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LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS

LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS. Julia Yeomans. Rudolph Peierls Centre for Theoretical Physics University of Oxford. Lattice Boltzmann simulations: discovering new physics. Binary fluid phase ordering and flow

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LATTICE BOLTZMANN SIMULATIONS OF COMPLEX FLUIDS

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  1. LATTICEBOLTZMANNSIMULATIONS OF COMPLEXFLUIDS Julia Yeomans Rudolph Peierls Centre for Theoretical Physics University of Oxford

  2. Lattice Boltzmann simulations: discovering new physics Binary fluid phase ordering and flow Wetting and spreading chemically patterned substrates superhydrophobic surfaces Liquid crystal rheology permeation in cholesterics

  3. Binary fluids • The free energy lattice Boltzmann model • The free energy and why it is a minimum in equilibrium • A model for the free energy: Landau theory • The bulk terms and the phase diagram • The chemical potential and pressure tensor • The equations of motion • The lattice Boltzmann algorithm • The interface • Phase ordering in a binary fluid

  4. The free energy is a minimum in equilibrium Clausius’ theorem Definition of entropy B A

  5. The free energy is a minimum in equilibrium Clausius’ theorem Definition of entropy B A

  6. isothermal first law The free energy is a minimum in equilibrium constant T and V

  7. The order parameter for a binary fluid nA is the number density of A nB is the number density of B The order parameter is

  8. Models for the free energy nA is the number density of A nB is the number density of B The order parameter is

  9. F Cahn theory: a phenomenological equation for the evolution of the order parameter

  10. Landau theory bulk terms

  11. Phase diagram

  12. Gradient terms

  13. Navier-Stokes equations for a binary fluid continuity Navier-Stokes convection-diffusion

  14. Getting from F to the pressure P and the chemical potential first law

  15. Homogeneous system

  16. Inhomogeneous system Minimise F with the constraint of constant N, Euler-Lagrange equations

  17. The pressure tensor • Need to construct a tensor which • reduces to P in a homogeneous system • has a divergence which vanishes in equilibrium

  18. Navier-Stokes equations for a binary fluid continuity Navier-Stokes convection-diffusion

  19. The lattice Boltzmann algorithm Lattice velocity vectors ei, i=0,1…8 Define two sets of partial distribution functions fi and gi Evolution equations

  20. Conditions on the equilibrium distribution functions Conservation of NA and NB and of momentum Pressure tensor Velocity Chemical potential

  21. The equilibrium distribution function Selected coefficients

  22. Interfaces and surface tension lines: analytic result points: numerical results

  23. Interfaces and surface tension

  24. N.B. factor of 2

  25. lines: analytic result points: numerical results surface tension

  26. Phase ordering in a binary fluid Alexander Wagner +JMY

  27. Phase ordering in a binary fluid Diffusive ordering t -1 L-3 Hydrodynamic ordering t -1 L t -1 L-1 L-1

  28. high viscosity: diffusive ordering

  29. high viscosity: diffusive ordering

  30. High viscosity: time dependence of different length scales L(t)

  31. low viscosity: hydrodynamic ordering

  32. low viscosity: hydrodynamic ordering

  33. Low viscosity: time dependence of different length scales R(t)

  34. There are two competing growth mechanisms when binary fluids order: hydrodynamics drives the domains circular the domains grow by diffusion

  35. Wetting and Spreading • What is a contact angle? • The surface free energy • Spreading on chemically patterned surfaces • Mapping to reality • Superhydrophobic substrates

  36. Lattice Boltzmann simulations of spreading drops:chemically and topologically patterned substrates

  37. Surface terms in the free energy Minimising the free energy gives a boundary condition The wetting angle is related to h by

  38. Variation of wetting angle with dimensionless surface field line:theory points:simulations

  39. Spreading on a heterogeneous substrate

  40. Some experiments (by J.Léopoldès)

  41. LB simulations on substrate 4 • Two final (meta-)stable state observed depending on the point of impact. • Dynamics of the drop formation traced. • Quantitative agreement with experiment. Simulation vs experiments Evolution of the contact line

  42. Effect of the jetting velocity Same point of impact in both simulations With an impact velocity t=0 t=10000 t=20000 t=100000 With no impact velocity

  43. Base radius as a function of time

  44. Characteristic spreading velocityA. Wagner and A. Briant

  45. Superhydrophobic substrates Bico et al., Euro. Phys. Lett., 47, 220, 1999.

  46. Two droplet states A suspended droplet q* A collapsed droplet q* He et al., Langmuir, 19, 4999, 2003

  47. Substrate geometry qeq=110o

  48. Equilibrium droplets on superhydrophobic substrates Suspended, q~160o Collapsed, q~140o On a homogeneous substrate, qeq=110o

  49. Drops on tilted substrates

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