tion, at any point of the fluid, depend only on the state of relative motion at that point, and not on the position of the axes of reference. When throughout a finite portion of a fluid mass we have §, n, all zero, the motion of any element of that portion consists of a translation and a distortion only. We follow Thomson in calling the motion in such cases 'irrotational,' and that in all other cases rotational.' 39. The value of the integral f(udx+vdy+wdz), or, otherdx dy dz wise, f(u И +v +w dads, taken along any line ABCD, is called the 'flow' of the fluid from A to D along that line. We shall denote it for shortness by I(ABCD). If A and D coincide, so that the line forms a closed curve, or circuit, the value of the integral is called the 'circulation' in that circuit. We denote it by I(ABCA). If in either case the inte gration be taken in the opposite direction, the signs of will be reversed, so that we have I(AD) = — I(DA), and I(ABCA) = — I (ACBA). It is also plain that I(ABCD) = I(AB) + I (BC) + I (CD). dx &c., ds Let us calculate the circulation in an infinitely small circuit surrounding the point (x, y, z). If (x + X, y + Y, z + Z) be a point on the circuit, we have, by (2), UdX+ VdY+ WdZ=d(UX+VY+WZ) + }d(aX2+bY2 + cZ3 +2ƒYZ+2gZX+2hXY) +$(YdZ–ZdY)+n(ZdX− XdZ)+$(XdY-YX). The first two lines of this expression, being exact differentials of single-valued functions, disappear when integrated round the cir * Thomson, On Vortex Motion. Edin. Trans. Vol. xxv., 1869. cuit. Again (YdZ - ZdY) is twice the area of the projection of the circuit on the plane yz, and therefore equal to 2ldS, where dS is the area of the circuit, and l, m, n the direction-cosines of the normal to its plane. The coefficients of n and give in the same way, on integration, 2mdS and 2ndS, respectively. Hence, finally, the circulation round the circuit is or, twice the product of the area of the circuit into the component angular velocity of the fluid about the normal to its plane. We have here tacitly made the convention that the direction of the normal to which l, m, n refer, and the direction in which the circulation in the circuit is estimated, are related in the same manner as the directions of advance and rotation in a right-handed screw *. 40. Any finite surface may be divided, by a double series of straight lines crossing it, into an infinite number of infinitely small elements. The sum of the circulations round the boundaries of these elements, taken all in the same sense, is equal to the circulation round the original boundary of the surface (supposed for the moment to consist of a single closed curve). For, in the sum in question, the flow along each side common to two elements Fig. 2. comes in twice, once for each element, but with opposite signs, and therefore disappears from the result. There remain then only the flows along those sides which are parts of the original boundary; whence the truth of the above statement. * See Maxwell, Electricity and Magnetism, Art. 23. Expressing this statement analytically we have, by (5), ♫♫2 (l§ + mn +ng) ds = [(udx + vdy+wdz).........(6), or, substituting the values of E, n, from Art. 38, = f(udx+vdy+wdz)............. (7); where the double-integral is taken over the surface, and the single-integral along the bounding curve. In these formulæ the quantities l, m, n are the direction-cosines of the normal drawn always on one side of the surface, which we may term the positive side; the direction of integration in the second member is then that in which a man walking on the surface, on the positive side of it, and close to the edge, must proceed so as to have the surface always on his left hand. The theorem (6) or (7) may evidently be extended to a surface whose boundary consists of two or more closed curves, provided the integration in the second member be taken round each of these in the proper direction, according to the rule just given. Thus, if the surface-integral in (6) extend over the shaded portion of the annexed figure, the directions in which the circulations in the several parts of the boundary are to be taken are shewn by the arrows, the positive side of the surface being that which faces the reader. The value of the surface-integral taken over a closed surface is zero. It should be noticed that (7) is a theorem of pure mathe matics, and is true whatever functions u, v, w may be of x, y, z, provided only they be continuous over the surface*. Irrotational Motion. 41. The rest of this chapter is devoted to the study of irrotational motion, as defined by the equations The existence and the properties of the velocity-potential in the various cases that may arise will appear as consequences of these equations. Considering any region occupied by irrotationally-moving fluid, we see from (6) that the circulation is zero in every circuit which can be filled up by a continuous surface lying wholly in the region, or in other words capable of being contracted to a point without passing out of the region. Such a circuit is said to be ' evanescible.' Again, let us consider two paths ACB, ADB, connecting two points A, B of the region, and such that either may by continuous variation be made to coincide with the other, without ever passing out of the region. Such paths are called 'mutually reconcileable.' Since the circuit ACBDA is evanescible, we have I (ACBDA) = 0, or since I (BDA) = — I (ADB), i.e. the flow is the same along any two reconcileable paths. A region such that all paths joining any two points of it are mutually reconcileable is said to be 'simply-connected.' Such a region is that enclosed within a sphere, or that included between two concentric spheres. In what follows, as far as Art. 53, we contemplate only simply-connected regions. 42. The irrotational motion of a fluid within a simply-connected region is characterized by the existence of a single-valued * It is not necessary that their differential coefficients should be continuous. The theorem (7) is attributed by Maxwell to Stokes, Smith's Prize Examination Papers for 1854. The proof given above is due to Thomson, l.c. ante. For other proofs, see Thomson and Tait, Natural Philosophy, Art. 190 (j), and Maxwell, Electricity and Magnetism, Art. 24. velocity-potential. Let denote the flow from some fixed point A to a variable point P, viz. P $ = ["(udx + vdy + wdz) (9). The value of has been shewn to be independent of the path within the region. position of P; let along which the integration is effected, provided it lie wholly Hence is a single-valued function of the us suppose it expressed in terms of the coordinates (x, y, z) of that point. By displacing P through an infinitely short space parallel to each of the axes of co-ordinates in succession, we find i.e. is a velocity-potential, according to the definition of Art. 22. The substitution of any other point B for A, as the lower limit in (9), simply adds an arbitrary constant to the value of p, viz. the flow from B to A. The original definition of 4 in Art. 22, and its physical interpretation in Art. 26, leave the function indeterminate to the extent of an additive constant. As we follow the course of any stream-line the value of & continually increases; hence in a simply-connected region the streamlines cannot form closed curves. 43. The function with which we have here to do is, together with its first differential coefficients, by the nature of the case, finite, continuous, and single-valued at all points of the region considered. In the case of incompressible fluids, which we now proceed to consider more particularly, must also satisfy the equation of continuity, (5) of Art. 25, or as we shall write it, for shortness, at every point of the region. Hence & is now subject to mathematical conditions identical with those satisfied by the potential of masses attracting or repelling according to the law of the inverse square of the distance, at all points external to such masses; so that many of the results proved in the theories of Attractions, Statical Electricity, &c., have also a hydrodynamical application. |