An object moving through a medium collides with the particles that compose the medium and slows down as a result. The qualitative behaviour of the drag force depend on two criteria. The first criterion is whether the motion of the object relative to the medium is subsonic or supersonic. The second criterion is whether the size of the object is larger or smaller than the mean free path. We assume that the density of the medium is $ \rho $, the relative velocity between the object and the medium $ u $, the medium's speed of sound $ c $, the radius of the object $ R $ and the mean free path between collisions (of the particles comprising the medium) is $ \lambda $.

## Supersonic Motion Edit

If the motion of the particle is much larger than the speed of sound $ u \gg c $ then the thermal motion of the medium particles can be neglected, and the object can be thought of as colliding with stationary particles. Each particle that collides with the object is accelerated to velocity $ u $. Hence, the drag force is

$ F_d \approx R^2 \rho u^2 $

## Subsonic Motion Edit

### Epstein Drag Edit

In this case the object is smaller than the mean free path $ R \ll \lambda $. For a object at rest, the force on each hemisphere is $ R^2 \rho c^2 $. However, because the particle is moving then more particles are hitting the object harder from the front than from the lee side. The net force is therefore proportional to a fraction $ u/c $ of the force on each hemisphere, and the drag force is therefore

$ F_d \approx R^2 \rho c u $

### Stokes Drag Edit

In this case the object is much larger than the mean free path $ R \gg \lambda $. In this case the object creates a perturbation in the flow that propagates upstream. The velocity field near the object is different from the velocity field at infinity. Due to viscosity, the velocity vanishes on the surface of the sphere, and the velocity gradient scales as $ u/R $. Particles that hit the object originate from a distance $ \lambda $ from its surface, and hence their velocity is $ v \lambda / R $. Hence the drag force is

$ F_d \approx R \lambda \rho c u $