Coefficient of Drag: Low Velocity Ellipsoids

[omertech]

Hello,

I understood that in low velocities the standrad drag equation:
$$F_d=\frac{ρv^2C_dA}{2}$$
Could linearized to something like:
$$F_d=γv$$
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
$\mathrm{C_d}=\frac{24}{\mathrm{Re}}$
With the Reynolds number
$\mathrm{Re}=\frac{\rho v D}{\mu}$

Because A is the cross-sectional surface of the sphere, the force can be written as:
$F_d=3\pi \mu D v$, 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
$F_d=1.2\pi \mu (4+E) a v$.

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