ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 30 Number 7 July 2013 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIET Y Institute of Physics PUBLISHING CHIN. PHYS. LETT. Vol. 30, No. 7 (2013) 074702 An Immersed Boundary-Lattice Boltzmann Simulation of Particle Hydrodynamic Focusing in a Straight Microchannel * SUN Dong-Ke(孙东科)**, JIANG Di(ñ&), XIANG Nan(项楠), CHEN Ke(陈科), NI Zhong-Hua(X¥u)** Jiangsu Key Laboratory for Design and Manufacture of Micro-Nano Biomedical Instruments, School of Mechanical Engineering, Southeast University, Nanjing 211189 (Received 5 March 2013) An immersed boundary (IB)-lattice Boltzmamm method (LBM) coupled model is utilized to study the particle focusing in a straight microchannel. The LBM is used to solve the incompressible fluid flow over a regular Eulerian grid, while the IB method is employed to couple the bead-spring model which represents the fluid- particle interaction. After model validation, the simulations for hydrodynamic focusing of the single and multi particles are performed. The particles can be focused into the equilibrium positions under the pressure gradient and self-rotation induced forces, and the particle radius and Reynolds number are the key parameters influencing the focusing dynamics. This work demonstrates the potential usefulness of the IB-LBM model in studying the particle hydrodynamic focusing. PACS: 47.27.nd, 47.11.−j, 87.85.gf, 04.60.Nc DOI: 10.1088/0256-307X/30/7/074702 The particle hydrodynamic focusing, which takes ticle dynamics at finite particle Reynolds number, all place when a fluid with suspended particles flows in of the work that we are aware of has been confined a microchannel at a sufficiently high Reynolds num- to considering the particle motions including tank- ber, has been developed as one of the most utilized treading, swinging, and tumbling,[19−21] and fluid- techniques in microfluidics.[1] During the focusing, the structure interactions.[22] A good understanding on uniformly distributed particles in the fluid tend to mi- particle hydrodynamic focusing is still lacking. There- grate to certain positions by the balance of forces as- fore, it is necessary to investigate the complex hydro- sociated with the fluid inertia and the influence of dynamical interactions by the IB-LBM coupled model. channel walls. Various simulations were successfully The aim of this work is to study the hydrodynamic carried out to study the focusing mechanism in mi- focusing of particles in a straight microchannel by the crochannels by the traditional Navier-Stokes equation- IB-LBM coupled model. We first present the simu- based methods,[2−5] however the fluid-particle interac- lation methodology involving the LB model for fluids tion is not fully considered in detail in all the studies. flow, the bead-spring model for particles, and the cou- The lattice Boltzmann method (LBM),[6] for its pling method by IBM. After discussing the numerical advantages in efficiency and parallel scalability, has results, a summary closes this letter. gained increasing popularity in the last two decades We adopt the standard lattice Bhatnagar–Gross– as an alternative numerical approach for solving prob- Krook (LBGK) approach with a single relaxation time lems in particle-fluid systems.[7−9] Ladd[10;11] has suc- scheme[6] to solve the LB equation. Accordingly, the cessfully applied the LBM to model particle-fluid sus- evolution equation with a force term can be expressed pensions, and the various numerical studies on parti- as[23] cle dynamics in suspensions were performed following f (x + e ∆t; t + ∆t) − f (x; t) Ladd’s work.[12−15] However, a large number of grids i i i for the particles should be employed when modeling 1 eq = − [fi(x; t) − fi (x; t)] + ∆tFi(x; t); (1) accurately the physical boundaries of particles by the 휏 Ladd-type model, which limits the application of the where fi(x; t) is the particle distribution function model.[16] Recently, the immersed boundary method (PDF) representing the probability of finding a pseudo (IBM)[17] has emerged as a superior particle modeling fluid particle moving in the ith direction on the dis- [18] technique. Fogelson and Peskin have shown that crete lattice at position x and time t, ei is the discrete this method is especially suitable for the simulation moving velocity of the pseudo fluid particle, ∆t is the of the fluid-particle system. Feng and Michaelides[16] time step, and 휏 is the relaxation time. The force first developed the IB-LBM coupled model and ap- term Fi(x; t) is considered to be caused by the pres- plied it to simulate the sedimentation of a large num- sure gradient and the interaction from the particles. ber of particles in an enclosure. Although much The D2Q9 topology,[6] in which the two-dimensional progress has been made towards elucidating the par- (2D) space is discretized into a regular square lattice *Supported by the National Natural Science Foundation of China under Grant No 91023024, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No 20110092110003, and the National Science Foundation for Post-doctoral Scientists of China under Grant No 2012M511647. **Corresponding author. Email: [email protected]; [email protected] © 2013 Chinese Physical Society and IOP Publishing Ltd 074702-1 CHIN. PHYS. LETT. Vol. 30, No. 7 (2013) 074702 including nine velocities, is employed in the present and the membrane configuration xn is updated ac- eq [19] work. The equilibrium distribution function fi is de- cording to the local flow velocity uf fined by X up = u(xn)훿(xf − xn); (10) 2 2 f eq h ei · u (ei · u) u i fi = wi휌 1 + 3 2 + 4:5 4 − 1:5 2 ; (2) c c c where 훿(·) denotes a discretized Dirac delta function where u is the macroscopic flow velocity. The discrete with a finite support domain. Since the 훿-function is velocities ei are given by not restricted a priori and can be chosen freely obey- ing the basic properties to maintain momentum and 8(0; 0); i = 0; > angular momentum conservation,[17;20] we adopt the < (i−1) (i−1) [20] ei = (cos[ 2 휋]; sin[ 2 휋])c; i = 1; 2; 3; 4; common decomposition for the 2D model, >p : 2(cos[ (2i−9) 휋]; sin[ (2i−9) 휋])c; i = 5; 6; 7; 8; 4 4 훿(r) = 휑(x)휑(y); (3) where c = ∆x=∆t is the lattice speed, and ∆x is the lattice spacing. In the D2Q9 scheme, the force term {︂1 − jrj; jrj ≤ 1; 휑(r) = (11) Fi(x; t) in Eq. (1) can be expressed as 0; jrj ≥ 1: (︁ 1 )︁ h e − u e · u i To validate the coupled IB-LBM model, the flow pass- F = 1 − w 3 i + 4:5 i e · F ; i 2휏 i c2 c4 i ing over a circular cylinder, as shown in Fig. 1(a), is (4) taken as a benchmark to confirm that the model is working properly. The drag force coefficient C of the where wi are the weight coefficients given by w0 = 4=9, D cylinder can be accurately approximated by[25] w1;2;3;4 = 1=9 and w5;6;7;8 = 1=36, and F is the force on the fluid. The macroscopic fluid density 4휋 CD = ; (12) 휌, velocity u and kinematic viscosity 휈 can be ob- − 1 2 3 P P 1 ln 휑 2 − 0:738 + 휑 − 0:887휑 + 2:039휑 tained by 휌 = i fi, 휌u = i cifi + 2 F ∆t, and 1 1 2 2 2 휈 = 3 (휏 − 2 )c ∆t, respectively. where the solid fraction is defined as 휑 ≡ 휋R =L , R The particle is represented by the bead-spring (BS) is the radius of the cylinder, and L is the side length [24] model including N membrane Lagrange nodes, of the domain. In the simulation, we set R = 5, 6, which are connected to their neighboring Lagrange 7, 8, 9, L = 40, and 휏 = 0:8. The difference between nodes by springs. The total energy E of the BS model the pressures at the inlet/outlet of the channel is set contains three parts, the elastic energy El stored in the 2 to be ∆p = 0:002휌푐s. The LBM simulated data coin- stretch/compression spring, the bending energy Eb cides well with the analytical approximation solution stored in the bending springs, and the area energy Ea as shown in Fig. 1(b). representing the incompressibility. These three kinds of energies can be expressed as (a) Per iodic boundary t 4.0 in ou p (b) p 3.5 Simulated results 1 N (︁ln − l0 )︁ X Analytal approic ximation El = Kl ; (5) 3.0 2 n=1 l0 D N C 2.5 1 X 2 (︁휃n − 휃n;0 )︁ Eb = Kb tan ; (6) 2 n=1 2 2.0 2 1 (︁s − s0 )︁ 1.5 Ea = Ka ; (7) Pressure boundary, 2 s0 Periodic boundary Pressure boundary, 5.0 6.0 7.0 8.0 9.0 R where l is the length of the nth spring element, 휃 de- n n Fig. 1. notes the angle of the bending element, s is the surface (a) Boundary conditions and (b) drag coefficient CD as a function of the cylinder radius R for the flow area of the membrane, and N is the total number of passing over a circular cylinder. spring elements; l , 휃 and s denote the references of 0 n;0 0 Next, the simulation of the hydrodynamic focus- length, bending angle, and surface area, respectively.