Linear Drag (Stokes’ Law)

A particle drag model suitable for low Reynolds number flows is Stokes’ Law. This drag model is linear in the relative velocity \(\vec{V}\).

\[\vec{F}_d = -3 \pi d^2 \mu_f \vec{V}\]

Quadratic Drag (High Reynolds Number Solid Spheres)

At higher Reynolds numbers a drag model that is quadratic in the relative velocity \(\vec{V}\) is:

\[\vec{F}_d = -C_d \frac{\pi}{8} d^2 \cdot \frac{1}{2} \rho_f |\vec{V}| \vec{V}\]

The value \(C_d = 0.42\) is the high Reynolds number limit of equation 8 of Loth.

This equation applies for solid spheres at high Reynolds numbers. Additional physics relevant to droplets are missing from this model.

References

• E. Loth, “Quasi-steady shape and drag of deformable bubbles and drops,” International Journal of Multiphase Flow, vol. 34, no. 6, pp. 523–546, Jun. 2008, doi: 10.1016/j.ijmultiphaseflow.2007.08.010.