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In fluid dynamics, Epstein drag is a theoretical result for the drag force exerted on spherical objects with high Knudsen numbers in a gas that obeys kinetic theory. This may apply, for example, to sub-micron droplets in air, or to larger spherical objects moving in gases more rarefied than air at STP. Note that while they may be small by some criteria, the spheres must nevertheless be much more massive than the species (molecules, atoms) in the gas that are colliding with the sphere, for Epstein drag to apply.

Statement of the law

The magnitude of the force on a sphere moving through a rarefied gas, in which the diameter of the sphere is of order or less than the collisional mean free path in the gas, in the case of purely specular, elastic collisions, is

${\displaystyle F={\frac {4\pi }{3}}a^{2}nm{\bar {c}}u}$

where ⁠${\displaystyle a}$⁠ is the radius of the spherical particle, ⁠${\displaystyle n}$⁠ is the number density of gas species, ⁠${\displaystyle m}$⁠ is their mass, ${\displaystyle {\bar {c}}}$ is the arithmetic mean speed of gas species, and ⁠${\displaystyle u}$⁠ is the relative speed of the sphere with respect to the rest frame of the gas. The force acts in a direction opposite to the direction of motion of the sphere.

For mixtures of gases (e.g. air), the total force is simply the sum of the forces due to each component of the gas, noting with care that each component (species) will have a different ${\displaystyle n}$, a different ${\displaystyle m}$ and a different ${\displaystyle {\bar {c}}}$. Note that ${\displaystyle nm=\rho }$ where ${\displaystyle \rho }$ is the gas density, noting again, with care, that in the case of multiple species, there are multiple different such densities contributing to the overall force.

The net force is due both to momentum transfer to the sphere due to species impinging on it, and momentum transfer due to species leaving, due either to reflection, evaporation, or some combination of the two. Additionally, the force due to reflection depends upon whether the reflection is purely specular or, by contrast, partly or fully diffuse, and the force also depends upon whether the reflection is purely elastic, or inelastic, or some other assumption regarding the velocity distribution of reflecting particles, since the particles are, after all, in thermal contact - albeit briefly - with the surface. All of these effects are combined in Epstein's work in an overall prefactor "${\displaystyle \delta }$". Theoretically, ${\displaystyle \delta =1}$ for purely elastic specular reflection, but may be less than or greater than unity in other circumstances. Using this prefactor, the Epstein drag may be rewritten as

${\displaystyle F=\delta {\frac {4\pi }{3}}a^{2}nm{\bar {c}}u}$

For reference, note that kinetic theory gives $${\displaystyle {\bar {c}}={\sqrt {{\frac {8}{\pi }}\cdot {\frac {k_{\mathrm {B} }T}{m}}}}.}$$ For the specific cases considered by Epstein, ${\displaystyle \delta }$ ranges from a minimum value of 1 up to a maximum value of 1.444. A careful empirical measurement, for example, for melamine-formaldehyde spheres in argon gas, gives ${\displaystyle \delta =1.26\pm 0.13}$ as measured by one method, and ${\displaystyle \delta =1.44\pm 0.19}$ by another method, as reported by the same authors in the same paper (Liu et al 2003, see below).

In his paper, Epstein also considered modifications to allow for nontrivial ${\displaystyle {\rm {Kn}}^{-1}}$. That is, he treated the leading terms in what happens if the flow is not fully in the rarefied regime. Also, he considered the effects due to rotation of the sphere. Normally, by "Epstein drag," one does not include such effects.

References

• Epstein, Paul S. (1924), "ON THE RESISTANCE EXPERIENCED BY SPHERES IN THEIR MOTION THROUGH GASES", Phys. Rev., 23: 710–733
• Liu, Bin; Goree, J.; Nosenko, V.; Boufendi, L. (2003), "Radiation pressure and gas drag forces on a melamine-formaldehyde microsphere in a dusty plasma", Phys. Plasmas, 10: 9–20, doi:10.1063/1.1526701