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To find the total rate of dissipation of energy we must multiply this by 2πr sin Orde dr, and integrate with respect to 0 between the limits 0 and π, and with respect to r between the limits a and ∞. The final result is

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Now if P be the force which must act on the sphere in order to maintain the motion, the rate at which this force works is PV, whence

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In the case of a sphere of mean density o falling under the action of gravity, P is the excess of the weight oa'g over the buoyancy πра3g, so that the terminal velocity is given by

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For a globule of water, of 001 in. radius, falling in air, Stokes finds*,

V = 1.59 inches per second.

* The value of μ employed in this calculation was deduced from Baily's experiments on pendulums, and is about half as great as that found by Maxwell (Art. 178). If we accept this latter value, the above value of V must be halved.


For a sphere of one-tenth this size, the value of V would be one hundred times less. The viscosity of the air is therefore amply sufficient to account for the apparent suspension of clouds, &c.

185. If we suppose that there is a finite amount of slipping at the surface of the sphere, the conditions (37) must be modified. Regarding the sphere as in motion, and the fluid at rest at infinity, the conditions to be satisfied when r = a are

R=Vcos 0, and 2μ [f] =ẞ(+V sin 0),

where is the coefficient of sliding friction.

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These give

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Bal a

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Substituting these values in (42), we obtain for the resistance

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When B∞ this reduces to (43). Whatever the value of ẞ may

be, P always lies between 4πμα V and 6πμαν.



WHEN, as in the present subject, and in the cognate theory of Elastic Solids, we study the changes of shape which a mass of matter undergoes, we begin by considering the whole mass as made up of a large number of very small parts, or 'elements,' and endeavouring to take account of the motion, or the displacement, of each of these.

If we inquire however to what extent this ideal subdivision is to be carried, we are at once brought face to face with questions as to the ultimate structure and properties of matter. We have, in the text, adopted the hypothesis of a continuous structure, in which case the subdivision is without limit.

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The truth of this hypothesis is, however, not probable. It is now generally held that all substances are ultimately of a heterogeneous or coarse-grained structure; that they are in fact built up of discrete bodies or molecules,' separated, it may be, by more or less wide intervals. These bodies are far too minute to admit of direct observation, so that we are almost wholly ignorant of their nature and of the manner in which they act on one another. It would therefore be futile to attempt to form the equations of motion of individual molecules, and even if the equations could be formed and integrated the results would not be directly comparable with observation, nor would they even be of interest from the point of view of our present subject. For our object is not to follow the careers of individual molecules, which cannot be traced or identified, but to study the motions of portions of matter which, though very small, are still large enough to be observed, and

* Thomson and Tait, Natural Philosophy, Art. 675.

which therefore necessarily consist each of an immense number of molecules.

In order then to establish the fundamental equations in a manner free from special hypothesis as to the structure of matter we adopt the following conventions.

We suppose the 'elements' above spoken of to be such that each of their dimensions is a large multiple of the average distance (d, say,) between the centres of inertia of neighbouring molecules, and also of the average distance (8, say,) beyond which the direct action of one molecule on another becomes insensible*. The latter proviso is necessary in order that the mutual forces exerted between adjacent elements shall be sensibly proportional to the surfaces across which they act. Observation shews that we may suppose the dimensions of an element to be at the same time so small that the average properties of the constituent molecules vary regularly and continuously as we pass from one element to another. We shall, in what follows, understand by the word 'particle' or 'element' a portion of matter whose dimensions lie within the limits here indicated. The properties of an element surrounding any point P may then be treated as continuous functions of the position of P.

The 'density' at any point of the fluid is now to be defined as the ratio of the mass to the volume of an elementary portion surrounding that point.

The 'velocity at a point' is defined as the velocity of the centre of inertia of an elementary portion taken about that point. This is of course quite distinct from the velocities of the individual molecules, which may, and in all probability do, vary quite irregularly from one molecule to another.

By the 'flux' across an ideal surface situate at any point of the fluid we shall understand the mass of matter which in unit time crosses unit area of the surface, from one side to the other. Matter crossing in the direction opposite to that in which the flux is estimated is here reckoned as negative.

The flux across any surface at any point is equal to the product of the density (p) into the velocity (q, say,) estimated in the direction of the normal to the surface. This is sufficiently obvious if the fluid be supposed continuous, or even if it be molecular, provided that in the

* It is supposed that in gases d is large compared with 8; the reverse is probably the case in liquids and solids.

latter case we assume the velocities of all the molecules within an element to agree in magnitude and direction. To prove the statement when the velocity is supposed to vary quite irregularly from one molecule to another, we have only to suppose the molecules contained in an elementary space to be grouped according to their velocities, so that the velocities of all the members of any one group shall be sensibly the same in magnitude and direction. Let p, q, be the values of the density, and of the velocity in the direction of the normal to the given surface, corresponding to the first group alone, P,, 9, the corresponding values for the second group, and so on. The part of the flux due to the first group is P121 that due to the second is p22, and so on, so that the total flux in the given direction is

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which is, by the definition of the symbols, = pq.

Hence the flux across any portion of a surface every point of which moves with the fluid is zero.

The foregoing considerations and definitions apply alike to solids and to fluids. But if we attempt to follow the motion of an element we are met by a difficulty peculiar to our present subject. We cannot assume that the molecules which at any instant constitute an element continue to form a compact group throughout the motion. On the contrary the phenomena of diffusion shew that such an association of molecules is gradually disorganized, some molecules being continually detached from the main body, whilst others find their way into it from without. Thus although the matter included by a small closed surface moving with the fluid is constant in amount, its composition is continually changing. It is true that in liquids this process is exceedingly slow, and might fairly be neglected when regarded from our present point of view. In the case of gases it must however be taken into account, in consequence of the much greater mobility of their molecules.

The phrase 'path of a particle,' often used in the text, must therefore now be understood to mean the path of a geometrical point which moves always with the velocity of the fluid where it happens to be. In the case of a liquid this will represent with considerable accuracy the path of the centre of inertia of a definite portion of matter.

The effect of the foregoing definitions is to replace the original (molecular) fluid by a model, made of an ideal continuous substance, in which only the main features of the motion are preserved. The correspondence of the model to the original is however as yet merely

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