A lost steady state still holds the film

The film that heats itself finds a cliff. An oil film sheared at a fixed stress, with a viscosity that falls exponentially as it warms, has a steady temperature only while the Nahme–Griffith group $\delta = b\tau^2H^2/k\mu_0$ stays below 0.87846. Above that, the heating feeds on itself and there is no steady state at all. It is the Frank–Kamenetskii fold of thermal explosion theory, and the essay leaves open the question of what a film actually does once it has gone over it, which is a question about time rather than about steady states.

This essay answers it. The answer is not the one the word “cliff” suggests. A film just past the fold behaves, for a long time, almost exactly as if the fold had not been crossed.

The film in time #

The film is the same one, now allowed to change. With the temperature excess scaled by the temperature over which the viscosity falls by a factor of $e$, distance by the half-gap $H$ and time by the time heat takes to conduct across it, $H^2/\kappa$, the energy balance is

$$\frac{\partial\theta}{\partial t} = \frac{\partial^2\theta}{\partial x^2} + \delta\,e^{\theta}, \qquad \theta(\pm 1, t) = 0,$$

with the film starting at the wall temperature and the stress switched on at $t = 0$. The last term is the dissipation: at fixed stress the shear rate is $\tau/\mu$, the dissipation is $\tau^2/\mu$, and with $\mu$ falling as $e^{-\theta}$ it rises as $e^{\theta}$. Steady solutions are those of the earlier essay, and exist only for $\delta < \delta_c$.

The time unit is short in a bearing and long elsewhere. For a 25-micron oil film, with a half-gap of 12.5 microns and a thermal diffusivity of $8 \times 10^{-8}\,\mathrm{m^2/s}$, it is 1.9 milliseconds. For a millimetre of molten polymer in an extruder it is about ten seconds, and for a millimetre-thick zone of crushed rock on a slipping fault, about one.

The equation is marched by Crank–Nicolson for the conduction, with the dissipation explicit and a time step that shrinks as $e^{-\theta}$ so that the last stage of a runaway is followed rather than stepped over. A film is declared to have run away when its hottest point passes $\theta = 14$, beyond which the remaining time is less than a millionth of a diffusion time.

Lingering where the fold was #

Below the fold the film does the expected thing: its centre warms over a few diffusion times and settles on the cool steady solution. At 0.98 of the critical group the settled centre excess is 0.97, and at 0.999 it is 1.13, nearly the fold’s own 1.19.

Past the fold the first few diffusion times look identical. At 1.001 of the critical group the centre climbs to 1.19, flattens, and stays within five per cent of that from 13 diffusion times to 41, as if it had found a steady state. Then it leaves, and within a couple of diffusion times it has run away. At 1.01 the plateau is shorter and the runaway comes at 16.3; at 1.1, at 4.4.

What the film is lingering near is the steady state that no longer exists. At the fold the stable cool solution and the unstable hot one merge and vanish together. Just past it, neither exists, but the equation remembers where they were: the rate of change of the temperature, which is exactly zero on a steady solution, is still very nearly zero in that neighbourhood. The film is not held there; it drifts through a region where there is almost nothing to push it. The phenomenon is sometimes called the fold’s ghost, and it is the rule for every fold, not a feature of this film.

One over the square root #

Near a fold, whatever the system, the dynamics along the one direction that matters reduce to a single equation: $\dot a = \mu + a^2$, where $\mu$ is the distance past the fold and $a$ is the departure from the fold’s state. For $\mu < 0$ it has two steady states at $\pm\sqrt{-\mu}$. For $\mu > 0$ it has none, and the time to go from large negative $a$ to large positive $a$ is $\int da/(\mu + a^2) = \pi/\sqrt{\mu}$. The delay past a fold grows as one over the square root of the overshoot, and nothing about the film has entered yet.

The film obeys it. Marched at overshoots of a thousandth and three thousandths, its runaway times are 52.4 and 30.5 diffusion times, a ratio that makes the exponent $-0.492$; the $-\tfrac12$ is approached only as the overshoot falls, because the finite climb to the plateau adds a constant. The prefactor — the delay times the square root of the overshoot — is 1.66 at a thousandth, and in a bearing it means that a film loaded one per cent past its fold keeps its temperature for 31 milliseconds before losing it, and a film loaded a tenth of a per cent past it keeps it for 99.

Far past the fold the square root gives way to a different law. With the stress several times the critical, the heating is so fast that conduction never has time to act, and the film behaves as if it were insulated: $\dot\theta = \delta e^{\theta}$, which runs away from rest in exactly $1/\delta$. At twenty times the critical group the march gives 0.0505 against 0.0500. Between the two regimes is a crossover at an overshoot of order one, which is where most real accidents would start.

Where the shearing goes #

For most of the delay the profile looks like a film settling: a smooth dome, highest in the middle, barely changing in shape as it rises. The runaway itself is a narrowing. Once the centre starts to leave, the middle heats fastest, thins fastest and — because at fixed stress the shear rate is the stress divided by the viscosity — takes more and more of the deformation. At 15.6 diffusion times the middle is shearing six times faster than the walls; at 16.25, twenty-three times, and climbing.

The earlier essay predicted this localisation from the steady problem’s absence, and the time march shows its timing: the band forms only at the very end. For ninety-odd per cent of the delay the shearing is spread across the film almost as uniformly as it would be at the fold, which is why a calculation that assumes uniform shear describes the ghost phase well and the runaway not at all. The same concentration, in solids, is the adiabatic shear band of machining and ballistic impact, and in rocks, the narrow principal slip surface of a fault.

A warning before the fold #

The fold casts a shadow on the side that does have a steady state, too. A film held just below its fold settles, but it settles slowly, and it answers any disturbance slowly.

The recovery rate is the magnitude of the least negative eigenvalue of the operator $\partial^2/\partial x^2 + \delta e^{\theta^*}$, linearised about the cool steady state $\theta^*$. It is found by Sturm-sequence bisection on the discretised operator, which is exact for a symmetric tridiagonal matrix. Far from the fold the heating barely matters and the rate is close to the slab’s conduction rate, $\pi^2/4 = 2.47$; at 0.3 of the critical group it is 2.12. Approaching the fold it falls as the square root of the distance: 0.34 at one per cent below, 0.11 at a tenth of a per cent, 0.036 at a hundredth of a per cent. The same normal form gives it — the stable state sits at $-\sqrt{-\mu}$ and relaxes at $2\sqrt{-\mu}$.

That is a measurable warning. A bearing whose film is close to its fold takes longer to return to temperature after a load step than one far from it, by a factor that grows without limit as the fold approaches. The same slowing has been proposed as a warning of approaching thresholds in ecology and climate, where it goes by the name critical slowing down, and it is available here because the fold is a fold. What it cannot say is which side of the fold a slow film is on: a film just below the fold and one just above it, early in its delay, both look sluggish, and only the second one is going to leave.

An overload can be taken back almost until the end #

Because the film spends its delay near the fold’s temperature, and because that temperature is a stable steady state of any stress a little lower, removing the overload during the delay simply leaves the film where a stable state is and it settles. The question is how late the removal can be. For an overload of one per cent, dropping the stress to 0.6 of the critical value at any time up to 16.05 diffusion times saves the film, against a runaway at 16.30: ninety-eight per cent of the delay is recoverable. For a tenfold larger overload the figures are 4.22 and 4.39, and the recoverable fraction is still ninety-six per cent.

The point of no return is where the rising centre crosses the hot, unstable steady state of the reduced stress — the other branch of the fold, which exists again once the stress is lowered. Below it the cooled film falls back; above it, even the reduced stress runs away. And since the film only climbs past the fold’s temperature near the end of its delay, the point of no return is near the end too.

That changes how a thermal limit should be read. A limit on the steady load is not a limit on the instantaneous load. A clutch engagement, a start-up transient, a brief jam in an extruder can all exceed the fold with no harm, provided they end within the delay, and the delay is known: one and two-thirds diffusion times divided by the square root of the fractional overload. The fold decides what can be sustained; the ghost decides how long an excursion beyond it may last.

Three films, three clocks #

The scaled answers become physical ones through a single time, $H^2/\kappa$, and that time differs by four orders of magnitude between the places this film appears. Setting them side by side shows which of them can use the delay and which cannot.

A journal-bearing film. Twenty-five microns thick, oil, a diffusion time of 1.9 milliseconds. The pressure the tapered gap builds is carried by a film whose thermal memory is a few milliseconds long, so a load spike lasting a shaft revolution at 3,000 rpm — twenty milliseconds, about ten diffusion times — is survivable only if it overshoots the fold by less than about two per cent. A bearing’s thermal limit is effectively instantaneous, and it is right to design it as a steady limit.

An extruder’s melt film. A millimetre of molten polymer between a screw flight and its barrel, with a thermal diffusivity near $10^{-7}\,\mathrm{m^2/s}$, has a diffusion time of about ten seconds. Its viscosity falls with temperature far faster than an oil’s, which puts it close to its fold in normal running, and its viscosity also depends on the shear rate, which moves the fold with the throughput. But a surge lasting a few seconds is a fraction of one diffusion time, and at an overshoot of ten per cent the delay is four and a half diffusion times, three-quarters of a minute. A polymer processor has time to react to an excursion, provided something is measuring the right temperature.

A slipping fault. A zone of crushed rock a millimetre thick, sheared at a stress set by the weight of the rock above it, is as close to a fixed-stress film as nature makes, and its diffusion time is about a second. Its heating runs away in the sense that matters for earthquakes — the frictional strength collapses as the zone heats — and the delay sets how long a slip episode must last before that happens.

The ordering is the useful part. The same equation, the same fold and the same square root decide all three, and the only difference is the length squared in the time unit. That is the reason the heat of a drag is a steady-state question in a bearing and a transient one nearly everywhere else.

The march, checked against the steady problem #

The time march and the steady problem are computed independently — one by marching a partial differential equation, the other by shooting the ordinary one from the mid-plane — and a film that settles must settle on the shooting solution. At a group of 0.8 the march settles at 0.74645 against 0.74646. The runaway time itself was checked against the grid: at 1.01 of the critical group, 60 points and 120 give 16.25 and 16.34, half a per cent apart.

What the picture cannot show #

The viscosity law is the exponential approximation. Real oils follow it over a range of a few tens of kelvin and depart from it beyond, which is exactly where a runaway ends. Everything after the centre passes a few scaled units is the model’s, not the oil’s.

The walls hold their temperature. In a machine the metal warms too, on its own time scale, and a wall that heats lowers the fold rather than the delay — which is a second, slower transient this model does not contain.

The stress is exactly fixed. A real drive softens as its load rises, and the fixed-stress idealisation is the one that makes the fold exist at all; a stiff drive has no fold. Most machines lie between.

One dimension. The film is uniform along the flow. A real film heats unevenly along a bearing pad, and the fold is reached first where the gap and the stress are least favourable.

The convention the numbers depend on #

$\theta$ is the temperature excess over the walls, divided by the temperature interval over which the viscosity falls by a factor of $e$. Distances are fractions of the half-gap, and times are multiples of $H^2/\kappa$ with $H$ the half-gap and $\kappa$ the fluid’s thermal diffusivity. The group $\delta$ is the Nahme–Griffith number of the earlier essay, and the overshoot is $(\delta - \delta_c)/\delta_c$. “Recovery” means that once the stress drops to 0.6 of its critical value the film settles within forty diffusion times.

Who found it, and when #

The steady fold is Frank-Kamenetskii’s, from his 1939 theory of thermal explosion, and Nahme applied the same exponential to viscous heating in 1940. Gruntfest showed in 1963 that a sheared liquid has a thermal runaway with the same structure, which is why the group sometimes carries his name. The inverse-square-root delay past a fold is the passage time of the saddle-node normal form and belongs to the general theory of bifurcations rather than to any one field; its appearance as a “bottleneck” or “ghost” is standard in that theory. The same slowing on the stable side was proposed as an early warning of catastrophic shifts in ecology in the 1980s and has been studied widely since.

The structure recurs across this subject wherever a steady solution meets another and both vanish. A drop hanging from a tube falls at a fold of its family of static shapes, and it too takes its time; the wall that heats itself makes the same source term in an unconfined flow and has no fold at all.

Still open: a drive that gives way #

Every figure here holds the stress fixed, and the earlier essay showed that holding the speed fixed removes the fold entirely. A real motor sits between: its torque falls as its speed rises, along a characteristic whose slope is set by the motor and the gearing. The calculation that follows puts that characteristic into the time march as a boundary condition — the stress now a function of the film’s own mean shear rate — and asks for the drive stiffness at which the fold first appears, how the delay past it depends on that stiffness, and whether a film with a compliant drive and a thermal fold can oscillate rather than run away. It is the thermal counterpart of the solutions that stop being chosen, where what a flow does depends on what is holding it.