# Coefficient of Drag: Low Velocity Ellipsoids

• omertech
In summary, the conversation discusses the standard drag equation and its linearization at low velocities, and the search for drag coefficients for prolate or tri-axial ellipsoids at low velocities in water. It is mentioned that for spheres, the drag coefficient at low velocities can be determined analytically using the Reynolds number, and for nonspherical particles, the drag depends on its orientation and aspect ratio. A formula for calculating the drag of a prolate particle with a specific orientation is also provided.
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 Cd) 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

Last edited:
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.

## 1. What is the coefficient of drag for a low velocity ellipsoid?

The coefficient of drag for a low velocity ellipsoid is a dimensionless number that represents the resistance to motion through a fluid. It is denoted by the symbol Cd and is determined by the shape, size, and orientation of the ellipsoid.

## 2. How is the coefficient of drag calculated for a low velocity ellipsoid?

The coefficient of drag for a low velocity ellipsoid can be calculated using the formula Cd = Fd / (0.5 * ρ * V^2 * A), where Fd is the drag force, ρ is the density of the fluid, V is the velocity of the ellipsoid, and A is the reference area.

## 3. What factors affect the coefficient of drag for a low velocity ellipsoid?

The coefficient of drag for a low velocity ellipsoid is influenced by several factors, including the shape, size, orientation, surface roughness, and density of the ellipsoid. Additionally, the viscosity and density of the fluid, as well as the velocity of the ellipsoid, also play a role in determining the coefficient of drag.

## 4. What is the significance of the coefficient of drag in aerodynamics?

The coefficient of drag is an important concept in aerodynamics as it allows scientists and engineers to predict and analyze the aerodynamic performance of a low velocity ellipsoid. It helps in designing efficient and streamlined objects that can minimize drag and improve overall performance.

## 5. How can the coefficient of drag be reduced for a low velocity ellipsoid?

The coefficient of drag for a low velocity ellipsoid can be reduced by altering its shape, size, orientation, and surface roughness. Additionally, using smooth and streamlined designs, as well as reducing the viscosity and density of the fluid, can also help in reducing drag and improving the overall aerodynamic performance.

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