The surface that moves with the flow

Every problem in this collection until now has put a rigid wall in the fluid. No slip, no flow through, and the surface goes where it is told. That is a boundary condition and it is a modelling choice, and the interesting question is what changes when it is dropped.

A liquid drop or a gas bubble has a surface the fluid outside can drag along — which is the same freedom a swimming sheet uses on purpose, arriving here as something the drop has no say in. The shear the outside applies is transmitted to the inside, which circulates; the interface moves; and the fluid outside is sheared less than it would be against a rigid wall of the same shape.

The consequence is a drag that is smaller, and the amount is exact.

Why it is a boundary condition and not a force #

Before the arithmetic, it is worth being precise about what has changed, because the usual summary — “a bubble has less drag because it is not solid” — hides the mechanism.

Nothing about the equations has changed. The fluid outside a bubble obeys the same Stokes equations it obeys outside a marble. What has changed is what is imposed where the fluid meets something else, and there are two conditions at a clean interface where a rigid wall has one.

At a rigid wall the velocity is given: it is the wall’s. At a clean interface the velocity is not given — it is whatever the two fluids agree on — and what is given instead is that the tangential stress must be continuous across it, since an interface with no thickness has no mass and cannot carry a net force. Two conditions replace one, and the extra unknown is the interface’s own motion.

That is why the answer depends on the viscosity ratio and on nothing else: the ratio is what decides how much motion a given stress buys on each side. And it is why the whole effect can be destroyed by a monolayer, since a monolayer does have a tangential stress of its own and restores the velocity-is-given condition without changing anything about either bulk fluid.

The drag #

Hadamard and Rybczynski solved it independently in 1911, and the answer is

$$F = 6\pi\mu a U \cdot \frac{\tfrac23 + \kappa}{1 + \kappa}, \qquad \kappa = \frac{\mu_{\text{in}}}{\mu_{\text{out}}}.$$

At $\kappa \to \infty$ the interior is infinitely viscous, the surface cannot move, and the factor is one — Stokes’ law, recovered exactly. At $\kappa = 0$ the interior offers no resistance at all and the factor is two-thirds.

The range is 2/3 to 1 and nothing lies outside it. There is no drop, of any material, in any liquid, that has less than two-thirds of a rigid sphere’s drag by this mechanism.

Where the heat goes, and the ten per cent #

This is a phase about dissipation, and the drop offers a split that a rigid sphere cannot.

The total dissipation is $F\!\cdot\!U$, exactly — the balance established for a translating sphere, now over two regions instead of one, and closing to a part in a hundred thousand on a shell integration out to two thousand radii. What is new is that it can be apportioned.

The share destroyed inside the drop works out at

$$\frac{D_{\text{in}}}{D_{\text{total}}} = \frac{\kappa}{(1+\kappa)(2+3\kappa)},$$

which is zero at both ends and has a maximum of 10.10 per cent at $\kappa = \sqrt{2/3} = 0.8165$. At $\kappa = 1$ it is exactly a tenth.

That ceiling is not obvious and it is worth saying why it exists. The interior circulation is driven by the exterior; its velocity scale is set by what the interface is doing, which is at most half the free stream; and its volume is small compared with the region of the exterior that is being disturbed, because the exterior disturbance extends for many radii. Both factors work against the interior, and the product of them never exceeds a tenth.

The rise speed that clean water does not deliver #

Put the drag law into a buoyancy balance and a clean air bubble in water should rise 49 per cent faster than Stokes’ law says — the factor is $1/(2/3 + \kappa)/(1+\kappa)$ evaluated at $\kappa = 0.018$, and it is a constant, so the whole line shifts rather than tilting.

They lie on the lower line. A bubble smaller than about a millimetre in ordinary water rises at Stokes’ speed, as though its surface were rigid — and the discrepancy was a genuine puzzle for decades before it was resolved.

What immobilises a surface #

The resolution is surfactant, and the argument is worth following because it is a beautiful piece of reasoning about a boundary condition.

Any real water contains trace surface-active material — parts per billion is enough. It adsorbs onto the bubble’s surface. As the bubble rises, the surface flows from the front to the back, and it carries the adsorbed material with it, which accumulates at the rear.

A gradient of surface concentration is a gradient of surface tension, and a gradient of surface tension is a stress in the interface: it pulls from where the tension is high towards where it is low, which is from the back towards the front, which is against the motion of the surface.

So the surface stops. Not because anything is holding it, but because the contamination it swept backwards is pulling it forwards, and the balance is reached when the surface velocity is essentially zero. The bubble then presents a rigid boundary to the fluid outside, and its drag is Stokes’.

The bubble’s boundary condition is set by a contaminant present at parts per billion, and the factor it costs is a third of the drag. That is as sharp an example as this subject offers of a boundary condition mattering more than an equation.

The energy version of the same argument #

The surfactant story can be told in this essay’s own currency, and doing so makes it sharper.

A clean bubble and a contaminated one of the same size, rising at their own terminal speeds, are both in a steady state, so in both cases the buoyancy power equals the total dissipation. The clean bubble rises faster, so it is releasing potential energy faster — 49 per cent faster — and therefore dissipating 49 per cent more per second.

That is the opposite of what “less drag” suggests. A clean bubble is not a lower-loss object; it is a faster one, and it makes more heat per unit time precisely because it goes faster. What it is is more efficient per unit height climbed: the energy released per metre of rise is the buoyancy times a metre either way, so both bubbles dissipate exactly the same amount of energy getting to the surface.

The mobility does not save energy. It saves time. Which is the right way round for the applications: a degassing furnace and a flotation cell care about how quickly bubbles arrive, not about how much heat they make on the way.

The surface that is rigid at one end #

The account above has two states — clean and contaminated — and a real bubble is usually in neither. The reason is in the mechanism itself: the surfactant is swept backwards, so it does not coat the bubble evenly. It piles up at the rear and leaves the front bare.

What that produces is a stagnant cap: a region around the rear stagnation point where the surfactant is dense enough for the Marangoni stress to hold the interface still, and a leading region where there is not enough to matter and the surface moves freely. The boundary between them sits at some angle from the rear, and that angle — not the presence or absence of surfactant — is what the drag depends on.

The problem is harder than either limit, because the boundary condition changes type partway along the surface: velocity given over the cap, stress given over the rest, with the dividing angle an unknown fixed by the surfactant’s own balance of convection against adsorption. It has an exact solution anyway, obtained by Sadhal and Johnson in 1983, and the drag it gives rises monotonically from the Hadamard–Rybczynski value at zero cap angle to Stokes’ law at a cap covering the whole sphere. Both limits of this essay are the two ends of one curve, and the intermediate cases are not a smudge between them but states a bubble genuinely occupies.

Three observations become explicable at once.

Rise speed varies continuously with water quality rather than switching between two values. Adding surfactant to a clean system slows the bubbles gradually, which is what the cap angle doing so predicts and what two discrete states cannot.

A bubble slows down as it rises. Released clean into water that is not, it starts near the mobile speed and decelerates over the first part of its travel while the cap grows to its equilibrium extent — so a measurement in a short column and one in a tall column disagree, and the disagreement is not experimental error.

And size decides which limit wins. A bubble’s surface area grows as the square of its radius while the surfactant available near it grows as the cube of the distance swept, so a large bubble presents more surface than the trace impurity can cover and keeps a bare front. That is the quantitative version of the paragraph below about large bubbles staying mobile, and it says the threshold is a comparison of two rates rather than a size anybody could look up.

So the honest statement of the boundary condition is not that it is one thing or the other. It is that the interface carries a field of its own, and where that field is strong enough the fluid outside sees a wall.

When the surface is mobile after all #

Three cases, and they are the cases where the effect is used.

Large bubbles. A bubble big enough has a surface area growing faster than the surfactant can adsorb onto it, and the leading portion stays clean. Bubbles above a few millimetres in water do show mobile-interface behaviour — along with a great deal else, since they are no longer spherical and no longer creeping.

Very clean systems. Distilled and degassed water, carefully handled, gives bubbles that rise at the Hadamard–Rybczynski speed. It is difficult to arrange and it is the experiment that settled the question.

Liquid metals. Molten metals have no surfactant chemistry of the kind water has, and drops in them behave as the theory says. Bubbles in liquid steel — which matters, since that is how steel is degassed — are genuinely mobile.

The same story in a suspension #

The rigidity assumption dropped here is the one Einstein’s coefficient is built on, and dropping it changes that number by exactly the same mechanism.

Taylor gave the emulsion version in 1932: the coefficient is not 5/2 but

$$\frac{5}{2}\cdot\frac{\kappa + \tfrac25}{\kappa + 1},$$

running from 1 for clean bubbles to 5/2 for rigid particles. A dilute foam is thickened by only $\phi$ rather than by $2.5\phi$ — and the same surfactant argument applies, so a real foam behaves like a suspension of rigid particles and is thickened by the full 5/2.

That is the same result twice, in two different measurements, with the same trace impurity deciding both.

What “mobile” means quantitatively #

The interface speed is worth a number, because the phrase “the surface moves” invites a picture of the surface travelling with the flow, and it does not.

At the drop’s equator, the surface moves at

$$\frac{U}{2(1+\kappa)}$$

as a fraction of the free stream. For a clean bubble that is very nearly half; for equal viscosities a quarter; for a rigid particle zero. The most a free surface ever moves is half the free-stream speed, and it does so only in the limit of an inviscid interior.

That ceiling explains the drag ceiling. The exterior fluid is sheared between the free stream and the surface, and halving the velocity difference at the wall does not halve the shear — the disturbance adjusts over the same length scale — so the drag falls by a third rather than by a half.

It also explains why the effect is so much smaller than people expect it to be. A bubble is, from the point of view of the fluid outside it, a slightly slippery sphere rather than a hole. The whole mechanism is worth 33 per cent, and the difference between a bubble and a lead shot of the same size falling in the same oil is entirely in the buoyancy, not in the drag.

What the picture cannot show #

The drop is spherical and stays spherical. That requires surface tension to dominate both viscosity and inertia — a small Weber number and a small capillary number. A drop moving fast enough deforms, and what it deforms into is not a tear.

The interface is clean, and almost none are. Everything above is what happens when the surface carries nothing. The surfactant argument is described here and not computed; doing it properly needs an adsorption isotherm and a surface transport equation, which is a different subject.

There is no Marangoni flow of the other kind. A temperature gradient along an interface also produces a surface-tension gradient and drives a flow, and a bubble in a temperature gradient migrates towards the warm side with no gravity involved at all. None of that is here.

The dissipation is the incompressible one. Both fluids here are treated as having no divergence anywhere, so the second viscosity contributes nothing — exactly rather than approximately, which is worth saying for a bubble, whose interior is a gas.

And the flow is creeping. The formula holds at Reynolds numbers well below one. The bubbles it is usually applied to are not, which is why the terminal-velocity figure marks the points that have left the regime.

Two places the third of the drag is worth money #

Flotation. Mineral processing separates valuable particles from waste by attaching them to rising bubbles, and the rate of the whole plant is the rate at which bubbles rise through the cell. A thirty-three per cent difference in rise speed is a thirty-three per cent difference in throughput, and it is decided by the water chemistry rather than by anything mechanical. Flotation cells are operated with deliberate surfactant, because a bubble that is too mobile does not hold its particle.

Steel degassing. Argon is bubbled through molten steel to strip dissolved gases, and the contact time is set by the rise speed. Liquid metals have no adsorbed organic layer, so the bubbles are genuinely mobile and the design uses the two-thirds factor rather than Stokes’ law — one of the few industrial processes where the clean-interface answer is the right one.

Between them those make the point that a third of a drag is not a rounding error in any process whose economics is a residence time.

Who found it, and when #

Hadamard and Rybczynski published within months of one another in 1911, unaware of each other’s work, and the result carries both names. The discrepancy with measured bubble speeds was noticed almost immediately and was not settled until Frumkin and Levich’s surfactant argument in the 1940s, with the experimental confirmation — clean bubbles rising at the predicted speed — coming later still.

The surprising connection is with a completely different kind of contamination problem. The mechanism here — a swept-along impurity accumulating at a stagnation region and generating a stress that opposes the sweeping — is the same mechanism by which a surfactant stabilises a foam, by which a tear film keeps an eye wet, and by which the Marangoni effect drives a droplet of alcohol across a water surface. In every case a vanishing amount of material at an interface governs the behaviour of a great deal of fluid, because an interface has no thickness and therefore no dilution.

Where the ladder goes next #

Beside this rung is Einstein’s coefficient, which is the same boundary condition in a bulk property, and a swimmer that cannot go backwards, which is the other essay in this collection about a surface that moves on purpose.

Below it is how many things a flow must be told, which is where the arithmetic of boundary conditions is set out, and the world with no inertia, which is the regime this all holds in.