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 UD[rho]/[mu] 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 [mu] multiplied by the rate of distortion, as [mu](du/dy); in this case the fluid has lubricating properties. But when the circumstances are such that UD[rho]/[mu] is greater than K, other solutions become possible, and the parallel flow becomes unstable, breaks down into eddying motion, and the resistance varies as [rho]u^n, which approximates to [rho]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 [rho] 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 [mu] 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:--
dp du dv dw \ 0 = -- - [mu]²u, &c., &c., 0 = -- + -- + -- | dx dx dy dz | > (1) / du dv \ | 0 = p_yx - [mu] ( -- + -- ), &c., &c. | \ dy dx / /
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
dp d²u dp dp d²w du dv dw \ 0 = -- - [mu]---, 0 = --, 0 = -- - [mu]---, 0 = -- + -- + -- | dx dy² dy dz dy² dx dy dz | > (2) du dw | 0 = p_yx - [mu]--, 0 = p_yz - [mu]--, p_xz = 0. | dy dy /
For the boundary conditions, putting f(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 \ dh | when y = h, u = U1, w = 0, v1, = U1 -- + V1 > (3) dx | when f(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,
1 dp h - y y \ u = ----- -- (y - h)y + U0 ----- + U1 --- | 2[mu] dx h h | > (4) 1 dp | w = ----- -- (y - h)y | 2[mu] 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 / dp\ d / dp\ | dh | --- ( h³ -- ) + --- ( h³-- ) = 6[mu] | (U0 + U1) -- + 2V1 | (5) 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 f_x and f_z for the intensities of the tangential stresses at the lower and upper surfaces:--
1 h dp \ f_x = [mu](U1 + U0) --- ± --- -- | h 2 dx | > (6) h dp | f_z = ± --- -- | 2 dx /
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, [mu] and [rho] 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 f(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 = R[theta], 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 (h0) 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 (h0); 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, h0, and [phi] the angular position of h0 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, h0, [phi]0, z0. 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.)
FOOTNOTES:
[1] _Mém. de l'Acad._ (1826), 6, p. 389.
[2] _Mém. des sav. étrang._ l. 40.
[3] _Mém. de l'Acad._ (1831), 10, p. 345.
[4] _B.A. Report_ (1846).
[5] _Cambridge Phil. Trans._ (1845 and 1857).
LUCAN [MARCUS ANNAEUS LUCANUS], (A.D. 39-65), Roman poet of the Silver Age, grandson of the rhetorician Seneca and nephew of the philosopher, was born at Corduba. His mother was Acilia; his father, Marcus Annaeus Mela, had amassed great wealth as imperial procurator for the provinces. From a memoir which is generally attributed to Suetonius we learn that Lucan was taken to Rome at the age of eight months and displayed remarkable precocity. One of his instructors was the Stoic philosopher, Cornutus, the friend and teacher of Persius. He was studying at Athens when Nero recalled him to Rome and made him quaestor. These friendly relations did not last long. Lucan is said to have defeated Nero in a public poetical contest; Nero forbade him to recite in public, and the poet's indignation made him an accomplice in the conspiracy of Piso. Upon the discovery of the plot he is said to have been tempted by the hope of pardon to denounce his own mother. Failing to obtain a reprieve, he caused his veins to be opened, and expired repeating a passage from one of his poems descriptive of the death of a wounded soldier. His father was involved in the proscription, his mother escaped, and his widow Polla Argentaria survived to receive the homage of Statius under Domitian. The birthday of Lucan was kept as a festival after his death, and a poem addressed to his widow upon one of these occasions and containing information on the poet's work and career is still extant (Statius's _Silvae_, ii. 7, entitled _Genethliacon Lucani_).
Besides his principal performance, Lucan's works included poems on the ransom of Hector, the nether world, the fate of Orpheus, a eulogy of Nero, the burning of Rome, and one in honour of his wife (all mentioned by Statius), letters, epigrams, an unfinished tragedy on the subject of Medea and numerous miscellaneous pieces. His minor works have perished except for a few fragments, but all that the author wrote of the _Pharsalia_ has come down to us. It would probably have concluded with the battle of Philippi, but breaks off abruptly as Caesar is about to plunge into the harbour of Alexandria. The _Pharsalia_ opens with a panegyric of Nero, sketches the causes of the war and the characters of Caesar and Pompey, the crossing of the Rubicon by Caesar, the flight of the tribunes to his camp, and the panic and confusion in Rome, which Pompey has abandoned. The second book describes the visit of Brutus to Cato, who is persuaded to join the side of the senate, and his marriage a second time to his former wife Marcia, Ahenobarbus's capitulation at Corfinium and the retirement of Pompey to Greece. In the third book Caesar, after settling affairs in Rome, crosses the Alps for Spain. Massilia is besieged and falls. The fourth book describes the victories of Caesar in Spain over Afranius and Petreius, and the defeat of Curio by Juba in Africa. In the fifth Caesar and Antony land in Greece, and Pompey's wife Cornelia is placed in security at Lesbos. The sixth book describes the repulses of Caesar round Dyrrhachium, the seventh the defeat of Pompey at Pharsalia, the eighth his flight and assassination in Egypt, the ninth the operations of Cato in Africa and his march through the desert, and the landing of Caesar in Egypt, the tenth the opening incidents of the Alexandrian war. The incompleteness of the work should not be left out of account in the estimate of its merits, for, with two capital exceptions, the faults of the _Pharsalia_ are such as revision might have mitigated or rendered. No such pains, certainly, could have amended the deficiency of unity of action, or supplied the want of a legitimate protagonist. The _Pharsalia_ is not true to history, but it cannot shake off its shackles, and is rather a metrical chronicle than a true epic. If it had been completed according to the author's design, Pompey, Cato and Brutus must have successively enacted the part of nominal hero, while the real hero is the arch-enemy of liberty and Lucan, Caesar. Yet these defects, though glaring, are not fatal or peculiar to Lucan. The false taste, the strained rhetoric, the ostentatious erudition, the tedious harangues and far-fetched or commonplace reflections so frequent in this singularly unequal poem, are faults much more irritating, but they are also faults capable of amendment, which the writer might not improbably have removed. Great allowance should also be made in the case of one who is emulating predecessors who have already carried art to its last perfection. Lucan's temper could never have brooked mere imitation; his versification, no less than his subject, is entirely his own; he avoids the appearance of outward resemblance to his great predecessor with a persistency which can only have resulted from deliberate purpose, but he is largely influenced by the declamatory school of his grandfather and uncle. Hence his partiality for finished antithesis, contrasting strongly with his generally breathless style and turbid diction. Quintilian sums up both aspects of his genius with pregnant brevity, "Ardens et concitatus et sententiis clarissimus," adding with equal justice, "Magis oratoribus quam poetis annumerandus." Lucan's oratory, however, frequently approaches the regions of poetry, e.g. the apotheosis of Pompey at the beginning of the ninth book, and the passage in the same book where Cato, in the truest spirit of the Stoic philosophy, refuses to consult the oracle of Jupiter Ammon. Though in many cases Lucan's rhetoric is frigid, hyperbolical, and out of keeping with the character of the speaker, yet his theme has a genuine hold upon him; in the age of Nero he celebrates the republic as a poet with the same energy with which in the age of Cicero he might have defended it as an orator. But for him it might almost have been said that the Roman republic never inspired the Roman muse.
Lucan never speaks of himself, but his epic speaks for him. He must have been endowed with no common ambition, industry and self-reliance, an enthusiastic though narrow and aristocratic patriotism, and a faculty for appreciating magnanimity in others. But the only personal trait positively known to us is his conjugal affection, a characteristic of Seneca also.
Lucan, together with Statius, was preferred even to Virgil in the middle ages. So late as 1493 his commentator Sulpitius writes: "Magnus profecto est Maro, magnus Lucanus; adeoque prope par, ut quis sit major possis ambigere." Shelley and Southey, in the first transport of admiration, thought Lucan superior to Virgil; Pope, with more judgment, says that the fire which burns in Virgil with an equable glow breaks forth in Lucan with sudden, brief and interrupted flashes. Of late, notwithstanding the enthusiasm of isolated admirers, Lucan has been unduly neglected, but he has exercised an important influence upon one great department of modern literature by his effect upon Corneille, and through him upon the classical French drama.
AUTHORITIES.--The _Pharsalia_ was much read in the middle ages, and consequently it is preserved in a large number of manuscripts, the relations of which have not yet been thoroughly made out. The most recent critical text is that of C. Hosius (2nd ed. 1906), and the latest complete commentaries are those of C. E. Haskins (1887, with a valuable introduction by W. E. Heitland) and C. M. Francken (1896). There are separate editions of book i . by P. Lejay (1894) and