﻿410 Lord Kelvin on the Resistance 



unyielding, the motion produced in the fluid from rest, by any 

 motion given to the disk, is determinately the unique motion 

 of which the energy is less than that of any other motion 

 possible to the fluid with the g'ven motion of the disk. We 

 suppose the disk to be very thin, and therefore the profile- 

 curvature at every point of its edge to be very great: there 

 is no limit to the thinness at which the proposition could 

 cease to be true ; so it still holds in the ideal ca^e of an 

 infinitely thin disk, when the fluid and its boundary fulfil 

 the ideal conditions of the enunciation. 



§ 3. But in nature every fluid has some degree of viscous 

 resistance to change of shape ; and any viscosity however 

 small (even with ideally perfect incompressibility of the fluid 

 and unyieldingness of the boundary) would prevent the in- 

 finitely great velocities at the edge of the disk which the 

 unique minimum-energy solution gives when the disk is 

 infinitely thin ; and would originate so great a disturbance 

 in the motion of the fluid that the resistance to the motion of 

 the disk would probably be very nearly the same whatever the 

 actual value of the viscosity, if not too great in comparison 

 with the velocity of the disk multiplied by the least radius of 

 curvature of the boundary of its area. No approach, ho vvever, 

 has hitherto been made towards a complete mathematical 

 solution of any case of this problem, or indeed of the motion 

 of a body of any shape through a viscous fluid, except when, 

 as in Stokes's original solutions for the globe and circular 

 cylinder, the motion is so slow that its configuration is the 

 same as it would be if it were infinitely slow, and when 

 therefore* the velocity of the fluid at every point is equal to, 

 and in the same direction as, the infinitesimal static displace- 

 ment of an elastic solid when a rigid body imbedded in it is 

 held in a position infinitesimally displaced from its position 

 of equilibrium, in the manner translational ly and rotationally 

 corresponding to the translational and rotational velocity 

 given to the rigid body in the fluid. 



§ 4. It has occurred to me, guided by the teaching of 

 William Froude regarding the continued communication of 

 momentum to a fluid by the application of force to keep a 

 solid moving with uniform translational velocity through it, 

 that an approximate determination of the resistance, which is 

 the subject of the present communication, may probably be 



* The equations for the steady infinitely slow motion of a viscous fluid 

 are identical with those for the equilibrium of an elastic solid. See 

 ' Mathematical and Physical Papers' (Sir W. Thomson), -vol. iii., 

 art, cxix. §§ 17. 18. 



