Application of the Lattice Boltzmann method with moving ...€¦ · with moving boundaries in a...

BasicsMoving particles

Coarse-grained blood model

Application of the Lattice Boltzmann methodwith moving boundaries in a coarse-grained

suspension model for hemodynamics

Florian Janoschek Jens Harting

Department of Applied Physics, Eindhoven University of Technology, The Netherlands

Institute for Computational Physics, Stuttgart University, Germany

3rd June 2009

Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics



Lattice Boltzmann methodMid-link bounce-back boundary condition

The lattice Boltzmann method

D3Q19 lattice, drawing [Hecht and Harting, 2008]

Definitionsdiscrete velocities cr

populations nr (x, t)

Hydrodynamic quantities

density ρ(x, t) =∑

r

nr (x, t)

velocity u(x, t) =

∑r nr (x, t)cr

ρ(x, t)





The lattice Boltzmann method

D3Q19 lattice, drawing [Hecht and Harting, 2008]

Time evolutionCollisionn∗r (x, t) =

nr (x, t) −nr (x, t) − neq

r (ρ(x, t),u(x, t))

τ(BGK) with equilibrium population neq

r

Advectionnr (x, t) = n∗r (x − cr , t − 1)





Mid-link bounce-back boundary condition

[Ngu

yen

and

Ladd

,200

2]Advection/boundary condition

nr (x, t) =

n∗r (x, t − 1) x − cr wall

n∗r (x − cr , t − 1) otherwise

with cr = −cr

noslip: u = 0 at boundary




Moving boundary conditionFree motion of particlesLubrication correction

Moving boundary condition

[Ngu

yen

and

Ladd

,200

2]

Steady state:nr = nr (x, t) = n∗r (x, t) = neq

r (ρ,u)

Conventional bounce-back condition

nr = nr

consistent with fluid at rest (u = 0)Moving boundary condition must be consistentwith fluid at speed u , 0.





Moving boundary condition

[Ngu

yen

and

Ladd

,200

2]

neqr (ρ,u) , neq

r (ρ,u) for u , 0Equilibrium population [Chen et al., 2000]

neqr (ρ,u) =

ραr

[1 + βcru︸︷︷︸

changes sign for r → r

+12β2(cru)2

−12βu2 + O(|u|3)

]

Moving wall boundary condition

nr (x, t) = n∗r (x, t − 1) + 2ραrβcru︸︷︷︸first order correction term

consistent with fluid speed u , 0





Freely moving particles

Particle configuration characterized bytranslational and rotational position ri and oi

Analytical particle surface defined by function

Γ(x − ri , oi) ∈

{[0; 1[ x within particle i[1;∞[ x elsewhere

discretized on the latticeForces Fi and torques Ti acting on eachparticleIntegration of equations of motion like in typicalMolecular Dynamics codes (here: leap-frogintegrator)

Fi → vi → xi

Ti → ωi → oi





Momentum balance

[Ngu

yen

and

Ladd

,200

2]

Velocity at boundary node xb

vb(xb) = vi +ωi × (xb − ri)

Momentum transfer per timestep ∆t = 1

∆p = (2nr + 2ραrβcrvb)cr

results inforce

Fb = ∆p/∆t = ∆p

and torque

Tb = (xb − ri) × Fb

on moving particle.





Fluid extinction

Fluid nodes xf covered by the particle turn intowall nodes.Fluid momentum is incorporated by theparticle.

ForceFf =

∑r

nr (xf)cr

TorqueTf = (xf − ri) × Ff





Fluid creation

Wall nodes xp released by the particle turn intofluid nodes.Equilibrium population neq

r (ρ,vp) for thesystem’s initial density ρ and the particle’ssurface velocity vp is created.Fluid momentum is taken from the particle.

ForceFp = −ρvp

TorqueTp = (xp − ri) × Fp





Lubrication correction

The fluid between particles close to contact is notresolved due to the finite lattice resolution.

Static effects→ assumption of equilibriumdistribution at particle nodesDynamic effects→ correction term forspherical particles [Ladd and Verberg, 2001]

Fij =

−6πρν(RiRj)

2

(Ri + Rj)2rij [rij(vi − vj)]

(1

rij − Ri − Rj−

1∆c

)with cut-off separation ∆c = 2/3More flexible method: [Ding and Aidun, 2003]




MotivationModelApplication on blood

Human blood

Applications for blood simulation

study of transport phenomenasupport for surgerydesign of lab-on-a-chip devices. . .

100

101

102

103

10−2 10−1 100 101 102 103

µ/µ

(c=

0)

γ [s−1]

data

[Chi

en,1

970]

Blood properties

ρblood ≈ 1.06ρH2O

≈ 55 vol. % plasma≈ 44 vol. % redblood cellsµ , const: shearthinning

[Eva

nsan

dFu

ng,1

972]





Existing blood models

Continuum modelsµ = const (Newtonian fluid)µ = µ(γ) (Casson,Carreau-Yasuda)

[Boyd et al., 2007]

Particulate models that resolveRBC deformability

[Nog

uchi

and

Gom

pper

,200

5]

[Noguchi and Gompper, 2005],[Dupin et al., 2007]

Drawbackno resolution of particulate effects

Drawbackcomputational cost





New phenomenological model

Similar to existing models

cell trajectories integrated by molecular dynamics codeplasma modelled as Newtonian fluid by lattice Boltzmann

New in this modelinteractions of deformable cells modelled solely by soft potentials(at the present completely repulsive)

cell-cellcell-wall

hydrodynamic coupling between cells and plasma

The goal

resolution of particulate effectscapability to simulate small macroscopic systems in 3D(∼ 1 mm3)





Cells and vessel walls: BPrH-Potential

0.00.20.40.60.81.0

0.0 0.5 1.0 1.5 2.0

φrH

r

Repulsive Hooke potential

φrH(r) =

{(1 − r)2 r < 10 r ≥ 1

Orientation-dependent energy and rangeparameters ε(oi , oj) and σ(oi , oj , rij)[Berne and Pechukas, 1972]

φBPrH(oi , oj , rij) = ε(oi , oj) φrH(rij/σ(oi , oj , rij))

Surface σ(oi , oj , rij) = rij closely resemblespositions of two ellipsoids or ellipsoid andsphere that are in touchSize parameters σ⊥, σ‖, and σr

Strength parameters ε and εr





Blood plasma: lattice Boltzmann

Blood plasma

LB3D (D3Q19, BGK)viscosity µ, density ρf

Interaction with blood cellssuspended rigid particles[Nguyen and Ladd, 2002]no lubrication corrections, touching andoverlapping is part of the modeldensity ρp, sizes R⊥ and R‖

Interaction with vessel wallsmid-link bounce-back


[Ngu

yen

and

Ladd

,200

2]




Unit conversion I

Factors for conversion between the dimensionless quantities of thesimulation and physical units

dx: physical distance between to lattice nodesdt : physical length of one lattice Boltzmann timestepdm: physical mass at one lattice site populated with unit density





Unit conversion II

Space

Sufficient resolution of RBCs→ choose dx = 23 µm.

TimeNumerical instability and increasing deviations between inputradii and effective hydrodynamic radii for τ , 1→ choose τ = 1.Still fluid viscosity ν = (2τ − 1)/6 should match physical plasmaviscosity

νdx2

dt= ν(p) = 1.09 · 10−6 m2

s.

This fixes dt = 6.79 · 10−8 s.Mass

Assume unit density ρf = 1 for the fluid.Given dx and known density of blood plasma results indm = 3.05 · 10−16 kg.





Plane shear flow: parameter search

c=cpf=45 %

Preliminary parameters

ρp = 1.07σ⊥ = 6, σ‖ = 2R⊥ = 5.5, R‖ = 1.833ε = 0.05

µ =F

A γ

2

3

4567

101 102 103 104

µ/µ

(c=

0)

γ [s−1]

ε

100

101

100 101 102 103

µ/µ

(c=

0)

γ [s−1]

modelChien (1970)

Consistency with literature where attractive forces are negligible





Channels: qualitative demonstration

c=cpf=42 %

Wall node potential parameters

size σr = 0.5strength εr = ?

0.00.20.40.60.81.0

0 5 10 15 20 25 30

v z/

max

r(v z

)

r [µm]

maxr(vz) [cm/s]1.525.53 · 10−4

02468

10121416

0 5 10 15 20 25 30v z

[cm/s

]

r [µm]

εr5 · 10−3

5 · 101

Behavior consistent with literature (Goldsmith and Skalak, 1975)





Junctions: qualitative demonstration

0.00.20.40.60.81.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Qi/

∑ jQj

t [s]

not narrowednarrowed

Observationsclogging visibleeffect of εr

red cells choose faster branch asin literature [Fung, 1981]





Macroscopic systems

System

0.32 mm3

10243 lattice sites2.3 · 106 particles

Resources1024 processes onXC2 (SSC Karlsruhe)1.9 h/1000 LB steps

Per process

10242 lattice sites2.2 · 103 particles





Conclusion and outlook

ResultsModel reproduces shear viscosity at high shear rates forplausible choice of parameters.Important effects in channels and junctions are reproduced atleast qualitatively.The code is able to simulate macroscopic systems.

Further workImplementation of attractive forcesImprovement of choice of parametersQuantitative investigation of behavior in channels and junctionsOptimization for sparse systemsEnlargement of LB timestep dt if possible





Bibliography I

Berne, B. J. and Pechukas, P. (1972).Gaussian model potentials for molecular interactions.J. Chem. Phys., 56(8):4213–4216.

Boyd, J., Buick, J. M., and Green, S. (2007).Analysis of the Casson and Carreau-Yasuda non-Newtonianblood models in steady and oscillatory flows using the latticeBoltzmann method.Phys. Fluids, 19:093103.

Chen, H., Boghosian, B. M., Coveney, P. V., and Nekovee, M.(2000).A ternary lattice Boltzmann model for amphiphilic fluids.Proc. R. Soc. Lond. A, 456:2043–2057.





Bibliography II

Chien, S. (1970).Shear dependence of effective cell volume as a determinant ofblood viscosity.Science, 168(3934):977–979.

Ding, E.-J. and Aidun, C. K. (2003).Extension of the lattice-Boltzmann method for direct simulation ofsuspended particles near contact.J. Stat. Phys., 112(3/4):685–708.

Dupin, M. M., Halliday, I., Care, C. M., Alboul, L., and Munn, L. L.(2007).Modeling the flow of dense suspensions of deformable particlesin three dimensions.Phys. Rev. E, 75:066707.





Bibliography III

Evans, E. and Fung, Y.-C. (1972).Improved measurements of the erythrocyte geometry.Microvascular Research, 4:335–347.

Fung, Y. C. (1981).Biomechanics. Mechanical Properties of Living Tissues.Springer, New York.

Hecht, M. and Harting, J. (2008).General on-site velocity boundary conditions for latticeBoltzmann.http://arxiv.org/abs/0811.4593, submitted for publication.

Ladd, A. J. C. and Verberg, R. (2001).Lattice-Boltzmann simulations of particle-fluid suspensions.J. Stat. Phys., 104(5/6):1191–1251.





Bibliography IV

Nguyen, N.-Q. and Ladd, A. J. C. (2002).Lubrication corrections for lattice-Boltzmann simulations ofparticle suspensions.Phys. Rev. E, 66:046708.

Noguchi, H. and Gompper, G. (2005).Shape transitions of fluid vesicles and red blood cells in capillaryflows.PNAS, 102(40):14159–14164.


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