In my last column, I mentioned that, at a high Reynolds number, drag is proportional to speed squared and that this statement can be justified by a dangerous combination of two back-of-the-envelope methods.1 

The argument's first ingredient, safe on its own, is dimensional analysis. The conclusion from dimensional analysis is that F, the magnitude of the drag force on an object moving through a fluid, is related to the object's speed v, to its characteristic size L, to its cross-sectional area Acs, to the fluid's density ρ, and to the fluid's kinematic viscosity ν by


where f is a dimensionless function. The left side is itself a first dimensionless group: the drag coefficient cd. On the right side, the argument of f is a second,...

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