A motor turns the runaway into a jump

The film that heats itself found that whether an oil film’s viscous heating runs away is not a property of the oil. Sheared at a fixed stress, a film whose viscosity falls exponentially with temperature has a steady state only while the Nahme–Griffith group $\delta$ stays below 0.87846; sheared at a fixed speed, it has one at every speed. A lost steady state still holds the film followed the first case past its fold in time, and found the ghost: a film that climbs to the temperature the vanished steady state would have had, lingers there, and only then runs away.

Both essays drove the film with an idealisation. Nothing holds a stress exactly or a speed exactly. An electric motor, a turbine, a hydraulic motor, a falling weight, a servo — each delivers a torque that depends on how fast it is turning, and the film’s shaft turns at whatever speed makes the motor’s torque equal the film’s resistance. This essay puts that dependence into the problem, in its simplest form — a torque falling in a straight line from its value at stall to zero at the motor’s no-load speed — and asks where between the two idealisations the fold appears, and what a film does when it does.

The film’s own characteristic #

The trick is to stop thinking of the film as heated by a stress and think of it as a machine with a torque–speed curve of its own, the way a pump has a head–flow curve.

Every steady state of the film is a Frank–Kamenetskii profile, set by its centre temperature $\theta_0$ in units of the temperature over which the viscosity falls by $e$. Each one carries a stress — the group $\delta$ is its square, in a natural unit $\tau^*$ — and lets one wall slide past the other at a speed. The local shear rate at a given stress is proportional to $e^{\theta}$, since the oil has thinned by that factor, so the speed is the stress times the mean of $e^{\theta}$ across the gap, $S$. In scaled units,

$$\hat\tau = \sqrt{\delta}, \qquad \hat V = \sqrt{\delta}\,S .$$

Both are known in closed form along the whole family: with $a = \operatorname{arccosh} e^{\theta_0/2}$, $\delta = 2a^2e^{-\theta_0}$ and $S = e^{\theta_0}\tanh a / a$.

The first figure draws that curve, and it has a shape anyone who has read a fan’s characteristic will recognise. While the film is cool the stress rises with speed, nearly in proportion — a Newtonian film, Couette flow. As the film warms it thins, and the stress rises more slowly, peaks, and then falls: past the peak a faster shaft is resisted less, because the oil it heats is thinner than the extra speed is fast. The peak is at $\hat\tau = \sqrt{0.87846} = 0.937$, and it is the fold of the first essay seen from a different angle. A drive that holds the stress fixed is a horizontal line on this plane. Below the peak it crosses the rising limb, the cool steady state, and the falling limb, the hot one the first essay called unstable. At the peak it touches, and above it it misses the curve entirely: no steady state, the runaway.

A motor is a sloping line. Its torque at stall, $\tau_s$, is where it meets the stress axis; its no-load speed is where it meets the speed axis. In the film’s units it reads

$$\hat\tau = \sqrt{\delta_s} - k\,\hat V, \qquad k = \frac{\tau_s/V_0}{\mu_0/2H},$$

where $\delta_s$ is the group at stall and $k$, the drive’s droop, is the only new number. It compares two slopes that are both stresses per unit speed: how fast the motor’s torque falls as it speeds up, and how fast the cold oil’s stress rises as it is sheared faster. A drive with $k = 0$ holds its torque whatever the speed; a drive with large $k$, holding a fixed $\tau_s/k$, is so much stiffer than the oil that it holds its speed.

Any droop removes the runaway #

The first thing the picture says is the most surprising, and it does not need a calculation. Along the falling limb, the film’s stress goes to zero as its speed goes to infinity — a very hot film is nearly inviscid — but it goes there slowly, like $1/\hat V$ multiplied by a logarithm. A straight line with any negative slope at all reaches zero stress at a finite speed. So a sloping line that starts above the curve must cross it again, somewhere out on the falling limb, however shallow the slope and however large the stall torque. Every motor with a finite no-load speed has a steady operating point at every load.

The second figure redraws the same states as the conventional bifurcation diagram — centre temperature against the drive’s load — and the difference between $k = 0$ and every other drive is the shape of the hot limb. At fixed stress the curve bends back at the fold and heads off to infinite temperature as the stress falls to zero: there is nothing on the far side, and the film that crosses the fold has nowhere to go. With a droop of only $k = 0.02$ the hot limb turns forward again at a centre temperature near 4.6 e-folds and rises from there with the load. The curve is an S. The middle of the S is still the unstable branch, but the top is a steady state the film can reach.

This is the thermal counterpart of what happens in a set of flows that stop being chosen: which states exist, and which are stable, depends on what is holding the system, not only on the system. The first essay’s claim — that fixed stress runs away and fixed speed does not — is the two ends of a family, and the family’s interior does not interpolate between them. It jumps.

A jump instead of a runaway #

The third figure marches the film from rest, the same Crank–Nicolson march the second essay used, but with the stress recomputed at every step from the motor’s line and the film’s current mean of $e^\theta$. As the film heats it thins, the shaft speeds up, and the motor’s torque falls — the group the film sees is $\delta_s/(1 + kS)^2$, and it falls as the film warms. For a drive with $k = 0.04$, two per cent below its fold, the film settles at 1.13 e-folds, on the cool branch. Two per cent above, it climbs slowly through the fold’s temperature for about ten diffusion times — the ghost, exactly as at fixed stress — and then steeply. At fixed stress that climb ends in a runaway at 11.2 diffusion times; under the motor it ends at 4.9 e-folds, and stays there.

The linger is the same saddle-node ghost because the fold is the same kind of fold: near its tip, a sloping load line and a flat one both cut the film’s curve in a small parabola, and the delay past the tip scales as the inverse square root of the overshoot in either case. What changes is the far side. The film does not escape to infinity. It lands.

Five per cent past the fold, a film marched from rest settles at a centre temperature of 4.987 e-folds; the hot operating point where the load line crosses the falling limb is at 4.988. Five per cent below, 0.9886 against the cool point’s 0.9884. The march knows nothing about operating points, and it arrives at them.

The number that decides whether there is a jump #

The S has two turning points, and they are the fourth figure’s two curves. The upper one is where the cool state disappears as the load rises — the fold the film jumps from. The lower one is where the hot state disappears as the load falls — the fold the film drops back from. Between them the film has two stable states, and which it is in is decided by its history.

Increase the droop and the two curves converge, until at

$$k_c = 0.073829, \qquad \delta_{s} = 1.268$$

they meet. That meeting point is a cusp, in the precise sense of catastrophe theory, and past it there is one steady state at every load: the S has straightened into a single rising curve. Geometrically $k_c$ is the steepest slope of the falling limb of the film’s own characteristic. A load line shallower than that can be laid across the falling limb to cut it twice, and with the cool crossing that makes three; a steeper line can cut the curve only once, whatever its intercept. The fold at $k = 0$ sits where the upper curve meets the stress axis, at 0.878458, which is the fixed-stress fold recovered as the limit of the family.

The lower curve, at $k = 0$, falls to zero load. That is the other half of why a fixed stress runs away: its hot branch does not turn back at any load, however small, so a film that has crossed the fold would have to have the load removed altogether to recover. For any droop, the lower fold is at a positive load, and the hot state can be left by reducing the load far enough.

Loaded up and back down #

The fifth figure tests the history. The load is raised slowly — 0.004 in stall group per diffusion time — through the fold and back. Under a soft drive, half the cusp’s droop, the film follows the cool branch up past the load at which the hot state first appears, and past the fold itself: the ramp and the ghost together carry it about five per cent beyond, to 1.08. Then it jumps four e-folds in a few diffusion times. On the way down it stays hot, following the hot branch well below the load at which it jumped, and falls back only at 0.77, near the lower fold. The loop is four e-folds tall and 0.3 wide in stall group.

Under a drive three times stiffer than the cusp the film traces one curve both ways, and the largest gap between the upward and downward paths is 0.003 e-folds — the lag of a finite ramp rate, and nothing else.

The loop has a practical consequence that the fixed-stress picture hides entirely. At fixed stress, a fold is crossed once and the film is lost. Under a soft drive a film that has jumped is hot but steady, and the operator who sees the temperature rise and backs the load off to where it was a moment before will find that it stays hot: the load has to be cut well below the one that caused the jump before the film cools. The hysteresis is the film remembering that it is thin.

Only a stalled motor can do it #

The droop $k$ is a comparison of two slopes and is not easy to picture, so the sixth figure translates it into what the motor is doing. At an operating point the motor’s speed as a fraction of its no-load speed is $kS/(1 + kS)$. At the upper fold of a soft drive the motor is barely turning — a few per cent of its no-load speed, dragged almost to stall by the viscous load. At the lower fold it is turning faster. The two branches meet at the cusp at a motor speed of 0.25000 of no-load, to the precision of the calculation. At the cusp the film’s characteristic has a logarithmic slope of $-\tfrac13$, and a straight line tangent to a curve at that slope intersects the speed axis at four times the tangent point’s speed. The calculation finds the quarter to six figures; it does not prove that it is exact.

That turns the cusp into a rule a designer can use without computing $k$. A drive can fold its film only if it is turning at less than a quarter of its no-load speed. An induction motor runs within a few per cent of its synchronous speed at full load, so its droop measured against a cold bearing film is of order tens, several hundred times the cusp; it cannot fold anything, and a bearing on such a motor behaves as the first essay’s fixed-speed film. So does anything driven through a stiff gearbox from such a motor. The drives that fold are the ones built to deliver a torque rather than a speed: a servo in torque mode, a stress-controlled rheometer, a hydraulic motor fed from a constant-pressure supply, a weight on a cord, and — the case the previous essay noted — a fault zone loaded by the rock above it. And any drive that is being dragged towards stall, whatever it was designed for.

What a hot state means for oil #

The hot operating points are steady solutions of the model, and they are at temperatures the model was not built for. Four to five e-folds of viscosity is, for a mineral oil whose viscosity falls by $e$ over about thirty kelvin, a rise of well over a hundred kelvin above the housing. Across that range the exponential law is a poor fit — real oils flatten out at high temperature, which would push the hot branch lower, and a multigrade oil’s viscosity also depends on the shear rate, which bends the film’s characteristic before any heating does — and the oil itself is oxidising. So the hot state should be read as “the film goes somewhere much hotter and stays there”, not as a prediction of its temperature.

That reading is still more useful than a runaway. A runaway predicts that the film’s temperature grows without limit in finite time and says nothing about what is left; the jump predicts a new steady state in which the bearing still turns, at a much lower torque, with a much thinner film. In a tapered gap that thinner oil carries far less load, and what happens to the bearing next is a question about the load, not the film.

What was checked #

The seventh figure is the ledger. The fold at zero droop is the Frank–Kamenetskii value, 0.878458. The cusp was found as the steepest slope of the falling limb by golden-section search, and checked the direct way: at three per cent below it both folds exist and at three per cent above neither does. The marched film, which uses no steady solution, lands on the steady operating point on both sides of a fold. And the ramp’s loop is four e-folds wide under the soft drive and three thousandths under the stiff one.

What the picture cannot show #

A straight characteristic. Real motors have curved torque–speed lines, and an induction motor’s has a peak — its breakdown torque — below which it is itself unstable. The film’s side of the picture is exact; the motor’s is the simplest line that has both ends.

No rotor inertia. The motor’s speed is taken to follow the film instantly. A heavy rotor adds a second clock, and with two clocks — the rotor’s spin-up time and the film’s diffusion time — the S-curve’s middle branch need not be the only unstable thing.

Walls at a fixed temperature, and a viscosity with one e-folding temperature. Both as in the first two essays; a bush that warms raises every curve, and a film with one wall insulated — nearer to the wall that heats itself — has its fold four times lower, which moves the film’s characteristic but not the construction; and a real oil’s flatter high-temperature viscosity moves the hot branch.

A planar film. The film is Couette flow between flat walls. A journal bearing’s film is thin compared with its radius, so the geometry is nearly planar, but its pressure and thickness vary round the shaft, and the fold’s position moves with the eccentricity.

The convention the numbers depend on #

Temperatures are excesses over the walls, in units of the temperature over which the viscosity falls by $e$. The stress unit $\tau^*$ is the one that makes the Nahme–Griffith group equal to one; the speed unit $V^* = 2H\tau^*/\mu_0$ is the speed at which a cold film would carry that stress. The droop $k$ is the motor’s stall stress divided by its no-load speed, against the cold film’s $\mu_0/2H$. Times are in thermal diffusion times across the half-gap.

Who found it, and when #

The fold at fixed stress is Nahme’s, from 1940, and Griffith’s, and Frank-Kamenetskii’s for the equivalent problem in thermal explosion. That the fold disappears at fixed speed was recognised in polymer processing, where the drives are stiff and the runaway rarely seen, and the same distinction shows up in rheometry, where a controlled-stress instrument and a controlled-rate one can disagree about the same sample once viscous heating matters. Putting the drive in as a line across the film’s own characteristic is the ordinary operating-point construction of pumps and fans, and of a compressor on its plenum.

Still open: a rotor that has to spin up #

The film and its motor share one speed, set instantly. The next calculation gives the rotor an inertia, so that the speed lags the torque by a mechanical time, and asks what happens when that time is comparable with the film’s diffusion time. On the S-curve’s middle branch the film’s thermal instability and the rotor’s speed are then two coupled clocks, and the question is whether a soft drive near its cusp settles, jumps, or — like a drop at its fold meeting a second degree of freedom — oscillates between the two branches without settling on either: a thermal relaxation oscillator built from a bearing and a motor.