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 + 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 \( \frac{\partial r}{\partial n}={r}_i{n}_i \).

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.

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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 )

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.

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 \( {\widehat{t}}_i \).

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.