Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit

( )2

29, 1 sin2 4UP a P ρθ θ∞

= + −

PARADOX OF D’ALAMBERT

surfacesphere

d ( , ) 0P a θ= − =∫F O

Viscosity = internal friction due to molecular diffusion, viscosity coefficient h:

Viscous force density: 2

visc visc η= − • ⇒ ∇Tf V

(incompressible flow!)

Equation of motion:

( ) 2Pt

ρ η∂ + • = − + ∇ ∂ V V V V

All flow quantities :

( )

3 3

2

0

3 1 3 1cos 1 , sin 12 2 4 4

3 cos,2

ra a a aV U V Ur r r r

U aP r Pa r

θθ θ

η θθ

= − + = − − −

= −

2

1 cos as sin

1 sin as sin

rV U rr

V U rr rθ

ψ θθ θ

ψ θθ

∂= + = →∞

∂

∂= − = − →∞

∂

( )11 1 21 2 31 3force ˆ ˆ ˆ ˆarea

momentum flux tensor

(Cartesian coordinates!)

ij ji

jiij

j i

T T T

T T

VVPx x

δ η

= = − • − + +

= =

∂∂= − + ∂ ∂

Tt n = x x x

Each particle transports local momentum to wall and “sticks”:

( cos )

d cosd

yp mV x

Vmx

θ

θ

= =

Momentum transferred per particle:

Each particle transports local momentum to wall and “sticks”:

2

d d =

d d dd cosd

y yx y

p Fn p

t O OVnmx

σ

σ θ

=

y-force per unit area:

Each particle transports local momentum to wall and “sticks”:

Average over all angles of incidence: 2 1cos

3θ =

d d dd 3 d d

yy

F nm V VtO x x

σ η ≡ ≡

( )

( )2 visc visc

surface element0

cos sin

= 2 sin cos sin |

D rr r

rr r r a

F dO T T

a d P T T

θ

π

θ

θ θ

π θ θ θ θ =

= − −

− + −

∫

∫

( )( )visc

†

-

P

P η

= +T I T

I = V + V

032 = = cos2

3 sin2

rrr

rr

u UT P P Pr a

u u UT rr r r a

θθ

ηη θ

η ηη θθ

∂= + −

∂

∂∂ = − − = ∂ ∂

( )2

0

20

0

2 sin cos sin

3 2 sin cos 62

D rr rF a d T T

Ua d P aUa

π

θ

π

π θ θ θ θ

ηπ θ θ θ πη

= − −

= − − =

∫

∫

For this particular flow at r=a:

Typical “ram pressure” : Projected area perpendicular to flow: Typical force = pressure x area = Drag coefficient =

21ram 2

2

21ram 2

DD 21

2

actual drag forcetypical drag force

P U

a

F U

FCU

ρ

π

ρ

ρ

=

=

=

= =

Very crude calculation for a sphere:

D 1/C Re∝ D constantC

In the stagnation point on symmetry axis flow comes to a standstill!

Bernoulli: 2 2st1 1st2 2constant PPU P P Uρ

ρ ρ+ = = ⇔ = +

In the stagnation point on symmetry axis flow comes to a standstill!

Net force: ( )

( ) 2 2 21st 2

Area Pressure difference front-back

= 8

K

P P U U Dπρ ρ

×

× − = =

Conclusion: 2 2Drag force constantK U Dρ= ×

• High Reynolds number UD/ν, low viscosity:

• Small Reynolds number UD/ν, large viscosity:

2 2

D 2 2

Drag force constant ,

Drag coefficient = constant

K U DKC

U D

ρ

ρ

= ×

=

D 2 2

Drag force constant , constantDrag Coefficient constant =

K UDKC

U D UD Re

ρνν

ρ

×

= ×

0t∂=

∂

Divergence product rule!

Combine mass cons + energy cons.:

Divergence chain rule!

Divergence product rule!

Bernoulli’s Law:

( )21

2 1is constant along stream lines

PV γγ ρ

≡ + +Φ−

Bernoulli’s Law:

( )21

2 1is constant along stream lines

PV γγ ρ

≡ + +Φ−

Adiabatic flow:

is constant along stream linesP γρ −

Astrophysical Application: Stellar and Solar Winds

Why is there a Solar Wind?

Escape velocity ~ Thermal Velocity in Solar Corona (T ~ 1 MK)

‘Something’ bends comet tails

2620 km/s

3 200 km/s

esc

bth

p

GMV

R

k TVm

= ≈

≈

Aurora: “something” acts as a medium supporting perturbations

which propagate from Sun to Earth

Solar wind velocity as measured by Ulysses satellite

The Parker Model

Assumptions: 1. The wind is steady and adiabatic

2. The flow is spherically symmetric

3. Neglect effect of magnetic fields and rotation star

b

p

sound speed: k TPmρ

b

p

sound speed: k TPmρ

There must be a sonic radius where flow speed = sound speed

2

212

4 ( ) ( ) constant

( ) constant( 1)

( ) ( ) constant

( )

r r V r M

PV r

P r r

GMrr

γ

π ρ

γγ ρ

ρ−

∗

= ≡

+ +Φ = ≡−

=

Φ = −

Conservation of mass in steady flow

Bernouilli: conservation of energy

Entropy is constant: Adiabatic Flow

Gravitational potential of a single star

2

area mass flux

4 constant= r V Mπ ρ× =

Steady flow in radial direction:

( ) ( ) ( )22

This is in spherical coordinates!

1 1 1sin 0sin sin

0 , 0

rr V V Vt r r r r

V Vt

θ φ

ρ

θ φ

ρ ρ θρ ρθ θ θ φ•

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂

∂= = =

∂⇔

V

( )22 2

1 0 r rAr V V

r r rρ ρ∂

= ⇔ =∂

dd = ( ) ( ) 0d

r r r rr

+ ∆ − = ∆ =

( )

( ) ( )

( ) ( )

212

2

1

d =d ( ) 01

d d 4 0

d d d 0

PV r

M r V

P P P γγ γ

γγ ρ

π ρ

ρ ρ γ ρ ρ− +− −

+ +Φ = −

= =

= − =

( ) ( ) ( )2 2 2d 4 ( ) ( ) 0 4 d d 2 d

d d d2

r r V r r V r V Vr r

V rV r

π ρ π ρ ρ ρ

ρρ

= = + + ⇔

= − +

Step 1: calculate density change

212 2

dd ( ) 0 d d( 1)

GMP PV r V V rr

γ γ ργ ρ ρ ρ

∗ + +Φ = ≡ + + −

( ) ( ) constantP r rγρ − = ( ) GMrr

∗Φ = −

d d d2V rV r

ρρ

= − +

2

dd d 0GMPV V rr

γ ρρ ρ

∗ + + =

( )2 2 2d d2 , s s sGMV r PV C C C

V r rγρ

∗ − = − ≡

Adiabatic sound speed

( )2 2 2d d2 , s s sGMV r PV C C C

V r rγρ

∗ − = − ≡

( )d d lnd ln d lnd ln

V VV rV r

= =

( )d d lnr rr=

( )2 2 2d ln 2d lns s

GMVV C Cr r

∗ − = −

sV C= ± 22cs

GMr rC

∗= =

Special velocity: sound speed (“Mach One”)

Special radius: critical radius

( )2 2 2d ln 2d lns s

GMVV C Cr r

∗ − = −

( )2 2 2d ln 2d lns s

GMVV C Cr r

∗ − = −

Accelerating wind solution: V > 0 and dV/dr > 0! Solution should remain regular at all radii!

Solution space for Parker’s Equation

( )2 2 2d ln 2d lns s

GMVV C Cr r

∗ − = −

Critical Point Condition:

2 at 2s c

s

GMV C r rC

∗= = =

Wind and Breeze Solutions

Special case: Isothermal Wind with constant temperature

constant

(case with 1 ; )

siP TC

P

ρ µ

γ ρ

= = =

= ∝

Accretion Solution

Bondi Accretion

( )2 2 2ln 2lns s

GMd VV c cd r r

∗ − = −

Critical Point Condition:

2 at 2s c

s

GMV c r rc

∗= = =

( ) ( ) rV r V r e= −

Isothermal Bondi Accretion

2 at 2s c

s

GMV c r rc

∗= = =

constant

(case with 1 ; )

siP TC

P

ρ µ

γ ρ

= = =

= ∝

1. Laval Nozzle (jet engines)

2

( ) ( ) ( ) constant

constant

= constant2 ( 1)

M z V z z

P

V P

γ

ρ

ρ

γγ ρ

−

= =

=

+ =−

Basic equations:

2. Astrophysical jets:

Stellar Winds and Jets: similarities and differences

• Steady flow Steady flow • Large opening angle Small opening angle

• Parker-equation Parker-type equation

• Flow geometry known Pressure known