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

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:

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.

Development of discrete models of medium Molecular dynamics (Alder, 1960) Kinetic Boltzmann equation (1872) 1964 Lattice Gas Automata Boltzmann equations with discrete set of velocities 1988 1997 Lattice Boltzmann Equation Chapman – Enskog expansion Macroscopic equations of hydrodynamics (Navier – Stokes equations)

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

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)

Lattice Boltzmann equation method (LBE) The discrete single-particle distribution functions Nk 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.

New general method of incorporating a 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 full derivative along the Lagrange coordinate at a constant density is equal to Thus, we obtained the Boltzmann equation in form

Exact difference method for lattice Boltzmann equation After discretization of Boltzmann equation in velocity space we have Here the changes of the distribution functions Nk due to the force F 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

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

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

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

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

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

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

Steady state of 1D phase transition layer Equation of state: Critical point for isothermal case and For specific function

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

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

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

Phase transition from metastable state with 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 Grid 160x160

Deformation and fragmentation of conductive vapor bubbles in electric field  = 0.5 t = 0 100 200 300 400 500 600 700  = 0.38 t = 100 200 300 400 500 600 700 770 850

Deformation and fragmentation of conductive vapor bubbles in electric field  = 0.2 t = 100 200 300 400 500 600 700 800 850

The droplet of higher permittivity in liquid dielectric under the action of electric field  = 1.41; E = 0.035 E = 0.1

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

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 LBE scheme, 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 EHD problems with free boundaries (systems with vapor bubbles and multiple components with different electric properties).

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

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

Stages of evolution of multiparticle system (N. N. Bogolubov) 1. Stage of initial randomizing (t  0). 2. Kinetic stage (0 << t < ). 3. Hydrodynamic stage (t > ). The local equilibrium was settled in small volumes. Even the exact information about single particle distribution function f(r, , t) is unnecessary. Only several first moments of it are enough to know

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.

Previous forms 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 u is used we have

Previous forms of body force term in LBE Method of undefined coefficients (MUC) (Ladd, Verberg, 2001) Its were found as 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.