Bulk stress distributions in the pore space of sphere-packed beds under Darcy flow conditions

Prediction of the aggregation rate of nanoparticles in porous media in the diffusion-controlled regime

The fate and aggregation of nanoparticles (NPs) in the subsurface are important due to potentially harmful impacts on the environment and human health. This study aims to investigate the effects of flow velocity, particle size, and particle concentration on the aggregation rate of NPs in a diffusion-limited regime and build an equation to predict the aggregation rate when NPs move in the pore space between randomly packed spheres (including mono-disperse, bi-disperse, and tri-disperse spheres). The flow of 0.2 M potassium chloride (KCl) through the random sphere packings was simulated by the lattice Boltzmann method (LBM). The movement and aggregation of cerium oxide (CeO2) particles were then examined by using a Lagrangian particle tracking method based on a force balance approach. This method relied on Newton's second law of motion and took the interaction forces among particles into account. The aggregation rate of NPs was found to depend linearly on time, and the slope of the line was a power function of the particle concentration, the Reynolds (Re) and Schmidt (Sc) numbers. The exponent for the Sc number was triple that of the Re number, which was evidence that the random movement of NPs has a much stronger effect on the rate of diffusion-controlled aggregation than the convection.

Computational Framework to Evaluate the Hydrodynamics of Cell Scaffold Geometries

The fluid dynamics of microporous materials are important to many biomedical processes such as cell deposition in scaffold materials, tissue engineering, and bioreactors. Microporous scaffolds are frequently composed of suspensions of beads that have varying topology which, in turn, informs their hydrodynamic properties. Previous work has shown that shear stress distributions can affect the response of cells in microporous environments. Using computational fluid dynamics, we characterize localized differences in fluid flow attributes such wall shear stress and velocity to better understand the fluid dynamics underpinning microporous device function. We evaluated whether bead packings with similar void fractions had different fluid dynamics as characterized by the distribution of velocity magnitudes and wall shear stress and found that there are differences despite the similarities in void fraction. We show that another metric, the average distance to the nearest wall, can provide an additional variable to measure the porosity and susceptibility of microporous materials to high shear stress. By increasing our understanding of the impact of bead size on cell scaffold fluid dynamics we aim to increase the ability to predict important attributes such as loading efficiency in these devices.

Hydrodynamic Dispersion in Porous Media and the Significance of Lagrangian Time and Space Scales

Transport in porous media is critical for many applications in the environment and in the chemical process industry. A key parameter for modeling this transport is the hydrodynamic dispersion coefficient for particles and scalars in a porous medium, which has been found to depend on properties of the medium structure, on the dispersing compound, and on the flow field characteristics. Previous studies have resulted in suggestions of different equation forms, showing the relationship between the hydrodynamic dispersion coefficient for various types of porous media in various flow regimes and the Peclet number. The Peclet number is calculated based on a Eulerian length scale, such as the diameter of the spheres in packed beds, or the pore diameter. However, the nature of hydrodynamic dispersion is Lagrangian, and it should take the molecular diffusion effects, as well as the convection effects, into account. This work shifts attention to the Lagrangian time and length scales for the definition of the Peclet number. It is focused on the dependence of the longitudinal hydrodynamic dispersion coefficient on the effective Lagrangian Peclet number by using a Lagrangian length scale and the effective molecular diffusivity. The lattice Boltzmann method (LBM) was employed to simulate flow in porous media that were constituted by packed spheres, and Lagrangian particle tracking (LPT) was used to track the movement of individual dispersing particles. It was found that the hydrodynamic dispersion coefficient linearly depends on the effective Lagrangian Peclet number for packed beds with different types of packing. This linear equation describing the dependence of the dispersion coefficient on the effective Lagrangian Peclet number is both simpler and more accurate than the one formed using the effective Eulerian Peclet number. In addition, the slope of the line is a characteristic coefficient for a given medium.

In vivo measurement of blood clot mechanics from computational fluid dynamics based on intravital microscopy images

Ischemia which leads to heart attacks and strokes is one of the major causes of death in the world. Whether an occlusion occurs or not depends on the ability of a growing thrombus to resist flow forces exerted on its structure. This manuscript provides the first known in vivo measurement of how much stress a clot can withstand, before yielding to the surrounding blood flow. Namely, Lattice-Boltzmann Method flow simulations are performed based on 3D clot geometries, which are estimated from intravital microscopy images of laser-induced injuries in cremaster microvasculature of live mice. In addition to reporting the blood clot yield stresses, we also show that the thrombus "core" does not experience significant deformation, while its "shell" does. This indicates that the shell is more prone to embolization. Therefore, drugs should be designed to target the shell selectively, while leaving the core intact to minimize excessive bleeding. Finally, we laid down a foundation for a nondimensionalization procedure which unraveled a relationship between clot mechanics and biology. Hence, the proposed framework could ultimately lead to a unified theory of thrombogenesis, capable of explaining all clotting events. Thus, the findings presented herein will be beneficial to the understanding and treatment of heart attacks, strokes and hemophilia.

Numerical accuracy comparison of two boundary conditions commonly used to approximate shear stress distributions in tissue engineering scaffolds cultured under flow perfusion

INTRODUCTION

Flow-induced shear stresses have been found to be a stimulatory factor in pre-osteoblastic cells seeded in 3D porous scaffolds and cultured under continuous flow perfusion. However, due to the complex internal structure of the scaffolds, whole scaffold calculations of the local shear forces are computationally intensive. Instead, representative volume elements (RVEs), which are obtained by extracting smaller portions of the scaffold, are commonly used in literature without a numerical accuracy standard.

OBJECTIVE

Hence, the goal of this study is to examine how closely the whole scaffold simulations are approximated by the two types of boundary conditions used to enable the RVEs: "wall boundary condition" (WBC) and "periodic boundary condition" (PBC).

METHOD

To that end, lattice Boltzmann method fluid dynamics simulations were used to model the surface shear stresses in 3D scaffold reconstructions, obtained from high-resolution microcomputed tomography images.

RESULTS

It was found that despite the RVEs being sufficiently larger than 6 times the scaffold pore size (which is the only accuracy guideline found in literature), the stresses were still significantly under-predicted by both types of boundary conditions: between 20% and 80% average error, depending on the scaffold's porosity. Moreover, it was found that the error grew with higher porosity. This is likely due to the small pores dominating the flow field, and thereby negating the effects of the unrealistic boundary conditions, when the scaffold porosity is small. Finally, it was found that the PBC was always more accurate and computationally efficient than the WBC. Therefore, it is the recommended type of RVE.

Computer simulation of the effect of deformation on the morphology and flow properties of porous media

We report on the results of extensive computer simulation of the effect of deformation on the morphology of a porous medium and its fluid flow properties. The porous medium is represented by packings of spherical particles. Both random and regular as well as dense and nondense packings are used. A quasistatic model based on Hertz's contact theory is used to model the mechanical deformation of the packings, while the evolution of the permeability with the deformation is computed by the lattice-Boltzmann approach. The evolution of the pore-size and pore-length distributions, the porosity, the particles' contacts, the permeability, and the distribution of the stresses that the fluid exerts in the pore space are all studied in detail. The distribution of the pores' lengths, the porosity, and the particles' connectivity change strongly with the application of an external strain to the porous media, whereas the pore-size distribution is not affected as strongly. The permeability of the porous media strongly reduces even when the applied strain is small. When the permeabilities and porosities of the random packings are normalized with respect to their predeformation values, they all collapse onto a single curve, independent of the particle-size distribution. The porosity reduces as a power law with the external strain. The fluid stresses in the pore space follow roughly a log-normal distribution, both before and after deformation.

Pore-scale dispersion: Bridging the gap between microscopic pore structure and the emerging macroscopic transport behavior

We devise an efficient methodology to provide a universal statistical description of advection-dominated dispersion (Péclet→∞) in natural porous media including carbonates. First, we investigate the dispersion of tracer particles by direct numerical simulation (DNS). The transverse dispersion is found to be essentially determined by the tortuosity and it approaches a Fickian limit within a dozen characteristic scales. Longitudinal dispersion was found to be Fickian in the limit for bead packs and superdiffusive for all other natural media inspected. We demonstrate that the Lagrangian velocity correlation length is a quantity that characterizes the spatial variability for transport. Finally, a statistical transport model is presented that sheds light on the connection between pore-scale characteristics and the resulting macroscopic transport behavior. Our computationally efficient model accurately reproduces the transport behavior in longitudinal direction and approaches the Fickian limit in transverse direction.

Analysis and improvement of Brinkman lattice Boltzmann schemes: bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media

This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination Λ. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of Λ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by "partial bounce-back" TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter Λ, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an "upscaling" algorithm for Λ, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

Numerical Study of Granular Scaffold Efficiency to Convert Fluid Flow into Mechanical Stimulation in Bone Tissue Engineering

Controlling the mechanical environment in bioreactors represents a key element in the reactors' optimization. Positive effects of fluid flow in three-dimensional bioreactors have been observed, but local stresses at cell scale remain unknown. These effects led to the development of numerical tools to assess the micromechanical environment of cells in bioreactors. Recently, new possible scaffold geometry has emerged: granular packings. In the present study, the primary goal was to compare the efficiency of such a scaffold to the other ones from literature in terms of wall shear stress levels and distributions. To that aim, three different types of granular packings were generated through discrete element method, and computational fluid dynamics was used to simulate the flow within these packings. Shear stress levels and distributions were determined. A linear relationship between shear stress and inlet velocity was observed, and its slope was similar to published data. The distributions of normalized stress were independent of the inlet velocity and were highly comparable to those of widely used porous scaffolds. Granular packings present similar features to more classical porous scaffolds and have the advantage of being easy to manipulate and seed. The methods of this work are generalizable to the study of other granular packing configurations.