Instabilities and Vortices Between Two Rotating Spheres 



Fig. 15 The critical Reynolds number for rotating 

 cylinders [ llj and the corresponding 

 measurennents for the spherical gap 



As far as theory is concerned we have treated three problems. 

 Without going into details we give a short summary, 



a. In case of fully laminar flow and a small relative gap width 

 the differential equations can be solved by using an approximation 

 method like that of v. Karman - Polhausen. The results are simple 

 expressions for the streamlines. Close to the walls these are loga- 

 rithmic spirals that fit very well to the experimental results (Fig. 4). 



b. Mode 1 -- with larger relative gap -- can also be treated 

 easily. For the region close to the fixed and the moving sphere esti- 

 mations can be used for the boundary layer thicknesses. For a first 

 approximation results for the boundary layer of a rotating disc [ 12] 

 can be used. Between these two boundary layers we have an already 



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