A bubble in water bottoms out at the speed of the slowest ripple

The size a drop is allowed took the Bond number as the ratio of gravity to surface tension on a stated length, and the capillary length as the size at which they balance. For water it’s about 2.7 millimetres. Its later essays followed static drops and their contact lines. A bubble rising through a liquid is a similar object with the weight pointed the other way, and one quite unlike a drop in having a very low density and hence a speed set entirely by the liquid.

A thousandth of a monolayer holds a bubble still found what controls the small ones: a mobile surface against a rigid one, with the difference between them made by a trace of contaminant. That essay’s scope ended at bubbles small enough to be spherical. Above that, a remarkable thing happens to the speed, and it needs no contamination, no viscosity and nothing about the wake to explain it.

Three curves in one #

The left of the figure is viscosity. A very small bubble in water has a Reynolds number below one, and the Hadamard–Rybczynski solution gives its speed, $gd^2/12\nu$, if its surface is mobile; contaminated, it behaves as a rigid sphere, $gd^2/18\nu$, Stokes’s own result. Both rise as $d^2$. As the Reynolds number grows, a clean bubble has an unusually low drag, since its surface doesn’t hold the fluid back. The flow around it is almost potential. Its dissipation comes from the potential flow’s own strain, and Levich calculated that in 1949, to find $C_D=48/Re$, a drag still inversely proportional to the Reynolds number at Reynolds numbers of hundreds. That’s a remarkable result. A rigid sphere has a boundary layer, a wake and a drag coefficient near 0.5 at a Reynolds number of a thousand, whereas a clean bubble has no separation until it deforms. Mei and his colleagues’ joining of these two limits is the clean curve’s ascending part.

So in clean water a bubble speeds up to about 0.3 metres a second by a diameter of 1.5 millimetres. Contaminated water, which is all water outside a very careful laboratory, gives a rigid-sphere curve, about half as fast at a millimetre. By three millimetres the two have run together, and no difference of cleanliness survives, which by itself says that the mechanism has changed.

The right of the figure is a different balance. A bubble at that size is no longer round. Dynamic pressure pushes on its nose and its base, and it has flattened into an oblate ellipsoid, wobbling in a zigzag or spiral path. Its drag now depends on its shape, and that shape depends on the balance of dynamic pressure against surface tension. At a Weber number of order one a bubble of a given volume has a given flatness, and it’s very hard to calculate. Yet the speed that results has a much simpler form than the mechanism leading to it.

Mendelson’s analogy #

In 1967 Harvey Mendelson pointed out that the rise speed of a large bubble in water is very nearly that of a surface wave whose wavelength is proportional to the bubble’s diameter. A surface wave of wavelength $\lambda$ on deep water has

$$c^2=\frac{2\pi\sigma}{\rho\lambda}+\frac{g\lambda}{2\pi}.$$

Set $\lambda=\pi d$, the half-circumference, and this becomes

$$U^2=\frac{2\sigma}{\rho d}+\frac{gd}{2}.$$

A later fit changes those coefficients only slightly, to 2.14 and 0.505. The argument is that a bubble’s upper surface is a free surface being displaced by the bubble’s motion, and that the speed of that displacement is the speed at which a free surface can respond at the bubble’s own scale. Nobody has shown this as a derivation. As an analogy, though, it’s unusually exact. It reproduces both ends: a small deformed bubble controlled by surface tension falls with size, and a large cap, in which the first term has gone, has $U=0.71\sqrt{gd}$. That is the spherical-cap law Davies and Taylor derived in 1950 from the pressure at a bubble’s nose, which is a genuine calculation, using an entirely different physical argument, with the same result.

Now the key step. A sum of a falling term and a rising one has a minimum, where the two are equal. For a wave, the minimum falls at $\lambda=2\pi\ell_c$, the wavelength Kelvin’s sheet picks, and its speed is

$$c_{\min}^2=2\sqrt{\frac{\sigma g}{\rho}}.$$

In water that’s 0.231 metres a second. A ripple can’t go any slower than this, whatever its length. That’s why a ship moving more slowly than that makes no waves at all, and why a fishing line in a slow stream shows a wedge of ripples upstream only above that speed. And the wave law for a bubble has a minimum at $d^2=2.14\sigma/(0.505\rho g)$, with

$$U_{\min}^2=2\sqrt{2.14\times0.505}\,\sqrt{\frac{\sigma g}{\rho}}=1.0395\;c_{\min}^2 .$$

So the slowest bubble rises at 1.02 times the slowest ripple’s speed, in any liquid. With the pure wave analogy, $\lambda=\pi d$, the factor is exactly one. It’s a striking coincidence of numbers from two experiments that seem to have nothing to do with one another. It’s why bubbles in water, across a size range from 2 to 10 millimetres, all rise at about 23 to 26 centimetres a second, which is a rule of thumb people are often surprised to learn. A bubble can’t go below that speed on this branch of the curve, however its size is chosen.

At the floor itself #

Three dimensionless numbers state where the floor lies, and each says something about the mechanism. At 5.6 millimetres the Eötvös number is 4.24, since by construction the floor comes at about two capillary lengths: gravity across the bubble and surface tension are nearly balanced, which is why the bubble is neither round nor capped but flattened. The Weber number is 4.3, so dynamic pressure and surface tension are also of comparable size. And the Reynolds number is about 1300, so viscosity’s direct contribution to the balance is small. That’s why the joined law’s viscous term hardly matters there: its viscous speed would be 8.7 metres a second, thirty-seven times the wave speed, and the viscous term only contributes one part in a thousand.

So the floor has no viscosity in it at all, and that explains why its speed barely depends on contamination. A clean surface and a dirty one have their difference from how the boundary layer works, and when the speed is no longer set by the boundary layer, it can’t matter. The whole of a monolayer’s dramatic effect applies at the other end of the curve, and a cap that slows a rising bubble has a different size to work at. It’s a useful rule: at around half a centimetre, water’s bubbles are the same in a clean laboratory and in a muddy pond.

It’s the same reason the slowest ripple is itself so insensitive. A surface wave has no viscosity in its speed, and the two terms making up its restoring force, one rising and one falling with size, are the same two. A wave doesn’t care what’s under it, as long as it’s deep compared with the wavelength. A bubble doesn’t seem to care either. In this way, a bubble’s speed shares the same two restoring forces with the wave drag of a ship, for which the gravity term alone sets a speed. A bubble has no hull, but its top surface has to get out of the way at the rate a free surface can.

A number a diver has used #

There’s a practical relic of all this. Old diving tables gave the advice that a diver should rise no faster than the smallest bubbles from the breathing valve. A bubble of a millimetre or two in water rises at about 0.3 metres a second, at the peak in the figure, which is 18 metres a minute, the ascent rate that the US Navy’s tables used for decades. The larger bubbles from a valve, at a few millimetres to a centimetre, are at the floor, about 14 metres a minute. And modern tables, more conservative, have brought the rate down to 9 or 10 metres a minute. It’s a reasonable coincidence and it’s not a derivation of anything, since nobody chose that rate on hydrodynamic grounds. Yet it’s a nice accident that a rule about nitrogen in a diver’s tissue rests on a speed fixed by the slowest ripple, and it’s a reminder of how narrow a range a bubble’s speed has. Whatever size came out of the valve, it was within a third of that number.

Other liquids #

That’s the floor. Whether a liquid’s bubbles actually come down to it depends on the viscous branch.

The peak exists when the viscous branch reaches the wave law above its floor. In water, the low viscosity lets a clean bubble overshoot. In ethanol the peak is much shallower, with a smaller Morton number’s margin. In the two viscous liquids, the viscous branch comes up from below and joins the wave law on its rising side. So their bubbles only ever speed up with size, and the common-sense rule holds all the way. Bubbles rising through honey don’t have a floor. They just rise more quickly when bigger.

What’s surprising is how little the floor varies among very different liquids. It’s set by $(\sigma g/\rho)^{1/4}$, and the fourth root allows a range of 0.175 to 0.236 metres a second across liquids whose viscosities differ by a factor of a thousand. Mercury has seven times water’s surface tension and fourteen times its density, and its slowest ripple travels only about a sixth more slowly. That’s why the number seems nearly universal: a few tenths of a metre a second covers almost anything a bubble rises in.

One line for each liquid #

Grace in 1973 organised the shapes and speeds of bubbles and drops into a single map, in terms of three numbers: a Reynolds number, an Eötvös number (which is the Bond number on the bubble’s diameter) and a Morton number, $Mo=g\mu^4/\rho\sigma^3$. Only two of the three are free.

The Morton number has a useful property, and it’s the reason the map works. It contains only the properties of the liquid: not the bubble’s size or speed. Each liquid is therefore a single curve on the map, and a bubble’s size moves along it. So comparing water with an oil is a matter of comparing two curves, and a measurement in one liquid can be used directly in another at the same Morton number. Water’s is extraordinarily small, $2.5\times10^{-11}$, because its viscosity is low and its surface tension is high. It’s near the end of the range of ordinary liquids. That’s why a bubble in water goes through such a rich sequence of regimes, whereas in a syrup at a Morton number of 31 there’s only one. A liquid’s Morton number is really a statement of how many regimes it has room for.

Leonardo’s bubble #

Just past the peak, something visible happens. A bubble of about 1.8 millimetres in clean water stops rising straight. It starts to zigzag or to spiral. Leonardo da Vinci drew it in the Codex Leicester, about 1510: a bubble rising in a spiral. For more than four centuries no one could say why, and the phenomenon has come to be called Leonardo’s paradox, as Prosperetti and his colleagues named it in 2004. The instability is a wake instability: at the critical size, bubble flattening produces enough surface vorticity to destabilise the wake, which then sheds a pair of trailing vortices, and those drive the bubble sideways. Mougin and Magnaudet computed that in 2002, showing it occurs at a critical shape rather than simply a critical Reynolds number. The bubble’s drag rises with that path, and the peak in the first figure is thus accompanied by a loss of straightness. Bubbles at the peak aren’t merely flattened. They’re on the point of changing their path.

So the joined law’s clean peak, 0.298 metres a second, is a smooth curve that has papered over a real change of state. It stands about 15 per cent below Duineveld’s measured peak of 0.35 metres a second in hyper-clean water, which is the sharpest such measurement. The peak in a clean liquid is hard to reach experimentally, since any tiny trace of contaminant erases much of it, and the measured peak depends on purification as well as size. At the floor, though, none of this matters: all the curves merge.

What the picture cannot show #

The path. The law gives one speed for each size, whereas in reality a bubble of 5 millimetres has a wobbling, spiralling path with a period of a few hundredths of a second, and its speed fluctuates over that path. What the figure calls the speed is the mean vertical rate over many oscillations. A bubble released in a different manner, by different nozzles, keeps memory of that release for a surprising distance, and that’s an additional source of scatter of ten per cent or more in measured speeds at a stated size.

The joined law. Combining the viscous and wave regimes as $1/U^2=1/U_v^2+1/U_w^2$ is a fit, not a derivation. It serves to draw one line through the regimes, and in the crossover the shape it has is a matter of convenience. The peak’s value depends on this, and the floor doesn’t.

Nearby walls and a crowd. A bubble near a wall feels it at several diameters, and a column of bubbles rises in the wake of those ahead, moving faster, by the same drafting that the follower beats in time with the wake has in another medium. A swarm has quite a different mean rise rate.

Mendelson’s reasoning. The wave analogy is only an analogy. It fits the data well, and a full justification of it remains unknown, especially near its minimum, in the region where surface tension and gravity both matter and the bubble’s shape is complicated. The Davies–Taylor cap is well understood on its gravity side alone. That two numbers agree doesn’t mean that a bubble “is” a wave.

The convention #

The bubble’s size is its volume-equivalent diameter: the diameter of a sphere of the same volume. All liquids are at about 20 °C, and the bubble’s own density and viscosity are ignored. A clean surface has no surfactant at all; a contaminated one is taken to behave as a rigid sphere, which is the limit for fully contaminated small bubbles. The coefficients in the wave law are 2.14 and 0.505, from Mendelson’s successors’ fit. The wave speed is on deep water, with no bubble in it. Capillary length is $\sqrt{\sigma/\rho g}$, the Eötvös number is on the diameter, and the Morton number is $g\mu^4/\rho\sigma^3$.

How it was checked #

The small-bubble limits are checked in a very viscous liquid, where both reduce to their exact solutions. The floor and ripple speeds are compared in closed form: the ratio is a pure number with no property of the liquid in it, and it comes out the same to all figures in all four liquids. The model refuses a bubble of no size and a liquid not stated.

Who found it, and when #

Leonardo’s drawing dates from about 1510. Hadamard and Rybczynski’s solutions are from 1911, Levich’s drag from 1949, Davies and Taylor’s cap from 1950, and Moore’s boundary-layer correction from 1963. Mendelson’s wave analogy is from 1967, Grace’s map from 1973, Duineveld’s measurements in hyper-clean water from 1995, Mei, Klausner and Lawrence’s drag law from 1994, and Mougin and Magnaudet’s computation of the path’s onset from 2002.

Still open: why the wave analogy should hold #

Nothing in the essay derives Mendelson’s law, and it can be tested more sharply than it has been. The next calculation would take a steady oblate bubble, with its shape computed from a boundary-integral potential flow and a Young–Laplace surface at each Weber number, and compute the speed at which its top surface’s curvature balances the dynamic pressure there. It would ask how closely that shape-resolved speed follows $U^2\propto\sigma/\rho d+gd$, whether the factor between the wave’s wavelength and the diameter really stays near $\pi$ across the floor, and so whether the 1.02 is a coincidence of two fits or the sign of a surface mechanism at work.