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"Introduction of Mechanics" states:

$F_v = −Cv$,

where $F_v$ is viscous force and $C$ is a constant that depends on the fluid and the geometry of the body. For objects of simple shape moving slowly through a gas at low pressure, C can be calculated from first principles. For a sphere of radius r moving at low speed through a common fluid like water or air, C = 6πη r.

How does C depends upon fluid and geometry of body?

Why is it necessary for the sphere to move slowly and at low pressure to be have C = 6πη r?

What's the first principle?

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The viscuos force given in the cited equation for an sphere moving in a viscous fluid is called Stoke's Law This drag force law holds for a sphere of radius r for laminary flow around the sphere at low velocities. At high velocity there will be turbulent flow and the law doen't hold any longer. The parameter $\eta$ is the so-called dynamic viscosity which is a property of the respective fluid. For shapes different to a sphere, Stokes law may be only a rough approximation when using half of the diameter.

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  • $\begingroup$ could you suggest be books or sites from where I can learn more about the relation of viscosity with the geometry of object? $\endgroup$
    – suiz
    Commented Mar 28, 2018 at 3:28
  • $\begingroup$ @suiz You can find a derivation of the Stokes formula in the book by Landau-Lifshitz "Fluid Mechanics". Even the calculation for a sphere by solving the Navier-Stokes equations is very involved. In a footnote there it is pointed out that the calculation can also be done for ellipsoids. General considerations of the drag in a viscous fluid regarding general body shapes is found in the book by Landau, Akhiezer, Lifshitz "General Physics". In particular, it is stated there that at low velocity the drag force of any body depends on its linear dimension (diameter). $\endgroup$
    – freecharly
    Commented Mar 28, 2018 at 14:15
  • $\begingroup$ @suiz There are empirical formula for the drag force with different body shapes usually giving a dependence on an equivalent diameter. There is an entry on that in the German Wikipedia (de.wikipedia.org/wiki/Sedimentationsgeschwindigkeit) and on equivalent diameters (de.wikipedia.org/wiki/Äquivalentdurchmesser) but no corresponding one sin the English Wikipedia. $\endgroup$
    – freecharly
    Commented Mar 28, 2018 at 14:16
  • $\begingroup$ @suiz Note: The text books by Landau et al. can be downloaded for free from archive.org $\endgroup$
    – freecharly
    Commented Mar 28, 2018 at 14:18

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