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So basically Stokes' law states that,

"The drag force on a spherical body of radius $r$ and velocity $v$ is $$F_{d}=6\pi \eta rv.$$

My question:

$(1)$ Can we derive Stokes' drag law from Navier-Stokes' equation?

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Yes, you can find a derivation in Landau and Lifshitz' textbook on Fluid mechanics in the Section on Flow with small Reynolds numbers in the Chapter on viscous fluids. The calculation is rather involved. Note that this is true only in the limit of a vanishing Reynold's number, the general calculation cannot be done analytically.

Hope this helps and tell me is you need more details.

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