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(k) It is remarkable that
(dy dß\ JJ
dß da dz \dz dx) dx dy, is the same for all surfaces having common curvilinear boundary; and when a, b, y are the components of a displacement from x, y, z, it is the entire tangential displacement round the said curvi. linear boundary, being a closed curve. It is therefore this that is nothing when the displacement of every part is non-rotational. And when it is not nothing, we see by the above propositions and corollaries precisely what the measure of the rotation is.
(1) Lastly, We see what the meaning, for the case of no rotation, of S(adac + Bdy + ydz), or, as it has been called, “ the displacement function," is. It is, the entire tangential displacement along any curve from the fixed point o, to the point P (x, y, z). And the entire tangential displacement, being in this case the same along all different curves proceeding from one to another of any two points, is equal to the difference of the values of the displacement functions at those points.
"Equation 191. As there can be neither annihilation nor generation of con tinuity."
of matter in any natural motion or action, the whole quantity of a fluid within any space at any time must be equal to the quantity originally in that space, increased by the whole quantity that has entered it and diminished by the whole quantity that has left it. This idea when expressed in a perfectly comprehensive manner for every portion of a fluid in motion constitutes what is called the "equation of continuity,” an unhappily
chosen expression. Integral 192. Two ways of proceeding to express this idea present equation of continuity. themselves, each affording instructive views regarding the pro
perties of fluids. In one we consider a definite portion of the fluid; follow it in its motions; and declare that the average density of the substance varies inversely as its volume. We thus obtain the equation of continuity in an integral form.
Let a, b, c be the co-ordinates of any point of a moving fluid, at a particular era of reckoning, and let x, y, z be the co-ordinates of the position it has reached at any time t from that era. To specify completely the motion, is to give each of these three rarying co-ordinates as a function of a, b, c, t.
Let da, 86, dc denote the edges, parallel to the axes of co-ordi- Integral
equation of nates, of a very small rectangular parallelepiped of the fluid, when continuity. t = 0. Any portion of the fluid, if only small enough in all its dimensions, must ($ 190, e), in the motion, approximately fulfil the condition of a body uniformly strained throughout its volume. Hence if da, 86, 8c are taken infinitely small, the corresponding portion of fluid must (8 156) remain a parallelepiped during the motion,
If a, b, c be the initial co-ordinates of one angular point of this parallelepiped : and a + da, b, c; a, b + 86, c; a, b, c + 8c; those of the other extremities of the three edges that meet in it: the co-ordinates of the same points of the fluid at time t, will be
Hence the lengths and direction cosines of the edges are respectively
doc (dx dy dz2\t
etc. (da * da' * da' (dx dy' da
+ \da2 das
The volume of this parallelepiped is therefore de dydz dxdy dz dxdydz dxdydz - dædydz dxdy dz) 8a8bốc da do dc da dc db db dc da db da dc dc da dbdc di da VOL. I.
or, as it is now usually written,
Integral equation of continuity.
da 86 dc.
Now as there can be neither increase nor diminution of the quantity of matter in any portion of the fluid, the density, or the quantity of matter per unit of volume, in the infinitely small portion we have been considering, must vary inversely as its volume if this aries. Hence, if p denote the density of the fluid in the neighbourhood of (x, y, z) at time t, and p. the initial density, we have dx
(1), da' da' da
= Po dx dy dz
dc' dc' de
193. The form under which the equation of continuity is equation of continuity. most commonly given, or the differential equation of continuity,
as we may call it, expresses that the rate of diminution of the density bears to the density, at any instant, the same ratio as the rate of increase of the volume of an infinitely small portion bears to the volume of this portion at the same instant.
To find it, let a, b, c denote the co-ordinates, not when t = (), but at any time t-dt, of the point of fluid whose co-ordinates are x, y, z at t; so that we have
according to the ordinary notation for partial differential coefficients; or, if we denote by u, v, w, the components of the velocity of this point of the fluid, parallel to the axes of coordinates,
X – a = udt, y -- b = vdt, 2-0= wdt.
Di Ferential Hence
equation of dc du dy do dz dro
continuty. 1 + da da da da
da da dc due
do dz doo
da + db dc This therefore expresses the ratio in which the volume is augmented in time dt. The corresponding ratio of variation of density is
da db dc Here P, U, V, w are regarded as functions of a, b, c, and t, and
Dp the variation of p implied in is the rate of the actual variation
dt of the density of an indefinitely small portion of the fluid as it moves away from a fixed position (a, b, c). If we alter the principle of the notation, and consider p as the density of whatever portion of the fluid is at time t in the neighbourhood of the fixed point (a, b, c), and u, v, w the component velocities of the fluid passing the same point at the same time, we shall have De de de de
1 de de dp de du do du
da db dc
0 , dt da db dc
Differential equation of continuity.
which is the differential equation of continuity, in the form in
which it is most commonly given. 194. The other way referred to above (8 192) leads immediately to the differential equation of continuity.
Imagine a space fixed in the interior of a fluid, and consider the fluid which flows into this space, and the fluid which flows out of it, across different parts of its bounding surface, in any time. If the fluid is of the same density and incompressible, the whole quantity of matter in the space in question must remain constant at all times, and therefore the quantity flowing in must be equal to the quantity flowing out in any time. If, on the contrary, during any period of motion, more fluid enters than leaves the fixed space, there will be condensation of matter in that space; or if more fluid leaves than enters, there will be dilatation. The rate of augmentation of the average density of the fluid, per unit of time, in the fixed space in question, bears to the actual density, at any instant, the same ratio that the rate of acquisition of matter into that space bears to the whole matter in that space.
Let the space S be an infinitely small parallelepiped, of which the edges a, ß, y are parallel to the axes of co-ordinates, and let x, y, z be the co-ordinates of its centre; so that x = a, y = 2B, 2+ y are the co-ordinates of its angular points. Let p be the density of the fluid at (x, y, z), or the mean density through the space S, at the time t. The density at the time t+ dt will be
dp p+ dt; and hence the quantities of fluid contained in the
dt space S, at the times t, and t + dt, are respectively paßy and +
Hence the quantity of Auid lost (there will of course be an absolute gain if be positive) in the time dt is
dt de aßydt.
"Steady motion” defined.
Now let u, v, w be the three components of the velocity of the fluid (or of a fluid particle) at P. These quantities will be functions of x, y, z (involving also t, except in the case of "steady motion"), and will in general vary gradually from point to point of the fluid ; although the analysis which follows is not restricted