Ronald J. Phillips

A constitutive equation for concentrated suspensions that accounts for shear‐induced particle migration

A constitutive equation for computing particle concentration and velocity fields in concentrated monomodal suspensions is proposed that consists of two parts: a Newtonian constitutive equation in which the viscosity depends on the local particle volume fraction and a diffusion equation that accounts for shear‐induced particle migration. Particle flux expressions used to obtain the diffusion equation are derived by simple scaling arguments. Predictions are made for the particle volume fraction and velocity fields for steady Couette and Poiseuille flow, and for transient start‐up of steady shear flow in a Couette apparatus. Particle concentrations for a monomodal suspension of polymethyl methacrylate spheres in a Newtonian solvent are measured by nuclear magnetic resonance (NMR) imaging in the Couette geometry for two particle sizes and volume fractions. The predictions agree remarkably well with the measurements for both transient and steady‐state experiments as well as for different particle sizes.

Hydrodynamic transport properties of hard-sphere dispersions. I: Suspensions of freely mobile particles

The hydrodynamic transport properties of hard-sphere dispersions are calculated for volume fractions (φ) spanning the dilute limit up to the fluid–solid transition at φ=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite-size effects are derived. The effects of using various levels of approximation in computing both the far- and near-field hydrodynamic interactions are also examined. The transport properties associated with freely mobile suspensions—sedimentation velocities, self-diffusion coefficients, and effective viscosities—are determined here, while the corresponding properties of porous media are determined in a companion paper [Phys. Fluids 31, 3473 (1988)]. Comparison of the simulation results is made with both experiment and theory. In particular, the short-time self-diffusion coefficients and the suspension viscosities are in excellent agreement with experiment.

A numerical calculation of the hydraulic permeability of three-dimensional disordered fibrous media

Hydraulic permeabilities of polymeric membranes and gels are of interest both for calculating fluid flow rates and hindered diffusion coefficients. We have calculated hydraulic permeabilities for monomodal and bimodal, periodic and random fibrous media. Hydrodynamic interactions between fibers are calculated by applying a numerical version of slender body theory to a collection of fibers in a cubic cell many Brinkman screening lengths in dimension. Results for random media are obtained by averaging over many ensembles of fibers. To account for the surrounding medium, the line distribution of point forces along the fiber axes are replicated throughout space by using the Ewald summation technique. Results for periodic media agree with previous theoretical results up to a fiber volume fraction of 50% for parallel flow and 40% for transverse flow. Hydraulic permeabilities calculated for three-dimensional, disordered media with monomodal and bimodal distributions of fiber radius are compared with existing theo...

Hindered transport in fibrous membranes and gels: Effect of solute size and fiber configuration

Abstract Hindered transport coefficients for a spherical macromolecule in a spatially periodic fibrous medium were calculated using two different methods. The first method is an effective medium approach based on Brinkmans equation and can be readily applied to disordered fibrous media. The second and more rigorous set of calculations makes use of generalized Taylor dispersion theory. Results from these two approaches are compared for two different spatially periodic lattices of bead-and-string fibers, which were chosen to illustrate the effects of inhomogeneity in the distribution of fibers. In addition, ratios of solute radius to fiber radius ranging from 0.5 to 5.0 were considered. Qualitative agreement between the two methods was obtained for each case studied, and quantitative agreement was obtained for volume fractions at which the hindering effects of the fibers were not too severe.

Hindered diffusion of spherical macromolecules through dilute fibrous media

Results are presented for the effect of solute–fiber hydrodynamic interactions on the hindered diffusion of a spherical macromolecule in random media comprised of cylindrical fibers. Hydrodynamic interactions are calculated by representing the sphere as a collection of point singularities and accounting for the fibers by using a numerical version of slender‐body theory. Electrostatic and other nonhydrodynamic interactions are neglected. The calculations show that the hydrodynamic mobility of the solute decreases in an exponential‐like fashion as the fiber volume fraction is increased. Also, at a given volume fraction, a medium of thinner fibers hinders solute transport more than a medium of thicker fibers. The results compare well with experimental data, both for protein diffusion in solutions of the polysaccharide Dextran and for protein diffusion in cross‐linked agarose gels.

Automated analysis of three‐dimensional xylem networks using high‐resolution computed tomography

Connections between xylem vessels represent important links in the vascular network, but the complexity of three-dimensional (3D) organization has been difficult to access. This study describes the development of a custom software package called TANAX (Tomography-derived Automated Network Analysis of Xylem) that automatically extracts vessel dimensions and the distribution of intervessel connections from high-resolution computed tomography scans of grapevine (Vitis vinifera) stems, although the method could be applied to other species. Manual and automated analyses of vessel networks yielded similar results, with the automated method generating orders of magnitude more data in a fraction of the time. In 4.5-mm-long internode sections, all vessels and all intervessel connections among 115 vessels were located, and the connections were analyzed for their radial distribution, orientation, and predicted shared wall area. Intervessel connections were more frequent in lateral than in dorsal/ventral zones. The TANAX-reconstructed network, in combination with commercial software, was used to visualize vessel networks in 3D. The 3D volume renderings of vessel networks were freely rotated for observation from any angle, and the 4.5 μm virtual serial sections were capable of being viewed in any plane, revealing aspects of vessel organization not possible with traditional serial sections.

A hydrodynamic model for hindered diffusion of proteins and micelles in hydrogels.

Recently, several papers have been published that address the important topic of hindered diffusion of macromolecules in hydrogels. Two papers in particular, by Johnson et al. (1996) and Clague and Phillips (1996), present models of hindered diffusion that account for hydrodynamic interactions, albeit in different ways. The model of Johnson et al. (1996) makes use of an effective medium approach based on Brinkmans equation, whereas Clague and Phillips (1996) explicitly calculate those interactions for a spherical solute suspended in a liquid-filled, three-dimensional medium of randomly placed cylindrical fibers.

Magnetic resonance imaging of concentration and velocity profiles of pure fluids and solid suspensions in rotating geometries

Steady and unsteady state velocity and concentration profiles are presented for a 40% by volume suspension of polymethyl methacrylate (PMMA) spheres in polyalkylene glycol (PG). The profiles were obtained by using the noninvasive technique of nuclear magnetic resonance imaging (MRI) in three experimental geometries: coaxial rotating cylinders (i.e., for generating wide‐gap Couette flow), coaxial cylinders in which a straight flight rotates with the inner cylinder and spans the annulus between the surfaces, and a single screw extruder. Concentration profiles document the presence of particle migration from high shear to low shear regions in the concentric cylinder apparatus and in the extruder. However, concentration gradients across the gap in the straight‐flight cylinder are not exhibited, indicating the relative importance of mixing in that geometry. Velocity profiles for the pure PG fluid and for suspension flows which remain well‐mixed agree quantitatively with profiles predicted for Newtonian fluids....

A constitutive model for microstructure and total stress in particulate suspensions

Constitutive equations for concentrated suspensions that explicitly account for the development of anisotropy in the microstructure are not generally available, even for relatively simple systems of hard spheres suspended in a Newtonian medium. Here, we use a directionally dependent mean-free path length and a truncated Cartesian tensor expansion to define a second-order structure tensor for systems of suspended particles. This tensor captures the principal nature of the microstructure. A semiempirical differential equation is developed for the structure tensor, with representation theorems being used to insure frame indifference. A separate equation is proposed to relate the stress tensor to the structure and rate of strain tensors. These coupled equations model structure and stress in both steady and time-dependent viscometric flows. Results from Stokesian dynamics simulations are used to demonstrate the utility of this modeling approach. The simulations were for monodisperse suspensions in an infinite ...

Particle migration and suspension structure in steady and oscillatory plane Poiseuille flow

A structure-tensor-based model is used to compute the microstructure and velocity field of concentrated suspensions of hard spheres in a fully developed, pressure-driven channel flow. The model is comprised of equations governing conservation of mass and momentum in the bulk suspension, conservation of particles, and conservation of momentum in the particle phase. The equations governing the relation between structure and stress in hard-sphere suspensions were developed previously and were shown to reproduce quantitatively results obtained by Stokesian dynamics simulations of linear shear flows. In nonhomogeneous, pressure-driven flows, the divergence of the particle contribution to the stress is nonzero and acts as a body force that causes particles to migrate across streamlines. Under steady conditions, the model predicts that the resulting migration causes particles to move to the center of the channel, where the concentration approaches the maximum packing for hard-sphere suspensions. In oscillatory f...