Introduction to Astrophysical Gas Dynamics

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Introduction to Astrophysical Gas Dynamics
Part 7

• Bram Achterberg
• a.achterberg_at_astro.uu.nl
• http//www.astro.uu.nl/achterb/aigdppt

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Diffusion and viscosity

• Diffusion and viscosity are transport phenomena
• Underlying physics
• TRANSPORT BY RANDOM
  WALK!
• DUE TO THE MOTION OF
  THE
• ATOMS OR MOLECULES IN THE GAS

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The one-dimensional random walk
Random sequence of forward/backward steps
of equal size
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Net distance travelled by a group of random
walkers (the Drunken Student Model )
Distance after N steps for an individual member
Average for the whole group
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Group average
Conclusion the center-of-mass of the group has
not moved (assuming they start all at x0 )
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Dispersion
Distance squared for a single group member
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Conclusion the group average, the dispersion is
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Definition diffusion coefficient ?
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Diffusion in two dimensions
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Diffusion in three dimensions
Random walk prescription
Isotropic random walk
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Tensor of averages
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Mean position
Mean position for a group of random walkers
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Dispersion of the group isotropic diffusion
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Diffusion in fluids the coffee model
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Diffusion flux and the diffusion equation
Low concentration
High concentration
Net flux
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Random thermal velocity ? Mean free path
? Particles transport a quantity C(x,t)
Flux of C for a small range in inclination angle
i
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Diffusive flux
Average over an isotropic distribution of
velocities
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Diffusive flux
Conclusion Diffusive Flux equals - (diffusion
coefficient) x (density gradient transported
quantity)
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Unit sphere
Unit sphere surface element is the same as solid
angle
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Average over the surface of the unit sphere!
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Balance equation
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Diffusive flux
Transport equation
Equation for the concentration C (x,t)
Diffusion equation
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The diffusion equation
Stepsize ? 0.25 After 5 steps All particles
start at x0
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The diffusion equation
Stepsize ? 0.25 After 10 steps All particles
start at x0
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The diffusion equation
Stepsize ? 0.25 After 20 steps All particles
start at x0
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The diffusion equation
Stepsize ? 0.25 After 10 steps All particles
start at x0 HISTOGRAM
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The diffusion equation
Stepsize ? 0.25 After 10 steps All particles
start at x0 ANALYTICAL SOLUTION DIFFUSION
EQUATION
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Diffusion equation
Fundamental solution
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Application diffusion of radiationin the Sun or
a star
Hot
Cold
Energy flux
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Thomson scattering in an electron gas
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Photon diffusion coefficient
Opacity
Photon energy density
Energy flux carried by photons
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Advection-Diffusion
For isotropic diffusion
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Viscosity diffusive transport ofmomentum
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Diffusive momentum flux
Momentum is a vector!
Momentum density
Viscous stress tensor
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Case of a pure shear flow withconstant density
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Viscous force density
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Viscous force density
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Viscous force density
Diffusion equation for the y-component of
momentum
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Conclusions

• Diffusion of molecules or atoms leads to momentum
• transport
• Diffusion equation for momentum leads to a
• viscous force density with

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Now in three dimensions!
Shear viscosity
Bulk viscosity
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A bunch of definitions (yawn)
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Why so complicated?
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Why so complicated the case ofuniform rotation
(gramaphone record)
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Equation-of-motion viscous fluid
Conservative version
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The Reynolds number
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The Reynolds number
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Energy equation for a viscous fluid
Internal energy per unit mass
True losses (external)
Free energy per unit mass
Dynamical friction