instant at all points of space occupied by the fluid; whilst for particular values of x, y, z they give the history of what goes on at a particular place. Now let F be any function of x, y, z, t, and let us calculate the rate at which F varies for a moving particle. This we shall denote by ƏF It the symbol being used to express a differentiation following the motion of the fluid. At the time t+dt the particle which at the time t was in the position (x, y, z) is in the position (x+udt, y+vdt, z+wdt), and therefore the corresponding value of Fis Since the new value of F for the moving particle is also ex dt, we have Ət 6. Let p be the pressure, p the density, X, Y, Z the components of the external impressed forces per unit mass, at the point (x, y, z) at the time t. Let us take a rectangular element having its centre at (x, y, z), and its edges dx, dy, dz parallel to the coordinate axes. The rate at which the x-component of the momen tum of this element is increasing is pdxdydz ди t; and this must be equal to the x-component of the forces acting on the element. Of these the external impressed forces give pda dy dz X. The pressure on the yz-face which is nearest the origin will be ultimately dp dda) dy dz, that on the opposite face (p + dp dr) dydz. The difference of these gives a resultant dp dx dx dxdydz in the di rection of x-positive. The pressures on the remaining faces are 7. We have thus three equations connecting the five unknown quantities u, v, w, p, p. We require therefore two additional equations. One of these is furnished by a relation between p and p, the form of which depends on the physical constitution of the particular fluid which is the subject of investigation. For the case of a gas kept at a uniform temperature we have Boyle's Law If we have a gas in motion of such a nature that we may neglect the loss or gain of heat by an element due to conduction and radiation, the relation is = where y 141 for air. In the case of an 'incompressible' fluid, or liquid, we have 8. The remaining equation is a kinematical relation between u, v, w, p obtained as follows. If V denote the volume of a moving element of fluid, we have, on account of the constancy of mass, a.pv Ət =0 Now the rate of increase of volume of a moving region is evidently expressed by the surface-integral of the normal velocity outwards, taken all over the boundary. If the region in question be that occupied by the matter which at time t fills the rectangular element of Art. 6, the parts of this surface-integral due to the two yz-faces are (u + ‡ du dx) dydz, and − (u – 4 du da) dydz, dx du 1 which give together dxdydz. Calculating in the same way dx Since we have also V = dxdydz, (6) becomes др du dy This is called the 'equation of continuity.' If the fluid be incompressible though not necessarily of uniform density, the value of p does not alter as we follow any element, др i.e. at =0, so that (7) becomes du dv dw dx + dy + dz = 0........ .(9). The expression du dv dw which, as we have seen, measures the rate of increase of volume of the fluid at the point (x, y, z), is very conveniently termed the ‘expansion' at that point. 9. There are certain restrictions as to the values of the dependent variables in the foregoing equations. Thus u, v, w, p, p are essentially single-valued functions. The quantities u, v, w must be finite, and in general continuous, though we may have isolated surfaces at which the latter restriction does not hold. If the fluid move so as always to form a continuous mass, a certain condition, given in Art. 10, must be satisfied at such a surface. The quantity p is necessarily continuous, and finite. It is also essentially positive, at all events in the case of ordinary fluids, which cannot sustain more than an infinitesimal amount of tension without rupture. Hence if in any of our investigations we be led to negative values of p, the state of motion given by the formulæ is an impossible one. At the moment when, according to the formulæ, p would change from positive through zero to negative, either the fluid parts asunder, or a surface of discontinuity is formed, so that the conditions of the problem are entirely changed. See Art. 94. The quantity p is finite and positive, but not necessarily continuous. 10. The equations, which have been obtained so far, relate to the interior of the fluid. Besides these we have, in general, to satisfy certain boundary conditions, the nature of which varies according to the circumstances of the case. F(x, y, z, t) = 0....................... ....... (10) Let be the equation to a surface bounding the fluid. The velocity relative to this surface of a particle lying in it must be wholly tangential (or else zero), for otherwise we should have a finite flow of liquid across the surface, which contradicts the assumption that the latter is a boundary. The instantaneous rate of variation of F for a surface-particle must therefore be zero, i.e. we have This must hold at every point of the surface represented by (10). At a fixed boundary we have dF = 0, so that (11) becomes = 0, dF dF dF И +v + w = dx dy dz or, if l, m, n be the direction-cosines of the normal to the surface, = If F-0 be the equation of a surface of discontinuity, i.e. a surface such that the values of u, v, w change abruptly as we pass from one side to the other, we have where the suffixes are used to distinguish the two sides of the surface. By subtraction we find 1 (u, — u2) + m (v, − v2) + n (w, — w ̧) = 0.........(13). - The same relation holds at the common surface of two different fluids in contact; and also, since in the proof of (11) no assumption is made as to the nature of the medium of which (10) is a boundary, at the common surface of a fluid and a moving solid. The truth of (13), of which (12) is a particular case, is otherwise obvious from the consideration that the velocity normal to the surface must be, in each of the cases mentioned, the same on both sides. 11. The equation (11) expresses the condition that if the motion be continuous the particles which at any instant lie in the bounding surface lie in it always. For (11) expresses that no fluid crosses the surface F= 0; and the same thing necessarily holds of every surface which moves so as to consist always of the same series of particles. If then we draw a surface parallel and infinitely close to F= 0, and suppose it to move with the particles of which it is composed, the stratum of fluid which is included between this and F= 0, and which in virtue of the continuity of the motion remains always infinitely thin, must always consist of the same matter; whence the truth of the above statement. It has been suggested that (11) would be satisfied if the particles of fluid were to move relatively to the surface F-0 in paths touching it each at one point only. The above considerations shew that this is not possible for a system of material particles moving in a continuous manner; although it would be so for mere geometrical points which might coincide with and pass through one another*. It is, indeed, difficult to understand how, in the case supposed, the particles which are receding from the surface are to keep clear of those which are approaching it. 12. In the above method of establishing the fundamental equations we calculate the rate of change of the properties of a definite * The student may take as an illustration the motion of a series of points given by the formulæ u = ±x, v=c, w=0, the upper sign in u being taken for points receding from the fixed boundary x=0, the lower for points approaching it. |