A pressed blade floats on its land and leaves half the gap behind

Taylor’s solution for a scraper held flat put the blade’s edge on the wall and let the wall slide paint into the corner. The paint pushed the blade off with a force that grew as the logarithm of the blade’s length over some small distance $\varepsilon$ at which the corner was ended, and as one over the square of the blade’s angle. The essay could say that a flat blade rides up on what it was meant to remove, but not by how much, because $\varepsilon$ was chosen from outside the problem: a gap under the blade, the wall’s slip, the paint’s own structure.

A real blade settles the question itself. It is pressed on with a load — a spring, a weight, an air cylinder — and it rises until the liquid it lets through pushes back exactly as hard. The gap is then an output, and so is the film the blade leaves on the surface, which is the number anyone using a blade actually cares about. This essay finds both, and finds that the answer depends less on the angle than on something the corner solution never had: the short flat land a real edge carries, by grinding or by wear.

A corner with a gap in it #

The geometry is Taylor’s corner with its apex lifted. The surface slides at speed $U$ towards the blade. The blade’s face meets it at an angle $\alpha$ and ends, at the bottom, in a flat land of length $L$ parallel to the surface at a height $h_0$ above it; behind the land the liquid leaves at ambient pressure, carried away on the surface, and up the face, a centimetre back, the corner opens into the bath at ambient pressure too. A flux $q$ per unit span passes through the gap. In the thin layer under the face and the land, Reynolds’s equation says the pressure gradient is whatever makes the dragged flow carry exactly that flux:

$$\frac{dp}{dx} = \frac{6\mu U}{h^2} - \frac{12\mu q}{h^3},$$

with $h$ the local gap. Where the gap is wide the first term wins and the pressure climbs towards the edge; where it is narrow the second wins and the pressure falls. With the pressure ambient at both ends, the flux is fixed, and the film left on the surface — moving at $U$ with nothing to shear it further — is $t = q/U$.

The equation is lubrication theory, and lubrication theory is the corner’s small-angle limit: it overstates the corner’s push by 0.6 per cent at ten degrees, by 14 per cent at forty-five and by more than a factor of two at sixty. The face’s logarithmic contribution is therefore scaled here by the ratio of Taylor’s exact force to the thin-wedge one at each angle, a correction of 1.01 at ten degrees, 1.15 at thirty and 2.16 at sixty, while the land, which is a genuinely thin film, needs none. How much that correction matters at a given angle depends on how much of the load the face carries, and the answer to that is the essay’s first surprise.

A blade with no land floats on a logarithm #

Take the land away first. The flux is then $Uh_0h_R/(h_0 + h_R)$, with $h_R$ the gap at the top of the face — almost exactly $Uh_0$, since the face opens to millimetres — so the film is the whole gap. The liquid passes under the edge as a plug moving with the surface, and the pressure that lifts the blade is the corner’s: it rises as one over the distance into the corner until the gap is a few times $h_0$ and then falls to zero at the edge. Integrated, the lift per unit span is the earlier essay’s $\mu U \ln(R/\varepsilon)$ times its angle factor, with $\varepsilon$ now about $h_0/\tan\alpha$.

That is the trouble. A load that has to be carried by a logarithm makes the gap an exponential of the load: $h_0 \approx R\tan\alpha\, e^{-W\tan^2\alpha/6\mu U}$, up to a constant. At ten degrees and a load of 100 N/m the exponent is small and the blade floats 109 µm up, riding on the ink as Taylor said it would. At twenty degrees it floats at 7.7 µm, at thirty at 0.05 µm — fifty nanometres. At forty-four degrees the gap that would carry the load is smaller than a ten-thousandth of a nanometre, which is to say that no film a liquid can form carries it. Above that angle the edge is in contact with the surface and the load is carried by the solids, with the film set by asperities and wear rather than by hydrodynamics.

A land holds a parallel film #

Now give the edge a land a tenth of a millimetre long. Under the land the gap is constant, so the pressure there falls linearly from the value the face has built up to ambient at the back. A falling pressure in a parallel gap pushes liquid backwards on top of the drag, so the flux through the land is more than half the gap times the speed: $q = \tfrac12 U h_0 (1 + \delta)$. The face, for its part, can only build a bounded pressure at its narrow end, of order $\mu U/(h_0\tan\alpha)$, because its gap widens away from the edge. Matching the two gives

$$\delta = \frac{1}{1 + 2L\tan\alpha/h_0},$$

which is small whenever the land is long compared with the gap divided by the face’s slope — a hundred-micrometre land and a micrometre gap at thirty degrees give $\delta$ of a hundredth. Then the film is half the gap, very nearly: at sixty degrees and 100 N/m the gap is 0.535 µm and the film 0.268, a ratio of 0.503.

The figure shows how the load divides. At twenty degrees the face still carries ninety per cent of it and the land is an appendage. At thirty the land carries 38 per cent, at sixty 81 per cent, its triangle of pressure — 1.6 MPa at its front, falling to nothing at its back — dwarfing the face’s thin spike. The steeper the face, the less the corner can hold and the more the land must, and that is the reason the uncertain correction for steep faces matters less than it looks: at sixty degrees, where lubrication overstates the corner twofold, the corner is a fifth of the load.

In the long-land limit the land carries everything, and its load is the pressure at its front, about $3\mu U/(h_0\tan\alpha)$, times half its length. Set equal to the load $W$ that gives

$$h_0 \approx \frac{3\mu U L}{2W\tan\alpha}, \qquad t \approx \frac{3\mu U L}{4W\tan\alpha},$$

a film that falls as one over the load and is proportional to the land. At sixty degrees and 100 N/m it predicts 0.22 µm against the full calculation’s 0.27, the difference being the corner’s fifth.

One over the load, not an exponential #

That difference in law is what a land is for. A blade without one at thirty degrees, pressed at 100 N/m, leaves 50 nm; pressed ten per cent harder it leaves 19 nm; ten per cent lighter, more than a hundred. Every fluctuation in the load — a roll that is not quite round, a cylinder whose pressure drifts, a blade holder that flexes — is multiplied into the film. With a land the same ten per cent changes the film by seventeen per cent at thirty degrees and eleven at sixty, close to the one-for-one of the long-land law. The film becomes a smooth, controllable function of the load, and the load becomes a useful dial rather than a hazard.

The price is thickness. At thirty degrees the land multiplies the film by more than thirty, from 0.05 µm to 1.7 µm, because a parallel film is a much better bearing than the tail of a corner: it carries pressure over its whole length instead of over a few gap-widths. A land trades cleaning for steadiness. That is the choice a coating knife makes on purpose — a long land, a film set by the land and the load, laid down as reliably as the film a plate takes with it from a bath, though by a different balance — and the choice a doctor blade on a printing press makes by grinding a short, deliberate land on a steep blade, to be steady without being thick.

Steeper still cleans better #

The earlier essay’s argument for a steep blade was the force: a flat blade’s corner pushes it off harder. The film confirms it, and measures it. At 100 N/m with a 0.1 mm land the film is 110 µm at ten degrees, 10 µm at twenty, 1.7 at thirty, 0.56 at forty-five and 0.27 at sixty. Turned round, as a load to reach a given film, the angle is worth an order of magnitude of force.

To leave half a micrometre of ink, a blade at ten degrees must be pressed with a kilonewton per metre — ten newtons on every centimetre of its length — and one at sixty with sixty newtons per metre, sixteen times less. Some of that is the land’s own law, which goes as $1/\tan\alpha$: from ten to sixty degrees that ratio alone is ten. The rest is the face, which at ten degrees still carries nearly all the load in its corner and holds the blade up as Taylor’s paint did.

There is a reason not to go all the way. Above about sixty degrees the blade’s face is no longer a thin wedge in any sense, the corner correction exceeds two, and the flow at the edge has room to form the separated eddies that a corner stirred from outside always holds. More practically, a blade at seventy or eighty degrees is a chisel: it is stiff against the load, so a small runout in the roll becomes a large change in the force, and its land wears fastest.

A floating blade rubs as hard as a dry one #

The surface pays for the film in drag, and the drag is a second thing the steep blade changes. Under the land the liquid is sheared at the surface’s speed over the gap — five metres a second across half a micrometre at sixty degrees, a shear rate of ten million per second — and the shear stress there is about $\mu U/h_0$, near a megapascal. Integrated over the land and the face, the drag at 100 N/m is 17 N/m at ten degrees and 104 N/m at sixty. Divided by the load, that is a friction coefficient: 0.17 at ten degrees, 0.37 at thirty, 0.61 at forty-five and 1.04 at sixty.

A coefficient of one is what a steel blade rubbing dry on a steel roll would show, and it is the reason a doctor blade’s drag cannot tell its user whether the blade is floating or touching. The two regimes give the same order of force by entirely different routes — viscous shear over a land a hundred and eighty gaps long in one, ploughing of asperities in the other — and the line between them is the angle and land at which the film the load needs reaches the roughness. The flat blade’s low friction is no comfort either: it is low because the gap is a hundred micrometres, the film it leaves is thick, and the blade is doing the job of a coating knife rather than a scraper.

The drag times the speed is the heat, and at sixty degrees it is 520 W for every metre of blade, all of it made in a film half a micrometre thick under a land a tenth of a millimetre long. Almost none of that stays in the ink: conduction across a gap that thin is fast, and the temperature rise of a sheared film contains no gap at all, only $\mu U^2/8k$, a few kelvin here. The heat goes into the blade and the roll, and the blade’s edge is where it concentrates, which is one reason edges wear in the way the next section describes. The face’s share of the drag is computed in the thin-wedge form, without the exact corner’s correction; at the steep angles the land carries most of it, as it carries most of the load.

The film grows as the edge wears #

A new doctor blade’s edge is sharp, or carries a land of a few tens of micrometres ground on purpose; in service the edge wears, and the land lengthens to a few tenths of a millimetre. The load is set by the holder and does not change. The film does.

At thirty degrees the film grows as the land’s length to the power 0.74, from 0.15 µm under a three-micrometre land to 11 µm under a millimetre; at sixty degrees it grows faster, nearer to proportion, because there the land already carries most of the load and its own law is linear in its length. That is the familiar complaint of printers — the image darkens as the blade wears, at an unchanged pressure — written as a law, and it says the cure is not more pressure, which would need a near-proportional increase to compensate, but a blade with a land ground short and a geometry that keeps it short as it wears, which is what lamella-tipped blades are for.

The calculation also says what not to read into a very short land. At sixty degrees and a three-micrometre land the film is nine nanometres, a few dozen molecules of a typical ink vehicle. No continuum film of that thickness under that pressure behaves as Reynolds’s equation says; the surface’s roughness and whatever slip the liquid has at the wall set the film there, and the number is a statement that hydrodynamics has run out, not a prediction.

The convention, and the load that is per metre #

Loads are per unit span — newtons per metre of blade length — and are scaled by $\mu U$, the only combination of viscosity and speed with those units; the worked case is an ink of 0.1 Pa·s on a roll at 5 m/s, for which $\mu U$ is 0.5 N/m and a load of 100 N/m is a dimensionless 200. Every film scales with the land at a fixed dimensionless load, so the figures carry to any ink and speed by that rule: a liquid ten times as viscous at the same speed and load behaves as this one at a tenth of the load. Angles are between the face and the surface, measured on the liquid’s side. The film is the thickness carried away on the surface, which, for a surface moving as a plug behind the blade, is the flux through the gap over the speed.

What the land model leaves out #

The land is exactly parallel. A worn land is usually a little tilted, and a land tilted a fraction of a degree so that it converges towards its back is a tapered pad in its own right, with the load a taper adds; one diverging is worse than parallel. The blade is rigid, and a real doctor blade is a thin steel strip that bends under the pressure it generates, which changes both the land’s tilt and the load’s line. The ink is Newtonian and isothermal, while printing inks are shear-thinning and the shear rates under a land — the speed over the gap, millions per second — are where their viscosity has fallen furthest, and the heat the shear makes in a film that heats itself lowers it further. The film behind the land is taken to leave as a plug, with no cavitation or splitting in the diverging gap behind it, and the surface is smooth; an engraved anilox roll is the opposite of smooth, and there the land’s film is the part on the lands of the engraving, while the cells are emptied by a different mechanism altogether.

Who measured blades and who solved them #

Taylor’s corner is of 1962. Reynolds’s equation for a parallel pad and a wedge is in his paper of 1886, and the parallel film’s need of a pressure gradient to carry more than half its gap times its speed is in every bearing text since. Doctor blades were studied as an elastohydrodynamic problem from the 1980s, notably for gravure and flexographic printing and for paper coating, where the bent-blade coater’s film, set by the blade’s load and its tip, was analysed by Greener and Middleman and by Pranckh and Scriven. The two-region model here — a corner face scaled to Taylor and a lubricated land matched to it — is a simplification of theirs, chosen to show the two laws side by side rather than to reproduce a particular coater.

How it was checked #

Without a land the computed flux is $Uh_0h_R/(h_0 + h_R)$ to rounding at three gaps. The pressure where the face meets the land is the same computed from either side to a part in $10^{14}$, and the land’s flux matches $\tfrac12Uh_0(1 + \delta)$ with the closed-form $\delta$ to a part in $10^5$ for a long face. The face’s load, integrated numerically over its pressure on a stretched grid, matches the closed form to a part in $10^7$. At sixty degrees the film is 1.006 times half the gap. The blade refuses an angle of zero or past a right angle, a negative land, a gap of zero, a negative load and a load so light it would lift the blade clear of the bath.

Still open: a land that bends #

Every number here is for a rigid blade. A doctor blade is a strip a fifth of a millimetre thick, clamped well back from its edge and bent onto the roll by its own load, and the pressure under its land, a megapascal or more, is enough to bend the land too. The next calculation would give the blade a stated stiffness, let the land’s tilt and the face’s angle be set by the bending the pressure causes, and ask whether the coupled blade settles at a film thicker or thinner than the rigid one — whether a flexible blade, which tilts its land open at the back as it bends, loses the parallel film’s steadiness, or finds a self-consistent tilt that keeps the film at the long-land law.