**5.4.4. Viscosity**

If there are shears in the velocity in a fluid, then viscosity will produce forces that act against these shears. A unit volume of the fluid will be subjected to a force given by

(5.45) |

where is the dynamic viscosity, and the bulk viscosity has been assumed to be zero. For an ionized plasma without a magnetic field, is given by

(5.46) |

where *m*_{i}, *n*_{i},
<*v*_{i}>_{rms}, and
_{i} are the
mass, number density, rms velocity, and
mean free path of ions, respectively. Like the thermal conductivity, the
dynamic
viscosity is independent of density and depends strongly on the temperature.
However, because of the dependence on the particle mass, the viscosity is
primarily due to ions, not electrons. The Reynolds number for flow at a
speed *v*
past an object of size *l* is defined as *Re*
_{g}
*vl* / ,
which can be written as

(5.47) |

where *M*
*v/c*_{s} is the Mach number and *c*_{s} is
the sound speed. As noted in
Section 5.3.4, this indicates that
the flow around moving galaxies in a
cluster is probably laminar, but not certainly so. The viscosity affects the
rate of heating of the intracluster gas by galaxy motions
(Section 5.3.4),
and the rate of stripping by interstellar gas in cluster galaxies
(Section 5.9).

As with thermal conduction, one can define a velocity scale length
*l*_{v} so that
the absolute value of the term in parentheses in equation (5.45) is
<*v*_{i}>_{rms} /
*l*_{v}^{2}.
Then, if the ion mean free path
_{i} is
shorter than *l*_{v}, equation (5.45)
applies. However, if *l*_{v} <
_{i}, then
equation (5.45) requires that the viscous
stresses exceed the ion pressure and that momentum be transported
faster than the thermal speed of the ions. This is not possible, and the
viscous stresses must saturate at a value comparable to the ion pressure.
To my knowledge, this effect has not been included in calculations of
astrophysical fluid flows. It should be particularly important in flows
around galaxies, where previous calculations, particularly those involving
Kelvin-Helmholtz instabilities, have applied equation (5.45) to flows
with very large shears.