In hydrodynamics, the Perrin friction factors are multiplicative adjustments to the translational and rotational friction of a rigid spheroid, relative to the corresponding frictions in spheres of the same volume. These friction factors were first calculated by Jean-Baptiste Perrin.
Contents
- Perrin S factor
- Translational friction factor
- Rotation friction factor
- Time constants for rotational relaxation
- References
These factors pertain to spheroids (i.e., to ellipsoids of revolution), which are characterized by the axial ratio p = (a/b), defined here as the axial semiaxis a (i.e., the semiaxis along the axis of revolution) divided by the equatorial semiaxis b. In prolate spheroids, the axial ratio p > 1 since the axial semiaxis is longer than the equatorial semiaxes. Conversely, in oblate spheroids, the axial ratio p < 1 since the axial semiaxis is shorter than the equatorial semiaxes. Finally, in spheres, the axial ratio p = 1, since all three semiaxes are equal in length.
The formulae presented below assume "stick" (not "slip") boundary conditions, i.e., it is assumed that the velocity of the fluid is zero at the surface of the spheroid.
Perrin S factor
For brevity in the equations below, we define the Perrin S factor. For prolate spheroids (i.e., cigar-shaped spheroids with two short axes and one long axis)
where the parameter
Similarly, for oblate spheroids (i.e., discus-shaped spheroids with two long axes and one short axis)
For spheres,
Translational friction factor
The frictional coefficient of an arbitrary spheroid of volume
where
and
The frictional coefficient is related to the diffusion constant D by the Einstein relation
Hence,
Rotation friction factor
There are two rotational friction factors for a general spheroid, one for a rotation about the axial semiaxis (denoted
for both prolate and oblate spheroids. For spheres,
These formulae may be numerically unstable when
for oblate spheroids.
Time constants for rotational relaxation
The rotational friction factors are rarely observed directly. Rather, one measures the exponential rotational relaxation(s) in response to an orienting force (such as flow, applied electric field, etc.). The time constant for relaxation of the axial direction vector is
whereas that for the equatorial direction vectors is
These time constants can differ significantly when the axial ratio
It may seem paradoxical that