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Also, I find two forms of strokes equation in different literature, one with the partial derivative of velocity w.r.t. time while another did not, so which is the correct one? Or they refers to different scenario?

I am very confused...could someone kindly help?

By the way, I still don't catch the rationale behind non-dimensionalizing the variables...and why certain variables are non-dimensionalized in that way, like the pressure in Stokes' approximation, which is made dimensionaless by defining P* = {P-P(infinity)}/(viscosity*velocity/length), why it is P-P-infinity? why not just P/(that thing)? Please help....