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Mesoscale modelling with the Lattice Boltzmann model

Mesoscale modelling with the Lattice Boltzmann model. Jonathan Chin. <j.chin@qmul.ac.uk>. Lattice Boltzmann Method. Fluids represented by fictional “particles” with discrete velocities, occupying sites on a discrete lattice. Particles described by real-valued distribution function

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Mesoscale modelling with the Lattice Boltzmann model

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  1. Mesoscale modelling with the Lattice Boltzmann model Jonathan Chin <j.chin@qmul.ac.uk>

  2. 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

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

  4. 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.

  5. 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.

  6. 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.

  7. Phase separation images Timestep 0 500 1000 2000 4000 8000 32000 50000

  8. 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.

  9. 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.

  10. Domain Growth Laws Porod Law structure Power law growth

  11. 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.

  12. Surface tension measurement • “Laplace’s 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.

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