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1 Kinetic Equations: From Newton to Boltzmann
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have arisen recently, namely in solid state physics (see Chapter 5), Bosonic and
Fermionic transport, granular flows (see Chapter 3), traffic modeling, chemo-
tactic cell motion (see Chapter 4), just to name a few.
Comment on the Image 1.1 Aircraft and in particular airfoil design and opti-
misation is a classic task for computational fluid dynamics (CFL). Typically, the
three-dimensional incompressible Navier–Stokes equations are used for accu-
rateresultswhileincompressibleEulercomputations(whichdisregardboundary
layer effects) or even irrotational flow simulations (zero vorticity) give in many
cases usable quantitative results. Compressibility effects start to play a role for
transonic flows with sufficiently high Mach number. The theoretical basis for
these macroscopic gas dynamics systems is the microscopic Boltzmann equa-
tion, but the involved numerical effort, due to the high dimension of the phase
space(threevelocitydirectionsplusthree spatialdirections, andtime!),does
usually not justify its application in industrial aircraft design. Recently however,
latticeBoltzmann equation simulations(formoredetails seebelow)havebeen
employed for airfoil simulations and turbulence modeling in different applica-
tions, with striking success [6].
Comments on the Images 1.2–1.6 The modeling of the formation and motion
of atmospheric clouds is often done by macroscopic fluid equations (Navier–
Stokes or Euler) incorporating the interaction of air with cloud particles like
water droplets, ice crystals or non-volatile aerosols, again on a macroscopic
basis[8].The main difficultyliesinthe phasetransitions from watervapor to
water droplets and then, in certain cases, to ice particles (multi-phase flow).
Boltzmann-type kinetic models [13], [2] and an associated moment system for
particle dynamics in clouds were introduced in [13]. A somewhat related kinetic
approach for cloud, wind, smoke, aerosol and pheromone kinetics employs lat-
tice Boltzmann equation (LBM) simulation. In the language of kinetic equations,
LBM equations are discrete-velocity Boltzmann equations, posed on a discrete
grid in position space. Actually, they can be regarded as a numerical discreti-
sation of the Boltzmann equation, with an appropriate collision term, and LBM
solutions are known to converge (in a certain scaling limit and when the grids
are refined) in a suitable sense to solutions of the Navier–Stokes equations. Al-
ternatively, LBM models can be seen as cellular automata, where the collision
process under consideration defines the redistribution of density values after
each time step. They have proven great flexibility in applications of complex
flows, involving complicated geometries, incorporation of chemical reactions,
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phase transitions etc. .
Interesting applications can be found in [15], [17], [6], [12].
Another interesting application of kinetic theory occurs in cloud micro-
physics. Typically, clouds contain water drops, which increase their masses due
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For more information we refer to
http://www.science.uva.nl/research/scs/projects/lbm_web/index.html