OF FLUIDS ON THE MOTION OF PENDULUMS. 



[55] 



true for all other pairs of values of r and t, y can in general be expanded in a convergent 

 series of sines of 6 and its multiples ; but the coefficients, instead of being constant, will be 

 functions of r and t, or in the particular case of steady motion, functions of r alone. Now 

 a very slight examination of the general equations will suffice to shew that the coefficients of 

 the sines of the different multiples of 9 remain perfectly independent throughout the whole 

 process, and consequently had we employed the general expansion, we should have been led 

 to the very same conclusions which have been deduced from the assumed form of y. 



47. If we take the impossibility of the existence of a limiting state of motion, which 

 has just been established, in connexion with the results obtained in Section III., we shall be 

 able to understand the general nature of the motion of the fluid around an infinite cylinder 

 which is at first at rest, and is then moved on indefinitely with a uniform velocity. 



The fluid being treated as incompressible, the first motion which takes place is impulsive. 



Since the terms depending on the internal friction will not appear in the calculation of this 



motion, we may employ the ordinary equations of hydrodynamics. The result, which is 



easily obtained, is 



Va? 

 Rdr + QrdO = dcp, where (p =- cosfl* (131) 



As the cylinder moves on, it carries more and more of the fluid with it, in consequence of 

 friction. For the sake of precision, let the quantity carried by the element dl of the cylinder 

 be defined to be that which, moving with the velocity V, would have the same momentum in 

 the direction of the motion that is actually possessed by the elementary portion of fluid which 

 is contained between two parallel infinite planes drawn perpendicular to the axis of the cylin- 

 der, at an interval dl, the particles composing which are moving with velocities that vary from 

 V to zero in passing from the surface outwards. The pressure of the cylinder on the fluid con- 

 tinually tends to increase the quantity of fluid which it carries with it, while the friction of the 

 fluid at a distance from the cylinder continually tends to diminish it. In the case of a sphere, 

 these two causes eventually counteract each other, and the motion becomes uniform. But in 

 the case of a cylinder, the increase in the quantity of fluid carried continually gains on the 

 decrease due to the friction of the surrounding fluid, and the quantity carried increases indefi- 

 nitely as the cylinder moves on. The rate at which the quantity carried is increased, decreases 

 continually, because the motion of the fluid in the neighbourhood of the cylinder becomes more 

 and more nearly a simple motion of translation equal to that of the cylinder itself, and there- 

 fore the rate at which the quantity of fluid carried is increased would become smaller and 

 smaller, even were no resistance offered by the surrounding fluid. 



* According to these equations, the fluid flows past the 

 surface of the cylinder with a finite velocity. At the end of the 

 small time t after the impact, the friction has reduced the 

 velocity of the fluid in contact with the cylinder to that of the 

 cylinder itself, and the tangential velocity alters very rapidly 

 in passing from the surface outwards. At a small distance s 

 from the surface of the cylinder, the relative velocity of the 

 fluid and the cylinder, in a direction tangential to the surface, 



is a function of the independant variables t', s, which vanishes 

 with * for any given value of f, however small, but which for 

 any given value of s, however small, approaches indefinitely to 

 the quantity determined by (131) as t vanishes. The commu. 

 nication of lateral motion is similar to the communication of 

 temperature when the surface of a body has its temperature 

 instantaneously raised or lowered by a finite quantity. 



