1Mesoscale 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

• Density of single component given by

• Momentum of single component given by

)()( xx i

ifm

ii

ifm cu

Time Evolution

• Advection step: particles move to adjacent sites.

• Collision step: distribution function relaxes to equilibrium value via BGK operator.

(eq)1'

iiii ffff

• 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.

222

(eq)

21

s

ii

ss

iiii c

cc

c

uu

c

ccnTf

iT

n u

• 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.

iigrav nnngF ccxxzx )()(ˆ)(

Fuv

/

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.

CT

Phase separation imagesTimestep 0 500 1000 2000

4000 8000 32000 50000

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.

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.

Domain Growth LawsPorod Law structure Power law growth

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.

Surface tension measurement

• “Laplace’s Law” states that the pressure drop across the interface of a bubble is inversely proportional to its radius.

Rp

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