A cap that slows a rising bubble stops a migrating one

The warm side pulls a bubble with no force on it computed Young, Goldstein and Block’s speed for a clean bubble in a temperature gradient. Surface tension is lower where the liquid is warmer, the surface is pulled towards the cold pole, and the bubble moves towards the heat at $|\sigma_T| G a/(2\mu)$, with no net force on it. That essay assumed the surface was clean. A thousandth of a monolayer holds a bubble still had already shown how rarely that holds: a rising bubble’s surface flow sweeps any trace of surfactant to its rear, the surfactant piles into a stagnant cap whose own tension gradient holds the surface still, and the drag climbs towards a rigid sphere’s.

A migrating bubble’s surface also flows, from the warm pole towards the cold one, so contamination collects at the cold pole in the same way. The question is what the cap costs. For a rising bubble the answer has a floor. The worst a cap can do is make the bubble a solid sphere, and a solid sphere rising under the same weight still rises, at two-thirds of the clean speed. For a migrating bubble there is no floor, because a solid sphere in a temperature gradient does not migrate at all. The pull acts on the surface, and a surface that cannot move passes no pull to the liquid. This essay computes how fast the migration speed falls between those ends, and finds that it falls far faster than the drag rises.

Two problems added together #

The flow is slow, so it is linear, and a migrating bubble with a cap can be built from two problems that are each easier.

The first is a bubble with a cap being towed through still liquid at some speed, with no temperature gradient. Over the cap the surface is held still; over the rest it carries no shear. That is exactly the calculation the stagnant-cap essay solved, and its drag in units of the clean bubble’s runs from one with no cap to one and a half with the cap over everything, on Sadhal and Johnson’s closed form.

The second is the same bubble held fixed in the temperature gradient. Over the cap the surface is still; over the rest the thermal pull, $S \sin\theta$, is applied as a tangential stress, with $S = 3|\sigma_T| G/2$ for a gas bubble. The liquid is set moving by the free part of the surface and pushes on the held bubble with a force towards the warm side. The flow is written as a series of decaying axisymmetric modes, each of which leaves the surface impermeable, and the coefficients are fitted by least squares at several thousand points: zero surface speed on the cap, the thermal stress off it. The push is read from the coefficient of the single mode that carries a force, the Stokeslet.

Added together, with the towing speed chosen so that the towing drag cancels the push, the two problems make a free bubble with no net force on it, migrating in the gradient with its cap. The migration speed is the push divided by the drag.

Both problems switch boundary condition at one circle of latitude. How many things a flow must be told counted the conditions a viscous flow needs at a surface: two components of velocity or stress at every point. On the cap both are velocities, and the surface is still and impermeable. Off it, one is a velocity, the impermeability, and the other is a stress, which is zero for the towed bubble and the thermal pull for the held one. The count stays the same everywhere round the bubble, and only the type of condition changes at the cap’s edge, where the stress is singular.

The migration speed falls far faster than the rise speed #

The first figure puts the two speeds side by side against the cap’s half-angle, measured from the cold pole for the migrating bubble and from the rear for the rising one, which is the same place on the bubble. Small caps cost both very little: a cap of 30 degrees leaves the rising bubble 97 per cent of its speed and the migrating one 92 per cent. After that they separate quickly. A cap of 60 degrees leaves the rising bubble 86 per cent and the migrating one 57. A cap over the whole cold hemisphere, 90 degrees, leaves the rising bubble 74 per cent and the migrating one 21. At 120 degrees the rising bubble has 68 per cent, within two points of a solid sphere, and the migrating one has four.

The rising bubble’s curve flattens onto its floor at two-thirds. The migrating bubble’s curve has no floor and goes to zero, and it gets close to zero long before the cap covers the bubble. By 120 degrees it is at four per cent with a quarter of the surface still free, and by 135 degrees at one per cent. A bubble whose cold half is held still, and whose warm half is entirely clean, migrates at a fifth of its clean speed.

Where the push comes from #

The speed is a push divided by a drag, and the drag is the familiar part. The figure’s rising curve is the reciprocal of the drag, and it never falls below two-thirds. The large losses therefore come from the push. With a cap of 90 degrees the drag has risen by 36 per cent, while the push has fallen to 29 per cent of its clean value.

Why the push falls so fast is clearest from a different route to the same number. A force without the flow that makes it used the reciprocal theorem of slow viscous flow to compute a sphere’s motion from the solution of a different problem. The same theorem applies here. It says the force a surface stress puts on a held body equals that stress weighted by the speed the body’s own surface would have if the body were towed, integrated over the surface. For the held migrating bubble, the stress is the thermal pull on the free part, and the weight is the towed bubble’s surface speed, which is zero on the cap and reduced in front of it.

The second figure plots that integrand round the bubble from the cold pole, and its area is the push. For a clean bubble it is $\sin^3\theta$, a broad hump centred on the equator with more than nine-tenths of its area between 40 and 140 degrees. Every factor in it peaks at the equator: the thermal pull is largest there, the towed surface moves fastest there, and the equator’s ring has the most surface. A cap of 45 degrees removes the low cold-side shoulder and lowers the rest a little. A cap of 90 degrees removes the whole cold half of the hump and, by holding the surface still at the equator, pulls the remaining half down towards zero at its cold end. What is left is a lopsided sliver on the warm side, where the pull is weaker.

That is the whole mechanism. The stagnant-cap essay found the rising bubble’s drag running ahead of the area covered, because a cap slows the free surface in front of it. Here the same slowing acts twice: once on the drag, which divides the speed, and once on the push, which multiplies it. And the push depends on the surface’s motion at the equator far more than the drag does.

The figure also checks the calculation. The series for the held bubble and the reciprocal integral of the towed bubble share no unknowns, and they agree to within 0.04 per cent of the clean push at every cap tried. Each converges slowly, as the inverse of the number of modes, because of the singular stress at the cap’s edge, so each is extrapolated from two truncations, as the drag was in the stagnant-cap essay.

The surface keeps running, more slowly #

The third figure shows the surface’s own motion relative to the bubble. On a clean migrating bubble the surface runs from the warm pole to the cold one at one and a half times the bubble’s speed at the equator, faster than the bubble itself. That is the engine: the surface drags the liquid next to it towards the cold pole, and the bubble moves the other way. With a cap the free surface still runs towards the cold pole, but it has to stop at the cap’s edge, and its speed falls everywhere. With a cap of 90 degrees the fastest point of the free surface has moved onto the warm side, near 118 degrees from the cold pole, and it runs at two-thirds of the clean bubble’s speed, under half of what the clean equator manages.

The figure is in units of the clean bubble’s speed. It shows that the surface flow and the migration it drives fall together: the liquid near the surface is still being dragged rearward, just much more weakly.

The comparison with a swimmer is close. A swimmer that cannot go backwards moves by deforming its own surface, and like the migrating bubble it carries no net force and moves only because its surface moves relative to it. A migrating bubble with a cap is a swimmer with part of its body paralysed. The intuition that a swimmer with half its surface still working should go at about half speed fails for the same reason here. The working half is the half nearer the warm pole, where the pull is weaker, and its motion is held back by the still half next to it.

What holds the cap still #

A stagnant cap needs a surfactant gradient strong enough to hold the surface still against everything that would move it. On a rising bubble that is the shear of the outer flow alone. On a migrating bubble it is the shear plus the thermal pull itself, which acts on the cap as on the rest of the surface and would drag the capped surface towards the cold pole if nothing resisted. The surfactant’s surface pressure, $RT\Gamma$ for a dilute layer, must rise from nothing at the cap’s edge towards the cold pole steeply enough to cancel both at every point.

The fourth figure integrates that balance along three caps. The natural unit is no longer the viscous stress times the speed, as it was for the rising bubble, but $S a = \tfrac32 |\sigma_T| G a$, which is half the difference in surface tension that the temperature makes between the poles. On that scale the thermal difference from pole to pole is two. The surfactant peaks at the cold pole at 0.91, 1.43 and 1.81 for caps of 45, 90 and 135 degrees.

The result has a simple reading. A cap stops being a small perturbation when the surface pressure its surfactant makes is comparable to the tension difference the temperature makes across the bubble. That difference is the entire driving force of the migration, and it is very small. Across a bubble twenty micrometres in radius in water, in a gradient of one kelvin per millimetre, it is nine micronewtons per metre, about an eight-thousandth of water’s surface tension. To hold the cold half of the surface still, a surfactant has to lower the tension at the cold pole by about three-quarters of that.

How little surfactant that is #

The fifth figure turns surface pressure into coverage for a concrete case: an air bubble twenty micrometres in radius in water, with no gravity, in three gradients, and a surfactant whose full monolayer holds 3.2 micromoles per square metre, as in the stagnant-cap essay. Clean, the bubble migrates at 1.5 millimetres per second in one kelvin per millimetre. It has lost half that speed at a mean coverage of 1.2 ten-thousandths of a monolayer, about one surfactant molecule for every eight thousand sites a full layer would fill.

The three gradients make the same curve shifted sideways. The coverage at which half the speed is lost is 0.6, 1.2 and 2.4 ten-thousandths of a monolayer, in proportion to the gradient. A stronger gradient pulls harder on the surface, so a given amount of contamination is swept into a smaller cap. Contamination is therefore relatively most damaging in the gentlest gradients, which are the ones a careful experiment prefers, because they keep the flow slow and the formula valid.

That explains a practical habit. Experiments on thermocapillary migration have generally used silicone oils and similar liquids rather than water. One reason is that the theory holds over a wider range of sizes in them. The other is that a liquid whose own surface tension is already low has few contaminants that lower it further, so its surface stays clean. Water, with the highest surface tension of the common liquids, is lowered by almost anything that reaches its surface. A water bubble that rises at three-quarters of the clean speed may be sitting in a gradient at a fifth of its clean migration speed, with the same cap doing both.

The checks #

The last figure lists what each part was checked against. The clean drop’s four-by-four solve reproduces Young, Goldstein and Block’s formula to fifteen digits at six viscosity ratios, and its far field carries no point force. The held bubble with no cap pushes with exactly a third of the force scale, and with the cap over everything it pushes with exactly nothing, as a solid sphere in a gradient must. The push with a cap agrees with the reciprocal theorem to 0.04 per cent of the clean value. The drag with a cap agrees with Sadhal and Johnson’s closed form to 0.06 per cent. The hover radius in silicone oil is the radius at which the two clean speeds match.

What the calculation leaves out #

The surfactant is insoluble and its cap is in equilibrium. A real bubble collects surfactant as it moves, by adsorption from the liquid, and its cap grows along its path. The cap computed here is the one the bubble carries once that has settled. The rate at which it settles is the open question of the stagnant-cap essay, and it applies here unchanged.

The cap’s edge is sharp. The stagnant-cap model holds the surface entirely still on one side of a circle and entirely free on the other. A real cap has an edge region where the surfactant thins out and the surface moves partly. That smooths the singular stress at the edge and moves the speed slightly towards the clean value at each cap angle. The model is the limit of a strongly surface-active surfactant, and at small coverage it is the right limit.

The surfactant does not change the temperature. The surface temperature is taken from conduction round a clean sphere. That holds because conduction is fast compared with the surface flow at small Marangoni number, and the cap changes only the flow.

The flow is creeping and the bubble is round. These are the conditions of the clean calculation, carried over. The world with no inertia is the regime, and the surface that moves with the flow set out the conditions on shape.

There is no gravity. On the ground a bubble in a vertical gradient feels both effects. Each was computed here with the other absent, and since the flow is linear they add, but the cap’s size depends on the total surface flow. A bubble near its hover radius moves slowly overall yet still has a vigorous surface flow, so its cap is set by the sum of the two flows rather than by the bubble’s net motion.

Still open: the speed when heat is carried round the surface #

Both this calculation and the clean one assume the Marangoni number is small, so that the surface temperature is set by conduction. For a bubble in water that fails above about thirty micrometres at one kelvin per millimetre. Past that size the surface’s own flow carries warm liquid from the warm pole towards the cold one. The temperature difference along the surface shrinks, and so does the pull. Measurements on larger bubbles in orbit have found them migrating more slowly than Young, Goldstein and Block’s formula. The calculation that would say by how much couples the Stokes flow to the convection of heat round the bubble and inside it, and solves both together as the Marangoni number grows. Its answer at large Marangoni number depends on a thin thermal layer at the surface, so it is a problem that needs a boundary layer rather than a series.