Equations of Motion. ly 



Note IV. 



On flic ))i()fio)i of (t Jlnid in n^hicJi (i circnhir cijlinder of in- 

 jinite loiyfh is slowly rotating. 



Whatever motion takes place in the tluid is clue to internal 

 friction, and is two dimensional, /. c, in the plane perpendic- 

 ular to the axis of the cylinder. 



Let the very natural supposition be made that the layer 

 of fluid contiguous with the surface of the cylinder moves 

 with the surface, then the concentric layers will have a motion 

 due to friction. There will be no motion parallel to the axis, 

 consequently ?(;=0; the only force acting along r is that due 

 to centrifugal force, but since the velocity of rotation is small 

 the products and squares of velocities may without sensible 

 error be neglected, consequently ?/ also is ecpial to 0. There- 

 fore to completely determine the motion of the fluid we need 

 consider only the second of the ecpiations for cylindrical co- 

 ordinates. 



Since the initial condition was one of rest, the conditions 

 of equilibrium demand that 



or ,'// d'r , ..dr , d'^r 



— = - r ho 1 



ot <>\ dr dr rdo- 



2dn ,. . . ., 



-; — disappearing, since // = U. 

 )-'dO 



disappearing, since this motion is two dimentional. 



dr- 



Since the angular velocity of rotation is small, squares and 

 products of the velocities may be omitted, hence — becomes 



— ; after the rotation has continued long enough for the ino- 

 dt 



tion of the fluid to become steady, — =0. This last is on the 



^ ilt 



supposition that the velocity of rotation is not accelerated; 

 and since the motion is symmetrical to the axis of the cylin- 

 der, becomes 0. The final form for the motion, where 



rdo' 



