# Aerodynamics (Lanchester)/Chapter 3

Chapter III.
The Hydrodynamics of Analytical Theory.
§ 57. Introductory.—The analytical treatment of hydrodynamic problems commonly involves an extensive application of the higher mathematics, the classic methods being those of Euler, Lagrange, Stokes, and others.
The importance and bearing of the mathematical demonstrations, in connection with the subject of the present work, is comparatively limited, but many of the results are of great consequence; the present exposition has therefore been restricted to a brief indication of the mathematical method, and a digest of those results which, from the present standpoint, are of the greatest interest. Where it has been found possible, a simple physical demonstration is given; in many cases the results of established investigation are taken for granted.
The present discussion opens with a recapitulation of the physical properties of fluids, which may be taken as a concise re-statement of essential definitions, sufficient to render back reference unnecessary. The chapter concludes with a critical argument on the practical deficiencies of the Eulerian and Lagrangian method, and on the theory of Discontinuous Flow.
The hypothesis of the initial discussion is strictly that of an inviscid fluid, and in general the condition of incompressibility is assumed. Up to a certain point the mathematical treatment, as usually applied, takes cognisance of compressibility, but generally speaking, the tangible results, so far as they concern our present subject, relate to the simpler conditions.
§ 58. Properties of a Fluid.—All fluids are characterised by certain definite physical properties. The property that may be ​said to constitute fluidity, and which distinguishes fluid from solid bodies, is inability to sustain stress in shear. A fluid in which this property is perfect is said to be inviscid, and in such a fluid a shearing strain, i.e., distortion, may take place without being accompanied by any corresponding stress: such a fluid must be regarded as hypothetical. All actual fluids possess viscosity; in a viscous fluid a stress in shear may exist, but is accompanied by a continually increasing strain; "stress in shear in a viscous fluid bears, in fact, the same relation to the rate of change of strain that stress in a perfectly elastic solid bears to the strain itself.
The remaining physical properties of a fluid are identical with those of a solid body, and comprise density and elasticity (volumetric). These two quantities are related to a third quantity—pressure—in so much that the density is a function of the pressure, the nature of which function is defined by the law of elasticity; thus in a perfect gas under isothermal conditions we have P/ρ = constant where ρ is density, and P pressure.
If we take the two extreme cases in the relation of ρ and P, so that, firstly, the elasticity be supposed zero, we shall have any finite pressure, however small, produce an infinite density, and the fluid becomes identical with the medium of Newton. If, secondly, we suppose the elasticity to be infinite, so that a change in P, however great, produces no change in the density, we have the case of an incompressible fluid. The latter assumption is that of our present hypothesis.
§ 59. Basis of Mathematical Investigation.—The Equations of Motion may be said to constitute the starting point of all analytical investigation; these are:—
(1) The Equation of Continuity, expressing the relation between the density of the fluid and the linear rate of change of flow in each of the co-ordinate directions of space; or, under the restriction that density is constant, the relation between the ​(linear) rate of change in the three co-ordinate directions amongst themselves.
The equation of continuity is based upon the fact that the inflow and outflow of any small element of space must balance, or must balance against the change of density if the fluid is compressible. The form of the expression for an incompressible fluid is—
${\frac {Du}{Dx}}+{\frac {Dv}{Dy}}+{\frac {Dw}{Dz}}=0,$
where $u,\ v,$ and $w$ represent the velocities in the directions of the three co-ordinate axes $x,\ y,$ and $z.$
(2) The Dynamical Equations expressing the relation in the direction of each of the three co-ordinate axes $x,\ y,\ z,$ for every small element of the fluid, between the rate of change in its momentum, the difference of pressure on its opposite faces, and the component of the extraneous force, if any.
The Extraneous Forces are usually represented in the three co-ordinate directions by the symbols $X,\ Y,\ Z,$ and denote forces acting from without on the fluid particles, such, for example, as the force of gravity. In the present branch of the subject these forces do not require to be considered.
Employing, as is customary, the symbol $DF/Dt$ to denote a differentiation following the motion of the fluid, it can be shown that
${\frac {DF}{Dt}}=$
${\frac {dF}{dt}}+u\ {\frac {dF}{dx}}+v\ {\frac {dF}{dy}}+w\ {\frac {dF}{dz}}.$
Now the rate of change of the $x$ momentum of any small element $\delta x\ \delta y\ \delta z$ is $\rho \delta x\ \delta y\ \delta z\ {\frac {Du}{Dt}},$ and this must be equal to the difference of the pressure force on its two faces, which is evidently $-{\frac {dp}{dx}}\delta x\ \delta y\ \delta z,$ (where $p$ is pressure). The minus sign is due to the fact that the momentum increase takes place in the direction of the pressure decrease. So that:—
$\rho \delta x\ \delta y\ \delta z\ {\frac {Du}{Dt}}=-{\frac {dp}{dx}}\delta x\ \delta y\ \delta z,$
or
${\frac {Du}{Dt}}=-{\frac {dp}{\rho dx}}.$
​Substituting from (1) for ${\frac {Du}{Dt}},{\frac {Dv}{Dt}},$ and ${\frac {Dw}{Dt}}$ we have:—
| ${\frac {du}{dt}}+u{\frac {du}{dx}}+v{\frac {du}{dy}}+w{\frac {du}{dz}}$ | $=-{\frac {dp}{\rho dx}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| ${\frac {dv}{dt}}+u{\frac {dv}{dx}}+v{\frac {dv}{dy}}+w{\frac {dv}{dz}}$ | $=-{\frac {dp}{\rho dy}},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| ${\frac {dw}{dt}}+u{\frac {dw}{dx}}+v{\frac {dw}{dy}}+w{\frac {dw}{dz}}$ | $=-{\frac {dp}{\rho dz}}.$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
In the steady state ${du}/{dt}$ is zero, and the equations become:—
$u\ {\frac {du}{dv}}+v\ {\frac {du}{dy}}+w\ {\frac {du}{dz}}=-{\frac {dp}{\rho dx}},\ {\mbox{etc.}}$
When there is no motion or motion of translation only in the fluid, the last three terms of the left-hand side of the equation are zero, and the equations become:—
${\frac {du}{dt}}=-{\frac {dp}{\rho dx}},\ {\mbox{etc.}}$
The further development and employment of these equations is outside the scope of the present work, but the physical significance can be gathered by comparison with §§ 60 and 88.
The mathematical superstructure founded on the above consists in the main of finding solutions to the equations of motion in a number of well-defined cases, and in the general development of the theory in its application to the motions of bodies of stated geometrical form under known boundary conditions.
§ 60. Velocity Potential ($\phi$ Function).—If a force be applied to a body initially at rest in a fluid, a circulation of the fluid is set up, the flow taking place along paths of curvilinear form by which the displaced fluid is conveyed from one side of the body to the other. We may regard the initial direction of flow, produced in this manner, as denoting a "field of force," the ​direction of the lines of force being everywhere that of the initial acceleration of the particles.
When such a system is initiated in a fluid from rest, at the instant the force is applied the surfaces of equal pressure are everywhere normal to the lines of force. This is not necessarily the case when the fluid is in motion, for we have then superposed pressure differences due to the change in velocity and direction of the particles which modify the pressure distribution.
Let us suppose that the applied force is impulsive, i.e., let it be considered to be an infinite force applied for an infinitely short time; then the form of flow generated will be that due to the initial application of the force, that is to say the field of flow will coincide with the field of force.
Now it does not obviously follow that this form of flow will be stable or permanent. In actual fluids, such as water or air, we know in fact that it is not so. It would, however, appear that in the ideal fluid of hypothesis any form of motion generated by an impulse in this manner will persist without change of form, and therefore the field of force and system of pressure by which the flow is generated may be taken as defining the form of flow for the steady state.
Under these circumstances it is evident that the motion will be the same whether generated by an impulse or by a finite force, since the continued application of the force to the body in motion will accelerate the field everywhere in the line of flow.
If we now examine the initial pressure system, then the velocity produced on the fluid from rest along any line of force after a brief interval of time will be, for any small difference of pressure, inversely as the mass per unit section, that is, inversely as the distance separating, the points at which the said pressure difference exists. Or if $\delta p$ is the pressure increment, and $\delta l$ the distance along the line of force, the velocity after a certain brief interval of time will be everywhere proportional to ${\frac {\delta p}{\delta l}}$ or, when the increments are taken as evanescent ​velocity varies as ${\frac {\delta p}{\delta l}}$, or, resolving into its three co-ordinate components, we have—
$u,\ v,\ w,=c\ {\frac {\delta p}{\delta x}},\ c\ {\frac {\delta p}{\delta y}},\ c\ {\frac {\delta p}{\delta z}},$
where $c$ is a constant.
In the above expression the density of the fluid and the magnitude of the applied force are involved in the constant $c.$ It is, however, evident that we may regard the form of flow as a matter of pure kinematics, since the existence of the flow is not dependent upon the pressure system by which it is generated. Consequently we may substitute for $p$ a function $\phi ,$ which has no dynamic import, and which is termed velocity potential, and we may write the expression—
$u,\ v,\ w,={\frac {\delta \phi }{\delta x}},\ {\frac {\delta \phi }{\delta y}},\ {\frac {\delta \phi }{\delta z}},$
the terms on the right-hand side of this equation being sometimes written with a minus sign.
In the foregoing illustration $\phi$ is a single-valued function, inasmuch as it can have a definite value assigned for every point in the field of flow.
§ 61. Flux ($\psi$ Function), $\phi$ and $\psi$ interchangeable.—In cases of fluid motion in which a velocity potential exists the lines of flow are, as pointed out, everywhere normal to the equipotentials, that is to say the surfaces of $\phi =$ constant. It can be shown analytically that if the curves of flow be plotted for equal increments of flux (that is, so that the amount of fluid that flows per unit time past any point and between two adjacent lines is constant), and the curves $\phi =$ constant be plotted over the same field, the two series of lines will divide the field into a number of similar elements whose ultimate form in the case of motion in two dimensions, when the units employed are sufficiently small, becomes square within any desired degree of approximation. Thus (Fig. 33) let $e\ e$ be two lines of flow, and $f\ f$ be two lines ​$\phi =$ constant; then the cell cut off, $a,\ b,\ c,\ d,$ will be approximately square, and if we choose to subdivide for intermediate values of flux and velocity potential as indicated, the cellules so formed will approximate still more closely, and the whole field may be regarded as ultimately built up of a number of such square elements.
Fig. 33. If in two-dimensional motion the successive increments of flux be represented by increments of a quantity $\psi ,$ it can be shown that the $\phi$ lines and the $\psi$ lines may be interchanged, the lines of equal flux becoming equipotentials, and vice versâ. The applied impulse will of course require to be different for the two systems.
When the motion takes place in three dimensions, the $\psi$ lines and $\phi$ surfaces still divide the fields into a number of rectangular elements, or cubes (Fig. 34), but the conjugate property no longer exists; the $\phi ,\ \psi ,$ functions are not interchangeable.
Fig. 34. The foregoing principles may be illustrated by the simple case of a source and sink.
§ 62. Sources and Sinks.—A source is a hypothetical conception, and may be defined as a point at which fluid is being continuously generated, and conversely a sink is a point at which fluid is supposed to disappear. Nothing actually resembling a source or sink is known to experience, the utility of the conception resting in its application to theory. A point source gives rise to three-dimensional motion; a line source gives rise to two-dimensional motion. A line source may also be described as a point source in two dimensions.
The field of flow from a source or towards a sink in an infinite expanse of fluid can be laid down from considerations ​of symmetry. The conditions require that it should be constituted of radial straight line flow equally distributed circumferentially in space. The field, in the case of motion in two dimensions, being shown plotted in Fig. 35 for a series of equal increments of flux, each line of flow will represent some definite value of the function $\psi ,$ any one of the lines being arbitrarily chosen as datum.
Fig. 35.
The equipotentials, $\phi =$ constant, will be a series of concentric circles whose radii form a geometrical progression.
If the functions $\psi$ and $\phi$ be interchanged, the diagram represents a cyclic motion round a filament, the radial lines becoming the equipotentials.
In the case of the source or sink, the velocity of the fluid at the origin is infinite, the whole flux having to pass through a region having no magnitude. In order to keep the problem within the range of physical conception it is customary in ​this and similar cases to suppose the source or sink to be circumscribed by a small closed curve, which in the case we have under consideration will be a circle. When we interchange the functions $\psi$ and $\phi$ the same considerations apply. In this case the space within the circular enclosure represents the section of a cylindrical filament, around which the cyclic motion of the fluid is taking place. The introduction of such an obstacle, i.e., a circumscribed area in a two-dimensional space or an infinite cylinder in a three-dimensional space, involves what is termed the connectivity of the region. Where no obstacle exists the region is said to be simply connected; where one or more such obstacles exist the region is multiply connected. The question involves certain points of definition.
§ 63. Connectivity.—It is possible to connect any two points in a region containing fluid by an infinite number of paths traversing the fluid. Such paths as can by continuous variation be made to coincide without passing out of the region are said to be mutually reconcilable.
Any circuit that can be contracted to a point without passing out of the region is said to be reducible.
Two reconcilable paths combined form a reducible circuit.
A simply connected region is one in which all paths joining any two points are reconcilable, or such that all circuits drawn within the region are reducible.
A doubly connected region is one in which two irreconcilable paths, and no more, can be drawn between any two points lying within it, so that any third path shall be reconcilable with the one or the other, or shall be in part reconcilable with one or the other, and in part reducible to the circuit formed by the two combined. (The latter portion of this definition is necessary to provide for the case of a third path being drawn making one or more circuits of the "obstacle.")
In general, multiply-connected regions, in which $n$ irreducible ​paths, and no more, can be drawn to connect any two points, are said to be $n$-ply connected.
A few examples may be given. The region internal or external to the surface of a chain link or an anchor ring is a doubly connected region; a simple electric circuit, either internal or external to the conductor, is a doubly connected region; on breaking the circuit both regions become simply connected. A lake containing two islands is a triply connected region,. The region surrounding a gridiron is $n$-ply connected where $n$ is the number of the bars.
§ 64. Cyclic Motion.—The subject of connectivity derives its importance chiefly from its relation to the class of fluid motions known as cyclic. In a simply connected region, for all motions having a velocity potential, the latter, $\phi$, is a single-valued function, having at every point in the system a definite assignable value, varying continuously from point to point throughout the system. When the region is doubly connected this manifestly may not be the case, for if there is a circulation around an irreducible circuit it is evident that if we follow the variation of $\phi$ round such circuit we shall on arriving at the starting point have two conflicting values. Thus, referring to Fig. 35 when the radial lines are taken to represent $\phi =$ constant, we are unable to assign a progressive series of values to the $\phi$ lines that will be consistent.
Under these conditions $\phi$ is termed a cyclic function, and its value depends upon the datum point chosen for its zero and the number of times the path of integration has been taken round a circuit.
A physical conception of velocity potential under these circumstances is somewhat difficult, but if we revert to the dynamical hypothesis and regard the velocity potential system as the pressure system by which the motion is generated, we encounter at once the same difficulty in another form. Before we are able ​to interfere either to start or to stop the fluid in cyclic motion, we must introduce some imaginary barrier in its path. In the special case in the figure this evidently requires to extend from the central core outward to infinity in order to intercept the whole of the flux. We are at liberty to select what position we like, circumferentially, for this barrier, and in choosing such position we fix the datum for the value of $\phi .$ If then we suppose a suitable impulse to be applied the $p$ of the impulse pressure system will be a single-valued function throughout the field, and $p$ defines $\phi$ for the subsequent motion. The barrier, however, cannot be maintained under the conditions of steady motion, and it is the withdrawal of the barrier that renders $\phi$ indeterminate. It is the complementary fact that the barrier temporarily renders the region simply connected, and on its withdrawal the cyclic conditions supervene.
The particular case of cyclic motion taken as an illustration is one of the most elementary simplicity. The degree of complexity of any cyclic system of flow depends primarily upon the boundary conditions. We shall have occasion to refer later to cyclic systems of greater complexity, but at present the complete solution of the equations of motion is only known in some few cases where the boundary conditions are simple.
Although in any case of cyclic flow, such as in the example given, the fluid is in circulation around a central island, and so as a whole possesses angular momentum and rotary motion in the ordinary acceptation of the words, such a form of flow (i.e., one that can be generated by an impulse and possesses a velocity potential) is in reality irrotational. The theory of rotation in fluids is of considerable importance, in view of the fact that it can be proved that if the motion of an inviscid fluid is irrotational at any instant of time, it will remain irrotational for all time; that is to say, it is impossible to produce or destroy rotation in an ideal fluid.
§ 65. Fluid Rotation. Conservation of Rotation.—Let us suppose a hollow circular cylindrical vessel filled with fluid to be set in ​rotation, about its axis: then if the fluid possessed viscosity it would, in course of time, acquire sensibly the same speed of rotation as the vessel, so that the whole system would revolve en bloc. With an ideal fluid, however, the rotation of the vessel might be continued indefinitely without imparting any motion to its contents.
Fig. 36. If we suppose substituted for the circular cylinder one of square or other irregular section, it might be imagined that rotation would be imparted to the fluid by the irregularity of the boundary surfaces; such, however, is not the case. An inviscid fluid offers no resistance to distortion, and consequently the containing vessel, however irregular its form, is unable to acquire a "purchase" on the fluid contents, and the fluid is not set in rotation. Conversely if we suppose the fluid to be in a state of rotation in a vessel or region, no matter what its form, such rotation will persist and the fluid will continue to rotate for an indefinite time.
The foregoing reasoning, although touching the essence of the matter, can hardly be regarded as rigid proof.
§ 66. Boundary Circulation the Measure of Rotation.—The study of rotation may be confined to two dimensions. Let a a (Fig. 36) represent a circular cylindrical vessel of radius r within which the fluid possesses a motion of pure uniform rotation.
Now, such rotation is shared uniformly over the whole area; therefore, if we suppose the area divided into a number of equal small elements, and represent the rotation of each by a circulation ​round its boundary (Fig. 37), then the circulation round each element will be equal, and that along all the lines common to two adjacent elements is equal and opposite, and therefore of zero value, so that circulation along the boundary alone remains. It is proved then that:—
Fig. 37. The sum of the circulations round the boundaries of the individual elements is equal to the circulation round the boundary of the region; that is to say, the rotation of the fluid within the region is measured by the circulation round its boundary. It is evident that this result is not confined to uniform rotation. Let us suppose that the fluid contain rotation unevenly distributed amongst its parts, so that it may be in part irrotational, and in parts the sense of rotation may be opposite to that in other parts, but so that the velocity (u v) is, throughout the region, a continuous function of x y; then if we suppose it be divided as before into a number of small elements so that each element shall be indefinitely small, then the rotation within each element is uniform, and by the preceding argument is measured by the circulation round its boundary; but since u v is a continuous function of x y, the flow along the boundary of each element is in the limit equal and opposite to that of the element adjacent to it, and the two cancel out, leaving only the circulation round the boundary. Hence for any region the sum of the rotation integrated over the surface is equal to the sum of the circulation integrated along its boundary.
§ 67. Boundary Circulation Positive and Negative.—Referring again to Fig. 36, let us suppose a boundary surface to exist at e e e dividing the region into two parts, and let e e e coincide with one of the lines of flow so that it will not interfere with the motion of the fluid; the boundary e e will thus be circular, and concentric to the boundary a a.
​Then if r be the radius of the whole enclosure a a, and n r be the radius of the region e e, and x the total rotation, the rotation within the region e e as measured by the circulation along its boundary will be n$^2$ χ, the remaining rotation in the region between the boundaries will therefore be χ —n$^2$χ that is to say, the circulation along the external surface of the boundary e e is equal and opposite in sign to that along its internal surface.
Now if we regard the rotation of the fluid mass as a matter of rigid dynamics, the motion in the path e e is the same in sense whether it takes place in the matter external or internal, and in general rotation is an algebraic quantity, measured plus or minus, according to whether it takes place counterclockwise or clockwise (the latter being taken minus by convention). It is evident, however, that circulation along a boundary (also an algebraic quantity) cannot be so measured, but is plus or minus according as the fluid flows towards the right or the left hand of an observer stationed on the boundary facing the fluid. Thus, in the simple case illustrated, let us suppose the rotation to be positive (counterclockwise), then to an observer stationed on the "mainland" the circulation will pass from left to right, and is reckoned positive. If the observer now take his stand on the "reef" e e, and face the outer basin, the circulation will pass from right to left, and is therefore negative. If he now turn about and face the inner basin, the circulation is from left to right, and is positive. Another method of defining the sense of a circulation is to suppose an observer swimming in the fluid to keep the boundary always on his right hand, then the direction in which he is swimming is positive and the opposite direction negative. The positive direction is indicated by arrows in the figure.
Rotation in a fluid as above defined is a conception apart from any quantity known in rigid dynamics, and owes its importance to certain propositions relating to fluid motion. It is a quantity that in a perfect fluid can undergo no change. Conservation of rotation is an absolute law in an inviscid fluid.
Thus, the total rotation in a region containing fluid is zero when the sum of the circulation taken over a complete circuit of the boundary is zero; also, the motion of a fluid is "irrotational" when the sum of the circulation round a complete circuit of the boundary of its each small element is zero.
§ 69. Rotation. Mechanical Illustration.—In order to clearly dissociate the idea of rotation in a fluid from that of circular motion by virtue of which it may possess angular momentum, we may imagine a region of uniform rotation, such as that we have been considering, to have its motion intercepted by a net- work of rigid boundaries suddenly congealed throughout the region. Then the boundary system will at the instant of its formation receive an impulsive torque, and angular momentum of the rotating mass will be given up, but the rotation within the meshes of the network will persist, the new conditions being those of the supposition in Fig. 37, the equal and opposite circulations along the boundaries in common being materialised. We can suppose a mechanical model constructed to represent this action. Let us imagine a frame mounted upon a shaft capable of revolution, and carrying a multitude of accurately balanced wheels mounted on frictionless bearings, these bearings being arranged parallel, and parallel to those of the main shaft. Let us suppose that the whole apparatus be initially rotating en bloc; then if we stop the motion of the frame each of the wheels will continue to spin with the same angular velocity as previously, and nothing that we can do with the frame will alter their rate of spin in the slightest. The frame corresponds ​to the network boundary system and the wheels with the fluid in the meshes.
§ 70. Irrotational Motion in its Relation to Velocity Potential.—We have above defined irrotational motion as follows:—
The motion of a fluid is irrotational when the sum of the circulation round a complete circuit of the boundary of its each small element is zero.
Assuming this definition, it can be shown that fluid in irrotational motion has a velocity potential.
Let (Fig. 38) the cell $a\ b\ c\ d$ be any small element of the fluid in which $a\ b$ and $c\ d$ are lines of flow and $a\ c$ and $b\ d$ are normals thereto. Fig. 38. Then since the motion in the line of $a\ c$ and $b\ d$ is nil, the circulation round the circuit is the sum of the circulations along $a\ b$ and $c\ d$, and since that motion is irrotational, this quantity is zero.
| Let | $u_{1}$ be the velocity of the fluid along $a\ b$ $u_{2}$ be the velocity of the fluid along $c\ d$ $x_{1}$ be the distance $a\ b$ $x_{2}$ be the distance $c\ d$ |
(For the sake of simplicity the axis of $x$ has been chosen in the direction of the flow.)
Then let us take two columns of the fluid along the lines $a\ b$ and $c\ d$ respectively, whose section is defined as $\delta y\times \delta z$, then if $\rho =$ density, we have masses of the two columns $x_{1}\ \rho \ \delta y\ \delta z$ and $x_{2}\ \rho \ \delta y\ \delta z$ respectively. But their velocities $u_{1}$ and $u_{2}$ are connected by the relationship $u_{1}\ x_{1}=u_{2}\ x_{2}$, or ${\frac {u_{1}}{u_{2}}}={\frac {x_{2}}{x_{1}}}$. The momenta of the two columns are therefore in the relation $x_{2}\ x_{1}\ \rho \ \delta y\ \delta z$ is to $x_{1}\ x_{2}\circ \delta y\ \delta z$, which are equal; consequently, if a certain force applied to any column for a time $t$ will bring it to rest, the same force applied for the same time to the other column will bring that to rest also. But the areas of the columns are equal; therefore to stop or to reproduce the motion ​of the fluid the pressure difference applied between the points $a$ and $b$ requires to be the same as that between $c$ and $d$, so that the normals $a\ c$ and $b\ d$ to the field of flow are equipotentials ($\phi =$ constant).
This demonstration may be taken as applied to every small element of the field, so that the proposition is proved.
Corollary: When a fluid has velocity potential its motion is irrotational.
§ 71. Physical Interpretation of Lagrange's ${\boldsymbol {\phi }}$ Proposition.—The foregoing proposition, taken in conjunction with that relating to the conservation of rotation, constitutes a demonstration of Lagrange's theorem that “If a velocity potential exist at any one instant for any finite portion of a perfect fluid in motion under the action of forces which have a potential, then, provided the density of the fluid he either constant or a function of the pressure only, a velocity potential exists for the same portion of the fluid at all instants before or after.”
This statement, save to a mathematician, is not very clear, as it is difficult to obtain a sufficiently close conception of velocity potential to be able to attach any physical meaning to its conservation. The inversion of the statement, however, obviates all difficulty; it then becomes: If the motion of any portion of a perfect fluid he irrotational at any instant of time, then, provided the density of the fluid he either constant or a function of the pressure only, the motion of the same portion of the fluid will he irrotational at all instants before and after.
§ 72. A Case of Vortex Motion.—The case of cyclic motion resulting from an interchange of the functions $\psi$ and $\phi$ in the source or sink system is one of particular interest. If (Fig. 35) we suppose the origin circumscribed by a line of flow, then we ​have a cyclic system in which the origin represents the axis of a cylindrical body of infinite length making the space round it a doubly connected region. The velocity of the fluid is everywhere inversely as the length of its path of flow, consequently if we suppose the cylinder be made smaller the velocity at its surface will be proportionately greater, so that in the limit if we suppose the cylinder to become evanescent the velocity becomes infinite. The circulation round any such evanescent filament is indeterminate, for it is equal to $\infty \times 0$. The physical signification of this is that we have a system of flow that may be regarded as rotational or irrotational according as we regard the cylinder as non-existent or merely evanescent. If we regard the cylinder as non-existent and the motion as rotational, then the rotation is measured by the circulation round any of the lines of flow (for the circulation round each is the same), so that the whole rotation must be supposed concentrated at the geometric centre.
Such a motion is known as vortex motion, and the system figured constitutes a vortex filament. It will be seen that if $r$ represent the radius of the path of flow and $v$ the corresponding velocity, $v\ r=$ constant, and if the angular velocity $\omega =v/r$ we have $\omega \ r^{2}=$ constant,—that is to say, for any circuit of flow the area $\times$ angular velocity is constant, which is the relation for vortex motion established generally by the theorem of Helmholtz and Kelvin. The discussion of this type of motion will be resumed later in the chapter.
§ 73. Irrotational Motion. Fundamental or Elementary Forms. Compounding by Superposition.—All known forms of irrotational motion can be regarded as being compounded from a limited number of different types. These are:—(a) Uniform motion of translation; (b) rectilinear motion to or from a point, i.e., sources and sinks; (c) cyclic motion (in multiply connected regions only).
Let us examine first the simple case of a fluid mass possessed only of a uniform motion of translation, and let us suppose that its ​motion is compounded of two component motions whose velocity and direction are known. Then it is evident that the two component motions can be compounded by drawing a parallelogram, which may either be regarded as a “parallelogram of velocities” if we take its elements to represent velocity, or a “parallelogram of forces” if we take its elements to represent the impulses by which the motion is produced. Thus, if we compound a north wind with an east wind having the same velocity, the result is a north-east wind having a velocity ${\sqrt {2}}$ times as great; and the forces that would produce the two air currents separately would produce the combined current if acting simultaneously.
Fig. 39. If we denote the strength of each superposed stream by a series of parallel lines, so that the flux or quantity of fluid passed per unit time is the same at every point between each adjacent line and its neighbour—that is to say, if we draw the lines of flow, $\psi =$ constant, for each component stream, then the distance separating any two adjacent lines will be inversely as the velocity, and the network formed by the superposed systems will give the parallelogram of velocities at every point. This method of compounding the two systems of flow is illustrated in Fig. 39, in which $a\ a\ a$ and $b\ b\ b$ represent the component streams, and $c\ c\ c$, drawn diagonally, gives the resultant flow. It is evident that the lines $c\ c\ c$ will quantitatively represent equal values of $\psi$, for the resultant flux across any line $d\ d$ drawn through ​intersections athwart the stream will be the sum of the components.
§ 74. The Method of Superposed Systems of Flow.—The conception on which the foregoing method has been based can only be applied so long as the fluid moves en masse, but it can be shown that the method is applicable to all cases of irrotational motion. If we confine our attention to the field of force developed at the instant of application of the component impulses, then it is clear that the resultant field can be obtained by the use of the parallelogram of forces as shown in the figure; there is, however, another, and perhaps more convincing, method of proof; this is the method of superposition.
Let us suppose that instead of two motions being superposed on one fluid current two fluid currents be superposed on one another. This is at first difficult, owing to the instinctive but wholly imaginary difficulty of regarding it as possible for two bodies to occupy the same space at the same time. To simplify ideas, let us suppose the motion to be two-dimensional, so that it may be fully represented on a plane surface; then if we represent one motion on one plane and another motion on a plane adjacent to it the two systems will be superposed; and further, if we take as many systems as we wish and represent them on as many adjacent planes they become superposed. And since a plane possesses no thickness, such superposed systems, however numerous, occupy no finite quantity of the third dimension, and in fact constitute but one plane.
Now, reverting to the argument, let us suppose that any two systems of fluid motion be superposed one on the other. Then so long as we can identify the particles belonging to each separate system (as supposing the streams to consist of different kinds of matter), the two systems must be regarded as separate; but if we imagine that we cannot distinguish the matter in the one stream from that in the other, then a flux across any imaginary barrier in one direction will neutralise an equal flux across the same ​barrier in the opposite direction, and it will only be possible to recognise the resultant flow; thus, as before, if Fig. 39 represent the two superposed streams by the lines a a a and b b b, the flux across the imaginary barrier line e f, due to the stream a, will be equal and opposite to the flux across the same line due to the stream b, consequently there is no resultant flux across the line e f, which is therefore one of the lines of flow of the resultant system. Likewise in the case of the other parallelograms, so that the field c c c is the resultant system.
Fig. 40.
We therefore see that the superposition of two independent streams has the same resultant as the superposition of two motions on one stream.
The foregoing constitutes the basis of a comprehensive method ​of plotting the field of flow for any finite combination of known systems. It is the geometrical equivalent of the analytical machinery employed in the mathematical solution of a vast number of cases, and as such it is due to Clerk Maxwell. Many compound systems of flow involve an infinite number of elementary components, such as some prescribed distribution of sources and sinks over certain lines and surfaces; the graphic method in such cases is not generally applicable, and the field requires to be plotted from the mathematical solution.
§ 76. Source and Sink, Superposed Translation.—We might (with certain reservations) regard such a combination of source and sink as a tube (Fig. 41) through which fluid is being pumped, the fluid entering the tube at B and emerging again at A. If we suppose such a tube to move longitudinally through the fluid in the same direction as that in which the fluid flows in its interior, or, that which is in reality the same, if we suppose the tube fixed whilst the fluid as a whole has a velocity of translation in the opposite direction, the system of flow undergoes considerable modification.
Fig. 42 gives the solution of such a case for a two-dimensional field. The source and sink system of Fig. 40 being superposed on a motion of translation, it is found that two distinct systems of flow result, internal and external respectively to a surface of oval form ; the internal system consists of a source and sink in a region bounded externally, and the external system gives the stream lines proper to an oval cylinder in motion through the fluid; it is evident that we may suppose such a body substituted for the internal system. The form of this oval represents the shape of a body that will give rise to the same external system of flow as the simple source and sink, and according as the flux of the motion of translation is greater or less in relation to that of the source and sink, the oval will be more or less elongated, the limiting conditions approximating to a line joining the foci on the one hand and to a circle on the other. The form of this oval is not an ellipse, being fuller towards the extremities, especially in cases where the ratio of major to minor axis is considerable.
§ 77. Rankine's Water Lines.—These curves and the whole external series have been closely studied by Rankine, the method of plotting here given being that employed by him. Rankine has pointed out the general resemblance of these curves to ships, water lines, and has given them the name “Oogenous Neoids.”
Fig. 42.
​Rankine says, referring to the practical employment of these curves:—
“The ovals are figures suitable for vessels of low speed, it being only necessary, in order to make them good water lines, that the vertical disturbance should be small compared with the vessel's draught of water. At higher speeds the sharper water lines more distant from the oval become necessary. The water lines generated by a circle, or ‘cyclogenous neoids,’ are the ‘leanest’ for a given proportion of length to breadth ; and as the eccentricity increases the lines become ‘fuller.’ The lines generated from a very much elongated oval approximate to a straight middle body with more or less sharp ends. In short, there is no form of water line that has been found to answer in practice that cannot be imitated by means of oogenous neoids.”
And further:—
“Inasmuch as all the water-line curves of a series, except the primitive oval, are infinitely long, and have asymptotes, there must necessarily be an abrupt change of motion at either end of the limited portion of a curve which is used as a water line in practice, and the question of the effect of such abrupt change or discontinuity of motion is one which at present can be decided by observation and experiment only. Now it appears from observation and experiment that the effect of the discontinuity of motion at the bow and stern of a vessel, which has an entrance and run of ordinary sharpness and not convex, extends to a very thin layer of water only; and that beyond a short distance from the vessel's side the discontinuity ceases, through some slight modification of the water lines, of which the mathematical theory is not yet adequate to give an exact account.”
§ 78. Solids equivalent to Source and Sink Distribution.—In the light of present knowledge it would appear that the particular case of flow under discussion is merely one of an infinite number of possible systems in which sources and sinks of different strengths ​are distributed along an axis (or axis plane for two-dimensional motion in a three-dimensional space), and it is an established proposition that any solid whatever, in motion in a fluid, may be imitated by an appropriate distribution of sources and sinks situated on its surface, and it follows that within certain limitations as to abruptness of contour, an equivalent exists for every stream line solid of revolution in point sources and sinks distributed along an axis, and for every cylinder of stream line section in line sources and sinks located on an axial plane. The distribution of sources and sinks that will produce any particular form is only known in a few special cases, such as those of the elliptical cylinder and ellipsoid, in which the number Fig. 43. is infinite. Any finite distribution can be investigated by the graphic method by repeated compounding of system on system; a comprehensive way of investigating cases of infinite distribution is at present lacking. It may be noted that in all cases the investigation commences with the source and sink system, the form of the corresponding solid being obtained as a resultant; the reverse process can only be effected by recognising the solid as belonging to some particular system, and consequently only certain solutions are possible.
It is evident that if we take any pair of Rankine's “oogenous neoids” and trim fore and aft to form water lines (Fig. 43), we can regard the process as equivalent to a number of sources in the region a a a, and sinks in the region b b b, in order to generate and absorb the stream flux that otherwise runs to ​infinity in either direction. It would be more proper to discover by trial some combination of sources and sinks that would give an easy termination to the form than to effect this by an arbitrary mutilation, for the true stream lines round the modified form could then be plotted. Beyond this there is no advantage in the one course over the other; the criterion in either case is the eye of the designer. In the hydrodynamic theory of an inviscid fluid, every conceivable body is of stream-line form, and the conditions that obtain in practice do not exist; it is therefore Fig. 44. useless to attempt to rationalise the ichthyoid or stream-line form by existing analytical theory.
The foregoing example illustrates the graphic method as applied to effecting the combination of motion in two dimensions; certain cases of motion in three dimensions may be solved by ​proceeding in the manner laid down by Rankine for a solid of revolution.
§ 79. Typical Cases constituting Solutions to the Equations of Motion.—Some typical cases of stream-line flow constituting the Fig. 45. solution to the equations of motion, for the forms of body specified, are given in Figs. 44 (cylinder), 45 (sphere), and 46 (elliptical cylinder). Fig. 46. It is scarcely necessary to remark that these forms of flow do not hold good for actual fluids.
In Fig. 47 are plotted the lines of flow for a lamina of infinite ​extent relatively to the “enclosure,” i.e., the fluid at infinity, and relatively to the body itself. In the present work the former are referred to as the lines of flow and the latter as the stream lines, the latter term being employed in all cases where the primary flow is superposed on a motion of translation. This is merely a matter of convenience in terminology, in which the present work differs from some of the standard text-books in which the term stream line is used more generally.
Fig. 47.
Fig. 48.
​case, the cyclic motion being supposed to take place about a filament of negligible diameter; the resultant motion is again found to consist of two distinct systems of flow, one internal and the other external to the surface $a\ a\ a\ a$; the field is plotted in full for equal increments of both $\psi$ and $\phi$.
It may be pointed out here that any systems that individually possess velocity potential must of necessity possess velocity potential in their resultant, for otherwise two irrotational systems would, in combination, possess rotation, which is manifestly impossible.
§ 80. Consequences of Inverting ${\boldsymbol {\psi }}$, ${\boldsymbol {\phi }}$ Functions in Special Case. Force at Eight Angles to Motion.—In Fig. 48 the curves of $\psi$ and $\phi$ if interchanged would obviously give the case of a source or sink, the flow being vertical instead of horizontal. In this inverted reading of the diagram we again find two systems of flow; the surface of separation $e\ e\ e\ e$ passes away to infinity, and has parallel asymptotes. It is frequently convenient when reading any flow diagram to temporarily suppose the functions inverted in this way.
A remarkable and important fact in connection with a cyclic system with superposed translation is the existence of a reaction or force at right angles to the direction of motion, such force in the case represented in Fig. 48 being an upicard force acting on the filament, that is to say, a downward force must be applied to the filament in order that the motion as a steady state should be stable. Where the fluid is bounded externally the force must be supposed to act between the external boundary and the filament or such other body as constitutes the inner boundary.
The necessity for this applied force may be demonstrated in several ways, but it is in the first place necessary to consider the distribution of kinetic energy and pressure in the region occupied by the field of flow.
§ 81. Kinetic Energy.—The expression for the kinetic energy of any dynamic system is ${\tfrac {1}{2}}mr^{2}$, where $m$ is the mass and $r=$ ​velocity. Applying this to the case of stream motion, let $b$ be the distance between stream lines demarcating some definite increment of $\psi$, then we know that $v\propto {\frac {1}{b}}$, or $v^{2}\propto {\frac {1}{b^{2}}}$, that is to say, the energy per unit mass, is, in two-dimensional motion, inversely as the area of the square elements cut off by the $\psi$, $\phi$ lines. But the mass of fluid contained in these elements is directly as their area, or varies as $b^{2}$, consequently the kinetic energy in each element is proportional to $b^{2}\times {\frac {1}{b^{2}}}$, which is constant; therefore:
The kinetic energy contained in each element cut off by lines of equal increment of $\psi$ and $\phi$ is constant.
In a $\psi$, $\phi$ diagram, such as Fig. 48, the total kinetic energy is thus measured by the total number of squares, and the kinetic energy in any circumscribed region is equal to the number of squares in that region. In order to give an absolute value to the energy on this basis it is necessary that the quantity of energy in some particular square element should be known.
The kinetic energy in the field of flow round a body in motion is imparted to the fluid when the body is started from rest, and is given up when the motion is arrested. The effect of the fluid motion is thus to add to the apparent inertia of the body, so that a given force requires to act through a greater distance to impart a given velocity than for the same body in vacuo. Not only has a force to act for a greater distance, but also for a longer time, which means that the body possesses in effect a greater store of momentum for a given velocity. In reality, however, such increase of momentum is only apparent; the momentum of the body and fluid system combined is actually less than that of the body at the same velocity in vacuo by the amount due to its fluid displacement. That is to say, if the body be of the same specific gravity as the fluid the total dynamic system possesses no momentum at all, whatever the velocity. This apparent paradox is accounted for by the fact that during the period of application of force ​to the body an equal and opposite force has to be applied to the external boundary of the fluid; thus, if a stream-line body of the same specific gravity as the fluid be started from rest from a boundary surface, during the application of the accelerative force there will be a region of diminished pressure in the neighbourhood, whose sum is in effect of equal value and opposite sign to the applied force. (Compare Chap. I., § 5.)
§ 82. Pressure Distribution. Fluid Tension as Hypothesis.—The distribution of pressure in the field of flow of a fluid in a state of steady motion can be ascertained immediately from the distribu- tion of kinetic energy if we assume the principle of work.
The change in the velocity of any element of the fluid in passing from one to another part of the field is due to the difference of pressure on its boundary surfaces, and consequently, on the principle of Torricelli (which follows from the assumption of conservation of mechanical energy), the difference of pressure between any two points is that of the difference of "head" corresponding to the values of the velocity at the two points. Thus if the pressure where the motion is nil be taken as zero, the pressure at every point in the field will be proportional to $-(v^{2})$.
Now a minus pressure constitutes a tension, a kind of stress that actual fluids can only support within very narrow limits; we may, however, by subjecting the whole field to a superposed hydrostatic pressure $p$ of sufficient magnitude, do away with minus pressure throughout the region, the condition being that for every point $p\ {\boldsymbol {-}}\ nv^{2}$ is positive, where n is a constant. The pressure under these circumstances becomes, where the motion is nil, equal to the applied hydrostatic pressure $p$.
The objection to the existence of a tension of any desired magnitude in the fluid is entirely based on the behaviour and properties of real fluids, with which we are not for the time being concerned; it is a mere matter of hypothesis and definition to provide that the ideal fluid shall support without cavitation any ​tension whatever, and it leads to some simplification from a physical point of view to make this assumption and deal with the tension system that results. The consequences are the same whether the superposed pressure be taken account of in the ideal fluid, or whether it be regarded merely as a mechanical detail necessary to carrying the theory into the realms of reality.
We already know that the kinetic energy varies everywhere as $v^{2}$, and we now have it that the tension also varies everywhere as $v^{2}$ (pressure and tension being the same quantity but of reversed sign), consequently the tension on the fluid is everywhere proportional to the kinetic energy, that is the total tension on each element of the $\psi$, $\phi$ plotting is constant.
In the interpretation of this and the corresponding result as to energy the two-dimensional diagram must be regarded as consisting of a slice of unit thickness, the energy increment being that contained in the element consisting of the volume cut off by adjacent surfaces, the corresponding tension being measured over the surface of the element.
§ 83. Application of the Theorem of Energy.—A simple example of the application of the energy theorem is found in the case of a circular cylinder of infinite length in steady motion in an infinite region containing fluid.
Let Fig. 49 represent the cylinder in cross-section with the external field plotted for $\psi$ and $\phi$ with respect to space. Let the cylinder be supposed to consist of a thin shell filled with fluid having the necessary motion of translation only; then let the $\psi$, $\phi$ lines be plotted for the fluid within the cylinder as shown. Now if we count the complete squares within a quadrant, internal and external to the cylinder, the number is equal; further, for every part of a square internal to the sin-face there is a corresponding part external to the surface, and these fractional squares may be made as unimportant as we please by choosing increments of $\psi$ and $\phi$ small enough, consequently the energy external to the ​cylinder is equal to the energy internal to the cylinder; that is to say:—
The energy in two-dimensional motion about a circular cylinder having a motion of translation through a fluid is equal to the energy of motion in the cylinder itself, for equal densities, or the energy internal and external are as the respective densities of the cylinder and the surrounding fluid. Fig. 49.
§ 84. Energy of Superposed Systems.—The superposition of systems of flow containing energy may in certain cases result in the addition of their separate energies of motion, but it is evident that this is the exception rather than the rule. The energy of two combined systems is given by the number of $\psi$, $\phi$ elements in the combined field.
In the special case, for example, of the superposition of two motions of translation at right angles, as along the axes respectively of $x$ and $y$, it is found that the energy of the combined field is the sum of the separate energies, a fact which is otherwise obvious (Eucl. 47, L). In general, it can be shown that, if on a general motion of translation he superposed any system of flow whose mean velocity in the direction of the translation is zero, the energy of the resultant is the sum of the energies of the component fields.
Let us suppose the translation to take place along the axis of $x$, and let the velocity of translation be $U$; let the $x$ component of the velocity of the superposed system be $u$ a variable in respect of $x$, $y$ and $z$. Then the mass of each small element of the fluid is $\rho \ \delta x\ \delta y\ \delta z$, and the energy of the combined field is ${\frac {1}{2}}\ \Sigma \ \rho \ \delta x\ \delta y\ \delta z\ (U+u)^{2}$, but $\Sigma \ \rho \ \delta x\ \delta y\ \delta z\ u$ is zero; we therefore have energy $={\frac {1}{2}}\ m\ U^{2}+\Sigma \ \rho \ \delta x\ \delta y\ \delta z\ u^{2}$, where $m$ is the total mass; which proves the proposition in respect of motion along the axis of $x$. But the energy of any components of the superposed motion in the direction of the axes of $y$ and $z$, which may be regarded as translations at right angles to the main motion, we have already seen also comply. Therefore the total energy is the sum of the components.
§ 85. Example: Cyclic Superposition.—An example may be given in two-dimensional motion in the case of the cyclic superposition (Fig. 48). We know that the energy contained in a case of cyclic motion around a cylinder or cylindrical filament in space is infinite, for the linear size of the $\psi$, $\phi$ squares forms a geometrical progression, and any finite number of such squares, however ​great, may be circumscribed by a circle of finite diameter; that is to say, no finite quantity of energy, however great, will cover the whole field. When the diameter of the filament becomes zero the energy internal to any line of flow also becomes infinite, but so long as we regard the motion as cyclic, we are not entitled to regard the filament as of zero diameter; it is legitimate to suppose the filament of very small diameter, so small as not, by its size, to affect the superposed motion of translation.
Now since a pure cyclic motion round a fixed filament does not result in any displacement of the fluid in translation, its mean velocity in each of the co-ordinate directions of space is zero. Consequently, if such a motion be superposed on one of pure translation, the energy of the combined system is the sum of the separate energies and is infinite. Moreover, this result is independent of the energy of the motion of translation (which is never available except to an external system), for if we take the fluid at rest, at infinity (in the $x$ and $y$ directions), and the filament to undergo the translation, the problem is unaffected, and we have proved that to generate a cyclic motion about a filament in motion (Fig. 48) requires the same quantity of energy as to generate the same cyclic motion about a filament at rest, and in both cases where the expanse of fluid is infinite the total energy required is infinite also.
§ 86. Two Opposite Cyclic Motions on Translation.—In the case of the superposition of a system consisting of two cyclic motions of equal value and opposite sign, such as that obtained by the interchange of the functions $\psi$ and $\phi$ in the source and sink system (Fig. 40), the energy is finite, for the system consists of a limited number of squares, and consequently the energy required to generate such a system about two filaments moving uniformly in space is also finite; the resultant stream lines of such a superposition are given in Fig. 50. Such a system possesses a plane of symmetry ${\mathit {A}}\;{\mathit {A}}$, and the motion of the fluid on either side of this plane will be in nowise affected ​by supposing a rigid boundary substituted for the fluid on the opposite side; Fig. 50 may therefore be read as representing the case of a cyclic motion round a filament in the neighbourhood of a plane boundary, superposed on a translation parallel to the boundary surface, and the energy required to produce such motion is finite. Fig. 50.
§ 87. Numerical Illustration.—As a numerical illustration and a check on the foregoing, the author has estimated the energy in the plotting given in Fig. 48, in the region included in the external system within the circular limit indicated, being one of the lines of flow of the cyclic component. The number of squares in the component motions was calculated from the diameter of the circular limit, and the number in the combined system counted, fractions being estimated by a planimeter. The results are as follows:—
| Cyclic component | 336 |
| Translation | 384 |
| Total (sum of above) | 720 |
| Total by measurement | 719.2 |
| Difference (evidently due to unavoidable error in measurement) | .8 |
§ 88. Fluid Pressure on a Body in Motion. — The pressure system about a body in motion in a fluid may be regarded as composed of two distinct component systems, i.e., the accelerative system, being that developed the instant a force is applied to a body at rest, which is essentially identical with the field of velocity potential, and the steady motion system, in which we have seen the fluid experiences a tension everywhere proportional to the energy density (compare "Dynamical Equations," § 59). The first of these is in evidence at the instant when the velocity is nil, as when a motion is started from rest or at the instant it is brought to rest; the second system belongs to the steady state when the disturbance is not subject to acceleration. For intermediate states when motion and acceleration are both present the two pressure systems are found compounded. A good illustration is to be found in the case of a body vibrating in a fluid under the influence of a spring, such as a vibrating rod. At the moment such a body is at the end of its motion, when the accelerative force is greatest, the pressure system is that due to the field of velocity potential; when it is in mid-stroke, that is when its velocity is greatest, the pressure system is that of steady state and follows the law already given.
The accelerative pressure system may (as has been already stated) be provided for, so far as the effect on the motion of the body is concerned, by the supposition of an appropriate addition to the mass, and the extent of this addition has already been given in certain typical symmetrical cases on the basis of the energy of the disturbance. When the impulse, as in the case of an oblique plane, is not in the direction of motion, it is not possible to account for the whole effect on the added mass basis, and in fact it is difficult to obtain a clear conception of the physical aspect of the problem in such unsymmetrical cases, and in general the solution is wanting. It would appear in the case of a plane possessing in itself no mass, that the motion on the application of a normal impulse borders on the indeterminate, for a tangential component, however small, would result in an ​indefinite velocity in an edgewise direction being superposed. It will be seen that in such a case as this the corresponding state of steady motion is unstable, unless a torque be supposed applied from without.
It is established that in the perfect fluid any body in steady motion, no matter what its form, experiences no resistance whatever in the direction of its flight, that is to say, the sum of the longitudinal components of pressure on its posterior surface is equal to the sum of those on the anterior surface. It is only in certain symmetrical cases, however, that the conditions of motion are stable without a force or forces applied to the body.
§ 89. Cases fall into Three Categories.—Taking the body and fluid as a combined system, cases fall naturally into three distinct categories: Firstly, those in which the fluid motion is in effect symmetrical, in which case the motion is in equilibrium without any applied force (this includes cases of unstable as well as cases of stable equilibrium). Secondly, cases in which the body is unsymmetrical and in which the motion involves the application of a couple or torque. Thirdly, cases in which cyclic motion is present and in which the motion involves a transverse force. Cases may occur which fall into both categories 2 and 3.
The first category has been sufficiently dealt with already in the present chapter and in Chap. I; the second is typified by the case of the inclined plane, and in a generalised form has been investigated by Kirchhoff, who has pointed out that there are three mutually perpendicular directions for any solid, in which, if it be set in motion and left to itself, the motion will continue indefinitely; in general it has been shown that one only of these directions is stable, the other two represent cases of unstable equilibrium. Generally speaking, a body having an aspect of greatest area such as an oblate spheroid, or a plane disc, tends to move “broadside on,” and if its motion at any time is disturbed it will oscillate about such natural “aspect of ​equilibrium,” unless a restraining couple of sufficient magnitude be applied.
The third category possesses a particular interest in relation to aerial flight. The transverse force is characteristic of cyclic motion and is found as a consequence of the superposition of a cyclic motion on a translation, as in Fig. 48. It is due to the greater tension on the upper than the under surface of any circuit, such as that of the solid of substitution, $\alpha \ \alpha \ \alpha$; this difference of tension is indicated by the numerical superiority of the $\psi \ \phi$ squares in the region adjacent to the upper surface.
The connection between cyclic motion and a transverse force can be independently established by taking the transverse force as hypothesis and proving cyclic motion as a consequence.
§ 90. Transverse Force Dependent on Cyclic Motion — Proof.—Let $A\ B$ (Fig. 51) be successive positions of the body or filament at the beginning and end of a short interval of time, to which the transverse force is applied. Let it be granted that the filament exert a force $F$ on the fluid at right angles to its direction of translation, and let us suppose that this force be sustained by a distributed system of forces, $f\;f\;f_{1}\ f_{1}$, etc., acting from the boundary of the region, and let the line $S\ S$ represent the mean position of the force $F$ during the period under consideration.
$F$
$f\ f$
$f_{1}\ f_{1}$
$S\ S$
$S\ S$
§ 91. Difficulty in the Case of the Perfect Fluid.—In any actual fluid there can be no difficulty. If, for example, we suppose a plane of infinite lateral breadth gliding edgewise through the fluid to have a force applied at right angles to the direction of motion, this force is borne immediately by the fluid, and the conditions necessary to the development of the cyclic system are fulfilled. In a perfect fluid, however, a plane can move without resistance in any aspect, and thus it is not possible to generate a difference of pressure between its two sides except for the period whilst the normal component of its velocity is undergoing acceleration. Being limited in this manner, the quantity of energy disposable for the production of cyclic motion would appear to be strictly limited, and consequently we may form the following conclusions:—
(1) In an infinite fluid where a cyclic motion, however weak, possesses infinite energy, it will be impossible to generate cyclic motion.
(2) In a finite region it would appear possible that cyclic motion may be induced by a body whose normal motion is accompanied by kinetic energy and which therefore exerts a pressure on the fluid while it is acquiring lateral motion under the influence of the applied force; a portion of the applied force being eventually borne by the cyclic motion developed.
(3) Assuming (2), the more limited the region the less the body will yield to the applied force in the production of the cyclic motion necessary to give rise to an equal and opposite reaction.
Let us take the case of a cylindrical body of fluid rotating en masse about its axis. Then we may regard such motion as approximately composed of a number of cyclic motions superposed, and with their internal boundaries removed. Let us assume the cylindrical space to be subdivided by a number of concentric cylindrical surfaces, such as the lines of flow of a cyclic system, and, beginning at the centre, let us suppose a cyclic system to be started about a filament so that the velocity at the surface of the filament is that of the rotation. Then, taking the next concentric surface and treating it as a boundary, let us suppose a further cyclic system to be superposed on the first so that the velocity at the surface in question becomes that of the rotation, and again with the next concentric surface, and so on; then by taking the concentric surfaces sufficiently close to one another the motion of the fluid in rotation can be approximated to any desired degree. So long as the boundaries be supposed to exist the system is a superposed series of cyclic motion; if the boundaries be supposed withdrawn the motion is one of uniform rotation.
Now let us suppose such a system superposed on a motion of translation. Each cyclic system will give rise to a transverse resultant force on its boundary so that we shall have forces acting throughout the fluid occupied by the rotation. It is here assumed that the fluid is constrained to follow the paths of motion as geometrically laid down as the result of superposition, and it is shown that such constraint involves forces acting from without distributed over the whole region occupied by the rotation, a thing which under the conditions of the hypothesis is impossible of achievement.
§ 93. Vortex Motion.—It is unnecessary in the present work to do more than give a general description of vortex motion and vortices, and discuss their properties so far as bearing on the present subject.
Reference has already been made to vortex motion in § 72, where the character of the motion in a vortex filament is dealt with, and it is shown that such a filament possesses rotation, and the relation area $\times$ angular velocity $=$ constant is established.
The most common form of vortex motion is found in the vortex ring, familiar from the easy manner in which such rings can be produced in smoke-laden air (the smoke being necessary to render the rings visible), either by ejecting tobacco smoke from the mouth or by employing a simple apparatus consisting of a box having a circular aperture on one side and a loose diaphragm on the other. Vortex rings of great size may frequently be seen when a salute is being fired from guns of large calibre.
The motion in a vortex ring resembles that of an umbrella ring being rolled on its stick, only the rotation is in the reverse direction—that is, as if the ring were being rolled inside a cylinder; the fluid is, so to speak, being eternally turned inside out, with a motion of translation superposed. The superposed translation is necessary to its equilibrium.
In real fluids the rotation is not concentrated at the axis as in the case discussed in § 72, but is distributed about the axial region or core. As a matter of convention in the perfect fluid, it is usual to suppose the core to be in a state of uniform rotation—that is, to have constant angular velocity, and the motion of the part external to the core to be cyclic and irrotational, there being no discontinuity at the surface of the core, the velocity $(u\ v\ w)$ being a continuous function of the position $(x\ y\ z)$.
By this convention the core behaves as a solid body, since in an inviscid fluid under no circumstances can its rotation be destroyed or transferred.
We might equally suppose the core to consist of a void space, a ​region of cavitation in fact, whose pressure is zero. Spiral vortices of this type occur when a screw propeller gives rise to cavitation, the void being filled with water vapour. In other cases the core is constituted by a region filled with some other fluid. We again find an example in the motion produced by a screw propeller when the tips of the propeller emerge from the surface and carry down with them into the water a quantity of air—such vortices may frequently be seen astern of a vessel when steaming under a light load.
A vortex cylinder or filament may be regarded as a portion of a vortex ring of infinite diameter; it can only exist either if infinitely long or if its ends terminate on boundary surfaces. Fig. 52.
A single straight filament in infinite space is theoretically stable without motion of translation; two such filaments in the neighbourhood of one another mutually interact, and are only stable with superposed motion.
The superposed motion proper to two filaments depends upon their relative position; parallel filaments of like rotation rotate round one another at a velocity proper, each to each, to the cyclic motion of its neighbour, as in Fig. 52 $(A)$; parallel filaments with counter-rotation are in equilibrium when possessed of motion of translation (Fig. 52 $(B)$); when two such vortices are equal to one another the combination is termed a vortex pair, and ​the direction of the translation is at right angles to the plane containing their centres.
A vortex filament in the neighbourhood of a plane boundary surface behaves as if it can see its own reflection, that is, as if such reflection were another vortex filament.
A vortex ring may be looked upon as the analogue of a vortex pair in three dimensions, i.e., the mutual interaction of its parts results in a motion of translation, the translation taking place at right angles to the plane of the ring.
Two concentric co-axial vortex rings tend to behave as two similarly rotating filaments, i.e., revolve round one another; the consequence of such a motion under the changed conditions is that the two rings alternately overtake and pass through one another, the process being repeated and going on indefinitely. Rings behaving in this way are sometimes said to play "leap-frog."
Groups of filaments or rings behave in a similar manner to pairs: thus a group of rings may play "leap-frog" collectively so long as the total number of rings does not exceed a certain maximum; congregations of vortex filaments likewise by their mutual interaction move as part of a concerted system, like waltzers in a ball-room; when the number exceeds a certain maximum the whole system consists of a number of lesser groups.
In general, beyond the special features above described, the motion and behaviour of vortices and vortex rings presents much in common with that of solid bodies; thus two vortex rings on impact bounce off from one another like two perfectly elastic solids, and we have the Vortex Atom theory first propounded by Lord Kelvin (Sir William Thomson), and subsequently extended by Professor J. J. Thomson (ref. "Motion of Vortex Rings," Macmillan, 1883).
§ 94. Discontinuous Flow.—Up to this point the assumption has been made that the continuity of the fluid cannot be broken. As a working hypothesis, the fluid has been defined as capable of sustaining stress in tension; it has at the same time been pointed out that the equivalent result may be obtained by supposing the fluid to be subjected from without to a hydrostatic pressure superior to the greatest negative pressure (tension) due to its motion at any point throughout the region.
We will now suppose that the fluid is not capable of sustaining tension, and that the external hydrostatic pressure is either wanting or is insufficient to prevent cavitation.
The importance of studying these conditions does not rest so much upon the possibility of actual cavitation, as upon the general resemblance of the resulting systems of flow to those encountered where real fluids are concerned. It is evident that the void regions in the examples we are about to discuss may be supposed filled either with some different fluid, or even with inert masses of the same fluid as that in which the motion is taking place.
Let the area of the efflux jet be $A$; let s be the “head” of liquid whose density is $\rho$; let $r$ be the efflux velocity.
Let it be assumed that the pressure within the vessel is everywhere due to the hydrostatic head,—that is to say, let us suppose that the motions of the fluid within the vessel do not affect the pressure on its surfaces.
Then $r={\sqrt {2\ gs}}$, and mass of fluid passing out per second $\rho Av=\rho A{\sqrt {2\ gs}}$, or,
Momentum per second $=r\times \rho A{\sqrt {2\ gs}}=2\ \rho Ags,$
which is the reaction on vessel due to the “recoil” of efflux.
But pressure per unit area on wall of vessel at level of aperture $=\rho gs$, or, if $\alpha =$ area of wall of vessel on which pressure is relieved,
$\alpha ={\frac {2\ \rho Ags}{\rho gs}}=2\ A.$
That is to say, on the above assumption the aperture in the wall of the vessel is twice the area of the resulting jet.
When the aperture is a simple hole in the wall of the vessel (Fig. 53, A), the assumption is not strictly accurate, for the pressure in the region surrounding the hole is less than that due to the hydrostatic pressure owing to the converging of the lines of flow, and consequently the actual hole is of less area than that over which the pressure is effectively relieved, and the jet contracts less than the simple theory would indicate.
§ 96. The Borda Nozzle.—The conditions of hypothesis are most nearly conformed to by the Borda re-entrant nozzle (Fig. 53, B), in which the aperture is furnished with a short tube extending inward. Such an arrangement ensures, as closely as is possible in practice, that the pressure on the walls of the vessel is unaffected by the motion of the fluid. Experimenting with a circular cylindrical nozzle, Borda (1766) obtained the result $\alpha =1.94\ A,$ which is in sufficiently close agreement with theory. It is more usual to invert this expression, writing $A=.515\alpha .$
The complete solution of the path of flow at the free surface ​has been effected, in the case of the Borda nozzle in two dimensions, by Helmboltz, and may be found in Lamb's “Hydrodynamics,” where the solution is also given in the case of a simple two-dimensional aperture; the calculated coefficient in the latter case is .611, which does not differ hopelessly from the experimental value, usually taken for two-dimensional flow to be about .635.
We may evidently suppose the efflux to take place into a vacuous region, or into one filled with air, or even from one vessel containing liquid into another containing the same kind of liquid; Fig. 53. the only obvious condition would appear to be that the pressure at all points on the surface of the jet should be constant. Such a system of flow bears a considerable resemblance to that which actually occurs in the case of any real fluid, but on the assumption of continuity it is not the form of flow given by mathematical theory in such a case. If the edges of the aperture are taken to be infinitely sharp, then the discrepancy can easily be accounted for, as the velocity at the sharp edge becomes infinite, and consequently an infinite hydrostatic pressure will be necessary to prevent cavitation, which is not possible; the conditions of hypothesis are therefore departed from. This, however, is not the full explanation, for the flow in practice closely resembles the ​efflux system, even when the edges are given quite an easy radius. Before discussing this difficulty further another example of motion with a free surface may be given.
§ 97. Discontinuous Flow. Pressure on a Normal Plane.—In Fig. 47 the stream lines are given for a normal plane on the assumption of continuity. We now have to deal with the same example under different conditions, the form of flow involving discontinuity; Fig. 54. a stream of infinite breadth impinges normally on a fixed plane, from the edge of which springs a free surface. The solution to this problem is only known in the particular case of two-dimensional motion where the plane is a lamina of infinite lateral breadth, and is, in the main, due to Kirchhoff. The form of the resulting free surface is given in Fig. 54, in which the direction of flow is taken as vertical. The pressure force for one unit width of the lamina is given by the expression ​$.44\ \rho \ V^{2}\ l$, where $\rho$ is density, $V=$ velocity, and $l=$ width of lamina in absolute units. The expression for mean pressure will therefore be:—
$P=.44\ \rho \ V^{2}$
Similarly the case of an inclined lamina has been investigated by Kirchhoff and Rayleigh, and the following are the expressions obtained:—
| ${\frac {P}{\beta }}={\frac {\pi \ \sin \beta }{4+\pi \ \sin \beta }}\ \rho \ V^{2},$ | (1) |
where $\beta$ is the angle of inclination.
For the position of the centre of pressure:—
| $x={\frac {3}{4}}\times {\frac {\cos \beta }{4+\pi \sin \beta }}\times l,$ | (2) |
where $l$ is the width of the plane and $x$ the distance forward of the geometric centre.
The following Table gives the result for different values of $\beta$ (1) pressure on plane in terms of normal pressure, and (2) the proportionate distance of the centre of pressure from the central line:—
| $\beta$ | (1) | (2) |
| 90° 70° 50° 30° 20° 10° 0° | 1.000 .965 .854 .641 .481 .273 .000 | .000 .037 .075 .117 .139 .163 .187 |
§ 98. Deficiencies of the Eulerian Theory of the Perfect Fluid.—The deficiencies of hydrodynamic theory have already been pointed out in several instances and partially discussed. The forms of flow that result from the assumption of continuity' and the equations of motion bear in general but scant resemblance to those that obtain in practice, and it is not altogether easy to account for the cause of the failure. ​If an actual fluid behaved anything like the ideal fluid of theory, the necessity for the ichthyoid form would not exist; any shape, however abrupt, short of producing cavitation, would give rise to stream-line motion and be destitute of resistance. The actual phenomenon of fluid resistance, discussed in the two previous chapters, is characterised by features which at present are not capable of complete elucidation by analytical means.
The principal characteristic in which the actual flow and the Eulerian form differ is as to the existence or otherwise of resistance to motion. In all cases discussed in Chap. I., with the exception of the “stream-line form,” the surface or “stratum” of discontinuity is an ever present feature which is closely related to the resistance encountered by the body in motion. It has been shown that the proneness to develop discontinuity increases the less the viscosity, and it is difficult to understand in what manner the tendency, which grows greater as the value of viscosity approaches to zero, should suddenly cease when zero is reached. This argument may be otherwise stated in the form: It is difficult to understand how a fluid that offers by hypothesis no resistance to shear can assume a rigidity in shear not possessed by a viscous substance.
§ 99. Deficiencies of Theory (continued). Stokes, Helmholtz.—In the year 1847 Stokes, discussing a particular hypothetical case of flow, was the first to suggest the possibility of a discontinuity or “rift” as a phenomenon connected with the motion of the perfect fluid. Helmholtz, writing in 1868 on the “Discontinuous Movements of Fluids” (Phil. Mag., XLIII.), pointed out the familiar instance of smoke-laden air escaping from an orifice as an example in which the motion is not at all in accordance with the hydrodynamic equation, the air moving in a compact stream instead of spreading out, as the theory of the perfect fluid requires. He remarks that such known facts cause physicists to regard the hydrodynamic equation as a very imperfect approximation to the truth, and that “divers and saltatory irregularities, which ​everyone who has experimented has observed, can m no wise be accounted for by the continuous and uniform action of [viscous] friction.”
This does not express the position of affairs one whit too strongly; in fact, before the date of the recent additions to the mathematical theory relating to discontinuous motion (largely initiated by Helmholtz himself), it might almost have been said that the hydrodynamic theory of the text-book had nothing at all to do with the motions of any known liquid or gas.
In the paper in question Helmholtz states that it is necessary to take account of a condition in the integration of the hydrodynamic equations, which had up till then been neglected. In the hydrodynamic equations, velocity and pressure are treated as continuous functions of the co-ordinates, but in reality there is nothing to prevent in a true inviscid fluid two layers slipping past one another with finite velocity. The author of the paper, referring to his previous work on gyratory movement, suggests that the surface of separation is a gyration surface, the conception being that the surface consists of an infinite distribution of lines of gyration at which the mass of fluid is vanishingly small (or evanescent), and the moment of rotation finite. It is pointed out that such a system involves a discontinuity, such as might be initiated by incipient cavitation, and under these circumstances the conditions of mathematical hypothesis are violated. The theory of discontinuous motions, such as outlined, is afterwards dealt with at some length, with results similar to those already given.
§ 100. The Doctrine of Discontinuity attacked by Kelvin.—The theory of discontinuity has been regarded by some authorities as a questionable innovation, and it has been violently attacked by Lord Kelvin in a series of articles to Nature in the year 1894, and so the subject has become a matter of controversy.
So far as the author is aware, this controversy has never been authoritatively settled; it is therefore necessary to give the ​matter more than passing attention and to discuss the subject from its controversial aspect.
In brief, Kelvin's objections appear to consist in the following: (1) Any system of discontinuous flow is inconsistent with his (Kelvin's) theorem of least energy, and therefore cannot exist. (2) That a surface of discontinuity in an inviscid fluid (whose physical continuity is unbroken) is essentially unstable and, if formed, will break up. (3) That in a real fluid possessed of viscosity a surface of discontinuity is impossible.
§ 101. Kelvin's Objections Discussed.—It is certainly true that the discontinuous system of flow violates Lord Kelvin's theorem; it is evident, however, that this theorem rests definitely upon the hypothesis of continuity, and it is precisely this hypothesis that Helmholtz has deliberately set aside. Consequently the objection is without weight.
In considering the behaviour of an inviscid fluid a certain ambiguity exists. Since rotation cannot be imparted to or abstracted from the fluid, there may be an infinite variety of possible forms of flow under given boundary conditions which are ordinarily excluded by hypothesis since they cannot be generated from rest. The Kelvin theorem of least energy is proved only for motions that can be generated from rest, and does not of necessity apply to motions that cannot be so produced.
It is conceivable that if a fluid possessed viscosity in a very small degree only, its motions, if generated and continued for a short period of time, would not sensibly depart from the Eulerian form, but if continued for a long time an entirely different system might eventually be evolved. On this basis, which supposes a cumulative change in the form of flow, the inviscid fluid may, after an infinite lapse of time, develop forms of flow quite foreign to the Eulerian theory, and such forms of flow will obviously be independent of Kelvin's theorem. The supposition of an infinite lapse of time merely constitutes an extension of the hypothesis of the perfect fluid, to simulate as far as possible the conditions ​obtaining in the case of the nearly inviscid fluid, discussed further in § 104.
On the second objection, i.e., the supposed instability of the surface of discontinuity in a perfect fluid, we are treading on very different ground, and reference should be made to Kelvin's article. There is certainly nothing to prevent the supposition of the momentary existence of a surface of discontinuity in an inviscid fluid, and it is difficult to see how it can be destroyed, in view of the fact that it contains rotation which by the theorem of Lagrange can never leave the infinitesimal film of fluid that initially constitutes the surface. It is certain that such a system of flow cannot break up into finite vortex rings, for if the rotation be distributed over a finite quantity of fluid in the core of such vortex rings, the theorem of Lagrange has been violated, and if the rotation be confined to a core that is vanishingly small the energy required to create one such ring is infinite.
§ 102. Discussion on Controversy (continued).—On the third objection, as to discontinuity in the case of the real fluid, it is unnecessary to dwell at length. Neither Helmholtz nor his followers could ever have supposed that the discontinuity exists as a surface under actual conditions, but rather as a stratum containing rotation. It has been elsewhere pointed out (§ 20), that in the case of the real fluid the conception of a surface of discontinuity must be looked upon as an abstraction of that which is essential in a somewhat complex phenomenon, and it is this fact that Kelvin appears to overlook; he points out that the surface will, if formed, break up at once into a series of vortex filaments, or vortex rings, and this view is in all probability correct; it may also be found practicable to assess the pressure reduction on the back of a plate on the basis of vortex theory, as suggested in Kelvin's article. It appears, however, to the author that all this may be considered in the light of an extension rather than a controversion of the Helmholtz theory.
In the course of his criticism Kelvin suggests certain cases of ​motion as constituting an absolute and patent disproof of the doctrine of discontinuity, which in reality do not seem capable of any such interpretation. One of the supposed cases is given in Fig. 55, which represents a projectile having a gap in its mid-body dividing it into two halves which are assumed to be rigidly connected; this has been indicated in the present reproduction by a stem or spindle.
Now it appears to the author that this example can be construed in favour, rather than otherwise, of the Helmholtz doctrine. Let us suppose the gap bridged initially by a telescopic sheath represented by the dotted line and the projectile set in uniform motion in a perfect fluid. Fig. 55. Next let us suppose the sheath to be withdrawn (by sliding it longitudinally), then we have a system of flow involving a surface of discontinuity, a system of flow alternative to that of the ordinary Eulerian theory, and contrary to the theorem of least energy, and one that has many points in common with that which obtains in practice.
§ 103. The Position Summarised.—We may summarise the possible causes of the departure from the theoretical Eulerian form of flow as follows:—
(1) The observed departure is due to viscosity, and:
(a) The departure is less the less the viscosity, as might be readily imagined (to harmonise with the Eulerian theory).
(b) The departure is greater the less the viscosity.
(2) The departure not necessarily connected with viscosity, and either:
(c) Due to cavitation (as suggested by Helmholtz).
(d) Due to compressibility (alternatively suggested by Helmholtz). ​(e) Due to imperfection of boundary conditions (as suggested by Kelvin).
(f) Defect of mathematical hypothesis concerning the nature of an inviscid fluid.
(g) The mathematical demonstration in error.
(h) The experimental observations in error.
(j) Some unaccounted physical conditions.
By a process of exhaustion we dispose of (g), (h), and (j) as highly improbable; (c) and (d) must be considered as insufficient in view of the fact that no cavitation is in general manifest, and a surface of gyration or discontinuity or vortex motion without an internal boundary involves rotation. Alternative (e), suggested by Lord Kelvin, does not seem cap)able of accounting for the facts known to experiment. It seems evident, under ordinary circumstances, that the boundary conditions are a sufficient approximation to theory.
§ 104. The Author's View.—The true explanation is probably to be sought in (1) (b). In all real fluids the influence of viscosity accounts for the departure; and the departure is greater the less the viscosity.
This seems paradoxical; it would appear to denote a sudden change in the behaviour of a fluid when viscosity becomes zero. Such a change would involve discontinuity in the physical properties of a substance, which is scarcely admissible; this paradox is only apparent, for the factor of time is involved in the production of the discontinuous system of flow, and, as will be subsequently shown, the continuity of behaviour extends to the fluid of zero viscosity.
The following conclusions may be formulated:
(2) That the discontinuous system may in a viscous fluid be regarded as arising by evolution from a motion initially obeying the mathematical equations.
(3) That in fluids possessing different values of kinematic viscosity the time taken for the evolution of the discontinuous system is greater when the kinematic viscosity is less, and vice versa.
(4) That the ultimate development of the discontinuous system of flow is more complete the less the value of the kinematic viscosity, and vice versa.
Taking the propositions in order:—
(1) Forces due to viscosity are proportional to velocity: when velocity is nil, such forces have no magnitude, consequently the initial direction of flow is unaffected by viscosity.
(2) In a viscous fluid it is established that the layer adjacent to the surface of a solid is adhesive, i.e., moves as part of the solid—that is to say, the viscous connection between fluid and solid is the same as that between two layers of fluid. Consequently when the flow has been established, there will be a layer of fluid next the solid more or less inert, which will only in a small degree partake of the motion of the dynamic system. Now the surface of the body possesses regions of greater and regions of less pressure, and this inert layer will be steadily pushed along the surface from the regions of greater pressure to those of less. Therefore, taking the typical case of a normal plane, the surface current of fluid so formed will be available to “inflate” the surfaces of hydrodynamic flow in the region of the edges, almost as if the edges of the plane were emitting fluid by volatilisation.
This inflation of the surfaces of flow in regions of least pressure can be conceived to continue until the combined inflated region becomes one whole, the “dead water,” occupying the space in the rear of the plane. Similarly for other forms of body.
(3) The less the viscosity the thinner the inert layer, and, ​other things being equal, the longer it will take to bring about a given degree of inflation.
(4) The viscous drag experienced between the live fluid and dead water tends to carry the latter away, and if the viscosity exceed a certain value, then, other things being equal, it is found in experience (notably in the case of an ichthyoid form) that the dead water may be ejected and carried away as fast as formed by the viscous drag of the surrounding current. Under these conditions it may be taken that viscosity by its direct drag prevents the surface current from flowing in opposition to the main stream, so that the surface current is consistently rearward, the result being an absence of dead water. The surface of discontinuity may be regarded as having coalesced with the surface film of the body. If the viscosity be sufficiently reduced, the surface of discontinuity will detach itself, and in general the less the viscosity the more complete will he the development of the discontinuous system of flow.
Let us now take the case of a fluid bordering on the inviscid. It is evident, firstly, that the change in the system of flow will be very slow; and, secondly, it would appear that the ultimate transformation of the system will be very complete.
Let us now go further and suppose the viscosity of zero value. Then, on the principle laid down in § 101, we may regard the ultimate condition as one involving discontinuity as investigated by Helmholtz and others, with the reservation that it will require an infinite time for its development.
It is by no means certain, however, that the Helmholtz system does actually represent the final form. Since the motion is a matter of infinitely slow development, it is probable that, in spite of the vanishing value of v, the fluid by which the dead water region is being developed will be set in motion just as in the case of a viscous fluid, the motion taking the form of a vortex ring on a core containing rotation, situated immediately in the wake of the plane or body. Such a system is quite in accord with hydrodynamic principles, but does not involve discontinuity and does not in itself give rise to resistance. It is a pregnant fact that, so long as the continuity of the system of flow is unimpaired, Fig. 56. the pressure distribution for uniform motion is that of § 88, and resistance other than that directly due to viscosity is absent.
§ 105. Discontinuity in a Viscous Fluid.—It has already been pointed out that the surface of discontinuity in a viscid fluid must begin to degenerate as soon as formed, owing to the fact that a finite velocity between adjacent layers would betoken an infinite tangential stress. We could suppose the degeneration to take the form of a thickening of the discontinuity so that it becomes a stratum of fluid with a velocity gradient. We can alternatively and with every appearance of probability suppose that the surface becomes a stratum of turbulence. The latter would certainly agree more closely with observation.
Suppose we adopt the suggestion of Lord Kelvin and regard ​the turbulence as initially taking the form of a series of vortex filaments following each other in rapid succession and acting as rollers between the live fluid and dead water (on this point Kelvin does not differ materially from Helmholtz), and if we represent the resulting system of flow as in Fig. 56, in which the motion is given diagrammatically relatively to the vortex rollers, so that the apparent motion of the fluid on the two sides is opposite, then, making certain assumptions, we can obtain some results from dimensional theory.
Let it be granted that for different values of $V$ and $\nu$ the size of the individual rollers may vary, but the form of the disturbance is homomorphous.
Let $\omega$ be the angular velocity of a roller taken at some stated point on some specified line of flow; then,
| | $\omega =(F)\ V,\ \nu .$ |
| As in § 38, let us write | $\omega ^{x}=V^{y},\ \nu ^{z}.$ |
| Dimensionally | ${\frac {1}{T^{x}}}={\frac {L^{y}}{T^{y}}}\ {\frac {L^{2z}}{T^{z}}}$ |
| Thence we have | $x=y+z,$ |
| and | $y={\boldsymbol {-}}2\ z.$ |
Taking $x=1,$ we obtain $y=2;\;z={\boldsymbol {-}}1.$
Hence the expression becomes $\omega =V^{2}\ \nu ^{-1}\ \times$ constant, or $V^{2}=\omega \nu \ \times$ constant. Taking $r$ for the radius of the roller at the point chosen, we can write this expression in the form,—
$r={\frac {\nu }{V}}\ \times$ constant.
§ 106. Conclusions from Dimensional Theory.—From the above expression the following conclusions may be drawn:—
In different fluids ceteris paribus the size (diameter) of the rollers will vary directly as the kinematic viscosity. Hence in an inviscid fluid the rollers will become of vanishingly small diameter, or the surface containing them will be a surface of gyration of Helmholtz, that is a surface of discontinuity.
In a fluid of given kinematic viscosity, the size of the rollers will vary inversely as the velocity, that is the velocity difference between the live stream and the dead water.
In a given fluid the frequency with which the vortices are generated will vary as the square of the velocity. It is probable that we have in this the origin of the “pitch note” that may be heard when a body is in rapid motion through the air, for example in the swish of a stick or the whistle of a projectile.
The foregoing conclusions are only strictly applicable so long as the vortex rollers are of small diameter compared to the body by which they are generated, for otherwise the motion will not be homomorphous, as required by hypothesis. It is probable that it is the relation between the size of the vortex rollers and that of the body that determines the point at which the discontinuous form of flow begins. Thus for velocities less than a certain minimum in any given fluid the value of $r$ will be so great that there is no room for the vortex to form; at a higher velocity it seems likely that a single vortex may be generated, which will follow in the wake of the body, as in § 104, and it will only be at velocities in excess of this that the vortices will detach themselves in accordance with the régime contemplated. The precise conditions must, however, be regarded as uncertain.
The ultimate fate of the vortices formed in the peripheral region of the wake is not altogether known; it would appear that they will break up into groups and sub-groups, after the manner described in § 93, till the whole wake of ““dead water” becomes a region of seething turbulence, the motion gradually becoming incoherent and dying out as the energy is absorbed in viscous strain.
* ↑ For the full mathematical treatment reference should be made to "Hydrodynamics," H. Lamb, Camb. University Press.
* ↑ An erratum published in volume 2 has been applied: "P. 85, line 8 from top, the cavity of the labyrinth of the ear is given as an example of a triply connected region; this is in error, delete." The original text reads: "The cavity of the labyrinth of the human ear is a triply connected region, as also is a lake containing two islands" (Wikisource contributor note)
* ↑ The mathematical demonstration of this important fact will be found in "Hydrodynamics" (H. Lamb, Cambridge), or reference may be made to the original investigation (Lagrange, "Oeuvres," T. IV., p. 714).
* ↑ The velocity potential may fall to zero in a portion of the fluid in the course of its motion without that portion of the fluid losing the attribute of velocity potential in the sense of Lagrange's theorem.
* ↑ Referring to the diagram to the right of Fig. 39, we have from geometrical considerations the normals g and j respectively proportional to the sides of the parallelogram h and k.
* ↑ Lamb, “Hydrodynamics,” pp. 56, 57 (3rd ed.).
* ↑ In the paper from which quotations are given it would appear that Prof. Rankine believed there to be some particular virtue in the forms derived from the special case of the simple source and sink system, that the stream lines of such a system constitute in fact natural water lines. In actuality ichthyoid or stream-line form is governed by conditions not yet amenable to rigid treatment, and the design of a stream-line form to work in a real fluid with a minimum of resistance is largely a matter of art. The underlying principles have been discussed in the previous chapters.
* ↑ “Principles Relating to Stream Lines,” The Engineer, October 16, 1868.
* ↑ An erratum published in Volume 2 has been applied: "P. 113, line 4 below Fig. 50, for 'were' read 'was.'" (Wikisource contributor note)
* ↑ For a full exposition of the theory of this branch of the subject, reference should be made to Lamb's “Hydrodynamics,” Chap. VI., and numerous references therein cited; also “Nat. Phil.,” Thomson and Tait, 313, 320.
* ↑ Conclusions (2) and (3) may be taken as provisional, pending proof or disproof on analytical lines. The inviscid fluid of Eulerian theory is a very peculiar substance on which, to employ non-mathematical reasoning. It is quite likely that in the inviscid fluid the dynamic conditions are satisfied without the production of cyclic motion under any circumstances.
* ↑ The present description of vortices and vortex motion is a bare statement of the most elementary facts of the subject. Most of that which is known will be found in the writings of Helmholtz, Kelvin, and J. J. Thomson, and a mathematical resumé, with copious references, in Lamb's "Hydrodynamics," Chap. VII.
* ↑ Gravity is assumed to be inoperative. In Fig. 54 the free surface only is an actual plotting; the stream lines are merely an indication of the character of the flow.
* ↑ The compressibility of a fluid does not enable it to evade Lagrange's theorem.
* ↑ Lord Kelvin in his article suggested the possibility of the boundary conditions being affected by the formation of bubbles at and in the region of sharp corners; but this cannot apply in the case of a gas.
* ↑ This is, as it evidently should be, the same expression as determined generally for homomorphous motion in § 38, $r$ being the linear dimension.