Instabilities and Vortices Between Two Rotating Spheres 



the Navier Stokes equations give t,^ ~ l/Re, the law for creeping flow. 

 Surprisingly this holds up to Re = 3.3* 10^ for the small gap and up 

 to Re = 600 for the larger one. Next to this regime follows one of 

 laminar boundary layer type with t,^ ~ l/vRe, Finally the turbulent 

 flow regime with t,^ ~ 1/ i/Re is reached after passing some possible 

 instable flow configurations in the transition region. In general we 

 have this behavior in all cases but quantitatively there are important 

 differences depending on the relative width of the gap. The reason 

 for this behavior is the multitude of the possible flow configurations. 



First we study the case of the small relative width of the gap. 

 For low Reynolds numbers (for instance Re =10) the streamlines 

 are concentric circles around the axis of rotation (Fig. 3). With 



Fig. 3 For small Reynolds number the streamlines 

 are concentric circles around the axis of 

 rotation, s = 5 mm, Re = R^co^/v =10 



increasing Reynolds number the streamlines change to spirals (Fig. 4) 

 Close to the rotating sphere the spirals are moving from the poles to 

 the equator but close to the fixed sphere the spirals are moving from 

 the equator to the poles. The inner and the outer spirals join and 

 fornn closed curves. With passing the critical Taylor number 

 Ta =41.3 Taylor vortices begin to develop close to the equator. It 

 is remarkable that the critical Taylor number here has the same 

 value as for the concentric cylinders. The axes of these vortices 

 have spiral form and end free in the flow field. (Fig. 5). From 



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