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

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

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

On weak plane Couette and Poiseuille flows of rigid rod and platelet ensembles

SIAM J. Appl. Math., Volume 66, Issue 4, 1227-1260, 2006.The article of record as published may be found at http://dx.doi.org/10.1137/04061934xFilms 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.This research was sponsored by Air Force Office of Scientific Research, Air Force Materials Command, grants F49620-02-1-0086 and F49620-03-1-0098; National Science Foundation grants DMS-0204243 and DMS-0308019; and the Army Research Office, Materials Division.This research was sponsored by Air Force Office of Scientific Research, Air Force Materials Command, grants F49620-02-1-0086 and F49620-03-1-0098; National Science Foundation grants DMS-0204243 and DMS-0308019; and the Army Research Office, Materials Division

On weak plane Couette and Poiseuille flows of rigid rod and platelet ensembles

Abstract. 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

Surface alignment control of nematodynamics

The primary study of this thesis is the response of the nematic director to pressure driven flow. Dynamic flow experiments using optical conoscopy and pressure gradient measurements are used to explore the physics behind the flow alignment seen to occur for some nematic liquid crystals. New research into the techniques and methods for aligning the director at a glass interface is also presented, the results of which are used towards the latter end of this thesis in the production of a highly novel flow cell. A bespoke technique for fabricating robust liquid crystal flow cells is also presented. The observation of flow alignment for the nematic liquid crystal 5CB is detailed for pressure driven flow via optical conoscopy when the director is initially aligned planar homogeneously at 45◦ to the direction of flow. The results of this experiment are compared to the theory of Ericksen and Leslie through a one dimensional dynamic model that provides simulated director profiles and corresponding simulated conoscopic images. Good agreement between the data and simulation is observed, whereby the director is seen to rotate to become parallel to the flow direction whilst exhibiting no net tilt distortion at all flow rates. The presence of small surface pretilt from a rubbed planar aligning polyimide layer and its effect on director rotation is also examined for cells that are rubbed in both the parallel and anti-parallel directions. The result observed is a striking difference in the mean director rotation when initially aligned close to normal to the direction of flow. The results of these experiments are also compared to the theory of Ericksen and Leslie through the one dimensional dynamic model. Good agreement is seen, highlighting the dramatic effect that a small amount of surface pretilt can have on the overall director orientation, whilst also demonstrating the need for caution when assuming that rubbed conventional alignment techniques provide true planar orientation. Two methods for producing intermediate or large pretilt angles at liquid crystal align- ment surfaces are also examined. Here, two recipes involving the commercial polyimides Nissan SE-1211, Nissan SE-130 and Nissan SE-4811 are experimentally investigated, with results showing the ability to tune the director pretilt angle as a function of the rubbing strength used to align the sample. The results also show an interesting depen- dance on the material upon which the aligning layer is deposited for the recipe involving Nissan SE-1211. Here, vastly different pretilt angles are observed for cells constructed with glass and indium tin oxide (ITO) layers. Finally, the large pretilt angles produced from the recipes mentioned above are also used to fabricate pressure driven flow cells exhibiting large pretilt angles on both sur- faces, constraining the director to align in a splayed state. When aligned parallel to the flow direction, experiments examining the valve-like nature of the director profile suggest that a preferential flow direction exists in what here is termed the ‘diode cell’. Measurements of the pressure gradient required to achieve a constant volumetric flow rate through the cell are compared for flow in both directions relative to the splayed di- rector profile. A striking difference is observed for flow ‘with’ the splay and ‘against’ the splay, leading to the realisation of a cell exhibiting a preferential flow direction through surface treatment. Again, results are compared to the theory of Ericksen and Leslie through the one dimensional dynamic model, showing good agreement.H