Dear Colleagues:
According to Dicmas book the shaft critical speed(fn) for Fixed-Fixed condition is fn (F/F) = 2.23 x fn(for Hinged/Hinged condition), for Fixed-Hinged condition fn (F/H) = 1.55 x fn (H/H), and according to L&B manufacturer fn (F/F) = 1,9 x fn (H/H) and fn (F/H) = 1.9 x fn (H/H)...
I think that the S/D formula was taken from the book "Swirling Flow Problems at Intakes" (J. Knauss) may be you can get more information from there, however I am not sure that this formula can be used for slurries.
I would ask for help to submersible slurry pump manufacturers like ITT Flygt...