# Encyclopedia of Microfluidics and Nanofluidics

Living Edition
| Editors: Dongqing Li

# Boundary-Element Method in Microfluidics

• Barbaros Cetin
• Besim Baranoğlu
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27758-0_121-6

## 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 semi-analytical 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 boundary-only 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 boundary-only 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 so-called 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 volume-based 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 particle-wall 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 pressure-driven 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 steady-state Stokes flow (assuming the inertial effects to be negligible) can be given as − ∇P + μ2 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 = pρ gx where x represents the coordinates, the Stokes equation can be rewritten as − ∇P + μ2 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 AV, δ ij /2 if A is on a smooth boundary, and 0 if AV. 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 $$\frac{\partial r}{\partial n}={r}_i{n}_i$$.

The RBEE is exact in the sense that, if the integral equations can be solved, the solution will be the exact solution to the actual model. Yet, even with many assumptions and simplifications both on geometry of the problem and the boundary conditions, the solution to RBEE is, if not impossible, very hard to obtain. Therefore, a numerical solution is attempted. For this, the boundary of the solution region is divided into a predefined number of elements. For each element, there exist a number of computational nodes (will be called only “nodes”). For simplicity, without losing generality, in this discussion constant elements are employed. For constant elements, the computational node is at the midpoint of the element, and all field variables associated with the element are assumed to have a constant value that equals to the value at the node of the element. A simplified mesh with constant elements can be seen in Fig. 1.

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 right-hand side (rhs) and all the unknowns to the left-hand side (lhs) of the equal sign. Therefore, for a selected couple (u n m , t n m )

which corresponds to the component in the n-direction of the m th element, if t n m is known, no action is required since the known/unknown is in the right place; but if u n m is known, then all terms containing this term should be swapped with the terms involving t n m , e.g., for each row, the corresponding equation is replaced with the equation below:
$$-{G}_{in}^{lm}\kern0.5em {t}_n^m=-{H}_{in}^{lm}\kern0.5em {u}_n^m$$

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 finite-sized 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)

In this approach, particles are assumed to be point particles, and the effect of the particle on the field variables is ignored; only the effect of the field variables on the particle is considered. The field variables are determined without the presence of the particles. The particle position x p can be determined, by integrating the particle velocity together with the initial position:
$${\mathbf{x}}_p(t)={\mathbf{x}}_o+{\displaystyle {\int}_0^t{\mathbf{u}}_p}\left(\tau \right) d\tau .$$
For a fixed frame of reference, the translational motion of a particle is governed by
$${m}_p\frac{d{\mathbf{u}}_p}{ dt}={\mathbf{F}}_{\mathrm{ext}},$$
where m p is the particle mass and F ext is the net external force. In a microfluidic environment, one source of the external force is the drag force on the particle. Depending on the application, on top of the drag force, electrokinetic forces, magnetic forces, or acoustic forces may need to be included. Depending on the particle, Brownian motion may also be included. In LTM, the analytical expression for a spherical particle for a given force is used. The drag force on a spherical particle is given by
$${\mathbf{F}}_{\mathrm{drag}}=6\pi \mu R\left(\mathbf{u}-{\mathbf{u}}_p\right),$$
at the creeping flow limit, which is known as the Stokes law, where R is the particle radius, u is the fluid velocity, and u p is the particle velocity (the analytical expressions for other forces can be found elsewhere [9]).

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 post-processing step.

#### Stress Tensor Approach

In this approach, the field variables are determined with the presence of the finite particle size. The resultant force on the particle can be obtained by integrating the appropriate stress tensor on the particle surface. In the case of a fluid flow, the resultant drag force on the particle can be determined by integrating the hydrodynamic stress tensor, which is given as
$$\underset{\bar{\mkern6mu}}{{\mathbf{T}}_h}=-p\underset{\bar{\mkern6mu}}{\mathbf{I}}+\mu \left(\nabla \mathbf{u}+\nabla {\mathbf{u}}^T\right)$$
over the particle surface. Once the resultant forces are given, Newton’s second law of motion can be integrated to get the new position of the particle. This approach can be applied for any geometry, and particle interaction with the field and particle-particle interaction can be included. The drawback of this method is that it is computationally expensive. As the particle moves in the microchannel, the meshes need to be updated from time to time. Although BEM can be used for both approaches (LTM and STA), at this point BEM introduces a unique advantage for STA since in BEM only the boundaries of the domain are meshed. For an undeformable body, only the mesh on the particle surface needs to be moved in accordance with the appropriate forces. Utilizing this fact, Dustin and Luo [10, 11] implemented BEM to simulate the particle trajectory within a microchannel under the action of electrophoretic and electroosmotically driven flow field.

### 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 steady-state 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 $${\widehat{t}}_i$$.

Considering the above discussion, three new unknowns can be introduced (in 2D): u 1 b , u 2 b , and ω. All nodal velocities on the particle boundary will be a function of these three new unknowns. Assume now that the nodes M to N (which makes up NM nodes) belong to the particle. Then, for a node m on the particle boundary, one can write the above rigid body motion equation with the emphasis that rr (m) gives the distance from the node to the particle center and $${\widehat{t}}_i\to {\widehat{t}}_i^{(m)}$$ is the unit tangent at the given node. With this, the equations can be modified for a given row (l, i) of the system of equations (which in fact belongs to the equation that is obtained by fixing the point A to A l and in the i-direction) which in fact corresponds to the (2 × (l − 1) + i)th row. The modification would be
$${\displaystyle \sum_{m=1}^{M-1}{H}_{ij}^{lm}}{u}_j^m+{\displaystyle \sum_{m=M}^N{H}_{ij}^{lm}}\left({u}_i^b+\omega {r}^{(m)}{\widehat{t}}_j^{(m)}\right)={G}_{ij}^{lm}{t}_j^m$$
which implies the addition of three columns for the three unknowns in the H matrix. For the elements of the first two columns, we have $${\displaystyle \sum_{m=M}^N{H}_{ij}^{lm}}$$ and for the elements of the third column, $${\displaystyle \sum_{m=M}^N{H}_{ij}^{lk}}\Big({r}^{(m)}{\widehat{t}}_j^{(m)}$$.

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

As a benchmark study, two-channel geometry and two-particle geometry are chosen as shown in Fig. 2. Fist geometry is a straight channel with a channel width, W, and a channel length, L. For the particle geometry, spherical (with a diameter d) and elliptical (with long-axis diameter, dl, and short-axis diameter, ds) particles are chosen, since many bioparticle-like bacteria and cells can be modeled with these two geometries. The particles are released initially from the location (xr, yr) and the particle simulations are stopped when the particles reach x = 450 μm. The geometric parameters used in the simulations are given in Table 1.
Table 1

Geometric parameters used in the simulations

 L 500 μm xr 50 μm W 100 μm yr 10, 20, 30, 50, 70 μm Lh 100 μm d 10 μm Wh 70 μm dl 10 μm ds 6 μm
In the simulations, the density of the fluid is taken as 1,000 kg/m3, and the viscosity is taken as 0.001 Pa.s which are the typical values for water (for bioparticle-based applications, the buffer solutions are water-based solutions). The particle is assumed to be buoyant which is the case for many bioparticles. At the inlet of the channel, uniform velocity of 300 μm/s is assigned. On the channel walls and the particle surface, no-slip boundary condition is applied, and zero pressure at the outlet of the channel is assigned. Typical mesh used in the simulations is shown in Fig. 3. 300 elements in the length direction, 50 elements in the width directions, and 32 elements on the particles are used for the straight channel. 250 elements on the top wall, 390 elements on the lower wall (finer mesh is implemented in the vicinity of the hurdle), and 50 elements on the inlet and the exit are used for the channel with the hurdle. The simulations were performed on an HP Z400 Workstation (Intel Xeon W3550, Quad core, 3.06GHz, 16GB RAM). Although the total time step taken for each particle depends on the released point, it was observed that typical run-time for the simulation of a single particle varies between 1 and 2 min.
The particle trajectory of 10 μm, spherical particle is illustrated in Fig. 4. The particles are released from yr = 10, 30, and 50 μm. For each time step of 0.25 s, the orientation of the particle is shown in the figure. To demonstrate the rotation of the particle, one point on the particle is marked with red dot. The x-velocity and the angular speed of the particles are given in Fig. 4-b. Since it is a spherical particle in the Stokes flow, the x-velocity and the angular speed of the particles are steady. As the particle moves closer to the centerline, it travels with a higher velocity due to the parabolic nature of the velocity profile. Therefore, the x-velocity of the article increases as yr increases. As the particle moves closer to the channel wall, the symmetry of the flow at the upper and the lower surface of the channel distorts which results in a higher rotational speed. As a result, the angular speed of the particle increases as yr decreases. Since the velocity on the upper half of the particles is higher than the lower half, the rotation of the particles is in the clockwise direction. Due to the symmetry of the flow field, the particles released from yr = 70 and 90 μm would have the same trajectory with a counterclockwise rotation.
The motion of an elliptical particle in a straight channel is shown in Fig. 5. The rotation of the particle can also be seen, and again it is in the clockwise direction. The same conclusions for spherical particle are also valid for the elliptical particle. However, the major difference is the x-velocity and the angular speed of the particle. Especially for the angular speed, there exists a steady periodic behavior due to the geometry of the particle. The magnitude of the angular speed depends on the instant orientation of the particle. Moreover, even x-velocity has a steady periodic behavior for the particle moving near the wall.
As a second benchmark problem, the motion of spherical and elliptical particles are analyzed in a channel with a hurdle at the middle. The results for spherical (Fig. 6) and elliptical particles (Fig. 7) are shown in the figures below. The rotation of the particles can also be realized. The particles realesed from 10 μm follows a streamline which is different than that of 10 μm after the hurdle due to hydrodynamic interaction of the particles at the corners and within the hurdle section. As the released location increases, this issue diminishes. Although, it is not simulated, the interaction of the particles with the corners has size dependence. So, the location of the particles after the hurdle depends on the size of the particle. BEM has clearly the ability to model this hydrodynamic interaction with the wall without any need for correction factor. The dependence of the equilibrium position after the hurdle on particle size is the key ingredient for the microfluidic devices for hydrodynamic separation of bioparticles. With the ability of BEM, this issue can be explored in details to come up with efficient microfluidics bioparticle separators.

## Future Research Directions

BEM is an effective computational tool for microfluidic simulations. Especially, computational tools to simulate motion of particles in a microchannel are very crucial for the optimum design of the microfluidic devices. The implementation of BEM for this kind of problems is relatively new in this field. Therefore, there are many possible future research directions which can be summarized as follows:
• Considering the electrokinetic phenomena occurring in many microfluidic-based 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 time-dependent 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 electroosmotic-based flow in which the flow profile is pluglike. For pressure-driven 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 particle-particle 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.

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