**LUBRICATION.** Our knowledge of the action of oils and other
viscous fluids in diminishing friction and wear between solid
surfaces from being purely empirical has become a connected
theory, based on the known properties of matter, subjected to the
definition of mathematical analysis and verified by experiment.
The theory was published in 1886 (*Phil. Trans.*, 1886, 177, pp.
157-234); but it is the purpose of this article not so much to
explain its application, as to give a brief account of the introduction
of the misconceptions that so long prevailed, and of the
manner in which their removal led to its general acceptance.

Friction, or resistance to tangential shifting of matter over matter, whatever the mode and arrangement, differs greatly according to the materials, but, like all material resistance, is essentially limited. The range of the limits in available materials has a primary place in determining mechanical possibilities, and from the earliest times they have demanded the closest attention on the part of all who have to do with structures or with machines, the former being concerned to find those materials and their arrangements which possess the highest limits, and the latter the materials in which the limits are least. Long before the reformation of science in the 15th and 16th centuries both these limits had formed the subject of such empirical research as disclosed numerous definite although disconnected circumstances under which they could be secured; and these, however far from the highest and lowest, satisfied the exigencies of practical mechanics at the time, thus initiating the method of extending knowledge which was to be subsequently recognized as the only basis of physical philosophy. In this purely empirical research the conclusion arrived at represented the results for the actual circumstance from which they were drawn, and thus afforded no place for theoretical discrepancies. However, in the attempts at generalization which followed the reformation of science, opportunity was afforded for such discrepancies in the mere enunciation of the circumstances in which the so-called laws of friction of motion are supposed to apply. The circumstances in which the great amount of empirical research was conducted as to the resistance between the clean, plane, smooth surfaces of rigid bodies moving over each other under pressure, invariably include the presence of air at atmospheric pressure around, and to some extent between, the surfaces; but this fact had received no notice in the enunciation of these laws, and this constitutes a theoretical departure from the conditions under which the experience had been obtained. Also, the theoretical division of the law of frictional resistance into two laws—one dealing with the limit of rest, and the other asserting that the friction of motion, which is invariably less in similar circumstances than that of rest, is independent of the velocity of sliding—involves the theoretical assumption that there is no asymptotic law of diminution of the resistance, since, starting from rest, the rate of sliding increases. The theoretical substitution of ideal rigid bodies with geometrically regular surfaces, sliding in contact under pressure at the common regular surface, for the aërated surfaces in the actual circumstances, and the theoretical substitution of the absolute independence of the resistance of the rate of sliding for the limited independence in the actual circumstances, prove the general acceptance of the conceptions—(1) that matter can slide over matter under pressure at a geometrically regular surface; (2) that, however much the resistance to sliding under any particular pressure (the co-efficient of friction) may depend on the physical properties of the materials, the sliding under pressure takes place at the geometrically regular surface of contact of the rigid bodies; and (3) as the consequence of (1) and (2), that whatever the effect of a lubricant, such as oil, might have, it could be a physical surface effect. Thus not only did these general theoretical conceptions, resulting from the theoretical laws of friction, fail to indicate that the lubricant may diminish the resistance by the mere mechanical separation of the surfaces, but they precluded the idea that such might be the case. The result was that all subsequent attempts to reduce the empirical facts, where a lubricant was used, to such general laws as might reveal the separate functions of the complex circumstances on which lubrication depends, completely failed. Thus until 1883 the science of lubrication had not advanced beyond the empirical stage.

This period of stagnation was terminated by an accidental
phenomenon observed by Beauchamp Tower, while engaged
on his research on the friction of the journals of railway carriages.
His observation led him to a line of experiments which proved
that in these experiments the general function of the lubricant
was the mechanical separation of the metal surfaces by a layer
of fluid of finite thickness, thus upsetting the preconceived ideas
as expressed in the laws of the friction of motion. On the publication
of Tower’s reports (*Proc. Inst. M.E.*, November 1883), it
was recognized by several physicists (*B.A. Report*, 1884, pp. 14,
625) that the evidence they contained afforded a basis for
further study of the actions involved, indicating as it did the
circumstances—namely, the properties of viscosity and cohesion
possessed by fluids—account of which had not been taken in
previous conclusions. It also became apparent that continuous
or steady lubrication, such as that of Tower’s experiments, is
only secured when the solid surfaces separated by the lubricant
are so shaped that the thickness at the ingoing side is greater than
that at the outgoing side.

When the general equations of viscous fluids had been shown
as the result of the labours of C. L. M. H. Navier,^{[1]} A. L. Cauchy,^{[2]}
S. D. Poisson,^{[3]} A. J. C. Barré de St Venant,^{[4]} and in 1845 of Sir
G. Gabriel Stokes,^{[5]} to involve no other assumption than that
the stresses, other than the pressure equal in all directions,
are linear functions of the distortional rates of strain multiplied
by a constant coefficient, it was found that the only solutions
of which the equations admitted, when applied to fluids flowing
between fixed boundaries, as water in a pipe, were singular
solutions for steady or steady periodic motion, and that the
conclusions they entailed, that the resistance would be proportional
to the velocity, were for the most part directly at
variance with the common experience that the resistances
varied with the square of the velocity. This discrepancy was
sometimes supposed to be the result of eddies in the fluid, but
it was not till 1883 that it was discovered by experiments with
colour bands that, in the case of geometrically similar boundaries,
the existence or non-existence of such eddies depended upon
a definite relation between the mean velocity (U) of the fluid,
the distance between the boundaries, and the ratio of the coefficient
of viscosity to the density (μ/ρ), expressed by UDρ/μ = K,
where K is a physical constant independent of units, which has
a value between 1900 and 2000, and for parallel boundaries
D is four times the area of the channel divided by the perimeter
of the section (*Phil. Trans.*, 1883, part iii. 935-982). K is thus
a criterion at which the law of resistance to the mean flow changes
suddenly (as U increases), from being proportional to the flow,
to a law involving higher powers of the velocity at first, but as
the rates increase approaching an asymptote in which the power
is a little less that the square.

This sudden change in the law of resistance to the flow of
fluid between solid boundaries, depending as it does on a complete
change in the manner of the flow—from direct parallel flow to
sinuous eddying motion—serves to determine analytically the
circumstances as to the velocity and the thickness of the film
under which any fluid having a particular coefficient of viscosity
can act the part of a lubricant. For as long as the circumstances
are such that UDρ/μ is less than K, the parallel flow is held stable
by the viscosity, so that only one solution is possible—that
in which the resistance is the product of μ multiplied by the
rate of distortion, as μ*du**dy*; in this case the fluid has lubricating
properties. But when the circumstances are such that UDρ/μ
is greater than K, other solutions become possible, and the
parallel flow becomes unstable, breaks down into eddying
motion, and the resistance varies as ρ*u ^{n}*, which approximates
to ρ

*u*

^{1.78}as the velocity increases; in this state the fluid has no lubricating properties. Thus, within the limits of the criterion, the rate of displacement of the momentum of the fluid is insignificant as compared with the viscous resistance, and may be neglected; while outside this limit the direct effects of the eddying motion completely dominate the viscous resistance, which in its turn may be neglected. Thus K is a criterion which separates the flow of fluid between solid surfaces as definitely as the flow of fluid is separated from the relative motions in elastic solids, and it is by the knowledge of the limit on which this distinction depends that the theory of viscous flow can with assurance be applied to the circumstance of lubrication.

Until the existence of this physical constant was discovered, any theoretical conclusions as to whether in any particular circumstances the resistance of the lubricant would follow the law of viscous flow or that of eddying motion was impossible. Thus Tower, being unaware of the discovery of the criterion, which was published in the same year as his reports, was thrown off the scent in his endeavour to verify the evidence he had obtained as to the finite thickness of the film by varying the velocity. He remarks in his first report that, “according to the theory of fluid motion, the resistance would be as the square of the velocity, whereas in his results it did not increase according to this law.” The rational theory of lubrication does not, however, depend solely on the viscosity within the interior of fluids, but also depends on the surface action between the fluid and the solid. In many respects the surface actions, as indicated by surface tension, are still obscure, and there has been a general tendency to assume that there may be discontinuity in the velocity at the common surface. But whatever these actions may be in other respects, there is abundant evidence that there is no appreciable discontinuity in the velocity at the surfaces as long as the fluid has finite thickness. Hence in the case of lubrication the velocities of the fluid at the surfaces of the solids are those of the solid. In as far as the presence of the lubricant is necessary, such properties as cause oil in spite of its surface tension to spread even against gravity over a bright metal surface, while mercury will concentrate into globules on the bright surface of iron, have an important place in securing lubrication where the action is intermittent, as in the escapement of a clock. If there is oil on the pallet, although the pressure of the tooth causes this to flow out laterally from between the surfaces, it goes back again by surface tension during the intervals; hence the importance of using fluids with low surface tension like oil, or special oils, when there is no other means of securing the presence of the lubricant.

The differential equations for the equilibrium of the lubricant are what the differential equations of viscous fluid in steady motion become when subject to the conditions necessary for lubrication as already defined—(1) the velocity is below the critical value; (2) at the surfaces the velocity of the fluid is that of the solid; (3) the thickness of the film is small compared with the lateral dimensions of the surfaces and the radii of curvature of the surfaces. By the first of these conditions all the terms having ρ as a factor may be neglected, and the equations thus become the equations of equilibrium of the fluid; as such, they are applicable to fluid whether incompressible or elastic, and however the pressure may affect the viscosity. But the analysis is greatly simplified by omitting all terms depending on compressibility and by taking μ constant; this may be done without loss of generality in a qualitative sense. With these limitations we have for the differential equation of the equilibrium of the lubricant:—

0 = | dp |
− μ∇^{2}u, &c., &c., 0 = | du |
+ | dv |
+ | dw | |

dx | dx |
dy | dz | |||||

0 = p_{yx} − μ dudy+dvdx
, &c., &c. |

These are subject to the boundary conditions (2) and (3). Taking
*x* as measured parallel to one of the surfaces in the direction of relative
motion, *y* normal to the surface and *z* normal to the plane of *xy*
by condition (3), we may without error disregard the effect of any
curvature in the surfaces. Also *v* is small compared with *u* and *w*,
and the variations of *u* and *w* in the directions *x* and *z* are small compared
with their variation in the direction *y*. The equations (1)
reduce to

0 = | dp |
− μ | d ^{2}u |
, 0 = | dp |
, 0 = | dp |
− μ | d ^{2}w |
, 0 = | du |
+ | dv |
+ | dw | |

dx | dy^{2} |
dy | dz |
dy^{2} | dx |
dy | dz | |||||||||

0 = p_{yx} − μdudy, 0 = p_{yz} − μdwdy, p_{xz} = 0. |

For the boundary conditions, putting ƒ(*x*, *z*) as limiting the lateral
area of the lubricant, the conditions at the surfaces may be expressed
thus:—

when y = 0, u = U_{0}, w = 0, v = 0 | |

when y = h, u = U_{1}, w = 0, v_{1}, = U_{1} dhdx + V_{1} | |

when ƒ(x, z) = 0, p = p_{0} |

Then, integrating the equations (2) over *y*, and determining the
constants by equations (3), we have, since by the second of equations
(2) *p* is independent of *y*,

u = | 1 | dp |
(y − h) y + U_{0} | h − y |
+ U_{1} | y | ||

2μ | dx |
h | h | |||||

w = | 1 | dp |
(y − h) y | |||||

2μ | dz |

Then, differentiating equations (4) with respect to *x* and *z* respectively,
and substituting in the 4th of equations (2), and integrating
from *y* = 0 to *y* = *h*, so that only the values of *v* at the surfaces may be
required, we have for the differential equation of normal pressure at
any point *x*, *z*, between the boundaries:—

d |
h^{3} | dp |
+ | d |
h^{3} | dp |
= 6μ (U_{0} + U_{1}) | dh |
+ 2V_{1} |

dx | dz |
dz | dz |
dx |

Again differentiating equations (4), with respect to *x* and *z* respectively,
and substituting in the 5th and 6th of equations (2), and
putting ƒ_{x} and ƒ_{z} for the intensities of the tangential stresses at the
lower and upper surfaces:—

ƒ_{x} = μ (U_{1} − U_{0}) | 1 | ± | h |
dp | ||

h | 2 | dx | ||||

ƒ_{x} = ± h2 dpdx |

Equations (5) and (6) are the general equations for the stresses at the boundaries at *x*, *z*, when *h* is a continuous function of *x* and *z*, μ and ρ being constant.

For the integration of equations (6) to get the resultant stresses and moments on the solid boundaries, so as to obtain the conditions
of their equilibrium, it is necessary to know how *x* and *z* at any point on the boundary enter into *h*, as well as the equation ƒ(*x*, *z*) = 0, which determines the limits of the lubricating film. If *y*, the normal to one of the surfaces, has not the same direction for all points of this surface, in other words, if the surface is not plane, *x* and *z* become curvilinear co-ordinates, at all points perpendicular to *y*. Since, for lubrication, one of the surfaces must be plane, cylindrical, or a surface of revolution, we may put *x* = Rθ, *y* = *r* − R, and *z* perpendicular to the plane of motion. Then, if the data are sufficient, the resultant stresses and moments between the surfaces are obtained by integrating the intensity of the stress and moments of intensity of stress over the surface.

This, however, is not the usual problem that arises. What is generally wanted is to find the thickness of the film where least (*h*_{0}) and its angular position with respect to direction of load, to resist a definite load with a particular surface velocity. If the surfaces are plane, the general solution involves only one arbitrary constant, the least thickness (*h*_{0}); since in any particular case the variation of *h* with *x* is necessarily fixed, as in this case lubrication affords no automatic adjustment of this slope. When both surfaces are curved in the plane of motion there are at least two arbitrary constants, *h*_{0}, and φ the angular position of *h*_{0} with respect to direction of load; while if the surfaces are both curved in a plane perpendicular to the direction of motion as well as in the plane of motion, there are three arbitrary constants, *h*_{0}, φ_{0}, *z*_{0}. The only constraint necessary is to prevent rotation in the plane of motion of one of the surfaces, leaving this surface free to move in any direction and to adjust its position so as to be in equilibrium under the load.

The integrations necessary for the solutions of these problems are practicable—complete or approximate—and have been effected for circumstances which include the chief cases of practical lubrication, the results having been verified by reference to Tower’s experiments. In this way the verified theory is available for guidance outside the limits of experience as well as for determining the limiting conditions. But it is necessary to take into account certain subsidiary theories. These limits depend on the coefficient of viscosity, which diminishes as the temperature increases. The total work in overcoming the resistance is spent in generating heat in the lubricant, the volume of which is very small. Were it not for the escape of heat by conduction through the lubricant and the metal, lubrication would be impossible. Hence a knowledge of the empirical law of the variation of the viscosity of the lubricant with temperature, the coefficients of conduction of heat in the lubricant and in the metal, and the application of the theory of the flow of heat in the particular circumstances, are necessary adjuncts to the theory of lubrication for determining the limits of lubrication. Nor is this all, for the shapes of the solid surfaces vary with the pressure, and more particularly with the temperature.

The theory of lubrication has been applied to the explanation of the slipperiness of ice (*Mem. Manchester Lit. and Phil. Soc.*, 1899).
(O. R.)