Marc S. Ingber

Particle migration in a Couette apparatus: Experiment and modeling

Suspensions comprised of neutrally buoyant spheres in Newtonian fluids undergoing creeping flow in the annular region between two rotating, coaxial cylinders (a wide-gap Couette) display a bulk migration of particles towards regions of lower shear rate. A series of experiments are performed to characterize this particle migration, including the influence of particle size, surface roughness, and volume fraction. Little, if any, effect of particle surface roughness is observed. An existing continuum diffusive-flux model [Phillips et al. (1992)] for predicting particle concentration profiles in monomodal suspensions is evaluated using the current series of experimental data. This model predicts a dependence of the migration rate on the square of the suspended particles’ radius, a2; whereas the present experiments indicate that systems with average particle volume fractions of 50% display a rate that scales with a3. Previous use of the diffusive-flux model has assumed constant values for diffusion coefficient...

Flow-aligned tensor models for suspension flows

Abstract Models to describe the transport of particles in suspension flows have progressed considerably during the last decade. In one class of models, designated as suspension balance models, the stress in the particle phase is described by a constitutive equation, and particle transport is driven by gradients in this stress. In another class of models, designated as diffusive flux models, the motion of particles within the suspension is described through a diffusion equation based on shear rate and effective viscosity gradients. Original implementations of both classes of models lacked a complete description of the anisotropy of the particle interactions. Because of this, the prediction of particle concentration in torsional flows in parallel plate and cone-and-plate geometries did not match experimental data for either class of models. In this work, the normal stress differences for the suspension balance formulation are modeled using a frame-invariant flow-aligned tensor. By analogy, the diffusive flux model is reformulated using the same flow-aligned tensor, which allows separate scaling arguments for the magnitude of the diffusive flux to be implemented in the three principal directions of flow. Using these flow-aligned tensor formulations, the main shortcomings of the original models are eliminated in a unified manner. Steady-state and transient simulations are performed on various one-dimensional and two-dimensional flows for which experimental data are available, using finite-difference and finite-element schemes, respectively. The results obtained are in good agreement with experimental data for consistent sets of empirical constants, without the need for ad hoc additional terms.

Modelling of concentrated suspensions using a continuum constitutive equation

We simulate the behaviour of suspensions of large-particle, non-Brownian, neutrally-buoyant spheres in a Newtonian liquid with a Galerkin, finite element, Navier–Stokes solver into which is incorporated a continuum constitutive relationship described by Phillips et al . (1992). This constitutive description couples a Newtonian stress/shear-rate relationship (where the local viscosity of the suspension is dependent on the local volume fraction of solids) with a shear-induced migration model of the suspended particles. The two-dimensional and three-dimensional (axisymmetric) model is benchmarked with a variety of single-phase and two-phase analytic solutions and experimental results. We describe new experimental results using nuclear magnetic resonance imaging to determine non-invasively the evolution of the solids-concentration profiles of initially well-mixed suspensions as they separate when subjected to slow flow between counter-rotating eccentric cylinders and to piston-driven flow in a pipe. We show good qualitative and quantitative agreement of the numerical predictions and the experimental measurements. These flows result in complex final distributions of the solids, causing rheological behaviour that cannot be accurately described with typical single-phase constitutive equations.

Stokes flow around cylinders in a bounded two-dimensional domain using multipole-accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N2) (iterative solvers) or O(N3) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well-posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill-conditioned systems of equations. The model system here is a two-dimensional wide-gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well-conditioned to allow the use of fast multipole methods. The direct approach results in severe ill-conditioning, to a point where application of the multipole method leads to non-convergence of the solution iteration. Copyright

A boundary element approach for parabolic differential equations using a class of particular solutions

Abstract Two boundary element methods are developed for solving a class of parabolic differential equations. The methods avoid performing time-consuming domain integrations by approximating a “generalized forcing function” in the interior of the domain with the use of radial basis functions. An approximate particular solution can be determined, and the original problem can be transformed into a homogeneous problem. Numerical examples demonstrate some advantages and disadvantages of the two methods.

A parallel implementation of the boundary element method for heat conduction analysis in heterogeneous media

Abstract A direct boundary element formulation is implemented on a parallel computer using MPI (message passing interface). This implementation solves potential problems, specifically heat conduction problems in heterogeneous media. Single-zone problems, where at least one boundary condition is known at every node on the boundary, produce dense, asymmetric coefficient matrices which are relatively easy to distribute and solve in parallel. Multi-zone problems, where no boundary conditions are specified at nodes on the heterogeneity interfaces, produce coefficient matrices which contain some unpopulated blocks. The parallel data distribution of the coefficient matrix for the multi-zone case is performed so that the populated blocks are evenly distributed across the processors to balance the computational load. Several application problems are solved to find the effect of various input parameters on the effective thermal conductivity. The numerical results agree well with experimental data and analytical models over the volume fractions and thermal conductivity ratios considered.

Shape optimization of acoustic scattering bodies

Shape optimization of acoustic scattering bodies is carried out using genetic algorithms (GA) coupled to a boundary element method for exterior acoustics. The BEM formulation relies on a modified Burton-Miller algorithm to resolve exterior acoustics and to address the uniqueness issue of the representation problem associated with the Helmholtz integral equation at the eigenvalues of the associated interior problem. The particular problem of interest considers an incident wave approaching an axisymmetric shaped body. The objective is to arrive at a geometric configuration that minimizes the acoustic intensity captured by a receiver located at a distance from the scattering body. In particular, the acoustic intensity is required to be minimum as measured proportional to the integral of the product of the potential and its complex conjugate over a volume of space which models the receiver. This is opposed to the more traditional measure of the potential at a single point in space.

A numerical study of three‐dimensional Jeffery orbits in shear flow

We perform numerical simulations of rods and spheroids undergoing Jeffery orbits in a variety of shear flows. The numerical simulations are based on the boundary element method, which allows for the accurate modeling of the problem geometry. We compare the period of rotation for spheroids and rods, both far from walls and very close to walls. We find that the wall effects in three dimensions are minimal, even for flow in gaps not much larger than the longest dimension of the particle. We also show that two‐dimensional simulations grossly overpredict the wall effects seen in three dimensions. Results are similar for both linear and nonlinear shear flows. We also briefly look at the orbital motion of a particle in close proximity to another particle, and show that, again, there is very little effect on the period of rotation, although the resulting centroid trajectories are very different from that of an isolated particle.

Parallel multipole BEM simulation of two-dimensional suspension flows

Abstract The motion of a large number of cylinders of circular cross-section in various two-dimensional flows is studied using a completed double-layer indirect boundary integral formulation. This type of formulation, with specified velocity boundary conditions, results in a Fredholm integral equation of the second kind. Discretization of this type of equation produces linear systems which are generally well-conditioned and suitable for iterative solution. However, the equilibrium and rigid body motion equations necessary to close the system disrupt the diagonal dominance of the matrix, resulting in high condition numbers. A preconditioner based on the known structure of the matrix is used to significantly reduce the condition number, to a point where the number of iterations to achieve a solution is independent of the number of particles in the system. Under these conditions, the solution of a highly populated N×N matrix is proportional to N2. The computational cost quickly becomes excessive as the number of particles in the system increases. This has been the main drawback in using the Boundary Element Method for studying suspension flows. The operation count required to solve an N×N linear system can be drastically reduced to O (N log N) by using a first-shift multipole formulation, in conjunction with a matrix-free iterative linear equation solver. Even with such a reduction in the operation count, dynamic simulation of systems involving large numbers of particles can still only be solved using large-scale parallelization. It is therefore important to establish the multipole technique is suitable for parallelization. It is shown here that this is the case, provided that the implementation of the multipole method is appropriate. Having established that large scale dynamic simulations can be performed in acceptable times, the results of several such simulations are presented. Analysis is performed on the simulation results, to illustrate the usefulness of the particle-level simulation approach in the investigation of suspension flow.

Grid redistribution based on measurable error indicators for the direct boundary element method

Abstract The quality of solution obtained by using the boundary element method is dependent on how the boundary is discretized. This particularly true when the geometry is complex or the field variables are singular at certain points on the boundary. In regions along the boundary where the field variable or its normal derivative has large gradients, a finer discretization is necessary in order to improve the quality of the solution. Most rules of grid optimization for the boundary element method are related to minimizing error in an appropriate boundary error norm. The error norm should be measurable, reliable, and easy to compute. In this paper, a suitable error norm is defined, and the procedure for its computation is presented. Further, a strategy for grid redistribution is examined. In this scheme, the number of nodal points remain constant, but these points are repositioned, reducing the size of the elements in regions of large error.