Astrophysical Fluid Dynamics: I. Hydrodynamics

Winter School on Computational Astrophysics, Shanghai, 2018/01/29

source: J. Stone

Xuening Bai (白雪宁)

Institute for Advanced Study (IASTU) &

Tsinghua Center for Astrophysics (THCA)

General references

n Landau & Lifshitz, Fluid Mechanics (Vol. 6 of Course of Theoretical Physics), 1987, 2nd English Edition

n Shu, F. H., The Physics of Astrophysics, Vol. 2: Gas dynamics, 1992, University Science Books

n Ogilvie, G. I., Lecture notes on Astrophysical Fluid Dynamics, 2016, Journal of Plasma Physics, vol. 82, Cambridge Univ. Press

n Spruit, H. C., Essential Magnetohydrodynamics for Astrophysics, 2013, arXiv:1301.5572

n Kulsrud, R., Plasma Physics for Astrophysics, 2005, Princeton University Press

2

Note: here we only consider Newtonian (non-relativistic) fluid dynamics

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

3

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

4

What is a fluid?n Fluid is an idealized concept in which the matter is described

as a continuous medium with macroscopic properties varying with position.

5

n It can be applicable to 3 of the 4 states of matter: liquid, gas, and sometimes, plasmas.

a “fluid element”

l sufficiently large to contain numerous particles to define macroscopic properties

l sufficiently small compared with the length scales of interest (can be considered as infinitely small)

A “local” physical quantity can be thought to be associated with a “fluid element” that is:

Validity of the fluid approximationn Generally requires the molecular mean free path to be much

smaller than scales of interest.

6

At microscopic level, this also means that collisions are sufficiently frequent so that particle distribution functions approach Maxwellian.

Size scale particle mean free pathAir ~106cm ~10-6 cmSolar interior ~1010cm ~10-8 cm (Coulomb)Solar wind ~AU~1013 cm ~1013 cm (Coulomb)Galactic center ~1013 cm ~1018cm (Coulomb)Galaxy clusters ~Mpc~1024cm ~1022 cm (Coulomb)

When fluid approximation fails, kinetic treatment is generally required.

Astrophysical fluid/gas dynamicsn Differs from “laboratory” and/or engineering fluid

dynamics in the relative importance of certain effects.

7

l Compressibilityl Gravityl Magnetic fields

l Radiation forcesl Relativistic effects

l Viscosityl Surface tensionl Presence of solid

boundaries

Generally important

Sometimes important

Generally unimportant

sometimes invoked to mimic turbulence

How to describe a fluid?

8

Macroscopic physical quantities are moments of the distribution function:

Occasionally, one considers multi-species, requiring one copy of the above quantities for each species => multi-fluid.

Density: ⇢(x) = m

Zd3vf(x,v)

Velocity: v(x) =

Rd3vvf(x,v)Rd3vf(x,v)

Pressure: P(x) = m

Zd3v(v � v)(v � v)f(x,v)

(mass)

(momentum)

(energy)

For isotropic DF, pressure is a scalar: P(x) = P (x)I

Note: with frequent collisions, f becomes Maxwellian, which can be exactly specified by the above 3 quantities.

How to describe a fluid?

9

Eulerian view:Follow physical quantities at fixed locations.

Temporal changes of a quantity Q are described by a partial time derivative:

Lagrangian view:Follow physical quantities as one travels along the flow.

Temporal changes of a quantity Q are described by a “Lagrangian” or “convective” time derivative:

@Q

@t=

Q(r, t+ �t)�Q(r, t)

�t

DQ

Dt=

Q(r + �r, t+ �t)�Q(r, t)

�twhere �r = v�t

=

✓@

@t+ v ·r

◆Q

Equation of continuity

10

Pick a volume V bounded by a closed surface S. The total mass contained in the volume is:

M =

Z

V⇢dV

This mass changes due to masses flowing through the surface S:

This must hold for any volume elements, which gives the continuity equation:

dM

dt=

Z

V

@⇢

@tdV = �

I

S⇢v · dS = �

Z

Vr · (⇢v)dV

@⇢

@t+r · (⇢v) = 0

mass flux

Two remarks

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The continuity equation can be rewritten into

@⇢

@t+r · (⇢v) = 0

In general, conservation law for a quantity A is expressed as

@

@t(density of A) +r · (flux of A) = 0

Source and sink terms, if present, enter onto the right hand side.

D⇢

Dt= �⇢r · v

The compression term:

compression or dilationr · v < or > 0 :

r · v ⌘ 0 : flow is incompressible

Equation of motion

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Pick a fluid element within volume V bounded by a closed surface S, with total mass M. Newton’s 2nd law of motion over this volume gives:

g

PdSThe force involves pressure:

F P = �I

SPdS = �

Z

VrPdV

For an infinitesimal volume, we arrive at Euler’s equation:

⇢Dv

Dt= ⇢

✓@v

@t+ v ·rv

◆= �rP + ⇢g

@v

@t+ v ·rv = �rP

⇢or

Additional forces/acceleration (e.g., gravity) can be added to the right:

MDv

Dt= F

@v

@t+ v ·rv = �rP

⇢+ g = �rP

⇢�r�

Thermal energy equation

13

With the continuity equation + Euler equation, the system is not closed unless pressure is known.

For the special case of a barotropic fluid (which is often adopted), the equation of state (EoS) is given by P=P(𝜌), which closes the system.

More generally, the EoS also depends on temperature P=P(𝜌,T), a separate equation is needed to determine temperature evolution.

With thermodynamic relations, there are multiple equivalent forms for this equation.

Thermal energy equation

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For an ideal fluid without energy dissipation or exchange, any fluid element behaves adiabatically, which conserves entropy:

Entropy is related to internal energy ε via the first law of thermodynamics:

d✏ = Tds� PdV =P

⇢2d⇢ ⇢

D✏

Dt=

P

⇢

D⇢

Dt= �Pr · v

Source terms incorporating additional irreversible processes, e.g. thermal conduction, viscous dissipation can be added to the right hand side.

⇢

@✏

@t+ (v ·r)✏

�= �Pr · v

s: specific entropy (entropy per unit mass)

Ds

Dt=

@s

@t+ (v ·r)s = 0

(=0)

Thermal energy equation

15

For ideal gas, we have:

⇢✏ = ncV T

P = nkT

P = (� � 1)⇢✏

⇢D✏

Dt= �Pr · v

1

� � 1

✓DP

Dt� P

⇢

D⇢

Dt

◆= �Pr · v

� = cP /cV = (cV + k)/cVwhereratio of specific heat

From the thermal energy equation

We obtain:

It further reduces to@P

@t+ v ·rP + �Pr · v = 0

which is a commonly used form of the thermal energy equation.

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

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Hydrostatics

17

rP = ⇢g

For a fluid at rest in a gravitational field, the mechanical equilibrium of the fluid (hydrodynamic equilibrium) is described by:

l Widely applicable to astrophysical systems, e.g., stellar interior, atmospheres.

l Commonly serve as initial conditions for stability analysis, and for setting up numerical simulations.

Accurate solutions requires knowledge about temperature T (thermodynamics) with an equation of state P=P (𝜌,T).

In simple cases one may consider an isothermal equation of state P=𝜌cs2.

For a one-dimensional problem along the z direction, we have

d⇢

dz= ⇢g(z)

which is directly integrable once g(z) is known.

isothermal sound speed (constant)

Hydrostatics: simple exercises

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Assuming isothermal EoS, solve for the following 1D hydrostatic problems:

1. Planetary atmosphere with g(z)=-g0=constant.

2. Vertical structure of an accretion disk: g(z)=-ΩKz.

c2sd⇢

dz= �⇢⌦Kz

c2sd⇢

dz= �⇢g0 ⇢ = ⇢0e

�z/H0

⇢ = ⇢0e�z

2/2H2

0

where H0=cs2/g0.

where H0=cs/ΩK.

Scale height

Scale height

Q: how thick is the Earth’s atmosphere?

Q: how thick is the solar nebula?

Disk aspect ratio:H

R⇡ cs

⌦KR=

cs

vK

Vorticity

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Consider: v ⇥ ! ⌘ v ⇥ (r⇥ v) =1

2rv2 � (v ·r)v

@v

@t+ v ·rv = �rP

⇢�r�Combined with Euler’s equation:

@v

@t+r

✓1

2v2 + �

◆+

rP

⇢� v ⇥ ! = 0We arrive at:

Define vorticity: ! ⌘ r⇥ v

Physically, it describes the rotation/circulation of the fluid element.

Taking a curl, we obtain the vorticity equation:

@w

@t= r⇥ (v ⇥w) +

1

⇢2r⇢⇥rP

Vorticity

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The vorticity equation reads:

@w

@t= r⇥ (v ⇥w) +

1

⇢2r⇢⇥rP

baroclinic term (vorticity generation)

If the gas is barotropic, meaning P=P(𝜌), then the barotropic term vanishes, and hence we simply have:

@w

@t= r⇥ (v ⇥w) +

1

⇢2r⇢⇥rP

This is related to Kelvin’s circulation theorem:

C

This is exactly analogous to the concept of magnetic flux freezing, which will be proved in the next lecture on MHD.

� ⌘I

Cv · dlThe velocity circulation:

along a closed contour is conserved as the contour moves with the fluid.

dl

Bernoulli’s theorem

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@v

@t+r

✓1

2v2 + �

◆+

rP

⇢� v ⇥ ! = 0Recall:

Assuming a steady flow:@

@t= 0

For an ideal fluid (no dissipation): dh = Tds+ V dP = dP/⇢

enthalpy

Restrict this equation along the flow streamline in steady state:

v ·r✓1

2v2 + �+ h

◆= 0

=0 along the flow

=> The Bernoulli constant B ⌘ 1

2v2 + �+ h is conserved along streamlines.

Bernoulli’s theorem: simple applications

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B ⌘ 1

2v2 + �+

ZdP

⇢Bernoulli constant:

water flow from the faucet De Laval nozzel

Loop drive

Compressibility

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Consider a flow in which fluid variables (e.g. P, 𝜌, v) vary over some characteristic scale L and time T (with L~vT). We can compare the magnitude of terms in the Euler equation:

Note that where cs is the sound speed, we find:�P ⇠ c2s�⇢

�⇢

⇢⇠ v2 + ��

c2sThe flow can be considered to be largely “incompressible” if v<<cs and gravitational potential does not vary greatly along streamlines.

r · v = 0

For incompressible flow, the continuity equation is simplified to:

Incompressible flow

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Fluid equations are substantially simplified for a compressible flow.

Continuity equation becomes: r · v = 0

Euler’s equation becomes:@v

@t+ v ·rv = �r(P/⇢)�r�

For homogeneous background medium, it implies 𝜌=const. (not true otherwise)

Kelvin’s circulation theorem applies, and the system can simply be closed with the vorticity equation alone.

B =1

2v2 +

P

⇢+ �Bernoulli’s theorem is simplified to

Methods for solving incompressible fluid equations are generally very different from those for solving compressible equations. We will focus on the latter.

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

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Conservation laws:

26

@⇢

@t+r · (⇢v) = 0

We already have derived the equation of mass conservation:

Given the fundamental importance of momentum and energy conservation, they should be applicable to fluids as well.

In general, conservation law for a quantity A is expressed as

@

@t(density of A) +r · (flux of A) = 0

Source and sink terms, if present, enter onto the right hand side.

Momentum conservation

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@⇢

@t+r · (⇢v) = 0

@v

@t+ v ·rv = �rP

⇢�r�

@

@t(⇢v) = ⇢

@v

@t+

@⇢

@tv = �⇢v ·rv �rP �r · (⇢v)v

Starting from the continuity equation and the Euler equation with no external force:

It is straightforward to obtain:

= �r · (⇢vv + P I) ⌘ �r ·⇧

momentum flux density tensor

With external forces, source terms can be added on the right:

@⇢v

@t+r · (⇢vv + P I) = ⇢g

identity tensor

The energy equation

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Energy density has contributions from internal, kinetic and gravitational energies.

⇢D

Dt

✓1

2v2◆

= ⇢v · Dv

Dt= �⇢v ·r�� v ·rP (kinetic)

Assuming static gravitational potential (can be relaxed)

⇢D�

Dt= ⇢v ·r� (gravitational)

⇢D✏

Dt= �Pr · v (internal)

Summing over the above: ⇢D

Dt

✓1

2v2 + �+ ✏

◆= �r · (Pv)

With further manipulation using continuity equation:

@

@t

⇢

✓1

2v2 + �+ ✏

◆�+r ·

⇢v

✓1

2v2 + �+ ✏

◆+ Pv

�= 0

energy flux densityenergy density

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

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Origin of viscosity

y

x

Exchange of x momentum due to thermal motion in y and molecular collisions gives a momentum flux:

Random motion of molecules in a fluid leads to momentum exchange at the scale of molecular mean free path.

Consider a background shear flow:

⇡xy ⇠ ⇢vth

✓dvxdy

�mfp

◆This momentum flux attempts to reduce background shear.

In general, viscosity should be proportional to rate of strain, and acts to reduce velocity gradients and drive the system towards uniform motion. 30

𝜆mfp

Viscous stress

31

Recall the equation of momentum conservation in an ideal fluid:

Viscosity should contribute to the momentum flux tensor, which becomes:

As we have seen, 𝝅 should be proportional to velocity gradients. In addition, it should vanish in uniform rotation. The most general tensor satisfying these is

dynamic viscosity

shear viscosity (traceless) bulk viscosity

second viscosity

viscous stress tensor

Navier-Stokes equation

32

When viscosity does not vary appreciably in the fluid,

Recall the viscous stress:

This is the Navier-Stokes equation, which becomes considerably simpler for an incompressible fluid:

where is the kinematic viscosity.

From our analysis earlier:

(more detailed analysis would give an additional factor 1/3)

When is viscosity important

Re ⌘ V L

⌫viscosity is dominant when Re~1 or less.

Order of magnitudes from the Navier Stokes equation:

~V2/L ~V2/L ~𝜈V/L2~cs2/L

Define the Reynolds number:

For flow speeds not far from sound speed, V~vth, hence Re is large when L>>𝜆mfp.

Note:

Even with large Re for the bulk flow, viscosity can still play important roles at small scales (e.g., at boundaries, discontinuities, turbulent dissipation).

Energy dissipation in viscous fluid

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Viscosity corresponds to internal friction and leads to irreversible energy dissipation.

One can follow the procedure and update the energy conservation equation:

Recall energy conservation in ideal gas:

where

In the process we can find

where

stress tensor

rate of viscous dissipation

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

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Linear waves

36

@⇢

@t= �v ·r⇢� ⇢r · v

Consider perturbations on top of a homogeneous fluid at rest (𝜌=𝜌0, v=v0=0), and assuming isothermal EoS (P=𝜌cs

2) for simplicity.Recall the general fluid equations:

@v

@t+ v ·rv = �c2sr⇢/⇢

Note, for adiabatic gas, dP = 𝛾(P/𝜌)d𝜌, and we have the same equation except that cs

2= 𝛾(P/𝜌).

To first-order, perturbation equations read (Eulerian form):

@�v

@t= �c2s

r�⇢

⇢0

@�⇢

@t= �⇢0r · �v

Linear waves

37

Consider 1D problem, and also restrict velocity to that dimension, perturbation equations becomes:

This is a wave equation, with general solution:

where R, L are arbitrary functions.

Disturbances propagate to right and left at velocity cs.

Q: for a incompressible fluid, are there sound waves?

Eigenmode analysis

38

Any disturbances can be decomposed into a series of Fourier modes.

Pick up one mode with wavenumber k, and look for solutions of the form

Recall the perturbation equations (again restricting to 1D):

We arrive a system of linear algebraic equations. Non-trivial solutions exist only when its determinant vanishes:

Dispersion relation

Eigenmode analysis

39

Eigenvector of a given mode must satisfy the linear equations:

+

Eigenvector gives the relative amplitude and phase of perturbations among different physical quantities.

Group velocity: Vg =@!

@k= csPhase velocity: Vp =

!

k= cs

Recovering the original form of perturbation by taking the real part:

This describes the propagation of a single mode of sound waves, with

Vg=Vp: sound wave is non-dispersive

Wave analysis: general approach

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@P

@t+ v ·rP + �Pr · v = 0

@v

@t+ v ·rv = �rP

⇢

@⇢

@t+r · (⇢v) = 0

Starting from the full equations to obtain linearized fluid equations (now also include the energy equation).

We still consider 1D (i.e., wave propagation in a given direction), but include all vector components (so that the analysis is complete):

Wave analysis: general approach

41

Define a vector of the primitive fluid variables:

The linearized equations can be cast into the form (we omit subscript “0”):

W ⌘ (⇢, vx, vy, vz, P )T

We obtain AW 1 =!

kW 1

Conduct eigenmode analysis assuming variables are of the form ei(kx�!t)

Clearly, the eigenvalues of matrix A give the speeds of all possible wave modes.

where@W 1

@t+A

@W 1

@x= 0

�P

Wave analysis: general approach

42

�P

The matrix A has 5 eigenvalues:

They correspond to 5 waves: what are they?

� = (vx � cs, vx, vx, vx, vx + cs)

where cs =p

�P/⇢(adiabatic sound speed)

Looking at the corresponding eigenvectors of each mode:

cs cs

cs cssound

entropysoundvortical

RecallW ⌘ (⇢, vx, vy, vz, P )T

l Two for sound waves (forward / backward propagating)

l Two vortical modes (perturbations in vy, vz advected along x)

l One entropy wave (with density but no pressure perturbations)

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

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Non-linear steepening of sound waves

44

Recall that in an ideal gas, sound speed is given by cs =✓�P

�⇢

◆

s

=

s�P

⇢

In linear analysis, this is considered to be constant (homogeneous background).

To the next order, however, cs is slightly larger in higher-density regions.

=> the crest of a wave propagates faster than the leading/trailing edge

Slightly non-linear sound waves will steepen to form a discontinuity: shock

Shocks and discontinuities

45

Fluid equations are PDEs: discontinuous solutions do not hold in a classical sense.

Physically, more fundamental are the integral form of the conservation laws, which do admit discontinuous solutions.

@

@t

Z(Density of A) dV +

Z(Flux of A) · dS = 0

Microscopic dissipation (e.g., viscosity) is important across the discontinuities.

l Shock is one type of discontinuity, which usually forms when the flow speed becomes supersonic.

l There are other types of discontinuities as we shall see.

Surface of discontinuity

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We work in the co-moving frame with the surface of discontinuity.The fluxes of mass, momentum, energy across this surface must be continuous.

1 2

x

𝜌1,v1, P1 𝜌2,v2, P2

Mass conservation:

Energy conservation:

Momentum conservation:

where we have used

Tangential discontinuity

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1 2

x

𝜌1,v1, P1 𝜌2,v2, P2Mass conservation:

Momentum conservation:

If there is no mass flux through the surface:

P1=P2, while vy and vz can be discontinuous by any amount.

For discontinuous tangential velocities, the interface is subject to the Kelvin-Helmholtz instability:

The special case with continuous tangential velocities:

, all other quantities are continuous.

This is called a contact discontinuity.

Shocks jump conditions

48

If the mass flux across the surface is non-zero, we obtain the shock jump conditions:

Mass conservation:

Energy conservation:

Momentum conservation:

This is a set of 3 algebraic equations with 3 unknowns, which are called “Rankine-Hugoniot” conditions.

shock front

upstream (1) downstream (2)

x

𝜌1,v1, P1 𝜌2,v2, P2

Supernova remnant shock

shock velocity: ~103-4 km/s

4’ ~ 2.4pc

Tychoshock front

downstream (shocked)upstream (unshocked)

Note: if we are in the shock frame, the upstream flows into the shock at the observed shock velocity.

Shock (sonic) Mach number is defined as

(based on upstream flow properties)

Chandra

Example: a strong shock

50

For a strong shock, the upstream kinetic energy >> thermal energy:

shock front

upstream (1) downstream (2)

x

𝜌1,v1, P1 𝜌2,v2, P2

It is straightforward to obtain:

substantial kinetic energy from upstream converted to thermal energy downstream

For monoatomic gas, 𝛾=5/3, leading to:

compression ratio

Dissipation at shock front

51

For an ideal fluid, the shock is infinitesimally thin and the dissipation process is not captured in fluid equations.

shock front

upstream (1) downstream (2)

x

𝜌1,v1, P1 𝜌2,v2, P2

On the other hand, most astrophysical shocks are “collisionless”, where particle mean free path >> characteristic scale of the shock. Need additional dissipation resulting from kinetic effects.

In reality, molecular viscosity starts to dominate once the shock thickness reaches about molecular mean free path:

Re~1 at the scale of shock thickness

Outline

n Formulation

n Fluid basics

n Conservation laws

n Viscosity

n Linear waves

n Shocks and discontinuities

n Example: Bondi accretion

52

Application: Bondi Accretion

53

Consider an object of mass M adiabatically accreting spherical symmetrically from a uniform medium of gas (density=𝜌0, temperature=T0) in a steady state.

Question: what is the accretion rate?

Bondi (1952)

Euler’s equation:

Mass conservation:

Adiabaticity: Ds

Dt=

D

Dtln

P

⇢�= 0

where the constant K can be determined by T0.

(we recovered Bernoulli theorem)

Application: Bondi Accretion

54

Bondi (1952)

This is called the “sonic point”. If it is achieved at radius r=rs, then the RHS must also vanish at this radius, hence

From the Euler equation:

and using: ,

We obtain:

Note: cs is a function of 𝜌, which is related to v through mass conservation.

This is an ODE and can be integrated from infinity except when v=cs.

rs =GM

2cs(rs)2

Possible types of solutions

55

M ⌘ v

cs Outer boundary condition:

v→0 at infinity

=> Types 1, 3 are viable.

There are a family of solutions as one varies until some maximum value corresponding to Type 1 solution.

M

Inner boundary condition:

For a sufficiently compact object, expect supersonic accretion flow.

=> Choose the type 1 solution with unique accretion rate.

Property of the solution

56

Bondi accretion rate:

This may also be obtained from order-of-magnitude considerations:

Characteristic radius where ambient gas is strongly affected by the central object: rb '

GM

cs(1)2(Bondi radius)

Order-of-magnitude accretion rate: M ⇠ 4⇡r2b⇢(1)cs(1) = 4⇡G2M2 ⇢(1)

c3s(1)

Additional remarks

57

l Exactly the same set of equations apply to describe spherical winds (simply reversing v), which corresponds to the Parker wind solution(Parker, 1958).

l Boundary conditions play an important role in determining the type of solutions.

l When solving steady-state problems, one always encounters critical points once flow speed reaches some characteristic wave speed.

l If we solve the time-dependent fluid equations, (e.g., running hydro simulations), one generally circumvents this mathematical difficulty and the system can relax to the desired steady-state solution.

Summary

n Fluid: describes continuum medium on macroscopic scales.

n There are multiple forms of fluid equations, with the conservation form being the most fundamental.

n Viscosity attempts to smear out velocity gradients, and is an important source of dissipation.

n An ideal fluid supports sound waves (compressible) and entropy waves (related to contact discontinuity).

n Non-linear nature of fluid equations leads to shocks and discontinuities, with Rankine-Hugoniot jump conditions.

n Bondi accretion: spherical accretion flow that transitions from subsonic to supersonic.

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