PROBLEM SET #3A AST242

Figure 1. Two concentric co-axial cylinders each rotating at a different

angular rotation rate. A viscous fluid lies between the two cylinders.
1. Couette Flow

A viscous fluid lies in the space between two infinitely long, concentric co-axial cylin-
ders of radius r1 and r2 > r1 . The cylinders rotate with angular speeds of Ω1 and Ω2 ,
respectively. We assume a high fluid viscosity. Such flow becomes unstable to formation
of Taylor vortices if Ω1  Ω2 .

(a) Using the φ component of the Navier Stokes equation in cylindrical coordinates,
show that in steady state the fluid rotates with tangential velocity
B
(1) uφ = Ar +
r
where constants
Ω2 r22 − Ω1 r12
(2) A =
r22 − r12
(Ω1 − Ω2 )r12 r22
(3) B =
r22 − r12
Assume that ur = uz = 0 and that the flow is steady state and axisymmetric.

(b) Assuming an incompressible flow find the difference in pressure between the inner
and outer cylinder for the limit r2 − r1 much smaller than r1 or r2 . Use the radial
1
2 PROBLEM SET #3A AST242

component of the Navier Stokes equation. To do this expand A and B in terms of

a where a = r2 − r1 and a  r1 . Then ∆P (the pressure difference) is ∆P ∼ a ∂P
∂r .

In cylindrical coordinates the components of the Navier Stokes equations are

u2φ 1 ∂2 ∂2
 
∂ur ∂ur uφ ∂ur ∂ur 1 ∂p 1 ∂ ∂ 1
+ ur + + uz − = − +ν (r ) + 2 2 + 2 − 2 ur
∂t ∂r r ∂φ ∂z r ρ ∂r r ∂r ∂r r ∂φ ∂z r
2 ∂uφ
−ν 2
r ∂φ
1 ∂2 ∂2
 
∂uφ ∂uφ uφ ∂uφ ∂uφ uφ ur 1 ∂p 1 ∂ ∂ 1
+ ur + + uz + = − +ν (r ) + 2 2 + 2 − 2 uφ
∂t ∂r r ∂φ ∂z r rρ ∂φ r ∂r ∂r r ∂φ ∂z r
2 ∂ur
+ν 2
r ∂φ
1 ∂2 ∂2
 
∂uz ∂uz uφ ∂uz ∂uz 1 ∂p 1 ∂ ∂
+ ur + + uz = − +ν (r ) + 2 2 + 2 uz
∂t ∂r r ∂φ ∂z ρ ∂z r ∂r ∂r r ∂φ ∂z

Note the non-trivial extra terms arising from the coordinates.

2. Dynamic and kinematic viscosity

The dynamic and kinematic shear viscosities, µ and ν are related by µ = ρν where ρ
is the density. Kinematic theory allows us to estimate that ν ∼ vλ where v is a thermal
velocity and λ is the mean free path. It may be useful to define ρ = mn where m is
the mass of the particles and n is the number of particles per unit volume. Show that
the dynamic shear viscosity is not dependent on the particle number density, n, if the
temperature and collision cross section don’t vary.

3. Accretion and Excretion Disks

By combining the φ component of the Navier Stoke’s equation and conservation

of mass in cylindrical coordinates it is possible to show that the basic equation that
describes viscous evolution of an accretion disk around a point mass is
 
∂Σ 3 ∂ 1/2 ∂ 1/2
(4) = R (νΣR )
∂t R ∂R ∂R
where Σ(R, t) is the disk surface density (mass per unit area) as a function of radius
and time. Here ν is the kinematic viscosity (units cm2 /s).

(a) Consider a steady state disk with ∂Σ ∂t = 0. In class we found Ṁ ∼ 3πΣν for a
steady state Keplerian disk. Show that when Σν is constant (and independent of
radius) the equation above gives a steady state solution. Are there other situations
that would give a steady state solution?
 PROBLEM SET #3A AST242 3

The more general equation for angular momentum transport can be written
∂j 1 ∂ 1 ∂ dΩ
(5) + (juR ) = (R3 νΣ )
∂t R ∂R R ∂R dR
where the angular momentum per unit area j = RΣuφ . The second term on the
left is the angular momentum flux caused by radial transport. Here we have not
assumed Keplerian rotation (and consequently the equation contains a factor of
dΩ/dr). The term on the right hand side can be thought of as a torque per unit
area ∂j∂t . With vanishing viscosity, ν = 0, we expect that there is no angular
momentum transport or uR = 0.

(b) Find a condition on the disk that would lead to excretion rather than accretion.
This means the radial velocity uR is positive instead of negative.

Recently excretion disks have been considered for circumstellar disks externally
truncated by photo-evaporation (proplyds) and for our solar system (work by Steve
Desch). Bill Ward and Robin Canup have considered them in the context of a
primordial circum-Jovian accretion disk.

(c) Consider the radial component of the Navier Stokes equation in steady state.
2
∂uR uφ ∂uR uφ 1 ∂p ∂Φ
(6) uR + − =− −
∂R R ∂φ R Σ ∂R ∂R
We can neglect the viscous term because we expect the velocity shear only gives
a strong viscous force in the φ direction, not in the radial direction. We do not
expect strong dependence on φ so derivatives with respect to φ can be dropped.
Show that in the limit of uφ  uR
R ∂Σ
(7) u2φ = vc2 + c2s
Σ ∂R
where cs is the sound speed and vc is the velocity of a particle in a circular orbit
not affected by the gas (Keplerian velocity);
r
∂Φ
vc ≡ R .
∂R
The mean tangential velocity is slightly below that of the velocity of a particle in
a circular orbit if the density drops with increasing radius.

4. Head winds in the minimum mass solar nebula

Many papers refer to a surface density that is called the “Minimum mass solar neb-
ula.” This is estimated from the masses and spacing of the 4 giant planets in our Solar
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system. The gas density

 −3/2
R
(8) Σgas = 2400 g cm−2
1 AU
The dust density
 −3/2
R
(9) Σdust = 10 g cm−2
1 AU
The density of solids (ices) is 3-4 times that of the dust. The above are by Hyashi, C.
1981, Prog. Theor. Physics Supp. 70, 35.

(a) Use equation 7 to estimate the difference between the circular velocity of a particle
in a circular orbit about the Sun and the gas in the minimum mass solar nebula.
Use a disk aspect ratio of h/R = 0.1 and give your velocity difference in units of
the circular velocity.

(b) Using an exponential scale height and h/R = 0.1, estimate the gas density, ρgas , in
the midplane at R = 1 AU.

Figure 2. A planetesimal moving in a disk is moving faster than the am-

bient gas so it feels a headwind. The drag force caused by this headwind
exerts a torque on the planetesimal causing it to spiral inwards.

The drag force on a planetesimal that is embedded in a gas disk

1
(10) FD = CD ρgas πs2 v 2
2
where CD is a drag coefficient, ρgas is the gas density, v is the difference in velocity
between the gas and the planetesimal and s is the radius of the planetesimal. (Note
the drag force depends on the area of the object; here A = πs2 ). A planetesimal
may be orbiting at the Keplerian speed, however equation 7 implies that when
the surface density of the gas drops with increasing radius then the gas is moving
slower than the Keplerian speed. Consequently a planetesimal in a circular orbit
 PROBLEM SET #3A AST242 5

would feel a headwind. This headwind would remove angular momentum from
the planetesimal causing it to spiral inwards. The torque can be estimated from
the drag force. The angular momentum of the planetesimal is mRvc where m is
the mass of the planetesimal, R is the radius from the star and vc the Keplerian
velocity.

(c) Show that a slowly in-spiraling planetesimal in a nearly circular orbit at radius r
that is drifting inwards at a speed Ṙ looses angular momentum at a rate
vc
(11) L̇ ∼ m Ṙ
2
where vc is the Keplerian velocity of a particle in a circular orbit. Using the torque
caused by the drag force show that
 
R ρd svc
(12) tinspiral = ∼ 2
Ṙ ρgas C Dv
where ρd is the density of the planetesimal.

(d) Assuming drag coefficient CD ∼ 1 and planetesimal density ρd ∼ 1 g cm−3 estimate

a timescale in years for the inspiral R/Ṙ of a meter sized planetesimal in the mini-
mum mass solar nebula at 1AU. Your timescale should be shorter than a thousand
years, presenting a challenging problem for current planetesimal formation models.

5. Radial temperature profiles for accretion disks

Consider an accretion disk with an accretion rate sufficiently high that its thermal
structure is due to energy viscously dissipated in the disk.
(13) νΣΩ2 ∼ σSB T 4
where ν is the viscosity,
p
(14) Ω= GM/R3
the angular rotation rate, M the mass the central object, Σ the mass surface density,
σSB the Stefan-Boltzmann constant. The above temperature is that of the surface if
the disk is optical thick, otherwise it is approximately the average temperature. The
quantities Σ, ν, T, Ω can vary with radius R.

(a) If the disk is optically thin how would its temperature scale with radius? In other
words T is proportional to R to what power? Assume a steady accretion rate
independent of radius Ṁ ∼ 3πΣν and use equations 13, 14. (Do not assume a
minimum mass Solar nebula.)

(b) If the disk is optically thick but the disk opacity κ is independent of temperature
and density how would its mid-plane temperature scale with radius? Assume vis-
cous forces dissipate energy in the disk mid-plane but the surface temperature is
6 PROBLEM SET #3A AST242

set by equation 13. Assume an α disk with α independent of radius and viscosity
set by the properties of the gas in the midplane. Assume Ṁ ∼ 3πΣν. The opacity
of the disk relates the mid-plane and surface temperatures. It may be useful to
remember that the sound speed cs ∝ T 1/2 , how hydrostatic equilibrium relates
cs , h, Ω where h is a scale height and Ω is the angular rotation rate (equation 14)
and how viscosity ν is defined for an α disk.