Basics Moving particles Coarse-grained blood model Application of the Lattice Boltzmann method with 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
Basics Lattice Boltzmann method Moving particles Mid-link bounce-back boundary condition Coarse-grained blood model The lattice Boltzmann method Definitions discrete velocities c r populations n r ( x , t ) Hydrodynamic quantities � density ρ ( x , t ) = n r ( x , t ) r � r n r ( x , t ) c r velocity u ( x , t ) = ρ ( x , t ) D3Q19 lattice, drawing [Hecht and Harting, 2008] Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Lattice Boltzmann method Moving particles Mid-link bounce-back boundary condition Coarse-grained blood model The lattice Boltzmann method Time evolution Collision n ∗ r ( x , t ) = n r ( x , t ) − n r ( x , t ) − n eq r ( ρ ( x , t ) , u ( x , t )) τ (BGK) with equilibrium population n eq r Advection n r ( x , t ) = n ∗ r ( x − c r , t − 1 ) D3Q19 lattice, drawing [Hecht and Harting, 2008] Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Lattice Boltzmann method Moving particles Mid-link bounce-back boundary condition Coarse-grained blood model Mid-link bounce-back boundary condition Advection/boundary condition n ∗ r ( x , t − 1 ) x − c r wall ¯ n r ( x , t ) = n ∗ r ( x − c r , t − 1 ) otherwise [Nguyen and Ladd, 2002] with c ¯ r = − c r noslip : u = 0 at boundary Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Moving boundary condition Steady state: n r = n r ( x , t ) = n ∗ r ( x , t ) = n eq r ( ρ, u ) Conventional bounce-back condition n r = n ¯ r consistent with fluid at rest ( u = 0 ) Moving boundary condition must be consistent [Nguyen and Ladd, 2002] with fluid at speed u � 0 . Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Moving boundary condition n eq r ( ρ, u ) � n eq r ( ρ, u ) for u � 0 ¯ Equilibrium population [Chen et al., 2000] n eq r ( ρ, u ) = + 1 2 β 2 ( c r u ) 2 − 1 � � 2 β u 2 + O ( | u | 3 ) ρα r 1 + β c r u ���� changes sign for r → ¯ r [Nguyen and Ladd, 2002] Moving wall boundary condition n r ( x , t ) = n ∗ r ( x , t − 1 ) + 2 ρα r β c r u ¯ � ����� �� ����� � first order correction term consistent with fluid speed u � 0 Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Freely moving particles Particle configuration characterized by translational and rotational position r i and ˆ o i Analytical particle surface defined by function � [ 0 ; 1 [ x within particle i Γ( x − r i , ˆ o i ) ∈ [ 1 ; ∞ [ x elsewhere discretized on the lattice Forces F i and torques T i acting on each particle Integration of equations of motion like in typical Molecular Dynamics codes (here: leap-frog integrator) F i → v i → x i T i → ω i → ˆ o i Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Momentum balance Velocity at boundary node x b v b ( x b ) = v i + ω i × ( x b − r i ) Momentum transfer per timestep ∆ t = 1 ∆ p = ( 2 n ¯ r + 2 ρα r β c r v b ) c ¯ r results in [Nguyen and Ladd, 2002] force F b = ∆ p / ∆ t = ∆ p and torque T b = ( x b − r i ) × F b on moving particle. Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Fluid extinction Fluid nodes x f covered by the particle turn into wall nodes. Fluid momentum is incorporated by the particle. Force � F f = n r ( x f ) c r r Torque T f = ( x f − r i ) × F f Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Fluid creation Wall nodes x p released by the particle turn into fluid nodes. Equilibrium population n eq r ( ρ, v p ) for the system’s initial density ρ and the particle’s surface velocity v p is created. Fluid momentum is taken from the particle. Force F p = − ρ v p Torque T p = ( x p − r i ) × F p Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Moving boundary condition Moving particles Free motion of particles Coarse-grained blood model Lubrication correction Lubrication correction The fluid between particles close to contact is not resolved due to the finite lattice resolution. Static effects → assumption of equilibrium distribution at particle nodes Dynamic effects → correction term for spherical particles [Ladd and Verberg, 2001] F ij = � � − 6 πρν ( R i R j ) 2 1 − 1 ( R i + R j ) 2 ˆ r ij [ˆ r ij ( v i − v j )] r ij − R i − R j ∆ c with cut-off separation ∆ c = 2 / 3 More flexible method: [Ding and Aidun, 2003] Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Motivation Moving particles Model Coarse-grained blood model Application on blood Human blood Applications for blood simulation Blood properties ρ blood ≈ 1 . 06 ρ H 2 O study of transport phenomena ≈ 55 vol. % plasma support for surgery ≈ 44 vol. % red design of lab-on-a-chip devices blood cells . . . µ � const: shear 10 3 thinning µ/µ ( c = 0 ) 10 2 data [Chien, 1970] [Evans and Fung, 1972] 10 1 10 0 10 − 2 10 − 1 10 0 10 1 10 2 10 3 γ [ s − 1 ] ˙ Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Motivation Moving particles Model Coarse-grained blood model Application on blood Existing blood models Continuum models Particulate models that resolve RBC deformability µ = const (Newtonian fluid) µ = µ ( ˙ γ ) (Casson, [Noguchi and Gompper, 2005] Carreau-Yasuda) [Boyd et al., 2007] [Noguchi and Gompper, 2005], [Dupin et al., 2007] Drawback Drawback no resolution of particulate effects computational cost Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Motivation Moving particles Model Coarse-grained blood model Application on blood New phenomenological model Similar to existing models cell trajectories integrated by molecular dynamics code plasma modelled as Newtonian fluid by lattice Boltzmann New in this model interactions of deformable cells modelled solely by soft potentials (at the present completely repulsive) cell-cell cell-wall hydrodynamic coupling between cells and plasma The goal resolution of particulate effects capability to simulate small macroscopic systems in 3D ( ∼ 1 mm 3 ) Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Motivation Moving particles Model Coarse-grained blood model Application on blood Cells and vessel walls: BPrH-Potential Repulsive Hooke potential 1 . 0 0 . 8 � 0 . 6 ( 1 − r ) 2 φ rH r < 1 ˜ φ rH ( r ) = 0 . 4 ˜ 0 r ≥ 1 0 . 2 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 Orientation-dependent energy and range r parameters ε (ˆ o i , ˆ o j ) and σ (ˆ o i , ˆ o j , ˆ r ij ) [Berne and Pechukas, 1972] o j ) ˜ φ BPrH (ˆ o i , ˆ o j , r ij ) = ε (ˆ o i , ˆ φ rH ( r ij /σ (ˆ o i , ˆ o j , ˆ r ij )) Surface σ (ˆ o i , ˆ o j , ˆ r ij ) = r ij closely resembles positions of two ellipsoids or ellipsoid and sphere that are in touch Size parameters σ ⊥ , σ � , and σ r Strength parameters ¯ ε and ¯ ε r Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
Basics Motivation Moving particles Model Coarse-grained blood model Application on blood Blood plasma: lattice Boltzmann Blood plasma LB3D (D3Q19, BGK) viscosity µ , density ρ f [Nguyen and Ladd, 2002] Interaction with blood cells suspended rigid particles [Nguyen and Ladd, 2002] no lubrication corrections, touching and overlapping is part of the model density ρ p , sizes R ⊥ and R � Interaction with vessel walls mid-link bounce-back Florian Janoschek, Jens Harting LBM with moving boundaries for coarse-grained hemodynamics
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