Mesoscale%20modelling%20with%20the%20Lattice%20Boltzmann%20model

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Mesoscale modelling with the Lattice Boltzmann
model

• Jonathan Chin

ltj.chin_at_qmul.ac.ukgt
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Lattice Boltzmann Method

• Fluids represented by fictional particles with
  discrete velocities, occupying sites on a
  discrete lattice.
• Particles described by real-valued distribution
  function
• Density of single component ? given by
• Momentum of single component given by

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Time Evolution

• Advection step particles move to adjacent sites.
• Collision step distribution function relaxes to
  equilibrium value via BGK operator.

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• The equilibrium distribution is a function only
  of density and velocity at a given site.
• Weighting factors chosen to ensure isotropic
  hydrodynamics.
• Bounce-back boundaries give non-slip walls.

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• Velocity is perturbed to take account of
  collisions with other components, and forcing
  term.
• Force term includes gravity and Shan-Chen
  immiscibility force.
• Immiscibility force proportional to
  single-component density gradient, repelling
  differing components.

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Spinodal decomposition

• Two fluids may mix above some temperature
• A mixture quenched below this temperature will
  separate into its component fluids this process
  is called Spinodal Decomposition.
• Simulation performed of demixing process using
  periodic boundary conditions.

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Phase separation images
Timestep 0
500
1000
2000
4000
8000
32000
50000
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Growth Regimes

• Very early time exponential growth in structure
  factor as interfaces form according to a
  Cahn-Hilliard model.
• Early-time hydrodynamic regime dominated by
  viscosity.
• Late-time inertial hydrodynamic regime domains
  grow as two-thirds power of time.
• Very late time turbulent mixing regime postulated
  but not seen.

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Early-time Cahn-Hilliard Growth

• Circularly-averaged structure factor S(k) retains
  shape but grows exponentially in magnitude as
  domains form.
• Although the model does not define any free
  energies, it produces results in agreement with
  free-energy models of phase separation.

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Domain Growth Laws
Porod Law structure
Power law growth
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Breakdown of scaling

• Many theories assume that a phase-separating
  system contains a single length scale evolving in
  time.
• Simulation shows regime with multiple length
  scales due to competition between different
  growth mechanisms.

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Surface tension measurement

• Laplaces Law states that the pressure drop
  across the interface of a bubble is inversely
  proportional to its radius.

• Bubbles simulated with different coupling
  constants to measure surface tension.