Instabilities and Vortioes Between Two Rotating Spheres 



[8] Schultz-Grunow, F. , "Der Relbungswidstand Rotierender 

 Schelben Im Gehause," ZAMM 15, 191-204, 1935. 



[9] Sawatzki, O, , "Das Stromungsfeld um eine Rotlerende Kugel," 

 Acta Mechanica, 9, 159-214, 1970. 



[10] Gregory, N. , Stuart, J. T. , and Walker, W. S. , "On the 

 Stability of Three-dimensional Boundary Layers with 

 Application to the Flow Due to a Rotating Disc," Phil. 

 Trans. Roy. Soc. A248, 155-199, 1955. 



[ 11] KlrchgaPner, K. , "Die Instabllitat der Stromung zwlschen zwei 

 rotierenden Zylindern gegeniiber Taylor- Wirbeln fiir 

 beliebige Spaltbreiten, ZAMP 12, 14-30, 1961. 



[ 12] Cochran, W. G. , "The Flow Due to a Rotating Disc," Proc. 

 Camb. Phil. Soc. A 140, 365-375, 1934. 



DISCUSSION 



L. van Wijngaarden 



Twente Institute of Technology 

 Ensohede^ The Netherlands 



The description of mode I reminds me of the result that 

 Batchelor [ 1956] derived for laminar flow with closed streamlines 

 of fluids with small viscosity: Thin boundary layers on solid 

 boundaries separated by a region of constant vorticity. This result 

 was derived for two- dimensioned flow. In your case the flow is three- 

 dimensional, but it might be that the same conditions which are neces- 

 sary for Batchelor's result hold in this case of mode I. 



REFERENCES 



Batchelor, G. K., "On steady laminar flow with closed streamlines 



at large Reynolds number," J. Fluid Mech. , Vol. 1 , p. 177, 

 1956. 



287 



