The film that heats itself

A journal bearing on a turbine shaft is dissipating four hundred kilowatts per square metre in a film twenty-five microns thick. That power density is comparable with a domestic hotplate and it is being deposited in a layer a third the thickness of a human hair.

The oil does not glow, or boil, or do anything dramatic, because the same thinness that concentrates the heating also puts the metal very close on both sides. What the oil does do is run about eighteen kelvin hotter than the housing it is in — which is a number nobody quotes, is a design constraint on every lubrication system ever built, and follows from three lines of algebra.

The rise, and the length that is not in it #

Take plane Couette flow between two walls held at the same temperature $T_0$, a distance $h$ apart, one sliding at $U$. Steady conduction with a uniform source gives

$$k\,T'' + \mu\left(\frac{U}{h}\right)^2 = 0 \quad\Longrightarrow\quad T = T_0 + \frac{\mu U^2}{2k}\,\frac{y}{h}\left(1 - \frac{y}{h}\right),$$

a parabola with peak

$$\Delta T = \frac{\mu U^2}{8k}.$$

There is no $h$ in it.

That deserves more than a note, because it contradicts the reflex. Halve the gap and four things happen: the shear rate doubles, so the dissipation per unit volume goes up fourfold; the volume being heated halves; and the distance to the wall halves, which doubles the conduction. Four factors of two, and they cancel exactly. A one-micron film and a one-millimetre film at the same sliding speed reach the same temperature.

What the rise does contain is the viscosity and the conductivity, and it is their ratio that decides whether a film is hot. Oil has a viscosity fifty times water’s and a conductivity a quarter of it, so an oil film runs two hundred times hotter than a water film at the same speed — which is why water-lubricated bearings exist and why they are used wherever the load allows.

The number that says whether it matters #

It is also why the arithmetic here differs from the unconfined case, where the heat a body makes is carried downstream and no control volume contains it. A film has nowhere downstream.

The group is the Brinkman number,

$$\mathrm{Br} = \frac{\mu U^2}{k\,\Delta T_{\text{ref}}},$$

the heat a flow makes against the heat it can conduct away, and it is the one dimensionless group in this subject that a reader is very likely never to have met. Formed on the film’s own peak rise it is exactly eight, which makes it uninformative; formed on a stated reference difference — the difference between the two walls, or between the oil and its sump — it is the ratio that decides whether the heating is a correction or the whole problem.

It belongs to the same family as the Péclet and Prandtl numbers and differs from both in one important way: it is the only one with a velocity squared in it. Doubling the sliding speed quadruples the rise, which is why lubrication problems change character with speed far more sharply than they change with load.

What changes when the oil notices #

Everything above treats the viscosity as a constant. Real oils do not oblige: a mineral oil loses about three per cent of its viscosity per kelvin, so an eighteen-kelvin rise halves it.

That is a feedback, and the interesting question is its sign. The answer is that it depends on what the machine is holding constant, and the two cases behave in opposite ways.

Held at a fixed shear rate — a stiff motor turning a shaft at a set speed — the dissipation is $\mu(T)\dot\gamma^2$. As the oil warms, $\mu$ falls, and the heating falls with it. The feedback is negative and the film settles. Write $\theta$ for the scaled temperature excess and the equation is $\theta'' = -\delta e^{-\theta}$, which has a solution for every value of $\delta$ there is.

Held at a fixed shear stress — a hanging weight, a pressure-driven seal, a clutch on a fixed torque — the dissipation is $\tau^2/\mu(T)$. As the oil warms, $\mu$ falls, and the heating rises. The feedback is positive. The equation is $\theta'' = -\delta e^{+\theta}$, and it is not the same problem at all.

The fold, computed #

The fixed-stress problem is the Frank–Kamenetskii equation, which arrives here from lubrication and arrives in combustion from a reacting solid, and it has a critical value.

Solve it by shooting from the mid-plane: pick a centre temperature, integrate outwards with unit $\delta$, and see how far the solution goes before it returns to the wall value. Rescaling says a slab of half-width one has $\delta = L^2$ with $L$ the half-width found, so the critical $\delta$ is the largest $L^2$ over all centre temperatures — a maximisation, not a root-find.

The answer is $\delta_c = 0.87846$, at a scaled centre excess of 1.1868. Above it the curve has no point on it. Below it there are two — a cool solution and a hot one, of which the hot one is unstable — which is why the criticality is a fold rather than a boundary and why a film that goes over it does not come back when the load is reduced slightly.

The group, in a bearing’s own quantities, is the Nahme–Griffith number

$$\mathrm{Na} = \frac{b\,\tau^2 H^2}{k\,\mu_0},$$

with $b$ the viscosity–temperature coefficient, $\tau$ the shear stress and $H$ the half-gap. It goes as the square of the stress and the square of the gap, so it moves very fast indeed across a range of machines.

Why the distinction is not academic #

A clutch and a gearbox are the same materials at the same speeds, and they fail differently, and this is why.

A clutch during engagement is a fixed-stress device: the plate is pressed at a set force and slips at whatever rate it slips at. So it has a fold, and clutch failure by thermal runaway — the friction material glazing, the temperature spiking, the torque collapsing — is a recognised mode with a critical energy attached to it.

A film in a bearing at rest is a squeeze film and has no fold because it has no steady state to lose. A hydrodynamic bearing driven by a stiff motor is a fixed-rate device. It has no fold, and its failure is by the film thinning until contact occurs, which is a slow and forgiving process with warning in it.

The same oil in the same gap, and one arrangement has a cliff in it and the other does not. Nothing about the fluid distinguishes them. What distinguishes them is what is on the other end of the shaft.

Where the runaway actually happens #

The fixed-stress branch says the film has no steady state above the fold. It does not say what the film looks like on the way there, and the answer is that the shearing stops being spread across the gap.

With no pressure gradient the shear stress is uniform across the film — that is a momentum balance and it holds whatever the temperature is doing. So the local shear rate is $\tau/\mu(T)$, and where the oil is hot it is thin and where it is thin it is being sheared hardest. The velocity profile is therefore not the straight line every figure above draws: it flattens near the cool walls and steepens through the warm middle.

And that is the same feedback again, now with a place attached to it. The middle shears faster, so it heats faster, so it thins further, so it takes an even larger share of the shearing. The deformation localises into a band whose width is set by how far heat can conduct out of it in the time the deformation takes — not by the gap, which has dropped out of the problem for a second time and for a different reason.

Which qualifies this essay’s headline result. $\mu U^2/8k$ has no gap in it because the shear was assumed uniform. Once it localises, the relevant length is the band’s own width, and the film has chosen it rather than the machine.

The phenomenon has different names in the disciplines that meet it. In a metal deformed fast enough that the heat cannot escape it is an adiabatic shear band, a few micrometres wide, and it is a standard failure mode in machining and in ballistic impact — the material fails along a line whose existence is thermal rather than metallurgical. On an earthquake fault the same argument concentrates slip into a zone millimetres thick, which is why exhumed faults show a single narrow principal slip surface where the models used to draw a wide sheared band.

Same equation, same feedback, and a length scale that the deformation manufactures for itself.

The other feedback, which points the same way #

There is a second temperature dependence in a real film and it has been left out of everything above, because including it changes the numbers and not the structure. It is worth naming so that the omission is deliberate rather than quiet.

The density falls with temperature, and in a bearing that matters because the film’s thickness is set by a balance of flow rates rather than by a fixed geometry. Warm oil is less dense, occupies more space, and is carried through the gap faster, so a hot bearing runs slightly thicker than a cold one at the same load — an effect of a few per cent that partially offsets the viscosity’s collapse.

The pressure dependence points the other way and is much larger. Oil’s viscosity rises roughly exponentially with pressure, doubling every fifty megapascals or so, and in a heavily loaded contact the pressures reach a gigapascal. That is the whole content of elastohydrodynamic lubrication: a contact that ought by hydrodynamic theory to have no film at all has one, because the oil in it has become a thousand times more viscous on the way through.

So a gear tooth contact has both feedbacks running at once and in opposite directions — pressure making the oil stiffer, temperature making it thinner — and which one wins decides whether the contact survives. The scuffing criterion that engineers actually use is a fitted combination of the two, and the fold computed above is the temperature half of it with the pressure half held fixed.

Where else the same equation turns up #

A polymer extrusion die — whose melt is also shear-thinning, so its viscosity depends on the question — is the industrial case in which viscous heating is not a correction but the design variable. A melt at two hundred pascal-seconds moving at half a metre a second through a millimetre gap makes twenty-five kelvin, and since a polymer’s viscosity falls faster with temperature than an oil’s, the flow rate at a fixed pressure is a strongly nonlinear function of the pressure. Extrusion dies are designed with the coupling in from the start.

A hypersonic boundary layer heats itself by exactly this mechanism, and it is one of this collection’s earlier essays: the wall that heats itself is this arithmetic with the confinement removed, so the heat is carried downstream instead of conducted sideways and the governing group is a recovery factor rather than a Brinkman number.

And a glacier — which is also a fluid with no single viscosity, since ice creeps as a power law with an index near a third — is a shear flow whose viscosity depends on temperature, and the same instability is one of the mechanisms proposed for surging. The numbers are absurd by bearing standards — a shear rate of one per year — and the group is not small, because the length is hundreds of metres and enters squared.

What the picture cannot show #

The walls are at a fixed temperature and they are not. A real bearing’s bush warms up, and the boundary condition that matters is conduction into a housing with its own resistance. Solving the film against a fixed wall temperature gives the rise within the film, which is the number computed here; the rise of the whole assembly above ambient is a separate and usually larger number.

The viscosity law is exponential and is a fit. $\mu = \mu_0 e^{-b(T-T_0)}$ is a good description over twenty or thirty kelvin and a poor one over a hundred; the standard oils are described by Vogel’s or Walther’s equation instead, and the critical $\delta$ is a property of the exponential and would move if the law were changed.

The dissipation is computed from the incompressible form. A film is not compressing, so the second viscosity contributes nothing to any number here — which is exact rather than approximate, and is worth stating because it is one of the few places on this site where the omission is deliberate rather than inherited.

The flow is laminar and one-dimensional. A film thick enough or fast enough to have Taylor vortices in it is mixing rather than conducting, and every number here is then an overestimate.

And nothing here is transient. The fold is a statement about the non-existence of a steady solution. What a film actually does above the critical value is a time-dependent problem this collection does not solve, and the answer is not “it gets infinitely hot” — it is that it leaves the regime the model describes, by boiling, by degrading, or by seizing.

One consequence for how a bearing is measured #

A last practical point, and it is the reason this essay is in a collection about flows rather than about machines.

A bearing’s performance is quoted as a load capacity at a speed, and the number is computed from Reynolds’ equation with a viscosity in it. Which viscosity? The oil’s viscosity at the sump temperature is the number on the datasheet, and it is wrong by a factor of two for the reasons above. The oil’s viscosity at the peak film temperature is right for the middle of the film and wrong at the walls.

The convention that has grown up is the effective temperature — an oil temperature chosen so that a constant-viscosity calculation reproduces the measured load — and it sits somewhere between the inlet and the peak. It is a fitted quantity and it is treated as a property of the bearing rather than of the oil, which is exactly the shape of thing this collection tries to avoid: a number that absorbs a physical effect nobody has computed.

That is the same objection this collection raises to any coefficient that absorbs a mechanism: an effective viscosity and an eddy viscosity are the same kind of object, and both are honest only while what they stand in for is understood.

What the arithmetic above buys is the ability to stop doing that. The rise is $\mu U^2/8k$, it needs no fitting, and it says which bearings need the correction and which do not: at a hundredth of a kelvin the effective temperature is the sump temperature, and at twenty-five it is not.

Who found it, and when #

Nahme worked out the coupled problem in 1940 and Griffith the year after, both for polymer processing rather than for bearings. The critical value they arrived at had already been computed by Frank-Kamenetskii in 1938 for a completely different problem — the spontaneous ignition of a reactive solid — and the two literatures did not notice for two decades.

The surprising connection is with that ignition problem, and it is exact rather than analogical. A reacting solid heats itself at a rate that rises exponentially with temperature; a stressed film heats itself at a rate that rises exponentially with temperature; and both are conducting the heat away through a slab. The two are the same equation with the same constant, so the temperature at which a haystack catches fire and the stress at which a gear tooth scuffs are the same number in different clothes, and neither community derived it from the other.

Where the ladder goes next #

Below this rung is the price of a gradient, which supplies the source term, and the sliding film, which is the geometry being heated. Beside it is the wall that heats itself, which is the same source term in an unconfined flow and produces a completely different structure.

Above it is the question this essay has to leave open — what a film actually does past the fold — which is a transient problem, and the honest position is that the model stops before the answer does.