Reading: Shu, Vol.2, Ch. 7
10.1 Viscosity
Experience shows that we have overlooked some interactions, when we wrote down the force
terms in the hydrodynamic equations.
What is missing is the ability of gases and Fuids to
sustain viscous stress. In homework we saw that for an isotropic distribution function, e.g. as
established in LTE, the kinetic pressure tensor collapses to a scalar isotropic pressure. However,
we must add stress terms on account of viscosity.
π
ij
=
P δ
ij
+
P
ij
(10
.
1)
with
P
ij
=

μ D
ij

β
(
~
∇ ·
~
V
)
δ
ij
D
ij
=
∂V
i
∂x
j
+
∂V
j
∂x
i

2
3
(
~
∇ ·
~
V
)
δ
ij
(10
.
2)
where
μ
and
β
are respectively called the shear and bulk coe±cients of viscosity.
The shear e²ect corresponds to a randomization of bulk velocity gradients, i.e. the transfer of
relative bulk kinetic energy to thermal energy. Viscosity plays an important role in contractive
Fows around central masses, such as the accretion of gas by galaxies, galaxy clusters, stars or
black holes.
Normally any mass point, that is falling toward a massive central object, will carry a non
vanishing angular momentum,
~
l
=
m~r
×
~
V
, with respect to the central mass. The centrifugal
force then scales as
~
F
∝

~
l

2
m r
3
V
φ
∝

~
l

m r
(10
.
3)
Two conclusions can be drawn:
•
The bulk velocity of the material will soon be very high, so the material could be radially
supported by centrifugal forces. Support in other directions must be e²ected by pressure, which
would be weaker if
v
th
< v
φ
. This explains why many structures have the geometry of a disk.
•
The material must loose angular momentum or it can not reach the central object.
10.2 Angular momentum transport
Let us study a cold accretion Fow around a gravitationally dominant central object in cylin
drical coordinate (
r, φ, z
), for it will have the shape of a thin disk.
As useful quantity is the
surface mass density,
σ
, that we can obtain by integration of the volume mass density. ³or an
axisymmetric system
σ
(
r, t
) =
Z
∞
∞
dz ρ
(
r, z, t
)
(10
.
4)
In the thin disk the velocity will depend, if at all, only weakly on
z
. We can therefore integrate
the Fuid equations over
z
and only use the midplane value of the velocity
~
V
=
~
V
(
z
= 0).
1