A Reynolds number is typically a defined quantity appropriate for a type of system, such as pipes, spheres, etc.  I know that a study of blending in conical flasks on a shaker table was done several years ago.  I don't remember where or even if the results were published.  The time required for blending was so short that measurements and correlations were difficult or even impossible.
 
With that as background, about the only relevant length dimension is the flask diameter, presumably measured at the widest point.  A rotational speed could be based on the number of cycles on the orbital shaker.  Density and viscosity are obvious physical property characteristics for the agitated fluid.  To be dimensionless, the diameter must be squared, times the rotational speed, times density and divided by viscosity.  At that point any similarity to the Reynolds number for impeller mixers ends.  Just as turbulent conditions in a pipe are defined by pipe Reynolds numbers greater than 4,000 and impeller Reynolds numbers greater than 20,000 for mixers, some value must be observed for conditions in a flask related to the definition of the flask Reynolds number.  The definition and values off the Reynolds number for a flask could be used to characterize other empirical observations.

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