The warm side pulls a bubble with no force on it

The surface that moves with the flow worked out Hadamard and Rybczynski’s drop: a sphere of one fluid moving slowly through another, its interface free to slide, so that the liquid outside drags the surface along and the drag falls to two-thirds of a rigid sphere’s. Among the things that essay set aside was a second kind of Marangoni flow. A gradient of surface tension along an interface drives the liquid next to it, and the gradient does not have to come from surfactant. Surface tension depends on temperature. Put a bubble in a liquid that is warmer on one side than the other and its surface is under more tension at the cold pole than at the warm one. The surface is pulled towards the cold pole, it drags the liquid next to it the same way, and the bubble moves the other way, towards the heat, with no gravity and nothing pushing it.

That is thermocapillary migration, and Young, Goldstein and Block calculated its speed in 1959 for the slow-flow case. The calculation is short and every step of it is a condition the previous essays have already used. What is worth having is not the formula alone but three things it says that a picture of a body being pushed through a liquid gets wrong: the bubble is not pushed, so its wake is nothing like a pushed body’s wake; a drop that conducts heat well hardly moves at all; and the size at which a bubble’s migration exactly balances its own buoyancy does not depend on how viscous the liquid is.

The temperature the surface actually sees #

The first step is the temperature, and it is a problem in conduction alone. Far from the bubble the liquid has a uniform gradient $G$. If the bubble’s own motion carries too little heat to matter — the condition is examined at the end — then the temperature inside and outside satisfies Laplace’s equation, and the solution that matches the uniform gradient at a distance is the gradient plus a dipole centred on the bubble. Inside, the temperature is again a uniform gradient, of a different size. Requiring the temperature to be continuous at the interface and the heat flux through it to be continuous fixes both, and the surface temperature comes out as

$$T_s = \frac{3}{2 + \kappa}\, G a \cos\theta ,$$

with $a$ the radius, $\theta$ the angle from the warm pole’s direction, and $\kappa$ the drop’s thermal conductivity divided by the liquid’s.

The figure plots that temperature from the cold pole round to the warm one, for four drops. A gas bubble, with $\kappa$ near zero, sees more than the gradient’s own difference across it — one and a half times as much — because heat cannot cross the bubble and has to flow round it, crowding the isotherms at the poles. A drop that conducts exactly as well as the liquid sees the undisturbed gradient, as if it were not there. A drop twenty times as conductive as the liquid short-circuits the gradient through itself, and its surface sees about a seventh of the difference. Conductivity therefore enters before any fluid moves at all: it decides how large a temperature difference, and so how large a tension difference, the surface has to work with.

A pull at the surface and nothing on the bubble #

Surface tension falls with temperature for almost every liquid: $\sigma = \sigma_0 + \sigma_T (T - T_0)$ with $\sigma_T$ negative — about $-0.15$ mN/m per kelvin for water, $-0.06$ for a silicone oil. Along the surface the tension therefore varies as $\cos\theta$, and its gradient is a tangential stress on the interface, strongest at the equator and zero at the poles, $S \sin\theta$ with $S = 3|\sigma_T| G/(2 + \kappa)$, pointing towards the cold pole.

The flow is then Hadamard and Rybczynski’s flow with a different condition in one place. Outside and inside, the flow is the Stokes flow round a sphere, written as stream functions with four unknown coefficients between them. The interface conditions are the ones the surface that moves with the flow used: no flow through the surface from either side, the same tangential velocity on both sides of it, and a balance of tangential stress. The only change is that the stress balance now has the thermal pull in it: the shear from the liquid outside, less the shear from the fluid inside, plus the pull, must sum to zero at every point of the surface.

Those are three conditions, or four counting the two sides’ impermeability separately, and the problem has five unknowns: the four coefficients and the bubble’s speed. What closes it is the condition that makes this a different problem from a falling drop. In a liquid with no gravity, nothing outside acts on the bubble, so the total force the liquid exerts on it must be zero. In the stream function, the force is carried by exactly one term — the Stokeslet, the term that grows as $r$ in the stream function and makes the velocity fall as $1/r$ — and setting its coefficient to zero is the fifth equation. Solved as a four-by-four linear system with that term removed, the speed lands on Young, Goldstein and Block’s

$$U = \frac{2\,|\sigma_T|\, G a}{(2 + \kappa)(2\mu + 3\mu')}$$

to the last digit for every viscosity ratio tried, towards the warm side. For a gas bubble it is $|\sigma_T| G a/(2\mu)$.

The figure shows what the zero-force condition does to the liquid. It draws streamlines in the frame where the liquid far away is at rest, for the same clean bubble in two situations. Rising under its weight, the bubble is a body with a net force on it, and the liquid round it is the field of a point force: every streamline is open, and all the liquid, ahead, behind and to the side, moves forward along the path. Migrating, with no net force, the bubble’s outer flow is the field of a potential dipole, and every streamline is a closed loop starting at the bubble’s front and ending at its back. Liquid is shoved aside ahead of the bubble and handed back behind it, and that is all. The bubble swims rather than being pushed: its surface does the work, as the surface of a swimmer that cannot go backwards does, and a swimmer at low Reynolds number is a force-free body for the same reason.

What having no force does a long way off #

The difference in the streamlines is a difference in how fast the disturbance dies away, and the second figure measures it. Along the path ahead of the bubble the liquid’s speed, as a fraction of the bubble’s own, falls as $1/r$ for the rising bubble and as $1/r^3$ for the migrating one. Ten radii ahead the rising bubble moves the liquid at a tenth of its own speed, the migrating one at a thousandth. A hundred radii ahead the two are a hundredth and a millionth.

The slow $1/r$ decay is the signature of a net force, and it causes most of the trouble in slow viscous flow. It is why a force without the flow that makes it could compute a sphere’s motion from the Stokeslet alone, and why a viscosity made of particles needed care with sums over distant spheres: the field of a point force reaches a long way. In two dimensions it reaches so far that the flow with no solution has no slow-flow solution at all. A migrating bubble has none of this. Its far field is the inviscid flow round a sphere — the same dipole as in a perfect fluid — and two migrating bubbles ten radii apart move each other at a thousandth of their speed, where two rising ones would move each other at a tenth. A cloud of bubbles migrating towards a heater in orbit behaves much more like a set of independent bubbles than a cloud of sedimenting particles does.

Two separate reasons a drop is slow #

The speed formula has two factors in its denominator, and they belong to different parts of the physics. The factor $2 + \kappa$ is the conduction: it is the reduction in the surface temperature difference that the figure on conductivity showed. The factor $2\mu + 3\mu'$ is the flow: the interior of a drop has to circulate when its surface moves, and a viscous interior resists being circulated, just as in Hadamard and Rybczynski’s drag.

The fourth figure plots the speed in units of $|\sigma_T| G a/\mu$, the speed scale the tension gradient sets, against the viscosity ratio for four conductivity ratios. A gas bubble migrates at half the scale. A drop as viscous as the liquid and as conductive migrates at 0.13 of it, about a quarter of the bubble’s speed. A drop a hundred times as viscous as the liquid moves at under a hundredth of the scale whatever its conductivity, because its surface is nearly rigid. A drop twenty times as conductive moves at under a twentieth of the scale whatever its viscosity, because it barely feels the gradient. The two factors simply multiply, and the figure’s curves are the same shape shifted down: the flow does not care why the surface pull is weak, only how weak it is.

This matters for which experiments work. A liquid-metal drop in an oil, or a water drop in a poorly conducting oil, is close to the conducting end, and the migration is slow for a reason that has nothing to do with viscosity. A gas bubble in any liquid is at the fast corner of the figure on both counts, which is why bubbles are the standard test of the theory.

Against gravity, and a radius with no viscosity in it #

On the ground, gravity does not go away, and the natural question is which effect wins. Buoyancy drives a clean bubble upward at Hadamard’s speed, $\rho g a^2/(3\mu)$, which grows as the square of the radius. Migration grows only in proportion to the radius. Small bubbles therefore go where the temperature sends them and large bubbles go where gravity sends them.

The fifth figure puts numbers on it for an air bubble in a 10 cSt silicone oil, a common working liquid for these experiments, in a gradient of one kelvin per millimetre. The migration line rises with slope one on the logarithmic axes and the buoyancy line with slope two. They cross at a radius of 9.8 µm, where both speeds are about 31 µm/s. Below that radius the temperature gradient wins.

The crossing radius is $3|\sigma_T| G/(2\rho g)$, and it contains no viscosity. That is not a coincidence of the numbers chosen. Both speeds are speeds against the same viscous drag on the same mobile surface, so the viscosity divides out of the comparison. A bubble balanced at 9.8 µm in a 10 cSt oil would be balanced at the same size in an oil a thousand times as thick. It would simply take a thousand times as long to settle into balance. The radius depends only on how strongly temperature changes the tension, on the gradient, and on the liquid’s weight. For a drop rather than a bubble, the viscosity ratio does enter, through the different ways the interior’s viscosity affects the buoyant and the thermal motions, but the outer viscosity still cancels.

For water in the same gradient the crossing is at 23 µm, larger because water’s tension changes more steeply with temperature. Water is also where the assumption of a clean surface is least safe, for reasons the next section and a thousandth of a monolayer holds a bubble still both bear on.

How the measurement stood bubbles still #

The radius independent of viscosity is also how the effect was first measured. Young, Goldstein and Block did not try to time bubbles moving sideways in a gradient. They heated a vertical cell of liquid from below, so that the warm side was down and the temperature pull acted downward against buoyancy, and they looked for bubbles that stood still. For a given gradient, bubbles larger than the balancing size rose and smaller ones sank towards the hot floor. Finding the gradient at which a bubble of a given size hovered measured the balance directly. It is a test of the theory that needs no clock, and it needs no knowledge of the viscosity.

The sixth figure shows the net vertical speed against radius for three gradients. Each curve dips below zero for small bubbles, which sink, and rises steeply for large ones, which float. It crosses zero at 4.9, 9.8 and 19.6 µm, doubling with the gradient. The balance sorts bubbles sharply: near the crossing the net speed changes sign steeply with radius, so a population of bubbles of mixed sizes separates into a rising half and a sinking half, and the size at which they divide is a clean threshold.

Heating from below has a hazard of its own. A liquid heated from below is unstable to convection once the gradient is large enough, and convection would carry bubbles far faster than either mechanism here. The experiment needs a gradient below that threshold, and a cell thin or viscous enough to keep it there. The threshold the walls decide computed when such a layer starts to overturn; a hovering-bubble cell is designed to stay on the quiet side of that number.

Where the formula stops holding #

Every step above assumed two things that the bubble’s own motion can undo. The flow was taken to have no inertia, which needs a small Reynolds number. The temperature was taken to be set by conduction alone, which needs the bubble’s flow to carry heat round its surface more slowly than conduction does: a small Marangoni number. Both can be computed in advance on the speed scale $|\sigma_T| G a/\mu$, and both grow as the square of the radius.

The last figure plots them for both liquids at one kelvin per millimetre. The Marangoni number is always the larger of the two, because both liquids carry momentum by viscosity faster than they carry heat by conduction, so it is the one that fails first. It reaches one near 120 µm in silicone oil and near 31 µm in water. The hovering bubbles in silicone oil, at ten micrometres or so, sit well inside the range where the formula holds. The water bubbles at the crossing radius do not: at 23 µm the Marangoni number is already about one half.

Past that number the bubble’s own surface flow carries warm liquid from the warm pole round towards the cold one, and the temperature difference along the surface shrinks. The pull that drives the migration weakens, and the bubble moves more slowly than the formula says. How much more slowly is not something the conduction solution can tell.

What the formula leaves out #

The surface is assumed clean. In water this is almost never true. A thousandth of a monolayer holds a bubble still found that a trace of surfactant swept to the rear of a rising bubble immobilises a cap there and raises the drag by up to a half. A migrating bubble sweeps its surface towards the cold pole just as a rising bubble sweeps it to the rear, so contamination collects at the cold pole. It does not only raise the drag here. It also removes the surface that the pull acts on, and that is a much larger effect.

The bubble is a sphere. Surface tension holds it round when the capillary number is small, as the drop that is not a tear required for a falling drop. The thermal pull does not change that at this order: the conduction-and-Stokes solution satisfies the balance of normal stress on an exact sphere, so the pull deforms nothing until inertia or convected heat enters.

The gradient is uniform across the bubble. A gradient that curves over the bubble’s diameter adds higher harmonics to the surface temperature. Only the first harmonic, the $\cos\theta$ term, contributes to the speed.

Nothing is soluble. A vapour bubble that condenses on its cold side and evaporates on its warm side moves by quite a different mechanism, and the latent heat changes the surface temperature it sees. That bubble is outside this calculation.

Still open: a contaminated surface in a gradient #

The obvious next calculation brings the two previous essays together. A bubble migrating in a temperature gradient in ordinary water will carry some surfactant, and its own surface flow will sweep that surfactant towards the cold pole into a stagnant cap, just as a rising bubble’s flow sweeps it to the rear. For a rising bubble, the cap can only cost a third of the speed, because the worst it can do is turn the bubble into a rigid sphere. For a migrating bubble there is no such floor. A rigid sphere in a temperature gradient does not migrate at all, because the pull has no free surface to act on. So the same cap that costs a rising bubble a fraction of its speed could stop a migrating one entirely. How large the cap has to be, and how little surfactant builds it, is the next calculation.