the latter form, which is semiconvergent, being suitable for large values of kr. *The proofs are analogous to those of Art. 194. Reference may also be made to Gray and Mathews, Treatise on Bessel Functions, p. 68. The function in question is there denoted by K。 (kr). The general interpretation would follow the same lines as in the case of the sphere (Art. 340). To calculate the force exerted by the fluid on the cylinder we have to integrate the expression with respect to the angular co-ordinate (9) from 0 to 2π. The products of plane harmonics of different orders will disappear in this process. The first term of (13) gives, when r is put equal to a, The second term contributes, on substitution from (11), μС. The third term vanishes identically, to our order of approximation. The final value for the resistance per unit length is therefore The investigation is subject, as in the case of Art. 340, to the condition that ka, or Ua/2v, is to be small*. It may be noted that the value of the expression in (15) does not vary rapidly with a. Thus for ka and, for ka = 26, 3.48μU. 20 = 10 we find 4.31μU, 344. Some interesting general theorems, relating to the dissipation of energy in the steady motion of a liquid under constant extraneous forces, have been given by Helmholtz and Korteweg. They involve the assumption that the inertia-terms in the dynamical equations may all be neglected. 1o. Considering the motion in a region bounded by any closed surface 2, let u, v, w be the component velocities in the steady motion, and u + u', v + v', w + w' the values of the same components in any other motion subject only to the condition that u', v', w' vanish at all points of the boundary 2. By Art. 329 (3), the dissipation in the altered motion is equal to [[{{(Pxx + P ́xx) (a + a') + + + (Pyz + P'yz) (ƒ + ƒ′) + + .} dx dydz, ...(1) ... *The above investigation is taken from the paper cited on p. 597. where the accent attached to any symbol indicates the value which the function in question assumes when u, v, w are replaced by u', v', w'. Now the formulae (2), (3) of Art. 326 shew that, in the case of an incompressible fluid, Pxxα' + Рyyb′ + PzzC' + Puzƒ' + Pzx9′ + Pxyh′ = P'xxα + P'yyb + P′szC + P'vzf + P'ex9 + P'xyh, ....(2) each side being a symmetric function of a, b, c, f, g, h and a′, b', c', f', g', h'. Hence, and by Art. 329, the expression (1) reduces to the form fffødx dydz + 2 ƒƒ§(Pxxa′ + Pyyb′ + PzzC′ + Pyzf' + Pzx9′ + Pxyh') dx dydz + fffø′ dx dydz. and by a partial integration, remembering that u', v', w' vanish at the boundary, this becomes where is a single-valued potential, the integral vanishes in virtue of the equation of continuity, by Art. 42 (4). Under these conditions the dissipation in the altered motion is equal to [[fødx dydz + [[fødx dydz, ..... ..(5) or 2 (F+F'), say. That is, it exceeds the dissipation in the steady motion by the essentially positive quantity 2F' which represents the dissipation in the motion u', v', w'. In other words, provided the inertia-terms may be neglected, the steady motion of a liquid under constant forces having a single-valued potential is characterized by the property that the dissipation in any region is less than in any other motion consistent with the same values of u, v, w at the boundary of this region. It follows that, with prescribed velocities over the boundary, there is only one type of steady motion in the region*. It has been pointed out by Rayleigh† that the integral (3) vanishes, and the dissipation is accordingly a minimum, under somewhat wider conditions. It appears from Art. 329 that in calculating the dissipation the term p in the values of Pxx, Puy, Pzz may be omitted, *Helmholtz, "Zur Theorie der stationären Ströme in reibenden Flüssigkeiten," Verh. d. naturhist.-med. Vereins, Oct. 30, 1868 [Wiss. Abh. t. i. p. 223]. "On the Motion of a Viscous Fluid," Phil. Mag. (6), t. xxvi. p. 776 (1913). where H is any single-valued function of x, y, z, subject of course to the condition v2H This implies that ...(8) 0. and conversely. Under this head are included the case of steady motion between parallel planes, where u = A+ Bz + Cz2, v = 0, w = 0 .(9) (Art. 330), and that of motion in circles between coaxal cylinders (Art. 333). It is to be noticed that there is now no necessity, so far as the truth of the theorem is concerned, that the motion represented by u, v, w should be small, or even that it should be dynamically possible as a steady motion, provided only that the relations (7) and the equation of continuity are satisfied. For instance, in the case of motion between concentric spheres discussed in Art. 334, the dissipation is necessarily greater than was there found, and the couple N required to maintain the motion must therefore exceed the value given by equation (10). 2o. If u, v, w refer to any motion whatever in the given region, we have 2Å = ƒƒƒė dx dydz = 2 §§§(£xxȧ + Pyyb + Pzzċ + Pyz† + Pzxġ + Pxyh)dx dydž, .... . . (10) since the formula (2) holds when dots take the place of accents. The treatment of this integral is the same as before. If we suppose that u, v, w vanish over the bounding surface 2, we find = PJ]](i + i + i) dxdydz + P [[[(Xử + Yi + Z)dxdydz, ........(11) in the case of a slow motion. When the extraneous forces have a single-valued potential the latter integral vanishes, so that F = p ƒƒƒ(ù2 + v2 + w2) dx dydz. .... .(12) This is essentially negative, so that F continually diminishes, the process ceasing only when i = 0, 0, w = 0, that is, when the motion has become steady. = Hence when the velocities over the boundary Σ are maintained constant, the motion in the interior will tend to become steady. The type of steady motion ultimately attained is therefore stable, as well as unique*. It has been shewn by Rayleigh† that the above theorem can be extended so as to apply to any dynamical system devoid of potential energy, in which the kinetic energy (T) and the dissipation-function (F) can be expressed as quadratic functions of the generalized velocities, with constant coefficients. If the extraneous forces have not a single-valued potential, or if instead of given velocities we have given tractions over the boundary, the theorems require a slight modification. The excess of the dissipation over double the rate at which work is being done by the extraneous forces (including the tractions on the boundary) tends to a unique minimum, which is only attained when the motion is steady ‡. * Korteweg, "On a General Theorem of the Stability of the Motion of a Viscous Fluid," Phil. Mag. (5), t. xvi. p. 112 (1883) tl.c. ante p. 604. Cf. Helmholtz, l.c. Periodic Motion. 345. We next examine the influence of viscosity in various problems of small oscillations. We begin with the case of 'laminar' motion, as this will enable us to illustrate some points of great importance, without elaborate mathematics. If we assume that v = 0, го = 0, whilst u is a function of y only, the equations (4) of Art. 328 require that p = const., and This has the same form as the equation of linear motion of heat. In the case of simple-harmonic motion, assuming a time-factor eit+e), we have Let us first suppose that the fluid lies on the positive side of the plane. xz, and that the motion is due to a prescribed oscillation of a rigid surface coincident with this plane. If the fluid extend to infinity in the direction of y-positive, the first term in (3) is excluded, and, determining B by the boundary-condition (5), we have The formula (7) represents a wave of transversal vibrations propagated inwards from the boundary with the velocity o/ß, but with rapidly diminishing amplitude, the falling off within a wave-length being in the ratio e-2, or 35. The linear magnitude B-1 is of great importance in all problems of oscillatory motion which do not involve changes of density, as indicating the extent to which the effects of viscosity penetrate into the fluid. In the case of air (v = ·13) its value is 21P centimetres, if P be the period of oscillation in seconds. For water the corresponding value is 072P. We shall have * Stokes, l.c. ante p. 575. L. H. 39 |