Frederick William Lanchester

“ Rotation. Irregular Distribution. Irrotation, —Definition.—The propositions connecting boundary circulation and rotation include all cases of rotation, so that we know that however much the rotation differs in different parts of the fluid, the algebraic sum of the rotation taken over the whole of the region is equal to the integration of the circulation along the boundary (reckoned plus or minus, according to the law laid down) . 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 ”

“ 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. ”

“ 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 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. ”

“ Physical Interpretation of Lagrange's 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.” ”

“ 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. ”

“ 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. ”

“ 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 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. ”

“ 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 being a continuous function of the position . ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 , 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. ”

“ 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. ”

“ 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 any one of the lines being arbitrarily chosen as datum. ”

“ 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. ”

“ 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. ”

“ 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, , 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 . ”

“ 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. ”