Hetero geneous strain. Displacement function. "Equation of con tinuity." equation of (k) It is remarkable that fjas {1 (dy - de) + m (da - dy) + n (de da)} dz dz dx dy is the same for all surfaces having common curvilinear boundary; and when a, ẞ, y are the components of a displacement from x, y, z, it is the entire tangential displacement round the said curvilinear 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 f(adx + ẞdy + ydz), or, as it has been called, "the displacement function," is. It is, the entire tangential displacement along any curve from the fixed point 0, 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. 191. As there can be neither annihilation nor generation 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 continuity. themselves, each affording instructive views regarding the properties 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. Το specify completely the motion, is to give each of these three varying co-ordinates as a function of a, b, c, t. equation of Let da, db, de denote the edges, parallel to the axes of co-ordi- Integral 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 Sa, 8b, Sc are taken infinitely small, the corresponding portion of fluid must (§ 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 + 8b, c ; a, b, c + dc; 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 re 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 varies. 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 Differential equation of db' db' db dx dy dz de' de' de which is the integral "equation of continuity." 193. The form under which the equation of continuity is 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= = 0, 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, z- c = wdt. and, as we must reject all terms involving higher powers of d than the first, the determinant becomes simply This therefore expresses the ratio in which the volume is augmented in time dt. The corresponding ratio of variation of density is Differential equation of continuity. if Dp denote the differential of p, the density of one and the same portion of fluid as it moves from the position (a, b, c) to (x, y, z) in the interval of time from t-dt to t. Hence Here p, u, v, w are regarded as functions of a, b, c, and t, and the variation of p implied in Dp of the density of an indefinitely small portion of the fluid as it Dp dp dap + u dt dt da + v drp + w dp Omitting again the suffixes, according to the usual imperfect notation for partial differential co-efficients, which on our new understanding can cause no embarrassment, we thus have, in virtue of the preceding equation, 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 (§ 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 xa, y± 1ẞ, zy 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 "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 |