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 μdudy; 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 ρun, which approximates to ρu1.78 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∇2u, &c., &c., 0 = | du | + | dv | + | dw | |
| dx | dx | dy | dz | |||||
| 0 = pyx − 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 2u | , 0 = | dp | , 0 = | dp | − μ | d 2w | , 0 = | du | + | dv | + | dw | |
| dx | dy2 | dy | dz | dy2 | dx | dy | dz | |||||||||
| 0 = pyx − μdudy, 0 = pyz − μdwdy, pxz = 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 = U0, w = 0, v = 0 | |
| when y = h, u = U1, w = 0, v1, = U1 dhdx + V1 | |
| when ƒ(x, z) = 0, p = p0 |
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 + U0 | h − y | + U1 | 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 | h3 | dp | + | d | h3 | dp | = 6μ (U0 + U1) | dh | + 2V1 |
| 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 ƒx and ƒz for the intensities of the tangential stresses at the
lower and upper surfaces:—
| ƒx = μ (U1 − U0) | 1 | ± | h | dp | ||
| h | 2 | dx | ||||
| ƒx = ± h2 dpdx |
