Coefficient of Drag

omertech

Hello,

I understood that in low velocities the standrad drag equation:
[tex]F_d=\frac{ρv^2C_dA}{2}[/tex]
Could linearized to something like:
[tex]F_d=γv[/tex]
I am looking for the drag coefficient(either γ or C_d) for either a prolate or a tri-axial ellipsoid at low velocities (less than 0.5 m/s) in water. I found some papers providing drag coefficients for relatively high velocities but none with drag coefficients for low velocities.

Best regards

bigfooted

For spheres, the drag coefficient at low velocities can be determined analytically, see e.g. the book of Clift, Grace and Weber - Bubbles, Drops and Particles or Happel and Brenner, Low Reynolds number hydrodynamics. It is
[itex]\mathrm{C_d}=\frac{24}{\mathrm{Re}}[/itex]
With the Reynolds number
[itex]\mathrm{Re}=\frac{\rho v D}{\mu}[/itex]

Because A is the cross-sectional surface of the sphere, the force can be written as:
[itex]F_d=3\pi \mu D v[/itex], which is known as Stokes' law.

The drag of a nonspherical particle depends on its orientation with respect to the mean flow.

For a prolate with aspect ratio E=b/a and oriented such that that the short axis with length a (from center to edge) is in the direction of the flow, the drag component is approximately
[itex]F_d=1.2\pi \mu (4+E) a v[/itex].

Note that when E=1, then 2a=D and Stokes' result is recovered. The derivation is for instance in Happel and Brenner's book.