B. P. Weinberg.

On the Study of the Hydrodynamics of the Steady Motion of a Viscous Fluid.

I. An Attempt at a Classification of Various Cases of Established Steady Motion of a Viscous Fluid.

1. Choice of the topic.

Since 1904, having devoted part of my own work and that of my collaborators (the list is given below) to questions directly concerning, or close to, the hydrodynamics of the steady motion of viscous fluids, and now beginning the development of other theoretical and experimental investigations in the same direction, which will form the subject of the subsequent articles in this series, I was faced with the question of precisely which cases should be selected for study.

When one works in an almost undeveloped field, presenting a broad range of diverse problems to be solved, then in choosing them one is usually guided either by considerations of possibility, or by considerations of desirability, i.e. one chooses either those problems which are the easiest—given the present state of mathematical knowledge or experimental technique—to solve, or those whose solution, though more difficult, is of the greatest interest from a theoretical or practical point of view.

In both cases, a systematic choice of problems must be preceded by a survey of the various cases subject to study and of the degree to which they have been studied at the present time, together with their classification, with as complete an analysis as possible of those theoretical and practical questions whose clarification may result from the solution of one or another problem. The present article represents an attempt in this direction with regard to the hydrodynamics of the established steady motion of a viscous fluid, and in a comparatively short time led directly to the development of the questions forming the contents of works Nos. 41, 42, 43, and 51 of the following list.

A similar attempt with regard to the methodology of studying the question, close to the hydrodynamics of steady motion, of the behavior of a solid body under the action of forces exceeding the elastic limit, is represented by works Nos. 28 and 30 of the same list. Investigations in this latter direction constitute another series of works, the first of which has already been printed and the second is printed below—Nos. 44 and 46 of the following list—and, in order to save space and in view of the closeness of the topics of both series, this list includes works relating both to the hydrodynamics of steady motion and to the behavior of a solid body beyond the elastic limit. The few works relating only to the second question are marked in this list with an asterisk. References to all these works will be made by indicating in parentheses the number of the work in bold type, with, where necessary, page numbers added in ordinary type.

1. B. P. Weinberg. “Some methods of determining the coefficient of internal friction of solids (preliminary communication).”—Journal of the Russian Physico-Chemical Society, 36, 47–48, 1904.
2. Boris Weinberg. “On Some Methods for Studying Viscosity of Solids.”—Proc. R. Soc. Lond., 19, 291–292, 472–474, 1905.
3. Boris Weinberg. “Über die innere Reibung des Eises.” Ann. d. Phys., 18, 61–91, 1905.
4. B. P. Weinberg. “A note on the influence of temperature on the internal friction of solids.”—Proceedings of Novorossiisk University, 105, 157–186, 1905.
5. B. P. Weinberg. “On the internal friction of ice.”—Journal of the Russian Physico-Chemical Society, 38, 185–225, 250–282, 289–304, 1906.

1. Boris Weinberg. “Über den Koefficienten der inneren Reibung des Gletschereises und seine Bedeutung für die Theorie der Gletscherbewegung.” — Zeits. f. Gletscherkunde, 1, 321–347, 1906.

2. Boris Weinberg. “Über die theoretische Möglichkeit der Existenz der flüssigen Kristallen.” — Phys. Zeits., 7, 831–832, 1906.

3. Boris Weinberg. “Über die innere Reibung des Eises II.” Ann. d. phys., 22, 321–332, 1907.

4. B. P. Weinberg. “On the Slipperiness of Ice.” — Journal of the Russian Physico-Chemical Society, 32, Part II, 164–165, 1907.

* 10. F. A. Kirillov. “Young’s Modulus as a Function of the Stretching of a Wire.” — Journal of the Russian Physico-Chemical Society, 39, 62–80, 1907.

* 11. B. P. Weinberg. “Comment on Kirillov’s Article ‘Young’s Modulus as a Function of the Stretching of a Wire.’” — Ibid., 81–82.

1. B. P. Weinberg. “On the Question of the ‘Quiet’ Flow of a Liquid through a Channel.” — Journal of the Russian Physico-Chemical Society, 42, 167–176, 1910.

2. Boris Weinberg. “Die wirbellose Strömung von Flüssigkeiten in Kanälen.” — Phys. Zeits., 11, 346–347, 1910.

3. Boris Weinberg. “Zur Theorie der Gletscherbewegung.” — Zeits. f. Gletscherkunde, 4, 308–310, 1910.

4. B. P. Weinberg. “On Investigations of Slow Deformations of Solids.” — Proceedings of the 2nd Mendeleev Congress, No. 3, 24–25, 1911.

5. B. P. Weinberg. “On the Study of Bodies with Large Values of the Coefficient of Internal Friction.” — Journal of the Russian Physico-Chemical Society, 44, 1–2, 1912.

6. B. P. Weinberg and I. Al. Smirnov. “Comparison of Certain Methods for Determining the Coefficient of Internal Friction of Var.” — Journal of the Russian Physico-Chemical Society, 44, 3–25, 1912.

* 18. Al. V. Ignatiev. “Determination of the Capillary Constant of ‘Solid’ Var from the Sizes of Drops.” — Ibid., 71–73.

1. B. P. Weinberg. “The Influence of Temperature on the Internal Friction of Var and Asphalt.” — Ibid., 201–229.

2. M. P. Gostyushin and P. Ya. Ledantyu. “The Resistance of a Viscous Body to Motion inside It of a Solid Body.” — Ibid., 241–251.

3. B. P. Weinberg. “Further Experiments on the Quiet Flow of a Viscous Liquid through a Channel.” — Ibid., 252–256.

4. S. I. Monstrov. “An Attempt to Determine Certain Mechanical Properties of Asphalt.” — Ibid., 492–502.

5. B. P. Weinberg. “Addendum to the Preceding Article.” — Ibid., 505–513.

6. A. A. Miloradov and I. A. Tolmachev. “Internal Friction of Asphalt.” — Ibid., 505–513.

7. B. P. Weinberg. “Study of Phenomena in Liquids under Homogeneous Shear (Preliminary Communication).” — Ibid., 514–515.

8. V. D. Kuznetsov. “Motion of a Plate in a Trough with a Viscous Liquid.” — Journal of the Russian Physico-Chemical Society, 45, 499–518, 1913.

9. V. D. Dudeckii. “Determination of the Coefficient of Internal Friction of Asphalt from Homogeneous Shear.” — Ibid., 519–533.

10. B. P. Weinberg. “Problems of Solid-State Physics and Progress on Certain of Its Questions in Recent Years.” — Ibid., Part II, 67–115.

* 29. B. P. Weinberg. “On the Internal Friction of Double Systems.” — Journal of the Russian Chemical Society, 45, 701–706, 1913.

1. Boris Weinberg. “Sur les méthodes d’étude des déformations lentes des solides.” — Prepared for print, 1913; to appear in the Transactions of the Institute of Applied Physics.

2. Boris Weinberg. “Quelques résultats de l’étude des substances douées de grandes valeurs du coefficient du frottement intérieur.” — Prepared for print, 1913; to appear in the Transactions of the Institute of Applied Physics.

3. Boris Weinberg. “Étude expérimentale du mouvement lamellaire des liquides visqueux.” — Prepared for print, 1913; to appear in the Transactions of the Institute of Applied Physics.

4. V. N. Alfimov and B. P. Weinberg. “On the Arrangement of Experiments on the Motion of Solid Bodies in Viscous Liquids.” — Journal of the Russian Physico-Chemical Society, 46, 66–74, 1914.

5. Yu. P. Reznikov. “The Coefficient of Internal Friction of Solutions of Dammar Resin in Turpentine as a Function of the Temperature and Concentration of the Solution.” — Ibid., 75–80.

6. B. P. Weinberg. “Probable Outlines of a Future Theory of the Deformation of a Solid Body (as a Preliminary Communication).” — Transactions of the Society of Naturalists and Physicians at Tomsk University for 1913, 175–187, 1915.

7. B. P. Weinberg. “Mechanical Properties of Clay.” — Prepared for print, 1915; abstract — Transactions of the 3rd Congress of the Russian Association of Physicists, 24, 1923; to appear in the Journal of the Russian Physico-Chemical Society.

8. B. P. Weinberg. “On the Effect of Admixtures of Neutral Solid Particles on the Internal Friction of Viscous Liquids.” — Prepared for print, 1915; abstract — Transactions of the 3rd Congress of the Russian Association of Physicists, 24–25, 1923; to appear in the Journal of the Russian Physico-Chemical Society.

9. B. P. Weinberg. “The Effect of a Tube on the Velocity of Fall within It of a Ball in a Viscous Liquid.” — Prepared for print, 1915; to appear in the Journal of the Russian Physico-Chemical Society.

10. B. P. Weinberg. “On the Resistance of a Viscous Liquid to the Motion within It of a Cylinder with Conical Ends.” — Prepared for print, 1915; to appear in the Journal of the Russian Physico-Chemical Society.

1. B. P. Weinberg. “The Established Regime of a Plastic Body under the Action of a Constant Force from the σ-Relaxation Point of View.”—Printed below.

2. B. P. Weinberg. “Steady Motion of a Viscous Fluid between Two Immovable Parallel and Intersecting Planes. Theoretical Part.”—Prepared for press, 1922; to appear in the Transactions of the Institute of Applied Physics.

3. B. P. Weinberg. “Two Theorems of the Hydrodynamics of Steady Motion.”—Prepared for press, 1922.

4. B. P. Weinberg. “Steady Flow of a Viscous Fluid between Two Coaxial Cylindrical Surfaces. Theoretical Part.”—Prepared for press, 1922; to appear in the Proceedings of the Russian Hydrodynamic Institute.

5. B. P. Weinberg. “On the Study of the Behavior of a Solid Body beyond the Elastic Limit. I. The Internal Friction of Iceland Spar and Rock Salt.”—Proceedings of the Tomsk Technological Institute, 43, No. 3, 39–46; 1923.

6. B. P. Weinberg. “Anamorphoses of Weight and Molecular Percentages and Their Application to the Internal Friction of Double Systems of Great Viscosity.”—Prepared for press, 1923; to appear in the Proceedings of the Institute of Physico-Chemical Analysis.

7. B. P. Weinberg. “The Internal Friction of Lead.”—Printed below.

8. B. P. Weinberg. “The Process of Raising and Lowering a Viscous Fluid in a Capillary.”—Printed below.

9. V. D. Kuznetsov. “The Influence of Temperature on the Internal Friction of Rock Salt.”—Proceedings of the Tomsk Technological Institute, 43, No. 3, 47–52, 1923.

10. V. D. Kuznetsov. “The Influence of Temperature on the Internal Friction of Paraffin.”—Prepared for press, 1923; to appear in the Transactions of the Institute of Applied Physics.

11. V. B. Weinberg and K. B. Weinberg. “Some Experimental and Theoretical Data on the Slipperiness of Ice.”—Prepared for press, 1923; to appear in the Transactions of the Institute of Applied Physics.

12. Yu. V. Troitsyn. “On the Applicability of Asphalt for the Joints of Water-Supply Pipes.”—Prepared for press, 1923; to appear in the Transactions of the Institute of Applied Physics.

2. Grounds for the Classification of Various Cases of Fluid Motion.

The most essential distinction among motions of a fluid is the difference between “steady,” or, otherwise, “parallel-stream,” “plastic,” “lamellar,” “irrotational” motion and “unsteady,” “vortical,” “turbulent” motion. Steady motion may be characterized as that in which the trajectories of individual elements of the fluid have no points of intersection either with one another or with themselves. The appearance of points of intersection may be regarded as the characteristic sign of the transition of steady motion into unsteady motion.

Another feature by which motions of a fluid are distinguished is based on the comparison of the velocities \(v\) of one and the same particle, or at one and the same point of space, at successive instants of time. Usually two main groups of motions are distinguished: established, or stationary, motions, in which the velocity at every given point of space remains constant throughout the entire time of motion, and non-established motions, in which the velocity at a given point changes during the motion. We shall somewhat refine this classification with respect to stationary motions and shall distinguish strictly stationary, quasi-stationary, and approximately stationary motions. By strictly stationary we shall mean those in which the velocity of each particle preserves its magnitude throughout the entire time of motion, and in which, all the more, the constancy of the velocity, both in magnitude and in direction, is preserved at every point of space. In quasi-stationary motion the velocity at every point of space remains constant, but—owing to the fact that, in place of a given particle which has changed its velocity in passing to other points of space, a new particle arrives, acquiring the velocity of the former one. Finally, in approximately stationary motions, both the velocities at a given point and the velocities of each particle change, but the former changes are much slower than the latter; or, more precisely, the change of velocity at a given point over a certain interval of time is very small in comparison with the differences between the velocity at the given point and the velocities at points distant from it by distances not exceeding the product of this velocity by the interval of time under consideration.

Of the four basic categories of motions that are obtained from the two principles of classification given above—namely, established steady, unestablished steady, established unsteady, and unestablished unsteady—in this series we shall deal exclusively with steady motions and, moreover, predominantly with established ones. I note that established steady motion of a fluid has been studied, both theoretically and experimentally, considerably less than established unsteady motion, despite the fact that the study of it, as the simpler phenomenon, ought to have preceded the study of the more complicated phenomenon of established unsteady motion. Likewise, unestablished steady motion has been studied almost not at all, whereas the more complicated unestablished unsteady motion has been the subject of a very large number of investigations.

As further bases of classification we shall take:

1) the number and form of the surfaces bounding the moving fluid,
2) the relative rest or relative motion and the character of the motion of these surfaces,

and also 3) the character of those constant forces that cause the motion of the fluid: line forces (surface tension on the free surface of the fluid), surface forces (external pressure on the free surface), or volume forces (for example, gravity).

For a detailed specification of the cases being classified, it is useful to keep in mind the constituent parts of a theoretical or experimental investigation of one or another particular case of motion. Such a task must consist in determining—depending on the prescribed system of external forces (pressure and surface tension acting on the free surface, and volume forces acting at all points of the fluid):

1) the distribution of velocities, rotations, and tangential and normal stresses at various points of the fluid,
2) the distribution of tangential and normal stresses at various points of the walls,
3) the form of the streamlines—the trajectories of individual particles of the fluid,
4) the quantity of fluid flowing through this or that part of a surface normal to the streamlines.

Since the task of the present article includes not only the clarification of the various cases of motion subject to study, but also the degree to which they have been studied, it is not superfluous to make one general observation (5, 196) connecting the hydrodynamics of steady motion of a viscous fluid with the theory of elasticity. The solution of any problem of the theory of elasticity concerning the deformations of an elastic solid when a definite system of forces is applied to the surfaces bounding it, provided that these forces do not cause volumetric compression at any point of the body, leads directly to the solution of the problem concerning a viscous fluid bounded by the same surfaces when the same system of forces is applied to them. For this it is sufficient, in the solution of the elasticity problem, to replace the angle of shear at a given point by the angular velocity of shearing (or, what is the same thing, by the velocity gradient at the given point), and the modulus of shear by the coefficient of internal friction. The reverse substitution may, from the solution of a problem concerning steady motion of a viscous fluid, give the solution of the corresponding problem in the theory of elasticity.

3. The Case of an Unbounded Fluid

Before proceeding to the classification of individual cases of established steady motion of a viscous fluid depending on the number and form of the surfaces bounding it, I shall point out that, when the fluid is unbounded and fills all space, such a motion is impossible.

Of great theoretical and especially practical interest would be the study of that unsteady complicated motion which must be presented by any—produced by some external action and then ceasing—single vortex, which, owing to the viscosity of the fluid, must gradually slow its rotation, and further by a system of vortices distributed in space. The practical significance of this question is evident from the fact that each of those vortices which, according to the theories of Prandtl, Zhukovsky, Kármán, and others,¹ are formed behind the surface of a solid body moving sufficiently rapidly in a fluid of sufficiently small viscosity, must represent a case of steady motion, since, if the distribution of velocities in it corresponded to the conditions for transition from a steady state to an unsteady one, or exceeded these conditions, then this vortex would inevitably have to break up into smaller ones, in each of which the motion would again be steady, or else would cause its fragmentation into still smaller ones, and so on.

The last remark shows that, for a complete theory of vortical motion and, in particular, of the resistance offered by a fluid in the case of such motion, it is inevitably necessary to consider the question indicated above concerning the absorption, owing to internal friction, of the energy of individual vortices and of systems of correctly arranged individual vortices, and also to elucidate those conditions concerning the distribution of velocities inside a viscous fluid which cause the appearance of a new vortex or the fragmentation of an already existing one.

4. Types and forms of surfaces bounding a fluid.

A fluid may border on solid, liquid, and gaseous bodies. A characteristic property of the surface of a solid body bounding a fluid is the possibility of the presence on it of tangential stresses—constant in steady motion and variable in unsteady motion.

Such a surface—depending on whether the system of forces acting on it will keep it at rest or impart to it a motion with a definite velocity—we shall call an “immovable wall” or a “moving wall,” in distinction from a “free surface,” a surface in contact with some gas, or with vapors of the same fluid, or with some other fluid. A free surface is characterized by the impossibility of the existence on it of tangential stresses, provided only that the fluid or gas with which the fluid under study is in contact is not in steady motion; in the latter case the surface of contact may, to a known degree, play the role of the surface of a wall.

Steady motion is possible only if not the whole surface of the mass of fluid under consideration is free, but part of it consists of walls, and in the presence of one or another system of forces acting on the fluid, the character of the motion of the fluid will be determined precisely by the number, form, and motion of these walls.

For simplicity we shall restrict ourselves in what follows to the case of completely wetted and absolutely solid, i.e. unchanging, walls.

Finally, a third kind of part of the surface bounding a given mass of fluid is possible—a kind which may be called a “free wall.” This will be such a part of a wall which, during the motion of the fluid, moves together with the layer of fluid adjoining it, without exerting any tangential forces upon it. In particular, such a part of a free surface which is displaced without changing its form and size may be regarded as a free wall.

In some cases the solutions of problems on the motion of a fluid between two fixed walls, between a fixed and a moving wall, and between a fixed and a free wall are connected with one another, as indicated in the third paper of this series (42).

As regards the form of the surfaces, we shall confine ourselves to the simplest and shall consider successively:

a) planes,

b) cylindrical surfaces—in particular, surfaces of right circular cylinders,

c) conical surfaces—in particular, surfaces of right circular cones,

d) spherical surfaces,

and various combinations of these.

We shall begin the discussion with planes; moreover, since the smallest number of planes that can bound a fluid (if, in the case of sufficiently large planes, the lateral surface may be neglected in comparison with the area of the planes) is equal to two, the simplest cases will be the motion of a fluid between two parallel planes and the motion between two intersecting planes, or, in other words, in a wedge.

5. Two parallel fixed planes.

If the walls are parallel and fixed, then steady motion can take place only in one of the directions parallel to these planes. The case of steady slow motion between fixed planes was studied both theoretically and experimentally by Hele-Shaw¹; and one of the cases of unsteady motion—the process of the rise or fall of a viscous fluid in a “capillary plane-parallel space”—was treated theoretically by me (47). Among cases in which a motion close to this occurs, one may mention brickmaking presses, the seepage of lubricant near the slide valve of a steam engine when it is at rest, the squeezing out of water from under the runner of a sledge (50), etc.

6. Two parallel planes in relative motion.

The motion of one plane relative to another, parallel to it, may be translational, rotational, or helical. Obviously, it is sufficient to analyze only the first two cases. The first of these, in turn, is divided into the motion of one plane parallel to both planes and motion perpendicular to them.

The motion of one plane parallel to another may occur in determining the coefficient of internal friction of solids from the shear of a parallelepiped² or of two contacting parallelepipeds (27), and may also clarify the role of lubrication and the magnitude of friction in the motion of a slide valve and various sliders.

The problem of the motion of a fluid in the case of the motion of one plane relative to the other in a direction perpendicular to both of them can be solved only for bounded dimensions—at least in one direction—of the planes; moreover, of course, the motion of the fluid between them will be only approximately stationary. The simplest case will be the bounding of mutually approaching or mutually receding planes by circumferences,

i.e. motion toward each other or away from each other of two circles or rings. The problem is thus reduced to finding the distribution of velocities, rotations, and stresses (tangential and normal), the forms of the streamlines, and the quantity of liquid flowing out of a shortening (or entering a lengthening) viscous cylinder, truncated cone, or torus with a rectangular or trapezoidal cross-section.

The case of a compressed disk may be of importance for determining the coefficient of internal friction of its material¹, although the phenomenon is also complicated by the presence of all-round compressions. Consideration of this case may provide an approach to phenomena under a flat heel when it is at rest relative to a heel-piece; to the squeezing out of liquid when more or less flat sheets, wetted or impregnated with liquid, are pressed; to the natural squeezing of stacks of layers or flat sheets separated by layers of liquid or even simply of air; to the diminution of the air layer between any flat horizontal support and an object with a flat base standing on it. A modification of the theory of compression of a disk by replacing the external pressure with body forces may give a theory of the spreading of a heavy viscous disk—in particular, a theory of the motion of ice in many polar glaciers, which are lateral branches of a spreading shield-like continuous ice cover (for example, in Greenland).

The rotational motion of one plane relative to another can be considered in exactly the same way only when their dimensions are finite; moreover, for the process to be stationary, the planes of rotation must be bounded by circumferences, and the axis of rotation—since the planes must remain parallel—must be perpendicular to them and pass through the centers of these circumferences. Thus the problem is reduced to determining the distribution of velocities, stresses, etc., in the twisting of a cylinder, a truncated cone, or a torus with a rectangular or trapezoidal cross-section.

Such “screw-like” twisting has been used by many authors to determine the coefficient of internal friction from the twisting of cylinders², cylindrical tubes (1, 2, 15, 46), and a torus³; it was proposed and tried for studying phenomena in liquids under nearly homogeneous shear (25), and may be useful for approaching phenomena under a flat heel and for the theory of disk brakes. But, on the other hand, the very fact of the difference of the angular velocities of shear in different coaxial layers of the twisted disk of viscous liquid can be used to clarify the conditions of transition from steady motion to unsteady motion, so far studied in detail only for “spiral-like twisting”—see § 11 (Couette’s work)—and for flow through capillaries (Reynolds’ “critical velocity”).

I note that, if in the study of steady motion between two parallel planes rotating relative to one another through very small angles of rotation one confines oneself to this case, then the prisms and rings may be bounded by arbitrary contours, for which the corresponding problem of the theory of elasticity has been solved—the Saint-Venant problem of the torsion of a prism. In particular, torsion of prisms was used by me and by V. D. Kuznetsov to determine the coefficient of internal friction of rock salt and paraffin (44, 48, and 49).

7. Two fixed intersecting planes.

Steady motion of a fluid between such planes is possible only if the streamlines are:

a) parallel to the line of intersection of these planes, or, in other words, to the edge of the wedge;

b) perpendicular to this edge.

For the case of a wedge angle equal to \(\frac{\pi}{2}\), an exact solution of the first problem can be obtained from the solution, given by Graetz\(^{1}\), of the problem of the flow of a viscous fluid in a tube of square cross-section, with the aid of the relation\(^{2}\) between the solution of the problem of the flow of a viscous fluid in a tube and the solution of the problem of the flow of a viscous fluid in a channel obtained by cutting the tube into two equal parts by any plane of symmetry parallel to the generators of the tube surface.

For the case of a small wedge angle, the approximate solution obtained by me (41) may be useful for the experimental study of the conditions of transition of tranquil motion into turbulent motion. The approximate solution set forth there as well (also for small wedge angles) of the second problem—the motion of a fluid in directions perpendicular to the edge of the wedge—can give the magnitude of the resistance experienced by a wedge in a stream of viscous fluid, and can also be used for the experimental elucidation of the already mentioned question of the critical conditions of fluid motion.

8. Two intersecting planes in relative motion.

Translational motion of the walls relative to one another may occur:

a) parallel to the edge of the wedge,

b) perpendicular to the bisecting plane of the wedge,

c) perpendicular to the edge of the wedge and parallel to the plane of one of the walls.

If the first case apparently has only theoretical interest, then the second may have some relation to pressing and to the process of sheets adhering to one another in a viscous fluid and even in air. The third case may find application to phenomena in the cleaning of liquid layers from plane surfaces and in the motion both of the lower and especially of the front parts of the runners of sledges or skates over the surface of ice, when between the runners and the ice, owing partly to the lowering of the melting temperature of ice with increasing pressure (9), and still more, apparently, owing to the work expended in overcoming frictional forces (50), a “natural” liquid lubricant is obtained from a layer of water which freezes again after the runners have passed, and also in the motion of the runners when there is incomplete—caused by the very presence of the lubricant—parallelism of their guiding planes\(^{3}\).

Rotational motion can, again, be considered only for walls whose contours are circumferences, i.e. of the form of a circle or a ring, and for the case of rotation about an axis perpendicular to the bisecting plane of the wedge and passing through the centers of these circumferences—a case which, for very small wedge angles, may have significance for the theory of plane—

\(^{1}\) Graetz, Zeits. f. Math. u. Phys., 25, 316—336, 375—404, 1880.

\(^{2}\) I found this relation in 1906 (5, 338, and 6, 363), but recently discovered that it had already been indicated in 1845 by Stokes, Trans. Cambr. Phil. Soc., 8, 286—319, 409—414, 1848; Math. & Phys. Pap., 1, 105, 1880.

\(^{3}\) Reynolds, Phil. Trans., 177, 157—234, 1886; Sommerfeld, Zeits. f. Math. u. Phys., 50, 97—155, 1903; Michell, ibid., 52, 123—137, 1905.

...of such a heel-piece when the axis of rotation is deflected from perpendicularity to the plane of the bearing. Such a restriction is superfluous when considering rotation about the edge of a wedge—a phenomenon which may likewise have significance for the theory of pressing, but which can yield only an approximately stationary motion of the fluid.

As for the third case of rotation—about an axis perpendicular to the axes of rotation in the two preceding cases, i.e. about an axis lying in the bisector plane and perpendicular to the axis of the wedge—then, however we might bound the surface of the walls, under this rotation they, the axes of which would also rotate in the bisector plane, must inevitably come into contact with one another, and then penetrate one another with their parts. Therefore this third case, since it concerns established motion, is not subject to consideration.

9. Cylindrical fixed surface.

Passing to cylindrical surfaces, let us begin with the simplest case: a single cylindrical surface enclosing the fluid within itself. The case of a solid cylindrical body situated inside an unbounded fluid is, in essence, the case of two surfaces, of which the second must be regarded as lying at infinity and represented, as needed, either in the form of a straight circular cylinder with an axis parallel to the generatrices of the given cylinder and with an infinitely large radius, or in the form of an infinitely distant plane parallel to, or forming an angle with, the generatrices. Therefore this case belongs either to § 10 or to § 13.

In the presence of a single cylindrical—in the particular case, prismatic—wall, only one stationary motion is possible: in a direction parallel to its generatrices.

This problem is soluble for all forms of cross-section of the cylindrical surface whose equation is of the form¹

\[ \varphi(x,y)=a(x^2+y^2)=C, \tag{1} \]

where the function \(\varphi(x,y)\) satisfies the equation

\[ \frac{d^4\varphi}{dx^4} +2\frac{d^4\varphi}{dx^2dy^2} +\frac{d^4\varphi}{dy^4}=0, \tag{2} \]

in particular, for all algebraic curves whose equations are obtained from the general form

\[ a_0+a_1x+a_2y+a_3xy+a_4(y^2-x^2)+a_5(y^3-3yx^2)+a_6(3y^2x-x^3)+\cdots -a(x^2+y^2)=C, \tag{3} \]

or may be expressed, in general, by the formulae

\[ \sum e^{mx}\left(A_m\cos my+A'_m\sin my\right)-a(x^2+y^2)=C, \tag{4} \]

\[ \sum e^{my}\left(B_m\cos mx+B'_m\sin mx\right)-a(x^2+y^2)=C, \tag{5} \]

as well as (5,340) for all sections having an axis of symmetry whose plane is perpendicular to the direction of motion of the fluid; the escape from the surface has the same velocity when the fluid moves in a pipe of any of the sections (1), (3), (4), or (5). The same applies to the Son-Retien problem.

In order to make clear the practical significance of the case of fluid motion under consideration, it is enough to recall the classical formula of Poiseuille and its varied applications.²

10. Two cylindrical stationary surfaces.

One surface must enclose the other; the generators of both surfaces must be parallel to one another, and the trajectories of the fluid particles must be parallel to the common direction of the generators.

Since I have not encountered a solution of this problem¹, I considered it useful to find its solution at least for the simplest case—two coaxial straight circular cylinders (43). This case may be of significance for clarifying the seepage of lubricant between a piston and a cylinder when they are at relative rest or in very slow motion (as, for example, in a hydraulic press), certain details of the extrusion of viscous and solid bodies into the gap between a piston and a cylinder (for example, the manufacture of macaroni and tubes for oil paints), and also in the motion of liquids or gases between coaxial tubes, as, for example, in Linde’s machine for obtaining liquid air. The solution found by me was applied by I. V. Grdina—on the basis of data on the internal friction of asphalt and the influence of temperature upon it (19, 22, 23, 24, and 27)—to the clarification of the question of the applicability of asphalt in the joints of water pipes.

In view of the possibility of incomplete coaxiality in many of the cases mentioned, it would be of interest to consider the motion of liquids between two straight circular cylinders with parallel, but noncoincident, axes.

One may think that a solution of the general problem of the motion of a viscous liquid between two cylindrical surfaces will prove possible, if these are two “equipotential” surfaces for the flow of a liquid inside such a cylindrical surface, for which the problem has a solution.

11. Two cylindrical surfaces in relative motion.

One surface must, as in the preceding case, enclose the other, and their generators must be parallel.

For simplicity we shall restrict ourselves to straight circular cylinders.

The case of coaxial cylinders (43) admits only two possibilities for their relative motion if the motion of the liquid is to be steady:

a) translational motion along the common axis;

b) rotational motion about the common axis.

The first case, in turn, splits into two:

α) the moving cylindrical surface is a free wall (for example, the flow of a viscous liquid down the inner wall of a cylindrical tube or along the outer surface of a cylindrical rod: the simplest water “pipes” in trees, the pouring of liquids into vessels with a narrow neck along a glass rod or a metal wire);

β) the moving wall exerts a tangential stress on the liquid (Segel’s method² for determining the coefficient of internal friction of solids; the lateral resistance experienced by a cylinder in moving inside a cylindrical tube—20 and 39—, in particular that experienced by a piston moving inside a cylinder; here also may be included the phenomenon occurring when cables are coated with a layer of insulation by drawing or pressing.

If the radius of the outer cylinder is infinite, or sufficiently large, then case β reduces to the motion of a cylinder along its axis inside a practically unbounded fluid (Hatschek’s method¹) for determining the relative viscosity of supercooled liquids; the correction is 20 and 38 for the resistance of a wire when cylinders are drawn through a viscous fluid).

The rotation of one cylinder relative to another, coaxial with it, causes a “spiral-like twisting” of the fluid between them, and the study of it may serve for determining the coefficient of internal friction of liquids²) and solids (17 and 19), and for clarifying the frictional forces between an axle and a bushing in the absence of forces acting on the axle perpendicular to its geometrical axis.

If the straight circular cylinders bounding the fluid have parallel axes, then all three mutually perpendicular translational motions of one of them relative to the other are possible, namely:

α) along the axis;

β) in the plane passing through both axes, perpendicular to the axis;

γ) perpendicular to the axis and to the plane passing through both axes.

Consideration of the first question may elucidate the phenomena of (strictly stationary) friction between a piston and a cylinder in the presence of a force perpendicular to the piston axis; consideration of the second question, the phenomena of (approximately stationary) squeezing-out of a layer of lubricant between a cylinder and an immovable, in the direction of the axis, piston in the same case, or between a shaft and an immovable bushing in the analogous case (this motion may explain the increase of frictional force at start-up after long standing); finally, consideration of the third question, the phenomena (quasi-stationary) in a layer of liquid lubricant at the front part of runners during motion over a flat ice surface (in this case the radius of the outer cylinder must be taken infinitely large). The second case—with initial coaxiality of the cylinders—is treated by Stokes³).

In the case of noncoaxial cylinders, stationary (and also quasi-stationary) motion under rotation of one of them is possible only when rotating about one of the axes parallel to the cylinder axes—in particular, about this very axis. Consideration of the latter case gives the solution of a question very important for the theory of machines: that of friction between a bushing and a shaft rotating inside it in the presence of a force acting on the shaft perpendicular to its axis.

The combination of such rotation with translational motion perpendicular to the axis and to the plane containing both axes gives rolling of one cylinder over a layer of lubricant on the surface of another—rolling with or without slipping, depending on the relation between the speed of translational motion and the angular speed of rotational motions. Among practical applications relating to the case of an infinitely large radius of one of the cylinders, I shall indicate the rolling-out of paints by a roller on a lithographic stone, of dough with a rolling pin, rollers for compacting and ramming pavements, crane trolleys, and so forth.

I note that the solution of the problem of the distribution of velocities and stresses in the case of rolling of one cylindrical surface over a layer of lubricant on another is closely connected with the problem of the motion of a viscous fluid between the same immovable surfaces under the influence of external pressure (42).

¹ Hatschek, Zeits. phys. Chem., 28, 17–39, 1899.
² Couette, Ann. chim. phys., 21, 433–551, 1890; Shvedov, Journ. de phys., 8, 341–359, 1889, 9, 34–46, 1890, and others; the theory was given already by Stokes (Trans. Cambr. Phil. Soc., 8; v. 1 in Math. Phys. Pap., 1, 102–103).
³ Stokes, Trans. Cambr. Phil. Soc., 8, 105–154, 1843; Math. Phys. Pap., 1, 35.
⁴ Reynolds, l. c.; Sommerfeld, l. c.

12. Three cylindrical surfaces.

The case of two cylindrical surfaces, one lying outside the other within an unbounded fluid, must in essence be regarded as the case of three cylindrical surfaces, one of which has an infinite radius. Among the various questions belonging here, the case deserving study is that of the translational motion of cylinders in a direction perpendicular to their axes, or their rotational motions about their axes, and moreover with velocities inversely proportional to their radii. The latter case may clarify certain details of the phenomena of rolling and rolling-out (squeezing of the rim, production of sheet glass, sheet metals, etc.).

More complicated cases may lead to the theory of lubrication in cylindrical toothed wheels, rotary wheels, roller cylindrical bearings, etc.

13. A cylindrical surface and a plane.

Since a plane may be regarded as a particular case of a cylindrical surface, at first sight it may seem that the question of the motion of a fluid bounded by a cylindrical surface and a plane has already been solved above in § 10 and 11. But there the discussion concerned two closed cylindrical surfaces; therefore, if the fluid is bounded by one cylindrical surface and by a plane situated at a finite distance from it, then, in order to represent the phenomenon as a limiting case, it is necessary to imagine, as the closing surface, yet another plane parallel to the first and situated at an infinitely great distance both from it and from the cylindrical surface; and to pass to the case considered in the preceding §§.

Meanwhile, another limiting case is possible: the case of motion of a fluid between a free cylindrical surface and the non-closed plane closing it, namely the case of motion of fluid along a channel whose bottom is this non-closed cylindrical surface, while the plane closing it will be the free surface of the fluid moving along this channel.

This problem, as already indicated in § 7, must be considered solved for all channel profiles, since they are obtained by cutting along the axis of symmetry the profiles of a tube, for which the problem of the flow of a viscous fluid through it has been solved.

The application of this connection has given the theory of the motion of ice in glaciers (5, 6, 14), and may also serve for determining the coefficient of internal friction of solid bodies (12, 13 and 21).

The converse case is also possible, when the immovable wall will be a segment of a plane bounded by two parallel straight lines, and the free surface will have the form of an unclosed cylindrical surface adjoining it, namely the case of a jet of fluid flowing down an inclined plane. In exactly the same way one may consider the flow of a jet of fluid along a segment of an unclosed cylindrical surface.

14. One conical surface.

In a steady state, only such a motion of a viscous fluid can occur in which the trajectories of the individual particles are straight lines passing, in their continuation, through their own pro—

¹ The case of a cylinder in an unbounded fluid was analyzed by Stokes (Trans. Camb. Phil. Soc., 9, 8—184, 1850; Math. Phys. Pap., 2, 62—67; see also E. Édouardes, Quant. Journ. of Math., 26, 70—78, 1893). The case of a cylinder between two parallel planes was considered by Lamb—see Hele Shaw, Trans. Inst. Nav. Archit., 40, 21—38, 1898.

motions through the apex of the given conical surface, while the surface itself is the lateral surface of a truncated cone.

Solving the problem even for the case of a right circular cone¹ could be of significance for introducing a correction for the conicity of a capillary in determining the coefficient of internal friction by Poiseuille’s method; for considering the motion of blood in narrowing blood vessels; for studying the process of the rise of liquids in conical capillaries and the motion of a drop of liquid—wetting or non-wetting—in a conical tube; for the theory of the outflow of a viscous liquid through a funnel, etc. From the values of the stresses on the walls of a narrowing cone during the flow of a viscous liquid along it, certain conclusions can be drawn concerning the rate of flow of a viscous liquid inside the cone formed by it itself, the process of formation of drops of very viscous liquids, and the process of drawing out glass, quartz, and similar threads.

15. Two conical immovable surfaces.

Let us confine ourselves to two right circular cones, one of which lies inside the other.

If their axes coincide, then, in the steady motion, the trajectories of all particles must intersect the circle along which these cones intersect; if the axes are not parallel, then they must intersect the ellipse along which these surfaces intersect. This problem may have some bearing on the question of conical regulators.

16. Two conical surfaces in relative motion.

A translational relative motion, when the cones are coaxial (the axes, as before, being restricted to two right circular cones), can be steady—and, moreover, not strictly steady but approximately—only if it is directed along their common axis, as will be the case, for example, in conical valves. In the absence of coaxiality of the cones, motions in other directions are possible—for example, approximately perpendicular to the axes of one of the cones. Thus, for example, if the axes of the cones are parallel, approximately steady motions are possible: α) of one of the cones translationally, β) perpendicular to a plane replacing the axes, and γ) in the plane containing both axes and perpendicular to their directions.

As for rotational motion, when the cones are coaxial it is possible only about their common axis (the cited problems of raising), while in the absence of coaxiality more varied cases are possible. Of greatest practical interest would be consideration of the case of rolling—without sliding or with sliding—of one conical surface over a layer of lubricant between it and another conical surface, both for two right circular cones (bearings with conical rollers) and for other conical surfaces used, for example, in toothed conical engagements.

17. A conical surface and a plane.

This case may be regarded as a special case of two conical surfaces, one of which is a right circular cone with an aperture angle equal to \(\pi\).

The consideration of the translational motion of a cone toward an infinitely distant plane would give a solution of the question of the resistance of a cone when it moves in an unbounded fluid.

18. Conical and cylindrical surfaces.

This case may be regarded as a particular case of two conical surfaces, in one of which the vertex lies at infinity.

The consideration of the translational motion of a right circular cone inside a right circular cylinder coaxial with it may give the value of the resistance encountered by the nose part of a body with a conical end inside a cylinder filled with a viscous fluid (as, for example, in the experiments of works 20 and 39).

19. Three conical surfaces.

This case would elucidate the motion of a fluid surrounding two conical surfaces lying outside one another, since here one must imagine a third conical surface (in particular, a plane or cylindrical one), which would be infinitely distant from the first two and would play the role of an immobile wall.

20. Two spherical surfaces in relative motion.

Passing now to completely closed surfaces, let us note first of all that, if a viscous fluid is bounded by one closed surface, then no stationary motion of it is possible (cf. § 39). Precisely the same is true in the case of two closed surfaces which bound a viscous fluid and one of which contains, of course, the other within itself; once the motion of this fluid—strictly or approximately—is established, it is possible only with relative displacement of these surfaces.

Restricting myself in the present article to spherical surfaces, I shall begin with the simplest case of two spherical surfaces in relative motion.

If these surfaces are concentric (example: the bearings of a Cardan joint), then the only possible motion is the rotation of one of the spherical surfaces about an axis passing through their common center². If the radius of the outer sphere is infinitely large, this will be the case of the rotation of a sphere inside an unbounded fluid².

If the spherical surfaces are not concentric, then two translational motions are possible:

a) along the line joining the centers of the spheres—approximately stationary,

b) perpendicular to this line—quasi-stationary.

The first case—with initial concentricity of the spheres—was treated by Stokes³, both for finite and for infinite radius of the outer sphere, and in the second variant has a number of varied applications. If the radius of the outer sphere is sufficiently large in comparison with the radius of the inner sphere and at the same time in comparison with its distance from the surface of this outer sphere, then the first case gives us the motion of a sphere toward a plane, also treated already by Stokes, and allowing, for example, a correction to be introduced for the influence of the bottom

in the fall of a sphere in a viscous fluid inside a sufficiently wide tube¹, and the second—the motion of a sphere over a lubricating layer on a plane, which has been analyzed by Lorentz², and may be relevant, for example, to the question of crushing paints with a pestle having a hemispherical head.

As for rotational motions, then, considering nonconcentric spherical surfaces, there are two cases of stationary motion of the fluid caused by the rotation of the inner sphere:

α) about a line passing through the centers of both spheres,

β) about a line perpendicular to the line joining the centers.

The motion caused by the rotation of the inner sphere will be quasi-stationary:

γ) about some line joining the center of the outer sphere with some point of the inner sphere, or with a point fixed to this sphere but not coinciding with its center,

δ) about some line perpendicular to the line joining the centers of both spheres, but not passing through the center of the inner sphere.

I shall also point to a combination of translations and rotations which would give the rolling of one sphere inside another—a case that may have a connection with the theory of ball bearings.

21. Three spherical surfaces.

If two spherical surfaces lie one outside the other, then, in order to consider the motion of the fluid surrounding them under their relative motion, one must assume that this fluid is enclosed in a sphere of infinitely large radius, so that what is in essence the case of three spheres will be present.

The relative translational motions to be considered are the following:

a) along the line passing through the centers of both spheres³,

b) along a line perpendicular to the line passing through the centers.

The rotational motions of one of the spheres relative to the other to be considered are the following:

a) about a line passing through the centers,

b) about a line passing through the center of the given sphere and perpendicular to the line passing through the centers,

c) about a line passing through the center of the second sphere but missing the center of the given sphere,

d) about a line passing neither through the center of the given sphere nor through the center of the second sphere.

22. Spherical and cylindrical fixed surfaces.

We shall restrict ourselves to a right circular cylinder and shall assume that its radius is greater than the radius of the sphere, whose center may lie:

a) on the axis of the cylinder,

b) off the axis of the cylinder.

The motion in both cases will be stationary—and moreover quasi-stationary—if the pressure gradient and the body force are directed parallel to the axis of the cylinder.

23. Spherical and cylindrical surfaces in relative motion.

If the center of the sphere lies on the axis of the cylinder, then the following three cases of stationary motion are possible:

a) the sphere moves translationally along the axis of the cylinder—a quasi-stationary motion, investigated experimentally and theoretically by Ladenburg for small values of the ratio of the radius of the sphere to the radius of the cylinder¹, and experimentally, for very varied values of this ratio, in the physical laboratory of the Tomsk Technological Institute (33 and 38); this case may be of importance in considering phenomena in Cardan couplings;

b) the sphere rotates about the axis of the cylinder;

c) the sphere rotates about an axis passing through its center and perpendicular to the axis of the cylinder.

The last two cases give a strictly stationary motion of the fluid.

In exactly the same way one may indicate the degree of stationarity of three mutually perpendicular translational motions (along a line parallel to the axis of the cylinder, along a line passing through the center of the sphere and perpendicular to the axis of the cylinder, and along a line passing through the center of the sphere and perpendicular to the two preceding ones) and of three rotational motions about these same three axes, or about axes parallel to them.

The consideration of these nine cases would be of special interest under the condition that the forces causing the motion of the ball are the same as those acting on all the rest of the fluid, i.e. that, apart from pressure, only body forces act on the free surface of the latter, the magnitude of which per unit volume of the moving sphere is the same as per unit volume of the fluid. In the case where the body forces are forces of gravity, such a restriction reduces to the consideration of the motion of a sphere which has the same specific weight as the fluid and moves only under the influence of the forces of internal friction in the layers adjacent to its surface.

Under the action of such forces only translational motions of such a “freely carried” ball are possible, and then only in a direction parallel to the axis of the cylinder. If, in this case, the center of the ball lies on the axis of the cylinder, then such translation will not be accompanied by any rotation; but if the ball is eccentric with its center, then the translation of the ball will be accompanied by its rotation about an instantaneous axis perpendicular to the axis of the cylinder and to the perpendicular from the center of the sphere to this axis—similar to the way a river rolls pebbles near the bottom.

24. Spherical and conical fixed surfaces.

We shall confine ourselves to a right circular (of course, truncated) cone and to the case when the surface of the sphere lies inside the cone.

Far from the sphere, the trajectories of the fluid particles must be straight lines passing through the vertex of the cone. Two cases are to be considered (both of quasi-stationary motion):

a) the center of the sphere lies on the axis of the cone (possible application—certain types of valves),

b) the center of the sphere lies outside the axis of the cone.

25. Spherical and conical surfaces in relative motion.

Here the same cases are possible as in the relative motion of a sphere and a cylinder (§23), with the difference that, instead of translation along a line parallel to the axis, or rotation about such a line, one must consider translation along a line, or rotation about a line, passing through the vertex of the cone. The types of motion in the nine cases analogous to those indicated in §23 will be the same—except for translation along the axis of the cone, or along a line passing through its vertex, which will be not quasi-, but approximately stationary.

26. One or several spheres and two planes.

Passing to a number, greater than two, of surfaces bounding a liquid and not situated at infinity, we shall not consider separately all possible kinds of relative motion of these surfaces, but shall chiefly contrast the cases of relative rest and relative motion.

If a sphere and two intersecting (in the particular case—parallel) planes are given, then the following categories of cases are possible:

a) the sphere and the planes are motionless—the motion of the liquid in the wedge (in the particular case between parallel planes) in which the motionless sphere is situated;

b) one plane and the sphere are motionless, the other plane is in motion;

c) the planes are motionless, the sphere moves;

d) one plane is motionless, the sphere and the other plane are in motion.

In the last two categories special attention should be paid to the cases when forces act on the sphere the same as on the rest of the liquid—a freely falling sphere in a liquid flowing between two intersecting or parallel planes, and to the motion of the liquid with a freely suspended sphere at rest relative to it when one plane moves relative to the other.

Of special importance is the approximate solution of such questions concerning mutual entrainment of fluids—for example, two identical regular rows of identical spheres, the centers of which are located on one straight line at equal distances from one another; and of liquids consisting of identical spheres, uniformly filling equal volumes and initially equidistant from one another; and of a volume filled with an equal number of identical spheres chaotically distributed through it, constituting the solution of the problem of the motion of a liquid through such a row, through a plane layer of spheres situated between two intersecting or parallel planes.

If such questions received even an approximate solution, this could elucidate the theoretical side of the influence of an admixture of neutral solid bodies on the coefficient of internal friction of viscous liquids (37) and the dependence of this coefficient on the ratio of the components of solutions of two viscous bodies of very different viscosity (22, 24, 26, 34, 45). The study of such questions experimentally and especially in theory could, in turn, serve as an approach to the theory of the action of forces exceeding the elastic limit, on solid bodies whose crystallites possess a higher limit of elasticity than the interlayer between them (35, 44).

If the bounding planes are infinitely distant, then one obtains a theory¹ of motion, in an unbounded viscous fluid, of a layer of equidistant and equal spheres, and of a volume uniformly or chaotically filled with identical spheres, under the action of equal and identically directed forces (perpendicular to the infinitely distant bounding planes). This theory may be very useful for elucidating the phenomena in the settling of a dense fog, in elutriation, etc. The study of the motion of a fluid through an unbounded layer or volume, uniformly or chaotically filled with identical spheres, would provide an approach to the theory of filters—in the form of plates or three-dimensional bodies—to the theory of the motion of ground waters through sandy soil, etc.

27. Several spheres and a conical (in the special case—cylindrical) surface.

Of the various possibilities here, the most important are those that were indicated in the preceding section, namely: a row of equidistant spheres arranged along one line and equal spheres; a layer of equidistant and equal spheres; and a volume uniformly or chaotically filled with equal spheres.

The following principal categories of cases are possible:

a) the conical (cylindrical) surface and the spheres are at rest;

b) the conical, cylindrical surface and the spheres are in relative motion.

The greatest theoretical and practical significance would attach to consideration of the case of a flow, through a cylindrical tube, of a fluid freely carrying along a row of equidistant and identical spheres arranged along the axis of the tube, or else an aggregate of identical spheres, very small in comparison with the radius of the tube, distributed uniformly or chaotically throughout the fluid. From such consideration one could obtain substantial data for explaining phenomena in the flow through pipes of pulp, emulsions, suspensions, liquid clays, unset cement, etc.

28. Conclusion.

From the fact that a negligible fraction of the cases indicated above, relating to the simplest geometrical forms of surfaces bounding the fluid, and therefore by no means hopeless under the present development of mathematics,² has been solved, it is evident how much remains to be done in the hydrodynamics of steady motion of a viscous fluid. At the same time, the study of these questions may, as was indicated in many places above, have great practical significance, in addition to the fact that the path toward a rational formulation of the theory of unsteady motion lies through the study of the corresponding cases of steady motion.³ I emphasize once again the importance of clarifying the conditions of transition from steady motion to unsteady motion, since only then will it be possible to apply the conclusions of the hydrodynamics of steady motion unerringly to actual cases, or else to know how accurate such an application is. A characteristic example in this respect may be considered the application of the theory of steady motion of a viscous fluid to the resistance of the medium to the motion of a pendulum body, since in the actual motion of pendulums (the same also applies to the oscillation of balances) there very probably occurs an unsteady motion of the air around the pendulum body.

Feb. 1922 / Oct. 1923.

II. The Process of Rise and Descent of a Viscous Liquid in a Capillary.

1. Choice of the topic.

After the experiment I had carried out¹ with Plateau’s cubes of wax, placed in a solution of saltpeter of the same density as themselves, had after ten years given a positive result (the elongated prisms approximated the form of a sphere), clearly revealing the presence in “solid” wax of forces of surface tension, and after A. V. Ignat’ev² had directly measured for it the capillary constant by the method of flat drops, it became desirable to trace the dependence of the coefficient of surface tension on the composition of solutions of one substance in another, the two bodies differing very greatly in magnitude of the coefficient of internal friction (wax and kerosene, dammar resin or rosin, and turpentine, etc.).

As first steps toward developing a methodology for determining the coefficient of surface tension of liquids of the most diverse viscosity, in the autumn of 1913 I proposed to the students S. M. Voinov and A. M. Mochalin, as a diploma project, to develop for liquids of fairly considerable, but not yet excessively great, viscosity \(\left(\eta\right.\) of the order of \(10^0—10^2 \dfrac{\text{g}}{\text{cm. sec.}}\)\()\) a method of rise in capillary tubes by observing the very process of rise or descent of the liquid in them. The height of the liquid must, obviously, tend asymptotically to a certain limiting value, corresponding to the equilibrium of the forces of surface tension and hydrostatic pressure. The formula I derived at that time—see § 3—for the height of rise \(h\) at the moment \(t\) after the beginning of the experiment showed the possibility of obtaining from a series of readings of \(h\) not only the value of the coefficient of surface tension \(\alpha\), but also the coefficient of internal friction \(\eta\), and the very careful observations of Voinov and Mochalin showed the applicability of this formula. Lack of time did not then allow me to develop a convenient method for finding from a series of values of \(h\) the final height \(H\)—the maximum in the case of rise and the minimum in the case of descent—and I turned to the final processing of these observations only in connection with the resumption of work on the hydrodynamics of steady motion.

In reviewing in Moscow in March 1923 the literature for the years 1916–1921, when Tomsk had been completely cut off from the rest of the scientific world, I found the work of Lucas (“Ueber Zeitgesetz des kapillaren Aufstieges von Flüssigkeiten,” Kolloid. Zeits. 23, 15—22, 1918), who arrived at a formula of the same type as I had and who had made a series of observations on liquids of comparatively small viscosity. Since this author’s method of processing is less complete than the one I use, I shall set out here both the observations of Voinov and Mochalin and the method of processing them that I applied, and shall give the results obtained from them.

2. Experimental arrangement.

The liquid under investigation was poured into a small porcelain cup almost to the top, and a vertically suspended capillary tube was lowered into it. The lower point of a pin with a micrometric screw was brought to the surface of the liquid until it touched this surface, as judged by the coincidence of this point with its mirror image. In a movable microscope with an ocular micrometer (1 division of the drum \(= 7.84 \cdot 10^{-2}\) mm.) the positions of the meniscus were observed at various moments (by the second hand

(of a pocket watch); moreover, when the meniscus, moving slowly, left the field of view, the microscope tube was quickly shifted upward or downward, and, for control, the vernier reading on the vertical column of the microscope was recorded. At the end of the experiment, readings of the vernier were taken when sighting on the meniscus, on the upper end (also sharpened) of the pin, and on the mark on the capillary. The distance from this mark to the lower end of the capillary, and the length of the pin of the micrometer screw, were measured by the same movable microscope.

At first it was intended to carry out observations simultaneously with two capillaries, transferring the microscope from one to the other, but preliminary experiments showed a lesser accuracy of the readings in this case. Therefore each series of observations was then made with only one of the two capillaries \((r_1 = 0.297\ \mathrm{mm},\ r_{II} = 0.504\)—by weighing a column of mercury).

The investigation covered 50%, 60%, and 65% solutions of dammar resin in turpentine; to prevent their evaporation, the entire apparatus was placed beneath a bell made of plane-parallel glass plates, under which there also stood an open cup with turpentine. In long experiments this precaution proved insufficient, since on the surface of the liquid in the cup and, apparently, in the capillary there formed, with time, a layer less mobile than the interior—probably owing to the difference in elasticity of the vapor of the resin solution in turpentine and of pure turpentine. This circumstance, and also the variation of temperature over the long interval of the experiment, convinced us still more strongly of the necessity, in determining the coefficient of surface tension of such viscous liquids, not to wait for many hours, or even several days, for the liquid to become established at its final height, but to find this height from observations of the actual process of rise or fall of the liquid.

I shall give, as an example, one of the main series with a comparatively small number of readings (to save space)—observations on the lowering of a 60% solution, 22.10.18. Temperature at the beginning and at the end of the experiment: 20°.5. Readings by the vernier before the experiment: on the pin = 54.7; on the mark = 48.5; on the lower edge of the tube = 9.2.

\(t\)	\(h\)	\(h\) reduced	Notes	\(t\)	\(h\)	\(h\) reduced	Notes
11 h 40 m	90.0	90.0	Reading by vernier = 33.3	11 h 58 m	67.0	−85.0
41	75.0	75.0		12 h 00 m	58.0	−94.0
42	59.5	59.0		04	44.0	−108.0
43	44.8	44.0		08	34.0	−118.0
44	32.5	32.0		12	27.5	−124.5
45	19.0	19.0		20	19.2	−132.8	reset, reading = 11.8
46	8.0	8.0	reset, reading = 25.8		91.0	−132.8	reset, reading = 11.8
	90.0	8.0	reset, reading = 25.8	28	87.0	−136.3
47	80.0	−2.0		36	85.0	−138.3
48	70.0	−12.0		44	84.0	−139.8
49	60.0	−22.0		1 h 00 m	83.5	−140.3
50	51.0	−31.0		16	83.0	−140.8
52	35.0	−47.0		32	83.0	−140.8	Readings on the vernier of the observer = 22.7; on the pin = 28.6
54	20.0	−62.0	reset, reading = 20.3	23.10.3 10 h 00 m	98.0	−140.8	Readings on the vernier of the observer = 22.7; on the pin = 28.6
	90.0	−62.0	reset, reading = 20.3
56	78.0	−74.0

With each of the three solutions, two series of observations were made on the rise and on the fall in each of the two capillaries (except for the 50% solution in the wider capillary), and some series were repeated.

Most of the graphs constructed leave much to be desired in the sense of the smoothness of their course—especially for the series of observations on descent.

3. Derivation of the formula.

If, in addition to the notation already introduced, we denote the density of the liquid by \(\Delta\), the acceleration due to gravity by \(g\), and the length of the immersed part of the tube by \(l\), then for the force of surface tension \(f_1\), which pulls the liquid upward in the capillary, for the force of gravity \(f_{II}\), which pulls it downward, and for the quantity of liquid \(\dfrac{dq}{dt}\) flowing upward (by Poiseuille’s formula), we obtain

\[ f_1 = 2\pi r \cdot \alpha \tag{1} \]

\[ f_{II} = \pi r^2 h \cdot \Delta g \tag{2} \]

\[ 2\pi r\alpha = \pi r^2 H\Delta g \tag{3} \]

\[ \frac{dq}{dt} = \frac{d(\pi r^2 h)}{dt} = \frac{\pi r^4}{8\eta}\cdot \frac{f_1 + f_{II}}{\pi r^2(h+l)} \tag{4} \]

where, in the case of the descent of the liquid, in formula (4) \(f_{II} > f_1\), and consequently \(\dfrac{dh}{dt} < 0\).

From (1), (3), and (4) we find

\[ v=\frac{dh}{dt} = \frac{r^2\pi^2\Delta g}{8\eta}\, \frac{(H-h)}{\pi r^2(h+l)} = \frac{r^2\Delta g}{8\eta}\, \frac{H-h}{h+l} \tag{5} \]

Denoting

\[ \frac{r^2\Delta g}{8\eta}=C \tag{6} \]

and separating the variables, we obtain

\[ C\,dt = \frac{h+l}{H-h}\,dh = -(H+l)\frac{d(H-h)}{H-h}-dh \tag{7} \]

whence

\[ C(t-t_0) = (H+l)\lg\frac{H-h_0}{H-h} - (h-h_0) \tag{8} \]

\[ h = H-(H-h_0)e^{-\frac{C(t-t_0)+h-h_0}{H+l}} \tag{9} \]

The form of equation (9) makes it impossible to find \(H\) from it from the observed \(h\), and, consequently, \(C\) also cannot be found from equation (8). In exactly the same way, equation (9) is inconvenient for testing the theory, since \(h\) cannot be found from it. Therefore, having convinced myself from several series that, if the observed final height is taken as \(H\), the right-hand sides of equation (8) give values approximately proportional to the time from the beginning of the experiment, I decided to use formula (5). The values of the velocity \(v\) of the change of height at various moments of time, determined from the experimental data, could, on the basis of this formula, be used to find—from a series of observations of the heights of rise at various moments—that final height \(H\) toward which these heights tend asymptotically, and from which one can find the coefficient of surface tension \(\alpha\), as well as the coefficient \(C\), which, serving as a kind of measure of the quantity of liquid flowing through the capillary under these conditions, can give the magnitude of the coefficient of internal friction \(\eta\).

Lucas, on the other hand, having derived formula (5), integrates it by means of a series, which proves to be both less convenient and less accurate than the method set forth below.

method of treatment, although it does represent a considerable step forward in comparison with this author's predecessors cited by him—Cameron & Bell (Bull. Bureau of Soils U. S. Dep. of Agriculture, 1908) and Wo. Ostwald (Kolloid Zeits., 2, Suppl. Heft II, 1908), who applied empirical formulae of an indicative type, giving an impossible value of \(H=\infty\) at \(t=\infty\).

If we denote by \(V_m\) and \(V_n\) the velocities at the moments when the heights of the liquid are equal to \(h_m\) and \(h_n\), then, introducing formula (11) and the notation \(K_{mn}\), we obtain from formula (5), successively,

\[ V_m=C\frac{H-h_m}{h_m+1},\quad V_n=C\frac{H-h_n}{h_n+1}\ldots\ldots\ldots\ldots\ldots\ldots (10), \]

\[ \frac{H-h_m}{H-h_n}=\frac{V_m(h_m+1)}{V_n(h_n+1)}=K_{mn}\ldots\ldots\ldots\ldots\ldots\ldots (11), \]

\[ H=\frac{h_mK_{mn}-h_n}{K_{mn}-1} =h_m+\frac{h_m-h_n}{K_{mn}-1}\ldots\ldots\ldots\ldots\ldots\ldots (12). \]

For the most accurate determination of \(H\), it is evidently necessary to choose such values of \(V_m(h_m+1)\) and \(V_n(h_n+1)\) whose ratio is as close as possible to 2; from the values of \(H\) obtained from various combinations of the products \(V_i(h_i+1)\), one must take the mean, and from this, by formula (3), find \(\alpha\). The value of \(H\) thus found then permits, from each pair of values \(h_i\) and \(V_i\), finding the value of the coefficient \(C\) by the formula

\[ C=\frac{V_i(h_i+1)}{H-h_i}\ldots\ldots\ldots\ldots\ldots\ldots (13), \]

whence, from the mean \(C\) of these values, we find [[unclear: continuation]] by formula (6).

4. The case of a plane-parallel capillary space.

In exactly the same way one can obtain a formula expressing the change in the height of rise of a liquid in the space between two parallel planes separated by a distance \(2a\). If their length, sufficiently large in comparison with the distance, is denoted by \(b\), then, instead of equations (1)—(4), using the formula for the quantity of liquid flowing between two planes\(^1\), we obtain

\[ f_I=2b\cdot\alpha\ldots\ldots\ldots\ldots\ldots\ldots (14), \]

\[ f_{II}=b\cdot 2ah\Delta g\ldots\ldots\ldots\ldots\ldots\ldots (15), \]

\[ 2b\cdot\alpha=2baH\Delta g\ldots\ldots\ldots\ldots\ldots\ldots (16), \]

\[ \frac{dq}{dt}=\frac{d(2abh)}{dt} =\frac{a^3}{3\eta}\cdot\frac{f_I-f_{II}}{2ab(h+1)}\ldots\ldots\ldots\ldots\ldots\ldots (17), \]

\[ v=\frac{dh}{dt}=\frac{a^2\Delta g}{6\eta}\cdot\frac{H-h}{h+1}\ldots\ldots\ldots\ldots\ldots\ldots (18). \]

The last formula differs from formula (5) only in the form of the proportionality coefficient, so that all subsequent treatment of this case differs from the case of a cylindrical capillary only in the value of this coefficient.

5. Treatment of observations.

In the manner indicated above, all the observations of Voinov and Mochalin were treated, the greatest difficulty being presented by obtaining the values of \(V\), which were found either from the graph—by drawing tangents—or by calcu-

\(^1\) See, e.g., 41.

by calculation from the reports of h. The first method gives little-satisfactory results in places where the curve rises upward very steeply or very gently, so that one has to draw various graphs with different scales of abscissae expressing the time. For calculation by the second method I took, as the value \(V_i\) at the moment when the height is equal to \(h_i\), the quantity

\[ \frac{2}{3}\cdot \frac{h_{i+1}-h_{i-1}}{t_{i+1}-t_{i-1}} + \frac{1}{3}\cdot \frac{h_{i+2}-h_{i-2}}{t_{i+2}-t_{i-2}} . \tag{19} \]

Moreover, in those cases in which the values of \(V\) did not show sufficient smoothness, I resorted to deriving averages from several neighboring values and to smoothing these averages according to Gauss¹.

As an example of the processing, I give the processing of the series presented in § 2; for this series the values \(V_i\) were determined both graphically—column \(v_i\)—and by formula (19)—column \(v'_i\). The values \(v_i\) are expressed in divisions of the micrometer per minute, and the values \(h\) (from the lower end of the pin) and \(l\)—in divisions of the micrometer.

Table 20.

\(i\)	\(h_i\)	\(V_i\)	\(V'_i\)	\(V'_i(h_i+l)\)	\(V''_i(h_i+l)\)	\(C\)	\(t_{\mathrm{obs.}}\)	\(t_{\mathrm{calc.}}\)
0	493.2	—	—	4—	—	—	0	0
1	478.2	16.8	15.5	9292	8574	49.1	1.00	1.03
2	462.7	14.6	14.9	8278	8434	42.5	2.00	2.12
3	448.0	13.8	13.7	8276	8219	44.4	3.00	3.22
4	435.7	12.8	12.9	7519	7577	43.4	4.00	4.20
5	422.2	12.0	12.1	6887	6944	43.1	5.00	5.01
6	411.2	10.3	10.7	5798	6023	39.0	6.00	6.26
7	401.2	10.0	10.1	5529	5584	39.9	7.00	7.20
8	391.2	9.90	9.93	5446	5391	41.8	8.00	8.17
9	381.2	8.85	8.83	4716	4972	39.7	9.00	9.21
10	372.2	8.60	8.33	4506	4364	41.1	10.00	10.20
11	356.2	7.40	7.04	4072	3880	42.8	12.00	12.14
12	341.2	6.40	6.42	3155	3165	40.1	14.00	14.22
13	329.2	5.60	5.79	2693	2784	40.4	16.00	16.14
14	318.2	5.00	4.87	2350	2288	42.5	18.00	17.97
15	309.2	4.06	3.75	1875	1728	40.1	20.00	20.13
16	295.2	2.92	2.91	1305	1300	39.9	24.0	24.1
17	285.2	2.14	2.04	935	891	41.2	28.0	27.8
18	278.7	1.45	1.22	624	525	38.5	32.0	31.3
19	270.4	0.830	0.763	350.4	322.1	44.4	40.0	38.6
20	266.4	0.412	0.426	172.2	178.1	44.2	48.0	46.6
21	264.4	0.235	0.188	97.8	78.2	(51.5)	56.0	54.8
22	263.4	0.104	0.062	43.2	25.7	(48.0)	64.0	60.2

Table (21).

\(m\)	\(n\)	\(K'm\)	\(K''m\)	\(H'\)	\(H''\)
1	9	1.925	2.24	(308.2)	(276.3)
2	10	2.098	2.192	(381.2)	(290.0)
3, 4, 5	11	1.968	1.862	267.7	273.2
6	12	1.903	1.669	257.0	303.7
7, 8	13	1.971	2.024	263.8	260.3
9, 10	14	2.044	1.962	257.2	262.0
11, 12	15	2.026	1.916	266.0	274.1
13, 14	16	1.949	1.932	264.7	267.0
15	17	1.939	2.001	261.3	259.6
16	18	2.474	2.092	263.6	267.5
17, 18	19	2.198	2.221	261.0	260.8
19	20	1.808	2.035	262.8	261.7
20	21	2.278	1.761	261.8	262.8
21	22	3.043	2.264	262.0	262.0

The mean values of \(H'\) and \(H''\), with the mean errors of the result and of a single determination, are equal to

\[ H'' = 262.5 \pm 0.8 \pm 2.7,\qquad H' = 264.2 \pm 1.2 \pm 4.3 . \tag{22} \]

Giving preference in this case to the values \(H'\), I computed, by formula (13), the values of \(C\) given above in Table (20), and, discarding the last two values, obtained on the average

\[ \overline{C}=0.42\pm 0.6\pm 2.5 . \]

With the aid of the values (22) and (23), I [[unclear: several words obscured]] calculated by formula (8), from the observed [[unclear: heights]], what such a height should have been, and the latter in Table (20), in the column “computed,” agrees quite well with the column “observed.”

6. Results of the treatment.

The results of treating all the final series are given in Table (24), in which are also entered the time of the experiment and that height—\(H_{\mathrm{obs}}\)—at which the liquid was established toward the end of the experiment. The heights are given in micrometer divisions, the values of the coefficient \(C\) in \(\dfrac{\text{div.}}{\text{min.}}\).

Table (24).

Solution	\(dh/dt\)	Capillary	\(H\) computed	Time of experiment	\(H\) observed	\(C\)	\(a\), dyn/cm	\(\eta\), g/(cm·sec)	\(\theta^\circ\)
65%	\(+\)	I	\(250.1 \pm 1.0\)	28 h	271.6	\(8.2 \pm 0.1\)	30.8	101	20.3
65%	\(-\)	I	\(273.7 \pm 0.9\)	24	269.6	\(22.0 \pm 0.5\)	30.8	34	20.5
65%	\(+\)	II	\(144.0 \pm 1.1\)	5	149.2	\(37.8 \pm 0.7\)	29.4	63.4	21.7
65%	\(-\)	II	\(149.7 \pm 2.1\)	21	149.2	\(16.7 \pm 0.5\)	29.4	83.7	(21.7)
60%	\(+\)	I	\(250.1 \pm 2.6\)	4	251.2	\(30.2 \pm 0.9\)	28.7	27	19.5
60%	\(-\)	I	\(262.6 \pm 0.8\)	22	262.4	\(42.0 \pm 0.6\)	28.7	14	20.5
60%	\(+\)	II	\(143.2 \pm 5.3\)	2	153.0	\(52 \pm 3\)	29.8	47	22.0
60%	\(-\)	II	\(156.3 \pm 1.5\)	1	154.3	\(133 \pm 2\)	29.8	18	(22.0)
50%	\(+\)	I	\(260.1 \pm 1.1\)	2	261.7	\(68 \pm 8\)	28.6	11.5	20.4
50%	\(-\)	I	\(266.0 \pm 3.7\)	4	266.2	\(313 \pm 46\)	28.6	2.5	20.2

When deriving the values of \(a\), the values \(H_{\mathrm{calc.}}\) from the raisings were given weights three times smaller than the values \(H_{\mathrm{calc.}}\) from the lowerings and than the values \(H_{\mathrm{obs.}}\).

Consideration both of this table and of those separate values of \(H\) and \(C\), the averages of which are given in it, leads to certain general remarks, confirmed also by the details of the observations of Voinov and Mochalin.

The value of the final height of the viscous liquid in the capillary, as obtained from observations of the process of raising, is noticeably smaller than the directly observed limiting height; and the value obtained from observations of the process of lowering is somewhat greater than the limiting height. A difference is also found between the limiting heights themselves in raising and in lowering, especially noticeable for a short duration of the experiment, but not disappearing—probably owing to the formation of a kind of “crust”—even if the observations are continued so long that the height has ceased to change.

Still sharper is the difference between the values of the coefficient \(C\), which is a measure of the rate of flow through the capillary when the pressure gradient is equal to unity (which is proportional to the ratio \(\dfrac{H-1}{h+1}\)). This coefficient is smaller—in some cases considerably so—in the process of raising than in the process of lowering, and apparently, without special precautions in setting up the experiment, cannot serve for determining from such experiments the coefficient of internal friction of the liquid. Indeed, other experiments (34) gave, for the same solutions, values of \(\eta\) equal to \(53\), \(12.5\), and \(1.19\ \dfrac{\mathrm{g}}{\mathrm{cm}\cdot\mathrm{sec}}\) at \(22^\circ\), and their differences from the values (24), which were obtained from the values of \(C\), cannot be explained solely by changes in concentration that may have occurred in the interval of time (about a year) between the two sets of experiments.

In the process of raising, the smaller values of \(H\) and \(C\) are connected with incomplete wetting of the capillary walls by the liquid—an effect directly observed in the microscope and also manifested in the lesser smoothness of the curves of raising as compared with the curves of lowering. If the contact angle at some instant is equal to \(\beta\), then, as is easily derived, formula (13) should have been replaced by the formula

\[ C=\frac{v_i\,(h_i+1)}{H\cos\beta-h_i}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (25) \]

which must give larger values of \(C\) than formula (13). This difference should be most pronounced in the middle parts of the experiments, when the liquid is still rising rather quickly and therefore has not had time to wet the walls sufficiently well, and when, at the same time, \(H-h\) is already not large and may be very appreciably larger than \(H\cos\beta-h\); then, as at the beginning of the experiment, the ratio \(\dfrac{H\cos\beta-h}{H-h}\) will differ only slightly from \(\cos\beta\), while at the end of the experiment the angle \(\beta\), owing to the slowness of the rise, will be close to \(0\). The course of the computed values of \(C\) for different instants agrees with what has been said.

In the process of lowering, however, there occurs (which was also clearly observed in the microscope) a very slow draining of the excess of liquid that has well wetted its walls from the walls\(^{1}\),—and this circumstance must greatly complicate the theory set forth above. Indeed, if there were no surface tension, then the surface of the liquid in the capillary, during its lowering, would have to assume the form of a paraboloid of revolution, gradually drawn downward,

\(^{1}\) The same thing is distinctly visible on wide tubes from which drops of 90% and 95% solutions of dammar in turpentine flowed out over a number of years (37).

whereas, if there were no internal friction, the surface of the liquid would at all times have the form of a hemisphere gradually descending downward. In reality, the form will be intermediate between spherical and paraboloidal, and without a special theoretical investigation based on the combined application of the formulas of Laplace and Poiseuille, it is impossible to say definitely how the deviation of the meniscus from the hemispherical form will affect the law of lowering. On the one hand, the forces of internal friction in the layer adhering to the walls will cause a retardation of the downward motion; on the other hand, in formula (1) for the force \(f_i\), instead of \(r\) one must take a somewhat smaller value \(\rho\), whereas for the force \(f_{II}\) the value given by formula (2) will remain. Since in this case \(v_i\), as well as \(H-h_i\), will be negative, the true value of \(C\), equal to

\[ C=\frac{v_i(h_i+1)}{-(h_i-H)} = -\pi r^2 \Delta g\, \frac{v_i(h_i+1)}{f_{II}-2\pi r\alpha} \ldots \ldots \ldots \ldots (26), \]

will be less than that calculated by formula (13), or by the equivalent formula

\[ C=-\pi r^2 \Delta g\, \frac{v_i(h_i+1)}{f_{II}-2\pi r\alpha} \ldots \ldots \ldots \ldots (27). \]

By the difference between formulas (26) and (27)—as also by the difference between formulas (1) and (25)—one may explain the larger values of \(C\) during lowering than during raising, as well as the fact that the values of \(C\) during lowering prove larger at the beginning of the experiments, when the layer on the walls, i.e. \(\rho\), has already had time to form but has not yet had time to drain appreciably; however, the excess—admittedly slight—of \(H_{\mathrm{calc}}\) over \(H_{\mathrm{obs}}\) during lowering remains unexplained.

From the theoretical and experimental results of the present work the following consequences follow:

1. The processes of raising and lowering a liquid in a capillary may be used for determining the coefficient of surface tension in those cases when, for one reason or another, one cannot wait long enough for the height to cease changing, or when there is reason to think that the cause of the cessation of changes in height depends on changes that have occurred in the surface layer.

2. In these cases it is preferable to use the process of lowering rather than that of raising.

3. The value of the coefficient of internal friction obtained from such observations can serve only as an indication of its order of magnitude; it is preferable to determine it by the usual methods of measuring the quantity of liquid flowing through the capillary.

4. It is desirable to investigate, theoretically and experimentally, the process of change in the form of the surface of a viscous liquid when it is lowered in a narrow tube.

23.10.23

1. E. T. Whittaker, Proc. Phys. Soc. 5, 1–8, 1921; see also Math. & Astr. Blest., 27, 137–153, 1922; Proc. Roy. Soc. Lond. A, 1923. ↩↩↩↩↩↩↩↩↩↩↩

2. Work No. 18 of the list, which is included in the preceding article and to which all subsequent references will be made. ↩↩↩↩↩↩↩↩↩

3. Cf. 32 and 33 and Bulletin of the Russian Hydrological Institute, No. 3, p. 4, 1922, and also § 3 of the present work. ↩↩↩↩