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 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)

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 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.

8. 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

9. 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

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 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).

14. Phase transition in 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 transition layer
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 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

21. Deformation and fragmentation of conductive vapor bubbles in electric field
 = 0.5
t =
 = 0.38
t =

22. Deformation and fragmentation of conductive vapor bubbles in electric field
 = 0.2
t =

23. The droplet of higher permittivity in 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 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).

26. 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

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

28. 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

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. 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

31. 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.