Question

Viscous drag proportional to $r$ or $r^3$?

Answer 1

When a spherical object is falling at terminal velocity through a fluid: W=U+FW=U+F, where WW is weight, UU is upthrust and FF is viscous drag. Rewriting, using Stoke's Law, we get:

43πr3ρobject.g=43πr3ρfluid.g+6πrηvt43πr3ρobject.g=43πr3ρfluid.g+6πrηvt

6ηvt=43r2ρobject.g−43r2ρfluid.g6ηvt=43r2ρobject.g−43r2ρfluid.g

vt=2r2(ρobject−ρfluid).g9η,vt=2r2(ρobject−ρfluid).g9η,

where rr is the radius of the object, ρρ is density, ggis the gravitational acceleration, ηη is the viscosity coefficient, and vtvt is the terminal velocity of the object.

Answer 2

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