Lecture 5: Kinetic theory of fluids

September 21, 2015

1 Goal

2 From atoms to probabilities

Fluid dynamics descrines fluids as continnum media (fields); however underconditions of strong inhomogeneities (shocks, boundary layers), the continuumpicture breaks down and the granular nature of matter is exposed. However,describing strongly inhomogenous fluids in terms of atoms or molecules is simplyunviable. It is then very convenient to take an intermediate point of view, a

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probabilistic one. What is the probability of finding a molecule at position ~rin space at time t with velocity ~v? This is precisely the Boltzmann probabilitydistribution function (pdf):

∆N = f(~r,~v; t)∆~r∆~v

where ∆N is the number of particles (atoms/molecules) in a volume ∆~r∆~v ofphase-space.

3 The Boltzmann equation

The BE is a continuity equation in phase space:

df

dt= ∂tf + vi∂xi

f + ai∂vif = C ≡ G− L

The lhs (Streaming) is a mirror of Newtonian mechanics, the rhs describesthe interparticle collisions (Collide). The Stream-Collide paradigm leads toAdvection-Diffusion in teh macrosciopic limit, but it is more general.

4 Global and local equilibria

The local equilibrium is defined by teh condition

C = 0

namely a dynamic balance between gain and loss terms, G = L. Inspectionof the Boltzmann collision operator shows, after making the molecular chaosassumption

f(1, 2) = f(1) f(2)

shows that the logarithm of the product f(1)f(2) is conserved in the collisionprocess. This means that it must be depend on the five collision invariants only

feq = A+Bivi + Cv2

By imposing the conservation of density, momentum and energy, one readilyderives the local maxwellian distribution

M(v, r; t) = An(r; t)e−ξ2(r;t)/2 (1)

whereξi(r; t) = (vi − u(r; t))/vth(r;t)

is the peculiar speed in units of the thernal speed, and A = (2πv3th is a normal-ization constant.

To be noted that the space-time dependence of the local equilibrium comesexclusively throught the macroscopic fields.

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This dependence is arbitrary, except that it must be slow in the scale of themean free path.

Global equilibria (thermodynamics) are attained when u = 0 (no flow) andvth = const.

The transient dynamics from local to global equilibria defines transport phe-nomena, which are key to pratical applications.

5 From Boltzmann to Navier-Stokes

Kinetic moments: they describe the statistic in velocity space Number density

n(r; t) =

∫f(r, v; t)dv (2)

Mass density

ρ(r; t) = m

∫f(r, v; t)dv = nm (3)

Mass current density:

Ji(r; t) = m

∫f(r, v; t)vidv ≡ ρui (4)

Energy density:

e(r; t) = m

∫f(r, v; t)v2/2dv (5)

TemperatureEnergy density:

nkBT (r; t) = m

∫f(r, v; t)(v − u)2/2 dv (6)

Momentum flux:

Pij = m

∫f(r, v; t)vivjdv (7)

Energy flux:

Qi = m

∫f(r, v; t)(v2/2)vidv (8)

Note that these quantities are statistical probes of velocity space. It s hap-pens that close enough to local equilibrium (low Knudsen) a few fields. density,velocity and temperature, capture the full story (hydrodynamics).

Project upon zeroth order basis (integrate over∫...dv):

∂tρ+ ∂iJi = 0 (9)

∂tJi + ∂jPij = 0 (10)

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∂tPij + ∂kQijk = ωP (Pij − P eqij ) (11)

where Qijk =∫fvivjvkdv.

Enslaving assumption: neglect the time derivative and replace the tripletesnors with its equilibrium expressions. This delivers:

Pij ∼ P eqij + τP∂kQeqijk (12)

By computing the equilibrium tensors

P eqij = ρv2thδij + ρuiuj

and

Qeqijk = Jiv2thδjk + Perm

6 Chapman-Enskog and higher order expansions

Formal solution of model Boltzmann equation (BGK):

∂f =1

τ(feq − f) (13)

f = (1 + τ∂)−1 feq

Formal series expansion ink ≡ τ∂ = τ∂t + vτ∂x(Knudsen number operator)

• Order 0 (Euler): f = feq

• Order 1 (Navier-Stokes):f = feq − kfeq

• Order 2 (Burnett):f = feq − kfeq + k2feq

• Order 3 (Super-Burnett):f = feq − kfeq + k2 + k3feq

• Order ∞ (Boltzmann): .

This expansion does not converge term by term: it is the series 1/(1 + z):each term is a growing power, alternating sign, all terms must be included toobtain a finite result. Mathematically: the small parameter upfronts the spacederivatives, if k << 1 all is well, otherwise no convergence.

Physically: boundary layers where k = O(1) require all terms of the sum-mation!

Going beyond the first order expansion (Navier-Stokes) typically leads toill-posed equations. That is why transport phenomena far from equilibrium arehard to handle.

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