On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

Films and molds of nematic polymer materials are notorious for heterogeneity in the orientational distribution of the rigid rod or platelet macromolecules. Predictive tools for structure length scales generated by shear-dominated processing are vitally important: both during processing because of flow feedback phenomena such as shear thinning or thickening, and postprocessing since gradients in the rod or platelet ensemble translate to nonuniform composite properties and to residual stresses in the material. These issues motivate our analysis of two prototypes for planar shear processing: drag-driven Couette and pressure-driven Poiseuille flows. Hydrodynamic theories for high aspect ratio rod and platelet macromolecules in viscous solvents are well developed, which we apply in this paper to model the coupling between short-range excluded volume interactions, anisotropic distortional elasticity (unequal elasticity constants), wall anchoring conditions, and hydrodynamics. The goal of this paper is to generalize scaling properties of steady flow molecular structures in slow Couette flows with equal elasticity constants [M. G. Forest et al., J. Rheol., 48 (2004), pp. 175–192] in several ways: to contrast isotropic and anisotropic elasticity; to compare Couette versus Poiseuille flow; and to consider dynamics and stability of these steady states within the asymptotic model equations

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. II: In-plane structure transitions

Nematic, or liquid crystalline, polymer (LCP) composites are composed of large aspect ratio rod-like or platelet macromolecules. This class of nanocomposites exhibits tremendous potential for high performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. Fibers made from nematic polymers have set synthetic materials performance standards for decades. The current target is to engineer multifunctional films and molded parts, for which processing flows are shear-dominated. Nematic polymer films inherit anisotropy from collective orientational distributions of the molecular constituents and develop heterogeneity on length scales that are, as yet, not well understood and thereby uncontrollable. Rigid LCPs in viscous solvents have a theoretical and computational foundation from which one can model parallel plate Couette cell experiments and explore anisotropic structure generation arising from nonequilibrium interactions between hydrodynamics, molecular short- and long-range elasticity, and confinement effects. Recent progress on the longwave limit of homogeneous nematic polymers in imposed simple shear and linear planar flows [R. G. Larson and H. Ottinger, Macromolecules, 24 (1991), pp. 6270-6282], [V. Faraoni, M. Grosso, S. Crescitelli, and P. L. Maffettone, J. Rheol., 43 (1999), pp. 829-843], [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett. 86 (2001), pp. 3184-3187], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17-37], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80-93], [M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech., 118 (2004), pp. 17-31], [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301] provides resolved kinetic simulations of the molecular orientational distribution. These results characterize anisotropy and dynamic attractors of sheared bulk domains and underscore limitations of mesoscopic models for orientation of the rigid rod or platelet ensembles. In this paper, we apply our resolved kinetic structure code [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853-870] to model onset and saturation of heterogeneity in the orientational distribution by coupling a distortional elasticity potential (with distinct elasticity constants) and anchoring conditions in a plane Couette cell. For this initial study, the flow field is imposed and the orientational distribution is confined to the shear deformation plane, which affords comparison with seminal [T. Tsuji and A. D. Rey, Phys. Rev. E (3), 62 (2000), pp. 8141-8151] as well as our own mesoscopic model simulations [H. Zhou, M. G. Forest, and Q. Wang, J. Non-Newtonian Fluid Mech., submitted], [H. Zhou and M. G. Forest, Discrete Contin. Dyn. Syst. Ser. B, to appear]. Under these controlled conditions, we map out the kinetic phase diagram of spatiotemporal attractors of a Couette cell film in the two-parameter space of Deborah number (normalized shear rate) and Ericksen number (relative strength of elasticity potentials). © 2005 Society for Industrial and Applied Mathematics

Microscopic-Macroscopic Simulations of Rigid-Rod Polymer Hydrodynamics: Heterogeneity and Rheochaos

Rheochaos is a remarkable phenomenon of nematic (rigid-rod) polymers in steady shear, with sustained chaotic fluctuations of the orientational distribution of the rod ensemble. For monodomain dynamics, imposing spatial homogeneity and linear shear, rheochaos is a hallmark prediction of the Doi-Hess theory [M. Doi, J. Polym. Sci. Polym. Phys. Ed., 19 (1981), pp. 229-243; M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, London, New York, 1986; S. Hess, Z. Naturforsch., 31 (1976), pp. 1034-1037. The model behavior is robust, captured by second-moment tensor approximations G. Rienäcker, M. Kröger, and S. Hess, Phys. Rev. E (3), 66 (2002), 040702; G. Rienäcker, M. Kröger, and S. Hess, Phys. A, 315 (2002), pp. 537-568; M. G. Forest and Q. Wang, Rheol. Acta, 42 (2003), pp. 20-46 and high-order Galerkin simulations of the Smoluchowski equation for the orientational probability distribution function (PDF) [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett., 86 (2001), pp. 3184-3187; M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17-37; M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80-93, and persistent up to critical thresholds of coplanar extensional flow M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301; M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newt. Fluid Mech., 118 (2004), pp. 17-31; S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E (3), 73 (2006), 061710] and magnetic fields [M. G. Forest, Q. Wang, H. Zhou, and R. Zhou, J. Rheol., 48 (2004), pp. 175-1921, as well as fluctuating shear rates [S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E (3), 73 (2006), 061710]. To be experimentally relevant, rheochaos of the Doi-Hess theory must persist amid heterogeneity observed in birefringence patterns [Z. Tan and G. C. Berry, J. Rheol., 47 (2003), pp. 73-104]. Modeling can further shed light on shear bands produced by hydrodynamic feedback which have thus fax eluded measurement. Some numerical evidence supports persistence: a one-dimensional (1D) study [B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy, and A. K. Sood, Phys. Rev. Lett., 92 (2004), 188301] with a second-moment tensor model and imposed simple shear; and a two-dimensional (2D) study [A. Furukawa and A. Onuki, Phys. D, 205 (2005), pp. 195-206] with a second-moment tensor model and flow feedback. Here we stage the micro-macro (Smoluchowski and Navier-Stokes) system so that monodomain rheochaos is embedded in a 1D simulation [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853-870] of a planar shear cell experiment with distortional elasticity. Longtime simulations reveal (i) heterogeneous rheochaos marked by chaotic time series in the PDF, normal and shear stresses, and velocity field at each interior gap height; (ii) coherent spatial morphology in the PDF and stress profiles across the shear gap and weakly nonlinear shear bands in each snapshot; and (iii) consistency between heterogeneous and monodomain rheochaos as measured by Lyapunov exponents and pointwise orbits of the peak orientation of the PDF but with enhancement rather than reduction in Lyapunov exponent values in the flow coupled, heterogeneous system. © 2007 Society for Industrial and Applied Mathematics

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. II: In-Plane Structure Transitions

Nematic, or liquid crystalline, polymer (LCP) composites are composed of large aspect ratio rod-like or platelet macromolecules. This class of nanocomposites exhibits tremendous potential for high performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. Fibers made from nematic polymers have set synthetic materials performance standards for decades. The current target is to engineer multifunctional films and molded parts, for which processing flows are shear-dominated. Nematic polymer films inherit anisotropy from collective orientational distributions of the molecular constituents and develop heterogeneity on length scales that are, as yet, not well understood and thereby uncontrollable. Rigid LCPs in viscous solvents have a theoretical and computational foundation from which one can model parallel plate Couette cell experiments and explore anisotropic structure generation arising from nonequilibrium interactions between hydrodynamics, molecular short- and long-range elasticity, and confinement effects. Recent progress on the longwave limit of homogeneous nematic polymers in imposed simple shear and linear planar flows [R. G. Larson and H. Ottinger, Macromolecules, 24 (1991), pp. 6270–6282], [V. Faraoni, M. Grosso, S. Crescitelli, and P. L. Maffettone, J. Rheol., 43 (1999), pp. 829–843], [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett., 86 (2001), pp. 3184–3187], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17–37], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80–93], [M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech., 118 (2004), pp. 17–31], [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301] provides resolved kinetic simulations of the molecular orientational distribution. These results characterize anisotropy and dynamic attractors of sheared bulk domains and underscore limitations of mesoscopic models for orientation of the rigid rod or platelet ensembles. In this paper, we apply our resolved kinetic structure code [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853–870] to model onset and saturation of heterogeneity in the orientational distribution by coupling a distortional elasticity potential (with distinct elasticity constants) and anchoring conditions in a plane Couette cell. For this initial study, the flow field is imposed and the orientational distribution is confined to the shear deformation plane, which affords comparison with seminal [T. Tsuji and A. D. Rey, Phys. Rev. E (3), 62 (2000), pp. 8141–8151] as well as our own mesoscopic model simulations [H. Zhou, M. G. Forest, and Q. Wang, J. Non-Newtonian Fluid Mech., submitted], [H. Zhou and M. G. Forest, Discrete Contin. Dyn. Syst. Ser. B, to appear]. Under these controlled conditions, we map out the kinetic phase diagram of spatiotemporal attractors of a Couette cell film in the two-parameter space of Deborah number (normalized shear rate) and Ericksen number (relative strength of elasticity potentials)

Stochastic semi-Lagrangian micro–macro calculations of liquid crystalline solutions in complex flows

A general method for the simulation of complex flows of liquid crystalline polymers (LCPs) using a stochastic semi-Lagrangian micro–macro method is introduced. The macroscopic part uses a spatial-temporal second order accurate semi-Lagrangian algorithm, where ideas from the finite element and natural element methods are mixed in order to compute average quantities. The microscopic part employs a stochastic interpretation of the Doi–Hess LCP model, which is discretized with a second order Richardson extrapolated Euler–Maruyama scheme. The new method is validated and tested using the benchmark problem of flow between rotating eccentric cylinders. In a decoupled analysis, a discussion on the sensibility of the scalar order parameter to the macroscopic flow is offered. For the coupled situation, the proposed method predicts disclinations at certain regions of the geometry, as well as an accentuated abatement of the flow as the strength of the micro–macro interaction increases. Further examples are provided at different Peclet and concentration numbers to gain insight on the behavior of complex flows of LCPs in the eccentric cylinder geometry. The generality and robustness of the method, as well as its accurate prediction of LCP behavior under complex flows are main features of the implementatio

Shapes and Dynamics of Blood Cells in Poiseuille and Shear Flows

The dynamics, shape, deformation, and orientation of red blood cells in microcirculation affect the rheology, flow resistance and transport properties of whole blood. This leads to important correlations of cellular and continuum scales. Furthermore, the dynamics of RBCs subject to different flow conditions and vessel geometries is relevant for both fundamental research and biomedical applications (e.g drug delivery). In this thesis, the behaviour of RBCs is investigated for different flow conditions via computer simulations. We use a combination of two mesoscopic particle-based simulation techniques, dissipative particle dynamics and smoothed dissipative particle dynamics. We focus on the microcapillary scale of several μm. At this scale, blood cannot be considered at the continuum but has to be studied at the cellular level. The connection between cellular motion and overall blood rheology will be investigated. Red blood cells are modelled as viscoelastic objects interacting hydrodynamically with a viscous fluid environment. The properties of the membrane, such as resistance against bending or shearing, are set to correspond to experimental values. Furthermore, thermal fluctuations are considered via random forces. Analyses corresponding to light scattering measurements are performed in order to compare to experiments and suggest for which situations this method is suitable. Static light scattering by red blood cells characterises their shape and allows comparison to objects such as spheres or cylinders, whose scattering signals have analytical solutions, in contrast to those of red blood cells. Dynamic light scattering by red blood cells is studied concerning its suitability to detect and analyse motion, deformation and membrane fluctuations. Dynamic light scattering analysis is performed for both diffusing and flowing cells. We find that scattering signals depend on various cell properties, thus allowing to distinguish different cells. The scattering of diffusing cells allows to draw conclusions on their bending rigidity via the effective diffusion coefficient. The scattering of flowing cells allows to draw conclusions on the shear rate via the scattering amplitude correlation. In flow, a RBC shows different shapes and dynamic states, depending on conditions such as confinement, physiological/pathological state and cell age. Here, two essential flow conditions are studied: simple shear flow and tube flow. Simple shear flow as a basic flow condition is part of any more complex flow. The velocity profile is linear and shear stress is homogeneous. In simple shear flow, we find a sequence of different cell shapes by increasing the shear rate. With increasing shear rate, we find rolling cells with cup shapes, trilobe shapes and quadrulobe shapes. This agrees with recent experiments. Furthermore, the impact of the initial orientation on the dynamics is studied. To study crowding and collective effects, systems with higher haematocrit are set up. Tube flow is an idealised model for the flow through cylindric microvessels. Without cell, a parabolic flow profile prevails. A single red blood cell is placed into the tube and subject to a Poiseuille profile. In tube flow, we find different cell shapes and dynamics depending on confinement, shear rate and cell properties. For strong confinements and high shear rates, we find parachute-like shapes. Although not perfectly symmetric, they are adjusted to the flow profile and maintain a stationary shape and orientation. For weak confinements and low shear rates, we find tumbling slippers that rotate and moderately change their shape. For weak confinements and high shear rates, we find tank-treading slippers that oscillate in a limited range of inclination angles and strongly change their shape. For the lowest shear rates, we find cells performing a snaking motion. Due to cell properties and resultant deformations, all shapes differ from hitherto descriptions, such as steady tank-treading or symmetric parachutes. We introduce phase diagrams to identify flow regimes for the different shapes and dynamics. Changing cell properties, the regime borders in the phase diagrams change. In both flow types, both the viscosity contrast and the choice of stress-free shape are important. For in vitro experiments, the solvent viscosity has often been higher than the cytosol viscosity, leading to a different pattern of dynamics, such as steady tank-treading. The stress-free state of a RBC, which is the state at zero shear stress, is still controversial, and computer simulations enable direct comparisons of possible candidates in equivalent flow conditions

Department of Applied Mathematics Academic Program Review, Self Study / June 2010

The Department of Applied Mathematics has a multi-faceted mission to provide an exceptional mathematical education focused on the unique needs of NPS students, to conduct relevant research, and to provide service to the broader community. A strong and vibrant Department of Applied Mathematics is essential to the university's goal of becoming a premiere research university. Because research in mathematics often impacts science and engineering in surprising ways, the department encourages mathematical explorations in a broad range of areas in applied mathematics with specific thrust areas that support the mission of the school

A Molecular Dynamics Study Of Polymer Chains In Shear Flows and Nanocomposites

In this work we study single chain polymers in shear flows and nanocomposite polymer melts extensively through the use of large scale molecular dynamics simulations through LAMMPS. In the single polymer chain shear flow study, we use the Lattice Boltzmann method to simulate fluid dynamics and also include thermal noise as per the \emph{fluctuation-dissipation} theorem in the system. When simulating the nanocomposite polymer melts, we simply use a Langevin thermostat to mimic a heat bath. In the single polymer in shear flow study we investigated the margination of a single chain towards solid surfaces and how strongly the shear flow influences this effect. In particular we also tried to study the effect of the polymer\u27s monomer size $a$ on its overall tendency to marginate. To this end, we studied polymer chains of length $N=16, 32$ in flows at multiple shear rates, $\dot{\gamma}$ and noted higher margination rates in the case of chains with larger radii monomers in comparison to smaller radii monomer chains. We quantified this behaviour and effect by considering various measures such as the distribution of the chain\u27s maximum extent into the flow, the distribution of its centre of mass normal to the surface as well as its radius of gyration in directions parallel and normal to the surface i.e $R_{x}, R_{y}, R_{z}$. In the second work, we looked at the effects of introducing nanorods into polymeric melts. We primarily focused on understanding the dispersion, orientation and conformational patterns exhibited by the nanorods and chains respectively. At lower concentrations, rods phase separated into distinct nematic clusters, while at higher concentrations they remained more isotropic and disordered. We noted that this behaviour is being driven by the system finding a trade-off between the entropic forces trying to create the isolated clusters and the enthalpic effects that work to improve mixing of the rods. We also noted that the rigid rods induced significant local conformational changes in the flexible chains in close proximity which in turn made the whole melt more ordered

Small amplitude oscillatory flows of nematic liquid crystal polymers

This dissertation presents two theoretical predictions of the behavior of solutions of nematic liquid crystal polymers when subjected to small amplitude flows that are oscillatory in time. First, we review theoretical models for predicting the behavior of nematic liquid crystals, including Leslie-Ericksen theory, which only attempts to capture the mean direction of molecular orientation, and Doi-Hess kinetic theory, which defines a probability density function on the unit sphere for the molecular orientation and also the mesocscopic orientation tensor models derived from it, which are the models that we will examine. In Chapter 2, we examine shear flow in the monodomain limit, in which there are no spatial gradients in molecular orientation, and we use multiple timescale perturbation analysis to capture very slowly developing effects in the dynamic moduli, similar to experimental observations. Then, in Chapter 3, we relax the monodomain restriction and examine the effect of heterogeneity in the molecular orientation and the choice of two special anchoring conditions for the orientation at the plates. We re- cover a Leslie-Ericksen-type prediction, formally connect imposed stress and imposed velocity boundary conditions in shear flow, and establish an equivalence at the level of the storage and loss moduli between shear flow and Poiseuille flow