BoundaryElement Method in Microfluidics
Synonyms
Definition
The boundary element method (BEM) is a numerical method for solving partial differential equations which are encountered in many engineering disciplines such as solid and fluid mechanics, electromagnetics, and acoustics.
Overview
Boundary element method (or boundary integral method) is a numerical tool that is well applied to many linear problems in engineering. There are several advantages of the boundary element method (BEM) over other numerical methods (such as finite element method (FEM)) and some of which are (i) discretization and modeling of only the boundary of the solution domain, (ii) improved solutions in stress concentration problems, (iii) accuracy of the solution regarding that it is a semianalytical method (the integral equation is obtained using exact solution of the corresponding linear problem; the numerical approach is necessary only for evaluating the resulting integrals), and (iv) the possibility of application to domains that extend to infinity in one or more dimensions. Especially, for problems that involve successive remeshing, the boundaryonly nature of the BEM presents a valuable property – since there is no domain discretization, remeshing is not an important issue. Also, the continuity and compatibility conditions within the solution domain are satisfied in an exact manner in the BEM (in the FEM, on the contrary, the continuity and compatibility conditions across the elements are satisfied in a numerical manner).
The major disadvantage of the BEM is that nonlinear or nonhomogeneous applications of the method are not common since the method requires predefined fundamental solutions, which are to be determined for the given specific problem, and for most nonlinear problems, there are no such fundamental solutions provided in literature. There are some techniques to overcome this disadvantage of the BEM, such as defining internal cells for volume integration (which destroys the boundaryonly nature of the BEM) or using dual reciprocity method (which requires a set of additional collocation points defined in the solution domain).
Considering fluid flow in microchannels, typical flow speed is low (resulting in very low Reynolds number) and the inertia forces are negligible (in magnitude) when compared with the pressure or the viscous forces; the flow can be considered as the socalled creeping flow [1]. The governing equation of the creeping flow is the Stokes flow, which is a linear partial differential equation suitable for solution with the BEM. The boundary element representation of such flow was first developed by Youngren and Acrivos [2] for the analysis of flow over 3D and axisymmetric rigid particles. Same authors then extended their work for similar flow over deformable bubbles [3], and later a similar study involving viscous drops was introduced by Rallison and Acrivos [4]. As a different approach, the motion of a rigid sphere moving normal to a deformable interface was investigated using boundary element method by Lee and Leal [5]. All boundary element formulations indicated above are direct, which means they are obtained using field variables which has physical meaning – like components of the velocity vector, traction on the surface, and pressure.
Manipulation of bioparticles within a microchannel is a key ingredient for many biomedical and chemical applications. A rigorous simulation of the particle motion including the effect on the particle presence on the field variables requires massive remeshing when finite difference, finite element, and finite volumebased numerical techniques are considered. Moreover, the determination of the forces induced on the particles requires the calculation of gradient of the field variables. Therefore, for an accurate calculation of gradient of field variables, fine mesh is required on the particle surface. This issue is problematic when a particle moves in the vicinity of the wall. The interaction of the particle with the wall when the particle is moving close to the wall is difficult to model. In many cases, it is compulsory to introduce a model to take into account the particlewall interaction [6, 7]. Since BEM does not require meshing within the flow field and the exact calculation of gradient of field variables, and considering the linearity of the governing equations, BEM is a preferable tool for the simulation of the bioparticles in a microchannel. In this entry, advantages and challenges of the BEM for the simulation of particle trajectory for a pressuredriven flow in a microchannel will be discussed, and some preliminary simulations results will be presented.
Basic Methodology
Boundary Element Formulation of Flow Field
The governing equations for the steadystate Stokes flow (assuming the inertial effects to be negligible) can be given as − ∇P + μ∇^{2} u + ρ g · x = 0
where p is the pressure, u is the velocity vector, and ρ g is the body forces. The material properties are the viscosity, μ, and the density, ρ, of the fluid. Defining a modified pressure as P = p − ρ g ⋅ x where x represents the coordinates, the Stokes equation can be rewritten as − ∇P + μ∇^{2} u = 0. The hydrodynamic stress tensor can be written as \( {\sigma}_{ij}={\delta}_{ij}p+\mu \left(\frac{\partial {u}_i}{\partial {x}_j}+\frac{\partial {u}_j}{\partial {x}_i}\right) \) from which the tractions on the boundary of the region can be obtained as t _{ i } = σ _{ ij } n _{ j } where n _{ j } are the components of the unit normal vector pointing to fluid side at the given point. The boundary element (BE) formulation in this case is straightforward [1], where for a fixed point A the integral equations in 2D can be given as \( {C}_{ij}\left(\mathbf{A}\right)\kern0.5em {u}_j\left(\mathbf{A}\right)=\underset{C}{\int }{G}_{ij}\left(\mathbf{A},\mathbf{P}\right)\kern0.5em {t}_j\left(\mathbf{P}\right)\kern0.5em \mathrm{d}S\underset{C}{\int }{H}_{ij}\left(\mathbf{A},\mathbf{P}\right)\kern0.5em {u}_j\left(\mathbf{P}\right)\kern0.5em \mathrm{d}S \) which is referred to as the reduced boundary element equation (RBEE). It is necessary at this point to note that for the above equation and for all equations, Einstein’s summation convention is in place which implies (if not explicitly defined otherwise) a summation over repeated index. In RBEE, C _{ ij }(A) = δ _{ ij } if A ∈ V, δ _{ ij }/2 if A is on a smooth boundary, and 0 if A ∉ V. The 2D first and second fundamental solutions of the Stokes flow, e.g., G(A, P) and H(A, P), that appear as kernels in the integrals in RBEE are given as \( {G}_{ij}\left(\mathbf{A},\mathbf{P}\right)=\frac{1}{4\pi \mu}\left[ \ln (r){\delta}_{ij}+{r}_i{r}_j\right] \) and \( {H}_{ij}\left(\mathbf{A},\mathbf{P}\right)=\frac{r_i{r}_j\partial r}{\pi r\partial n} \) where r = r(A, P) is the distance between the fixed point A and the varied point P and r _{ i } = r _{ i }(A, P) are the components of the unit vector directed from A to P. The directional derivative \( \frac{\partial r}{\partial n} \) can be obtained as \( \frac{\partial r}{\partial n}={r}_i{n}_i \).
For the constant elements, since the field variables u _{ i } and t _{ i } are assumed to be constant over the element, the integrals can be written as: \( \underset{C_k}{\int }{G}_{ij}\left(\mathbf{A},\mathbf{P}\right)\kern0.5em {t}_j\left(\mathbf{P}\right)\kern0.5em \mathrm{d}S={t}_j\left({\mathbf{A}}_k\right)\underset{C_k}{\int }{G}_{ij}\left(\mathbf{A},{\mathbf{P}}_k\right)\kern0.5em \mathrm{d}S \) and \( \underset{C_k}{\int }{H}_{ij}\left(\mathbf{A},\mathbf{P}\right)\kern0.5em {u}_j\left(\mathbf{P}\right)\kern0.5em \mathrm{d}S={u}_j\left({\mathbf{A}}_k\right)\ \underset{C_k}{\int }{H}_{ij}\left(\mathbf{A},{\mathbf{P}}_k\right)\kern0.5em \mathrm{d}S \) where C _{ k } is the portion of the boundary that corresponds to the kth element with the node A _{ k }. When the RBEE is rewritten at all nodes A _{ k } k = 1.. N, with \( {G}_{ij}^{lk}=\underset{C_k}{\int }{G}_{ij}\left({\mathbf{A}}_l,{\mathbf{P}}_k\right)\kern0.5em \mathrm{d}S \) and \( {H}_{ij}^{lk}=\underset{C_k}{\int }{H}_{ij}\left({\mathbf{A}}_l,{\mathbf{P}}_k\right)\kern0.5em \mathrm{d}S \) and u _{ j }(A _{ l }) = u _{ j } ^{ l } , t _{ j }(A _{ l }) = t _{ j } ^{ l } and considering the node on a constant element is on a smooth curve (line), one can obtain a linear set of equations which can be given as \( \frac{1}{2}{u}_i^l+{H}_{ij}^{lk}\kern0.5em {u}_j^k={G}_{ij}^{lk}\kern0.5em {t}_j^k \) which can be rewritten (if diagonals H _{ ii } ^{ ll } are augmented with \( \frac{1}{2} \)) as H _{ ij } ^{ lk } u _{ j } ^{ k } = G _{ ij } ^{ lk } t _{ j } ^{ k } .
This gives a 2N × 2N matrix equation which may be written as H · u = G · t which then is solved in view of the boundary conditions. The imposition of boundary conditions in boundary element method is a simple task, which is achieved only by swapping the columns of the coefficient matrices. The aim is to collect all the known terms on the righthand side (rhs) and all the unknowns to the lefthand side (lhs) of the equal sign. Therefore, for a selected couple (u _{ n } ^{ m } , t _{ n } ^{ m } )
This corresponds to a column swap of the (2 * (m − 1) + n)th column of the coefficient matrices (by taking the negatives of the columns).
After the necessary column changes, the system of equations take the form
K · x = L · b where now x represents the unknown vector and b represents the vector formed by the boundary conditions of the system. The solution can be obtained after evaluating the rhs multiplication, e.g., K · x = l.
Simulation of Particle Trajectory
Particle trajectory is the result of the interaction of the particle with the electric field and the flow field. To simulate the particle trajectories, there are two approaches. The first approach is the Lagrangian tracking method, which neglects the finite size of the particles and treats them as point particles and solves the field variables without the presence of the particles [8]. In this case, only the effect of the field variables on the particle is considered. The second approach is the stress tensor approach, which includes the size effect of the particle. In this approach, the field variables are solved with the presence of the finitesized particle, and the particle translates as a result of the interaction of the particle with the electric and flow field [8]. In each incremental movement of the particle, the field variables need to be resolved. The former approach is very simple and works good to some extent, and the latter approach is accurate yet computationally expensive.
Lagrangian Tracking Method (LTM)
For the particle size considered in this study, the characteristic time scale of acceleration period of the motion is in the order of 10^{−4} s [8] which is much smaller than the time scale of the variation of the field variables. Therefore, the acceleration term can be safely neglected, and the velocity of the particle can be derived by equating the total external force to zero.
This approach is very simple and however has some limitations. LTM is valid if the particle size is small compared to the device dimensions, and in this version, it is valid for spherical particles. For the Stokes law to be valid, the particle needs to be several diameters away from the solid boundaries and the other particles (to model the hydrodynamic interaction of the particle with the wall, a correction factor needs to be introduced [6]) Moreover, if the disturbance of the flow field is significant due to the presence of the particle (e.g., the number of particles may be high within the domain or the size of the particle may be comparable with the microchannel size), the validity of the LTM is questionable. Since LTM does not include the presence of the particle, the simulation of the flow field can be performed with any standard software which handles the solution of PDEs, and the trajectory of the particles can be obtained at the postprocessing step.
Stress Tensor Approach
Boundary Element Formulation of Particle Trajectory
For this purpose, an incremental time stepping is suggested. At each time increment, the fluid flow will be assumed to be steady. This assumption holds if the particle dimensions are sufficiently small when compared with the characteristic dimensions of the problem (e.g., channel wall dimensions) and the change of location of the particle within one time increment is not significant. In this case, the inertial terms in the fluid flow equation can be neglected, and a steadystate assumption can be made.
In the analysis, further assumptions can be stated as the particle is bouyant, e.g., the net force acting on it (as it freely moves within the medium) is zero and the mass of the particle and the time rate of change of its velocity are sufficiently small that we may assume the inertial effects in the motion of the particle may be neglected.
Also, within the framework of this analysis, the particle is assumed to be rigid.
Recall the matrix equations of the BEE and how the boundary conditions are imposed on the system. The imposition of such boundary conditions requires, at all points of the defined boundary, the definition of one and only one of the couples (u _{ n } ^{ m } , t _{ n } ^{ m } ) or at least a combination of the two. When the motion of a particle in the fluid is considered neither the velocity components, u _{ i }, nor the traction components, t _{ i }, nor a combination of the two is known at a given element m. Rather, it is known that since the particle moves freely (e.g., free of forces), the net force over the surface will be zero. Also, the motion is a rigid body motion; therefore, all points on the particle surface will have a velocity that can be expressed as \( {u}_i={u}_i^b+\omega r{\widehat{t}}_i \).
Here, u _{ i } represents the velocity at the particle boundary, u _{ i } ^{ b } are the velocity components of the particle center (translation), and ω is the angular velocity about the particle center. The distance between the particle center and the boundary point is given as r and the components of the counterclockwise normal to the position vector form the center of the particle to the corresponding node is denoted by \( {\widehat{t}}_i \).
At this point, one can easily judge that all traction components for the nodes belonging to the particle are unknown. Therefore, a column swap of columns for the nodes M to N is required.
To make the system of equations solvable, three more equations should be stated. These three equations are obtained from the equilibrium conditions, e.g., the total force and moment on the particle boundary should be zero. Thus, the integrals relating the tractions to the total force and moment should add up to zero. This will bring in the required three more equations.
As the system of equations are solved, the rigid body motion parameters for the particle center will be obtained. With this velocity and the predefined time step, the particle may be moved to its new calculated location.
Key Research Findings
Geometric parameters used in the simulations
L  500 μm  x_{r}  50 μm 
W  100 μm  y_{r}  10, 20, 30, 50, 70 μm 
L_{h}  100 μm  d  10 μm 
W_{h}  70 μm  d_{l}  10 μm 
d_{s}  6 μm 
Future Research Directions

Considering the electrokinetic phenomena occurring in many microfluidicbased devices, such as electroosmotic pumping, electrophoresis, and dielectrophoresis, the governing equation is the Laplace equation which is in many cases also a linear partial differential equation. Electrokinetic forces may also be included in the BEM model together with the flow field. Similarly, inclusion of the acoustic and/or magnetic forces is also possible since the governing equations are in the linear form for many microfluidic applications.

Although neglection of the acceleration term for the particle is a reasonable approach, the neglection of the timedependent term for the Stokes flow is problematic especially for nonspherical particles. Inclusion of the time derivative term for the flow field may result in more rigorous solution.

2D modeling is very common in microfluidic applications. Strictly speaking 2D modeling is appropriate for electroosmoticbased flow in which the flow profile is pluglike. For pressuredriven flows, a rigorous modeling requires 3D modeling. 3D modeling with BEM is a straightforward extension of the 2D model. Since the BEM discretizes only the boundary of the solution domain and no need for remeshing as the particle moves within the flow field, BEM can be implemented effectively even for 3D simulations.

In BEM, all the elements of the system matrices can be determined independently, and the resulting linear system of equations has a fully populated coefficient matrix. Referring to these issues, the algorithm is highly parallelable. Moreover, by exploring the nature of the linear system resulting in microfluidic applications, effective iterative numerical algorithms can be developed.

In many microfluidic applications, the number concentration of the bioparticles can be high, and the particleparticle interaction may be important when predicting the trajectory of particles. Introduction of many particles in a microchannel is a straightforward extension of the current model. Again with the advantages associated with BEM, simulation of many particles can be computed in an effective manner.

In the case of biomedical applications, bioparticles are highly deformable bodies. Although BEM is suitable for linear systems, nonlinear modeling has also been implemented for plasticity problems. Therefore, the inclusion of the deformation of the bioparticles is possible with BEM formulation which is very valuable for microfluidic applications.
CrossReferences
References
 1.Wrobel LC (2002) Boundary element method. Applications in thermofluids and acoustics, vol 1. Wiley, ChichesterGoogle Scholar
 2.Youngreen GK, Acrivos A (1975) Stokes flow past a particle of arbitrary shape: a numerical method of solution. J Fluid Mech 69:377–403CrossRefMathSciNetGoogle Scholar
 3.Youngreen GK, Acrivos A (1976) On the shape of a gas bubble in a viscous extensional flow. J Fluid Mech 76:433–442CrossRefGoogle Scholar
 4.Rallison JM, Acrivos A (1978) A numerical study of the deformation and burst of a viscous drop in an extensional flow. J Fluid Mech 89:191–200CrossRefzbMATHGoogle Scholar
 5.Lee SH, Leal LG (1982) The motion of a sphere in the presence of a deformable interface. Part 2: numerical study of the translation of a sphere normal to an interface. J Colloid Interface Sci 87:81–106CrossRefGoogle Scholar
 6.Al Quddus N, Moussa WA, Bhattacharjee S (2008) Motion of a spherical particle in a cylindrical channel using arbitrary Lagrangian–Eulerian method. J Colloid Interface Sci 17:20–630Google Scholar
 7.Ai Y, Joo SW, Jiang Y, Xuan X, Qian S (2009) Pressuredriven transport of particles through a converging–diverging microchannel. Biomicrofluidics 3:022404CrossRefGoogle Scholar
 8.Çetin B, Li D (2011) Dielectrophoresis in microfluidics technology. Electrophoresis 32:2410–2427, Special issue on dielectrophoresisCrossRefGoogle Scholar
 9.Bruus H (2008) Theoretical microfluidics. Oxford University Press, New YorkGoogle Scholar
 10.House DL, Luo H (2010) Electrophoretic mobility of a colloidal cylinder between two parallel walls. Eng Anal Bound Elem 34(5):471–476CrossRefzbMATHMathSciNetGoogle Scholar
 11.House DL, Luo H (2011) Effect of direct current dielectrophoresis on the trajectory of a nonconducting colloidal sphere in a bent pore. Electrophoresis 32:3277–3285CrossRefGoogle Scholar