di\ 



588 Mr. D. Coster on the Rotational Oscillation 



to establish the differential equation of the motion of the 

 liquid. 



Let p be the density, 



fj, the viscosity of the liquid. 

 a) the angular velocity of a cylindrical shell. 

 r the radius of the shell. 

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



then be F = r/&^- and the frictional couple on a cylindrical 



surface of radius r will be 27rrV^- • 



Or 



Taking a shell of thickness dr its equation of motion 



will be 



which reduces to 



p'day _ ~d' 2 (o 3d&> n s 



p-df-dr* + r§7 (1> 



For an infinitely long- time of vibration, i. e. for uniform, 

 rotation, (1) simplifies to 



0=£ + ?£ (2) 



air rdr v 



The solution of (2) is o>= — +c 2 , Ci and c 2 being constants 



r 2 



of integration. If the solid cylinder (radius R) rotates with 

 uniform speed D. in an infinite liquid, the result will be 



T? 2 

 co= -—#-, giving for the frictional couple, as is well known, 



the expression 



-lirfiWn (2') 



In order to arrive at a possible solution of (1) we have to 

 make our assumption regarding the motion of the liquid a 

 little more definite by assuming that the angular displace- 

 ment of each shell is represented by 



a r =f(r) coa \pt — <j>(r)\ (3) 



We may also consider (3) as the real part of the complex 

 function ue xpi , where u is a function of r the modulus of 

 which gives the amplitude of the oscillation and the argument 



the phase-shift <j>(r). Remembering that co = ^— equation (1) 



Qt 



may be reduced to 



p op-lPPl'^o (4) 



dr l rdr a v / 



