Physics. — "On the rotationnl osciUations of a cylinder in an 

 infinite incoiiipressible ii(/i(i(r. Bj- D. Costkr. (Conimiinicated 

 by Prof. J. P. Kuknen). 



(Communicated in the meeting of May 25, 1918) 



Tlie nietliod to be followed in I lie discussion of the problem will 

 be in the main the same as that used by Prof. Vehschaffkt.t in 

 the analogous case of the spheie^). We consider the rotational 

 swings about its axis of an infinitely long cylinder which executes 

 a forced vibration. Our object will be" to ascertain the motion in 

 the liquid which will establish itself after an infinite time (in practice 

 after a relatively short time *)) in order to compute the friclional 

 moment of forces exerted on the cylinder by the liquid. For the 

 sake of simplicity the calculations will be referred to a height of J cm. 



The motion of the cylinder may be represented by a z= a cos pt 

 wheie a is the angle of I'Otation. An obvious assumption to be 

 made is that the liquid will be set in motion in coaxial cylindrical 

 shells each of which will execute its oscillations as a whole. On 

 this assumption it is not diilicult to establish the differential equadon 

 for the motion of the liquid. 



Let Q be the density of the liquid. 

 H the viscosity of the liquid. 

 to the angular velocity of a cylindrical shell. 

 r the radius of the shell. 



The frictional force per unit surface of one of the shells will 



then be F ^ r(x ^ and the frictional couple on a cylindrical surface 



Or 



dtu 

 of radius r : 2.Tr r^ n -—. 

 Or 



Taking a shell of thickness dr its equation of motion will be 



duj d i dioï 



" </t dr I ' dr j 



which reduces to 



p dio d^o) 3 da> 



iy=T^-^ ^' (^) 



ft at . Or r Or 



1) Comp. Proceedings 18 p. 84U. Sept. 1915. Comm. Leiden l4Sb. 

 ') Comp. Comm. 1486. pag. 22 footnote. 



13 



Proceedings Royal Acad. Amsterdam. Vol. XXI. 



