In fluid dynamics, Epstein drag is a theoretical result, for the drag force exerted on spheres in high Knudsen number flow (i.e., rarefied gas flow).
[1] This may apply, for example, to sub-micron droplets in air, or to larger spherical objects moving in gases more rarefied than air at standard temperature and pressure.
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, in order for Epstein drag to apply.
The reason for this is to ensure that the change in the sphere's momentum due to individual collisions with gas species is not large enough to substantially alter the sphere's motion, such as occurs in Brownian motion.
The result was obtained by Paul Sophus Epstein in 1924.
high-precision measurements of the charge on the electron in the oil drop experiment performed by Robert A. Millikan, as cited by Millikan in his 1930 review paper on the subject.
[2] For the early work on that experiment, the drag was assumed to follow Stokes' law.
However, for droplets substantially below the submicron scale, the drag approaches Epstein drag instead of Stokes drag, since the mean free path of air species (atoms and molecules) is roughly of order of a tenth of a micron.
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, is where
is the relative speed of the sphere with respect to the rest frame of the gas.
encompasses the microphysics of the gas-sphere interaction and the resultant distribution of velocities of the reflected particles, which is not a trivial problem.
is found to be close to 1 numerically, and in part because in many applications, the uncertainty due to
For this reason, one sometimes encounters Epstein drag written with the factor
Forces acting normal to the direction of motion are known as "lift", not "drag", and in any case are not present in the stated problem when the sphere is not rotating.
[1] 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
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.
for purely elastic specular reflection, but may be less than or greater than unity in other circumstances.
is the accommodation coefficient, which appears in the Maxwell model for the interaction of gas species with surfaces, characterizing the fraction of reflection events that are diffuse (as opposed to specular).
(There are other accommodation coefficients that describe thermal energy transfer as well, but are beyond the scope of this article.)
In-line with theory, an empirical measurement, for example, for melamine-formaldehyde spheres in argon gas, gives
That is, he treated the leading terms in what happens if the flow is not fully in the rarefied regime.
[citation needed] As noted by Epstein himself,[1] previous work on this problem had been performed by Langevin[4] by Cunningham,[5] and by Lenard.
As mentioned above, the original practical application of Epstein drag was to refined estimates of the charge on the electron in the Millikan oil-drop experiment.
One application among many in astrophysics is the problem of gas-dust coupling in protostellar disks.
[10] Another application is the drag on stellar dust in red giant atmospheres, which counteracts the acceleration due to radiation pressure [11] Another application is to dusty plasmas.