Bruce Caswell

The Solution of Viscous Incompressible Jet and Free Surface Flows Using Finite Element Methods.

Abstract : The authors discuss the creation of a finite element program suitable for solving incompressible, viscous free surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. The authors conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related stick-slip problem and the tube inlet problem, both of which also contain stress singularities, are also given. (Modified author abstract)

A Multiscale Red Blood Cell Model with Accurate Mechanics, Rheology, and Dynamics

Red blood cells (RBCs) have highly deformable viscoelastic membranes exhibiting complex rheological response and rich hydrodynamic behavior governed by special elastic and bending properties and by the external/internal fluid and membrane viscosities. We present a multiscale RBC model that is able to predict RBC mechanics, rheology, and dynamics in agreement with experiments. Based on an analytic theory, the modeled membrane properties can be uniquely related to the experimentally established RBC macroscopic properties without any adjustment of parameters. The RBC linear and nonlinear elastic deformations match those obtained in optical-tweezers experiments. The rheological properties of the membrane are compared with those obtained in optical magnetic twisting cytometry, membrane thermal fluctuations, and creep followed by cell recovery. The dynamics of RBCs in shear and Poiseuille flows is tested against experiments and theoretical predictions, and the applicability of the latter is discussed. Our findings clearly indicate that a purely elastic model for the membrane cannot accurately represent the RBCs rheological properties and its dynamics, and therefore accurate modeling of a viscoelastic membrane is necessary.

Predicting human blood viscosity in silico

The viscosity of blood has long been used as an indicator in the understanding and treatment of disease, and the advent of modern viscometers allows its measurement with ever-improving clinical convenience. However, these advances have not been matched by theoretical developments that can yield a quantitative understanding of blood’s microrheology and its possible connection to relevant biomolecules (e.g., fibrinogen). Using coarse-grained molecular dynamics and two different red blood cell models, we accurately predict the dependence of blood viscosity on shear rate and hematocrit. We explicitly represent cell–cell interactions and identify the types and sizes of reversible rouleaux structures that yield a tremendous increase of blood viscosity at low shear rates. We also present the first quantitative estimates of the magnitude of adhesive forces between red cells. In addition, our simulations support the hypothesis, previously deduced from experiments, of yield stress as an indicator of cell aggregation. This non-Newtonian behavior is analyzed and related to the suspension’s microstructure, deformation, and dynamics of single red blood cells. The most complex cell dynamics occurs in the intermediate shear rate regime, where individual cells experience severe deformation and transient folded conformations. The generality of these cell models together with single-cell measurements points to the future prediction of blood-viscosity anomalies and the corresponding microstructures associated with various diseases (e.g., malaria, AIDS, and diabetes mellitus). The models can easily be adapted to tune the properties of a much wider class of complex fluids including capsule and vesicle suspensions.

Finite element simulation of viscoelastic flow

Abstract A finite element simulation has been carried out for a viscoelastic fluid of the single integral, memory type. In the current work, the memory kernel is chosen to be a single exponential in the time lapse (Maxwell model). However, the formulation is such that it can easily be generalized to more realistic models such as the BKZ theory. From the point of view of numerical analysis, differential models are appealing because they avoid the complexities of memory integrals. However, in these models the viscoelastic effect always enters through terms having the highest-order derivatives. The disadvantage of this situation for numerical analysis appears to be borne out in the experiences reported recently by several workers. In a memory integral formulation, the demand on differentiability of the velocity field is no greater than for the Newtonian fluid. The basic idea in the formulation is the approximation of the memory integral by a Laguerre numerical quadrature formula. The kinematical problem is the computation of the displacement vector from every node to the Laguerre points upstream along particle paths. Since this operation requires the velocity field to be known, the method is restricted to the calculation of non-linear effects as body forces. Thus, the equations being solved in any iteration are those of linear viscous flow with an arbitrary body force. In spite of this limitation, the method converges at dimensionless relaxation times greater than the largest values attained with formulations based on differential models. In the present paper, the method is illustrated with the die entry flow in which the fluid is forced through a four-to-one axisymmetric contraction.

Blood Flow and Cell-Free Layer in Microvessels

Please cite this paper as: Fedosov, Caswell, Popel and Karniadakis (2010). Blood Flow and Cell‐Free Layer in Microvessels. Microcirculation17(8), 615–628.

Quantifying the biophysical characteristics of Plasmodium-falciparum-parasitized red blood cells in microcirculation

The pathogenicity of Plasmodium falciparum (Pf) malaria results from the stiffening of red blood cells (RBCs) and its ability to adhere to endothelial cells (cytoadherence). The dynamics of Pf-parasitized RBCs is studied by three-dimensional mesoscopic simulations of flow in cylindrical capillaries in order to predict the flow resistance enhancement at different parasitemia levels. In addition, the adhesive dynamics of Pf-RBCs is explored for various parameters revealing several types of cell dynamics such as firm adhesion, very slow slipping along the wall, and intermittent flipping. The parasite inside the RBC is modeled explicitly in order to capture phenomena such as “hindered tumbling” motion of the RBC and the sudden transition from firm RBC cytoadherence to flipping on the endothelial surface. These predictions are in quantitative agreement with recent experimental observations, and thus the three-dimensional modeling method presented here provides new capabilities for guiding and interpreting future in vitro and in vivo studies of malaria.

Systematic coarse-graining of spectrin-level red blood cell models

We present a rigorous procedure to derive coarse-grained red blood cell (RBC) models, which yield accurate mechanical response. Based on a semi-analytic theory the linear and nonlinear elastic properties of healthy and infected RBCs in malaria can be matched with those obtained in optical tweezers stretching experiments. The present analysis predicts correctly the membrane Youngs modulus in contrast to about 50% error in predictions by previous models. In addition, we develop a stress-free model which avoids a number of pitfalls of existing RBC models, such as non-smooth or poorly controlled equilibrium shape and dependence of the mechanical properties on the initial triangulation quality. Here we employ dissipative particle dynamics for the implementation but the proposed model is general and suitable for use in many existing continuum and particle-based numerical methods.

Rheology, microstructure and migration in brownian colloidal suspensions.

We demonstrate that suspended spherical colloidal particles can be effectively modeled as single dissipative particle dynamics (DPD) particles provided that the conservative repulsive force is appropriately chosen. The suspension model is further improved with a new formulation, which augments standard DPD with noncentral dissipative shear forces between particles while preserving angular momentum. Using the new DPD formulation we investigate the rheology, microstructure and shear-induced migration of a monodisperse suspension of colloidal particles in plane shear flows (Couette and Poiseuille). Specifically, to achieve a well-dispersed suspension we employ exponential conservative forces for the colloid-colloid and colloid-solvent interactions but keep the conventional linear force for the solvent-solvent interactions. Our simulations yield relative viscosity versus volume fraction predictions in good agreement with both experimental data and empirical correlations. We also compute the shear-dependent viscosity and the first and second normal-stress differences and coefficients in both Couette and Poiseuille flow. Simulations near the close packingvolume-fraction (64%) at low shear rates demonstrate a transition to flow-induced string-like structures of colloidal particles simultaneously with a transition to a nonlinear Couette velocity profile in agreement with experimental observations. After a sufficient increase ofthe shear rate the ordered structure melts into disorder with restoration of the linear velocity profile. Migration effects simulated in Poiseuille flow compare well with experiments and model predictions. The important role of angular momentum and torque in nondilute suspensions is also demonstrated when compared with simulations by the standard DPD, which omits the angular degrees of freedom. Overall, the new method agrees very well with the Stokesian Dynamics method but it seems to have lower computational complexity and is applicable to general complex fluids systems.

Multiscale Modeling of Red Blood Cell Mechanics and Blood Flow in Malaria

Red blood cells (RBCs) infected by a Plasmodium parasite in malaria may lose their membrane deformability with a relative membrane stiffening more than ten-fold in comparison with healthy RBCs leading to potential capillary occlusions. Moreover, infected RBCs are able to adhere to other healthy and parasitized cells and to the vascular endothelium resulting in a substantial disruption of normal blood circulation. In the present work, we simulate infected RBCs in malaria using a multiscale RBC model based on the dissipative particle dynamics method, coupling scales at the sub-cellular level with scales at the vessel size. Our objective is to conduct a full validation of the RBC model with a diverse set of experimental data, including temperature dependence, and to identify the limitations of this purely mechanistic model. The simulated elastic deformations of parasitized RBCs match those obtained in optical-tweezers experiments for different stages of intra-erythrocytic parasite development. The rheological properties of RBCs in malaria are compared with those obtained by optical magnetic twisting cytometry and by monitoring membrane fluctuations at room, physiological, and febrile temperatures. We also study the dynamics of infected RBCs in Poiseuille flow in comparison with healthy cells and present validated bulk viscosity predictions of malaria-infected blood for a wide range of parasitemia levels (percentage of infected RBCs with respect to the total number of cells in a unit volume).

Wall shear stress-based model for adhesive dynamics of red blood cells in malaria.

Red blood cells (RBCs) infected by the Plasmodium falciparum (Pf-RBCs) parasite lose their membrane deformability and they also exhibit enhanced cytoadherence to vascular endothelium and other healthy and infected RBCs. The combined effect may lead to severe disruptions of normal blood circulation due to capillary occlusions. Here we extend the adhesion model to investigate the adhesive dynamics of Pf-RBCs as a function of wall shear stress (WSS) and other parameters using a three-dimensional, multiscale RBC model. Several types of adhesive behavior are identified, including firm adhesion, flipping dynamics, and slow slipping along the wall. In particular, the flipping dynamics of Pf-RBCs observed in experiments appears to be due to the increased stiffness of infected cells and the presence of the solid parasite inside the RBC, which may cause an irregular adhesion behavior. Specifically, a transition from crawling dynamics to flipping behavior occurs at a Youngs modulus approximately three times larger than that of healthy RBCs. The simulated dynamics of Pf-RBCs is in excellent quantitative agreement with available microfluidic experiments if the force exerted on the receptors and ligands by an existing bond is modeled as a nonlinear function of WSS.