When no extraneous forces act, we verify at once that these equations have the integrals ε2 + n2 + (2 = const., λέ + μη + νζ = const., ...... .(2) which express that the magnitudes of the force- and couple-resultants of the impulse are constant. 121. It remains to express έ, 7, 5, λ, μ, v in terms of u, v, w, p, q, r. the first place let T denote the kinetic energy of the fluid, so that In forces applied to taneously at time at which we have at time t, differs extraneous forces antially unaltered ch may moreover are replaced by where the integration extends over the surface of the moving solid Substituting the value of from Art. 118 (2), we get + Pp2 + Qg2 + Rr2 + 2P′qr + 2Q'rp + 2R′pq +2p (Fu+ Gv + Hw) + 2q (F'u + G'v + H'w) + 2r (F'u + G"v + H"w), .(2) Thus where the 21 coefficients A, B, C, &c. are certain constants determined by the transformations depending on Art. 118 (3) and on a particular case of Green's Theorem (Art. 44 (2)). These expressions for the coefficients were given by Kirchhoff. The actual values of the coefficients in the expression for 2T have been found in the preceding chapter for the case of the ellipsoid, viz. we have from Arts. 114, 115 with similar expressions for B, C, Q, R. The remaining coefficients, as will appear presently, in this case all vanish. We note that so that if a >b> c, then A <B < C, as might have been anticipated. The formulae for an ellipsoid of revolution may be deduced by putting b=c; they may also be obtained independently by the method of Arts. 104-109. Thus for a circular disk (a =0, b=c) we have A, B, C=pc3, 0, 0; P, Q, R=0, 1pc, 18 pc5...(6) The kinetic energy, T, say, of the solid alone is given by an expression of the form Hence the total energy T + T1, of the system, which we shall denote by T, is given by an expression of the same general form as (2), say + Pp2 + Qq2 + Rr2 + 2P'qr + 2Q'rp + 2R′pq + 2p (Fu + Gv + Hw) + 2q (F'u + G'v + H'w) + 2r (F'u + G′′v+ H"w), .(8) where the coefficients are printed in uniform type, although six of them have of course the same values as in (2) 122. The values of the several components of the impulse in terms of the velocities u, v, w, p, q, r can now be found by a well-known dynamical method*. Let a system of indefinitely great forces (X, Y, Z, L, M, N) act for an indefinitely short time on the solid, so as to change the impulse from (ξ, η, ζ, λ, μ, ν) to (ξ + δξ, η + δη, ζ + δζ, λ + δλ, μ + δμ, ν + δ»). The work done by the force X, viz. where u1 and u, are the greatest and least values of u during the time 7, И1 i.e. it lies between u1§§ and u1⁄2§§. If we now introduce the supposition that d§, dn, dl, da, du, dv are infinitely small, u, and u2 are each equal to u, and the work done is udέ. In the same way we may calculate the work done by * See Thomson and Tait, Art. 313, or Maxwell, Electricity and Magnetism, Part. IV. c. v. the remaining forces and couples. The total result must be equal to the increment of the kinetic energy, whence Now if the velocities be all altered in any given ratio, the impulses will be altered in the same ratio. If then we take since T is a homogeneous quadratic function. Now performing the arbitrary variation & on the first and last members of (2), and omitting terms which cancel by (1), we find Since the variations Su, Sv, Sw, Sp, Sq, dr are all independent, this gives the required formulae It may be noted that since έ, 7, 5, ... are linear functions of u, v, w, the latter quantities may also be expressed as linear functions of the former, and thence T may be regarded as a homogeneous quadratic function of ξ, η, ζ, λ, μ, ν. When expressed in this manner we may denote it by T`. The equation (1) then gives at once These formulae are in a sense reciprocal to (3). We can utilize this last result to obtain, when no extraneous forces act, another integral of the equations of motion, in addition to those found in Art. 120. Thus which vanishes identically, by Art. 120 (1). Hence we have the equation of energy T const. 123. If in the formulae (3) we put, in the notation of Art. 121, T=T+T1, .(5) it is known from the dynamics of rigid bodies that the terms in T1 represent the linear and angular momentum of the solid by itself. Hence the remaining terms, involving T, must represent the system of impulsive pressures exerted by the surface of the solid on the fluid, in the supposed instantaneous generation of the motion from rest. This is easily verified. For example, the x-component of the above system of impulsive pressures is by the formulae of Arts. 118, 121. In the same way, the moment of the impulsive pressures about Ox is 124. The equations of motion may now be written* * See Kirchhoff, l.c. ante p. 151; also Sir W. Thomson, "Hydrokinetic Solutions and Observations," Phil. Mag. Nov. 1871 [reprinted in Baltimore Lectures, Cambridge, 1904, p. 584]. If in these we write T = T + T1, and isolate the terms due to T, we obtain expressions for the forces exerted on the moving solid by the pressure of the surrounding fluid; thus the total component (X, say) of the fluid pressure parallel to x is and the moment (L) of the same pressures about x is* .(2) (3) For example, if the solid be constrained to move with a constant velocity (u, v, w), without rotation, we have The fluid pressures thus reduce to a couple, which moreover vanishes if i.e. provided the velocity (u, v, w) be in the direction of one of the principal axes of the ellipsoid Ax2 + By2 + Cz2 + 2A′yz + 2B′zx + 2C′xy = const. ...... .(5) Hence, as was first pointed out by Kirchhoff, there are, for any solid, three mutually perpendicular directions of permanent translation; that is to say, if the solid be set in motion parallel to one of these, without rotation, and left to itself, it will continue to move in this manner. It is evident that these directions are determined solely by the configuration of the surface of the body. It must be observed however that the impulse necessary to produce one of these permanent translations does not in general reduce to a single force; thus if the axes of co-ordinates be chosen, for simplicity, parallel to the three directions in question, so that A', B', C' = 0, we have, corresponding to the motion u alone, έ, n, ¿ = Au, 0, 0; λ, μ, ν = Fu, F'u, F'u, so that the impulse consists of a wrench of pitch F/A. The forms of these expressions being known, it is not difficult to verify them by direct calculation from the pressure-equation, Art. 20 (4). See a paper "On the Forces experienced by a Solid moving through a Liquid,” Quart. Journ. Math. t. xix. (1883). |