alone | whether it breaks |
Observation 6.4 Only the third row contains no material quantity. It is pure geometry——the same place as Paper 258’s solid angle, fixed by the shape of the source alone.
Observation 6.5 And $\ell_c$ is neither. It comes from material quantities, yet says nothing without something to compare——the number $2.7273$ mm means something only once a $2.7273$ mm object exists.
7. Roll Call and Closing
- Water gives $2.7273$ mm (Check 2.1): $\ell_c=\sqrt{\gamma/\rho g}$.
- Mercury has $6.6621$ times the $\gamma$ (Observation 2.2): and $0.7009$ times the $\ell_c$.
- Density is $13.5611$ times (Observation 2.3): only the square root acts.
- $4950$ in $\gamma$ becomes $10.4$ (Observation 2.4): the spread flattens.
- $\mathrm{Bo}=(L/\ell_c)^2$ (Check 3.1): exactly $1$ at $L=\ell_c$.
- Nobody’s size (Observation 3.3): it is the thing compared with.
- Capillarity gives $0.74$ m (Check 4.1): trees reach $100$ m.
- It would need $0.1488 \mu$m (Check 4.3): $134$ times narrower.
- An alveolus at $500$ Pa (Check 5.1): bare water gives $1456$ Pa.
- $2\pi R$ has no $\gamma$ (Check 6.1): whether it breaks is geometry.
Closing: A liquid carries one length, $\ell_c=\sqrt{\gamma/\rho g}$, and for water it is $\mathbf{2.7273}$ mm——mercury $1.9116$ mm, glycerol $2.2643$ mm, ethanol $1.6977$ mm, liquid helium $0.2625$ mm. Mercury has $\mathbf{6.6621}$ times the surface tension of water and a capillary length only $\mathbf{0.7009}$ times as long——because its density is $13.5611$ times greater, and $\ell_c$ takes only the square root of the ratio. A factor of $4950$ in $\gamma$ becomes a factor of $10.4$ in $\ell_c$. The length is a boundary——the Bond number $\mathrm{Bo}=(L/\ell_c)^2$ is exactly $1$ at $L=\ell_c$; below it surface tension wins, above it gravity does. So $\ell_c$ is nobody’s size——it is whom an object’s size is compared with. Capillarity is not what raises sap——in a $20 \mu$m vessel it lifts $\mathbf{0.74}$ m, and $100$ m would need a $0.1488 \mu$m tube, $134$ times narrower than any real one. In an alveolus what acts is not $R$ but $\gamma$——bare water gives $1456$ Pa, surfactant $\mathbf{500}$ Pa. What removes the instability is not the value but the fact that $\gamma$ is not a constant. One thing separates them——whether $\gamma$ enters the formula. The break-up threshold $2\pi R$ does not contain it and is pure geometry. Surface tension does not decide whether a column breaks, only how fast.
Revision Record
Second edition (2026-08-30): The subject of this paper has been replaced. The first edition, titled “Living Tissue Where $4\pi$ Appears, and Where It Does Not,” treated the solid angles of point, line and sheet sources and the diffusive reach around a capillary. That content duplicated Paper 258, “$4\pi$ Appears Only When the Source Is a Point”—— even the numbers ($24.1144 \mu$V, $111.0508 \mu$V, $316.78 \mu$m, $44.80 \mu$m) agreed, and Paper 258 has priority. The second edition stays in the same capillary setting but moves to a quantity 258 did not treat: the interfacial length $\ell_c$. All statements about solid angle have been removed from this paper; Paper 258 is the reference for them.
References
- de Gennes, P.-G., Brochard-Wyart, F. and Quere, D. (2004). Capillarity and Wetting Phenomena. New York: Springer.
- Jurin, J. (1718). An Account of Some Experiments Shown Before the Royal Society. Philosophical Transactions of the Royal Society, 30.
- Rayleigh, Lord (1878). On the Instability of Jets. Proceedings of the London Mathematical Society, s1-10(1).
- Dixon, H. H. and Joly, J. (1895). On the Ascent of Sap. Philosophical Transactions of the Royal Society B, 186.
- Clements, J. A. (1957). Surface Tension of Lung Extracts. Proceedings of the Society for Experimental Biology and Medicine, 95(1).
- Yamagishi, Y. (2026). Symmetry Fixes the Ratios, Dynamics Fixes the Scale [Paper 140]. Zenodo. 10.5281/zenodo.22081997
- Yamagishi, Y. (2026). $4\pi$ Appears Only When the Source Is a Point [Paper 258]. Zenodo. 10.5281/zenodo.22169011
- Yamagishi, Y. (2026). One Sea Holds Three Lengths, Spanning a Factor of $4.39\times10^4$ [Paper 291]. Zenodo. 10.5281/zenodo.22171479
- Yamagishi, Y. (2026). Only the Logarithm Adds [Paper 292]. Zenodo. 10.5281/zenodo.22171519
- Yamagishi, Y. (2026). How Many “Base” Units There Are Is a Promise [Paper 306]. Zenodo. 10.5281/zenodo.22172149
Reference
Yamagishi, Y. (2026). The Length That Divides Gravity from Surface Tension Is Nobody’s Size [Paper 309]. YOI: https://www.yamagishiyuuki.com/yoi/309
DOI: 10.5281/zenodo.22172337
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 309. This paper is a preprint and has not been peer reviewed.