The gap that carries the most

Nothing but the shape of the gap makes the case that a hydrodynamic bearing works because its film converges, and that neither the oil nor the speed is the mechanism. This essay takes the next step and asks how much it should converge — which turns out to be a question with an exact answer and almost no consequences.

The load separates, and what is left has no fluid in it #

Reynolds’ equation for a film of thickness $h(x)$ under a surface moving at $U$ is

$$\frac{d}{dx}\!\left(h^3\frac{dp}{dx}\right) = 6\mu U \frac{dh}{dx},$$

and for a linear film from $h_1$ to $h_2$ over a length $B$ it integrates in closed form. The load per unit width comes out as

$$W = \frac{6\mu U B^2}{h_2^2}\cdot\frac{\ln k - 2(k-1)/(k+1)}{(k-1)^2}, \qquad k = \frac{h_1}{h_2}.$$

Everything about the machine is in the prefactor. The bracket is a function of the convergence ratio alone — no viscosity, no speed, no length, no film thickness, no material.

So the taper that carries the most load is a pure number, and it is fixed before anything about the bearing is known. That is worth pausing on: a designer’s most consequential single geometric choice — the one that decides whether the film survives — is decided by arithmetic that cannot be influenced by the oil, the shaft speed, or the load itself.

Three bearings whose viscosities differ by four hundred thousand, whose speeds differ by three thousand, and whose loads differ by ten orders of magnitude all want exactly the same taper.

The pressure distribution shows what the taper is trading. A gentle convergence spreads the pressure and puts its peak well downstream; a steep one concentrates it. The peak always sits where the film thickness is the harmonic mean of the two ends, which is a second closed form and is checked below.

Why a converging gap builds pressure at all #

The mechanism deserves a paragraph, because the optimisation below is meaningless without it and it is often stated in a way that hides where the pressure comes from.

The film carries a flow rate that is the same at every station — the fluid has nowhere else to go. That flow is the sum of two parts: a Couette part, half the surface speed times the local thickness, which is the fluid dragged along by the moving wall; and a Poiseuille part driven by whatever pressure gradient exists. At the inlet the film is thick, so the drag flow alone would deliver more than the outlet can take; at the outlet it is thin, so the drag flow alone would deliver less.

The pressure is what reconciles them. It builds up in the middle to push fluid backwards against the drag near the inlet and forwards with it near the outlet, and the film thickness at which the two requirements change sign is the one where the drag flow exactly equals the through flow — which is the harmonic mean of the ends and is where the pressure gradient is zero, that is, where the pressure peaks.

A parallel film has no such conflict, so it builds no pressure and carries no load at all. That is why the bracket is exactly zero at $k = 1$ — a limit worth computing carefully, because expanding both of its terms to only second order gives 1/6 rather than 0 and produces a bearing that works without a taper.

The number, and how it has to be found #

Before optimising anything, the closed form is checked. Integrating Reynolds’ equation numerically — bisecting for the film thickness at the pressure maximum until the pressure returns to zero at the far end — reproduces the closed-form load to seven parts in a billion at four convergence ratios, and returns the harmonic mean to one part in ten billion without having been told about it.

Then the optimum. Setting the derivative of the bracket to zero and simplifying gives

$$\left(\frac{1}{k} - \frac{4}{(k+1)^2}\right)(k-1) = 2\left[\ln k - \frac{2(k-1)}{k+1}\right],$$

whose root is 2.1887048.

Finding it that way rather than by searching the load is not fastidiousness. The load is flat to parts in $10^{13}$ over the range, so a golden section on it returns whichever neighbour the floating-point arithmetic happened to favour; the derivative crosses zero with a slope of ordinary size, and bisecting on that gives every digit.

The number printed almost everywhere is 2.1889. That is not the maximum. It costs eight parts in a hundred billion of load, which is the flatness of this optimum stated in the bluntest possible way — and it is not a criticism of anybody, because at that level the difference between the two values is below every other approximation in the derivation.

Which is the actual finding #

Every convergence ratio from 1.99 to 2.43 is within one per cent of the best. Every one from 1.78 to 2.80 is within five. That is a range of nearly two to one on a quantity a machinist has to hold to microns, and inside it the aerodynamic — hydrodynamic — case is indifferent.

The reason is the one these essays keep arriving at. A maximum is a stationary point, so the objective is insensitive to its argument there, and knowing the objective to a precision $\varepsilon$ locates the argument only to $\sqrt{2\varepsilon/c}$ with $c$ the curvature. Half the significant figures are gone before anybody starts, and no amount of care recovers them. It is the same arithmetic that makes the elliptic wing’s optimum a band rather than a shape.

So the choice is made on something else #

The friction force has its own bracket, and the ratio of friction to load — the quantity that decides whether the bearing runs cool — is minimised at 2.47, thirteen per cent away from the load optimum. So there is no best bearing. There is a best bearing for the largest load and a different one for the least friction, and a designer picks somewhere between them or somewhere else entirely.

That the two optima are close is itself a fact worth having: a bearing cut for load runs about five per cent warmer than one cut for friction, and one cut for friction carries about half a per cent less load. Neither penalty is large, and that is because both curves are flat.

The friction is not what the load is #

There is a structural asymmetry between the two brackets that explains why their optima differ, and it is the same asymmetry that makes a bearing work.

The load comes from the pressure, which is a second-order effect: it exists only because the film converges, and it vanishes for a parallel film. The friction comes from the shear, which is a first-order effect: it exists for any film at all, converging or not, and for a parallel film of thickness $h$ it is simply $\mu U B/h$.

So as the convergence ratio rises, the load first grows — more wedge, more pressure — and then falls, because the mean film is getting thicker and the pressure that a given wedge produces scales as the inverse square of the thickness. The friction, meanwhile, falls monotonically, because a thicker mean film is a lower shear rate everywhere.

A monotone quantity divided by one with a maximum has its own minimum somewhere past the maximum, which is exactly where the friction coefficient’s optimum sits. The thirteen per cent gap between the two is not an accident of these numbers; it is the general shape of the trade, and it is why a bearing designer’s usual figure of merit is neither of them but the film thickness at the trailing edge — the quantity that decides whether the two surfaces touch, which is what the last of the oil is about.

And a real pad is not infinitely wide #

Everything above is for a bearing infinitely long across the flow. A real pad leaks out of both sides, and solving the two-dimensional Reynolds equation on a rectangle shows what that costs: a square pad carries 47 per cent of the infinite-width load, a pad twice as wide as it is long 74 per cent, and one four times as wide 90 per cent.

That is not a correction. It is the largest single number in the essay, and it dwarfs everything the optimisation was about — a bearing designed at the exactly optimal taper and made square carries less than half of what the closed form promises.

The leakage also moves the optimum, from 2.2 for a wide pad to 2.9 for a narrow one. The reason is local: the leakage is worst where the film is thickest, because the side flow goes as $h^3$, so thinning the inlet end pays for itself. A pure number for an infinitely wide bearing becomes a function of the aspect ratio for a real one, which is the general fate of pure numbers in this subject.

One more thing the bracket does not contain #

There is a quantity conspicuously absent from the whole calculation, and its absence is the reason lubrication works at all: the load the bearing is asked to carry.

Reynolds’ equation is linear in the pressure, so doubling the load does not change the pressure shape, only its scale — and the film thickness adjusts until the two match. A bearing under a heavier load runs on a thinner film at the same taper, with the pressure everywhere doubled and the distribution identical. The taper does not need to be re-optimised when the load changes, which is why a single number can be quoted for a family of machines and why the shape of the gap is the whole story rather than half of it.

The linearity fails when the film gets thin enough for the surfaces to interact, or hot enough for the viscosity to vary along the pad — and both of those are what actually limits a bearing, rather than anything in the optimisation above.

The other way a film builds pressure #

Everything above is steady, and the steadiness hides the second half of Reynolds’ equation. The full version has a term in $\partial h/\partial t$, and it is a mechanism in its own right: a film builds pressure when it converges in space — the wedge, which is this essay’s subject — and also when it thins in time, which needs no sliding at all.

Push two flat surfaces together and the fluid between them must escape sideways through the gap. The resistance to that escape goes as the inverse cube of the thickness, so as the gap closes it becomes enormously harder to close further, and under a constant load the thickness decays as the inverse square root of the time. The surfaces approach and never touch: no finite load squeezes a film to nothing in finite time.

That is why bearings survive things the wedge analysis says they should not. A big-end bearing passes through zero sliding velocity twice per revolution, where the wedge pressure is exactly zero — and the combustion load arrives near one of those instants. It is the squeeze film that carries it.

Which changes what a bearing’s load rating is about. A film survives a very large load applied briefly and fails under a moderate one held long, because what the squeeze term buys is time rather than capacity.

What a tilting pad does with all of this #

The commonest bearing of this kind in service does not have a fixed taper at all, and the reason is the essay’s own arithmetic.

A tilting-pad bearing supports each pad on a pivot and lets the film find its own convergence ratio. The pad rotates until the moment of the pressure distribution about the pivot vanishes, which fixes $k$ — and the value it settles at depends on where the pivot is placed along the pad, not on anything the designer has to hold to microns.

That is a good trade. A fixed-taper pad has its convergence set by manufacture and then changed by thermal distortion and by wear; a tilting pad re-finds its own at every speed and load. And because the optimum is flat, the pivot position does not have to be right either: pivot fractions between about 0.55 and 0.7 of the pad length all produce convergence ratios inside the one per cent band.

So the flatness that makes the exact optimum nearly worthless is what makes the practical design robust. Those are the same fact, and which of the two is worth saying depends on whether the reader is computing or building.

The one thing a pivoted pad cannot do is run backwards: pivot it downstream of centre and the equilibrium convergence is diverging, the pressure is negative, and the film collapses. That is a threshold rather than an optimum — the sign of a quantity rather than its stationary point — and it is located exactly, which is the distinction this essay is about.

What the essay is actually about #

Two things, and they pull in opposite directions.

The separation is real and it is the useful part. That the load factors into a dimensional prefactor times a bracket in one geometric ratio is why bearing design is tractable at all: it means the taper can be chosen once, for all bearings, and everything else scaled. The same separation is what lets the last of the oil be analysed without knowing which oil, and it is the ordinary payoff of a well-posed dimensionless reduction.

The optimum is a pure number of a particular kind. It is not an integer, not a ratio of small integers, and not expressible in closed form: it is the root of a transcendental equation, which is the same species as the eigenvalue that sets the onset of convection and quite a different species from the exact rationals of a sphere’s added mass. Knowing which kind a number is settles how far to trust it and how it will behave when the problem is perturbed — and a transcendental root moves smoothly under perturbation while a rational one either survives exactly or does not survive at all.

And the exactness of the optimum inside it is nearly worthless. The number is a genuine pure number and it is quoted to eight figures here because it can be, not because any of them matters past the second. What decides a real bearing’s taper is the manufacturing tolerance, the thermal distortion of the pad, the pivot position if it is a tilting pad, and above all the aspect ratio — and every one of those is a larger effect than the difference between 1.99 and 2.43.

What is not claimed #

Inertia is absent. Reynolds’ equation is the thin-film limit of the Navier–Stokes equations with the inertial terms dropped, which is justified by the reduced Reynolds number $Re\,(h/B)$ being small rather than by $Re$ itself being small — a distinction the exact theory drawn by viscosity makes for a Hele-Shaw cell and which applies here word for word.

Isothermal, isoviscous, laminar, and with no cavitation. All four fail in real bearings. The film heats up, which drops the viscosity by a factor that can exceed two along the pad; the diverging part of a full journal bearing cavitates, which is why the analysis there uses a half-Sommerfeld condition; and none of that is treated here.

A plane slider, not a journal. The geometry above is a flat pad over a flat runner. A journal bearing’s film is a wedge that closes and reopens round the circumference, and its optimum is stated in eccentricity ratio rather than in convergence ratio.

The two-dimensional solve is coarse. Sixty-four by sixty-four cells with successive over-relaxation, which is enough to establish the size of the leakage effect and the direction the optimum moves, and not enough to quote the moved optimum to more than two figures.

The convergence ratio is not the only geometry. A stepped pad — Rayleigh’s step, two parallel films of different thickness — carries more load than any linearly tapered one, by about fifteen per cent, and its own optimum is a different pure number. The essay’s arithmetic is about the best member of one family, and the family was chosen because it is what gets built rather than because it wins.

And the pad is rigid. A real pad deflects under the pressure it generates, which changes the film shape, which changes the pressure — and the coupled problem has an optimum of its own that this one does not bound.