Lattice Boltzmann Equation Method in Electrohydrodynamic Problems

1. Lattice Boltzmann Equation Method in Electrohydrodynamic Problems Alexander Kupershtokh, Dmitry Medvedev Lavrentyev Institute of Hydrodynamics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia

2. Equations of EHD Hydrodynamic equations: Continuity equation Navier-Stokes equation Here is the main part of momentum flux tensor Concentrations of charge carriers: Poisson’s equation and definitions:

3. Method of splitting in physical processes The whole time step is divided into several stages implemented sequentially: 1. Modeling of hydrodynamic flows. Lattice Boltzmann equation method (LBE). 2. Simulation of convective transport and diffusion of charge carriers. Additional LBE components (considered as passive scalars). 3. Calculation of electric potential and charge transfer due to mobility of charge carriers. 4. Calculation of electrostatic forces acting on space charges in liquid and incorporation these forces into LBE. 5. Simulation of phase transition or interaction between immiscible liquids using LBE method.

4. Development ofdiscretemodelsof medium Moleculardynamics (Alder, 1960) Kinetic Boltzmann equation (1872) 1964 Boltzmannequations with discrete set of velocities Lattice Gas Automata 1988 1997 Lattice Boltzmann Equation Chapman – Enskog expansion Macroscopicequations of hydrodynamics (Navier – Stokes equations)

5. Boltzmann equations with discrete velocities The discrete finite set of vectors ck of particle velocities could be used for Boltzmann equation at hydrodynamic stage For 1D Usually the populations Nk are used for each group of particles Hydrodynamic variables

6. Lattice Boltzmann equation method (LBE) The main idea is that time step must be so that One-dimensional isothermal variant (D1Q3) Two-dimensional variants (D2Q9) (D2Q13)

7. Lattice Boltzmann equation method (LBE) The discrete single-particle distribution functionsNk are used as variables Hydrodynamic variables Evolution equations of LBE method is the collision operator in BGK form (relaxation to the equilibrium state with relaxation time ). Viscosity Expansion in u is the body force term.

8. New general method of incorporatinga body force term into LBE Kinetic Boltzmann equation for single particle distribution function f(r,,t) Perturbation method For any equilibrium distribution function Hence From the other hand, the fullderivative along the Lagrange coordinateat a constant density is equal to Thus, we obtained the Boltzmann equation in form

9. Exact difference method forlattice Boltzmann equation After discretization of Boltzmann equation in velocity space we have Herethe changes of thedistribution functionsNkdue to the forceF are equal to the exact differences of equilibrium distribution functions at constant density The commutative property of body force term and the collision operator indicates the second order accuracy in time. The distribution function that is equilibrium in local region of space, is simply shifting under the action of body force by the value

10. Convective transport and diffusion of charge carriers Equations for concentrations of charge carriers: Method of additional LBE components with zero mass (passive scalars that not influence in momentum) Equilibrium distribution functions depend on concentrations of corresponding type of charge carriers and on fluid velocity . Diffusivities can be adjusted independently by changing the relaxation time

11. Calculation of electric field potential and charge transport due to mobility of charge carriers (conductivity) The time-implicit finite-difference equations for concentrations of charge carriers were solved together with the Poisson’s equation

12. Action of electrostatic forces on space charges in liquid The total charge density in the node was calculated from This equation takes into account both free space charge and charge density due to polarization. Electric field acted on this charge was calculated as numerical derivative of electric potential. Hence, we have the finite-difference expressions for electrostatic force

13. Phase transitionin 1D To simulate the phase transition, the attractive part of intermolecular potential should be introduced. For this purpose, the attractiveforces between particles in neighbor nodes was introduced (Shan – Chen, 1993).

14. Phase transitionin 2D The attraction between particles in neighbor nodes These attractive forces ensure also a surface tension.

15. Phase transitions for Chan-Chen models For isothermal models The equation of state: D1Q3 D2Q9 D3Q19 Critical point: For specific function and for

16. Steady state of 1D phase transitionlayer Equation of state: Critical point for isothermal case and For specific function

17. Phase transition in 2D Equation of state: And for specific function isotherms metastable states

18. Simulation of immiscible liquids The attraction between particles in neighbor nodes was introduced (Shan – Chen, 1993). Here we denote the components by the indexes and . The total fluid density at a node depends on densities of all components as Here The total momentum at a node Momentum of each component is The interaction forces change the velocity of each component as

19. Phase transition from unstable state(waves of higher density) Red– liquid in unstable state. LightRed– liquid. Black– vapour. Grid 160x160

20. Phase transition from metastable statewith different nucleuses Red – liquid in metastable state (G=0.6; 0 = 1.6) Black – vapour. LightRed - liquid 0 = 0.67 0 = 0.8 0 = 0.5 Small 0 = 0.4 Grid160x160

21. Deformation and fragmentation ofconductive vapor bubbles inelectric field  = 0.5 t = 0 100 200 300 400 500 600 700  = 0.38 t = 100 200 300 400 500 600 700 770 850

22. Deformation and fragmentation ofconductive vapor bubbles inelectric field  = 0.2 t = 100 200 300 400 500 600 700 800 850

23. The droplet of higher permittivityin liquid dielectric under the action of electric field = 1.41; E = 0.035 E = 0.1

24. Deformation of vapor bubble under the action of electric field due to “electrostriction” Permittivity

25. Conclusions A new method for simulating the EHD phenomena using the LBE method is developed: Hydrodynamic flows and convective and diffusive transfer of charge carriers are simulated by LBEscheme, as well as interaction of liquid components and phase transitions and action of electric forces on a charged liquid. Evolution of potential distribution and conductive transport of charge are calculated using the finite difference method. The exact difference method (EDM) is not an expansion but is a new general way to incorporate the body force term into any variant of LBE Simulations show great potential of the LBE method especially for EHDproblems with free boundaries (systemswith vapor bubbles and multiple components withdifferent electric properties).

26. Lattice Boltzmann equation methodwith arbitrary equation of stateZhang, Chen (Phys. Rev. E, 2003) Idea:to use theisothermal LBE method(T=T0) For mass and momentum conservation laws + Usual energy equation, that can be solved by ordinary finite-difference method. Hereenergy equation is written indivergentformand can be solved, for example, by Lax–Vendroff two-step method. The equation of statewas introduced by means the body forces acted on the liquid in the nodes here the potential is expressed through equation of state

27. Liquid boiling with free surface in gravity field t = 39.5 t = 41.5 t = 43.5 t = 45.5 Density distribution. Pr = 10, Re = 3·105

28. Stages of evolution ofmultiparticle system (N. N. Bogolubov) 1. Stage ofinitialrandomizing (t 0). 2. Kineticstage (0<< t< ). 3. Hydrodynamicstage (t>). The local equilibrium was settledin small volumes. Even the exact information about single particle distribution functionf(r,,t) isunnecessary. Only several first moments of it are enough to know

29. Publications 1) Kupershtokh A. L., Calculations of the action of electric forces in the lattice Boltzmann equation method using the difference of equilibrium distribution functions// Proc. of the 7th Int. Conf. on Modern Problems of Electrophysics and Electrohydrodynamics of Liquids, St. Petersburg, Russia, pp.152–155, 2003. 2) Kupershtokh A. L., Medvedev D. A., Simulation of growth dynamics, deformation and fragmentation of vapor microbubbles in high electric field// Proc. of the 7th Int. Conf. on Modern Problems of Electrophysics and Electrohydrodynamics of Liquids, St. Petersburg, Russia, pp. 156–159, 2003.

30. Previousforms of body force term in LBE Method of modifying the BGK collision operator (MMCO) (Shan, Chen, 1993) where u+ = F/. The deviation from EDM Methods of explicit derivative (MED) of the equilibrium distribution function (He, Shan, Doolen, 1998) The terms that are proportional to are absent at all. If the first order expansion of in uis used we have

31. Previousforms of body force term in LBE Method of undefined coefficients (MUC) (Ladd, Verberg, 2001) Its were foundas A=0, B=Du, and In method of Guo, Zheng, Shi (2002) the MUC was used in combination with MMCO where u* = u/ 2. The deviation from EDM for coefficients that were found by authors is equal to This method exactly coincide with the method of modification of collision operator at t = 0.5.