Non-rigid spheres

Non-rigid particles, e.g. liquid droplets, will deform in such a way that the drag will be reduced. The diameter calculated from the increased terminal velocity will therefore be greater than the true diameter D. It has been shown that the drag is [7]  

For the case of a gaseous bubble rising slowly through a liquid, in the laminar flow region, 7, 72 and thus  

This is identical to the drag force on a solid sphere at whose surface perfect slip occurs. 

Comparing with Stokes equation gives, for a droplet of diameter D  

Experimentally, small bubbles behave like solid spheres having terminal velocities closely approaching Stokes that, according to Levich [22] may be attributed to impurities at the interface. 

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For dispersions of non-rigid spheres (e.g. emulsions) the flow lines may be partially transmitted through the suspended particles, making k in Einstein s equation less than 2.5. 

Matsumoto et al. [1986] reported that in the cone-and-plate geometry, the storage, G , and loss, G , shear moduli of uniform, non-rigid spheres decrease monotonically with test time (or number of shearing cycles). G and G were observed to decrease by 4 decades, but steady state shearing for 15 seconds returned them to the initial values. Since the phenomenon depended on the rigidity as well as on the uniformity of shape and size, development of a structure during the dynamic test must be postulated. 

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