Active Polar Liquid Crystal Channel Flows: Analyzing the Roles of Nematic Strength and Activation Parameter

Suspensions of active polar liquid crystalline polymers (APLC) exhibit complex phenomena such as spontaneous flows, pattern formations and defects. Using the Kinetic Model, which couples the Smoluchowski Equation and the Navier-Stokes Equations, we conduct numerical simulations of APLC in a microfluidic channel to investigate the competitive effect among different material constants, such as the nematic concentration (the strength of the potential for nematic order) and active strength (the individual nano-rods strength of their individual movement) with and without a pressure gradient. Both Dirichlet and Neumann boundary conditions on the mathematical model are employed. Steady states, including isotropic and nematic states, as well as periodic states are observed. Spontaneous flows reveal interesting geometries in polarity vector orientation and nematic director orientation, such as flow reversals and banded structures with multiple regions within the channel boundaries.https://digitalcommons.odu.edu/gradposters2022_sciences/1007/thumbnail.jp

A Dynamic Model of the Choice of Technology in Economic Development

Abstract In this overlapping-generations model, there is unemployment in the manufacturing sector. Manufacturing firms engage in oligopolistic competition and choose technologies to maximize profits. With capital as fixed costs of production, increasing returns in the manufacturing sector exist. In the unique steady state, first, when individuals become more patient, the saving rate increases while the level of income of an individual decreases. Second, an increase in population or percentage of income spent on the manufactured good does not change steady-state technology while decreases the level of income of an individual. Third, an increase in the wage rate leads a manufacturing firm to choose a more advanced technology and the steady-state capital stock increases. Finally, an increase in the level of subsidy to technology adoption does not change steady-state technology

Shirking and Capital Accumulation under Oligopolistic Competition

In this infinite horizon model, unemployment results from the existence of efficiency wages. Consumers choose saving optimally and there is capital accumulation. Firms producing intermediate goods engage in oligopolistic competition and choose technologies to maximize profits. A more advanced technology has a higher fixed cost but a lower marginal cost of production. In the steady state, it is shown that an increase in population size or a decrease in the discount rate leads intermediate good producers to choose more advanced technologies and the wage rate increases. Interestingly, the equilibrium unemployment rate decreases with the size of the population

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

A Dynamic Model of the Choice of Technology in Economic Development

Abstract In this overlapping-generations model, there is unemployment in the manufacturing sector. Manufacturing firms engage in oligopolistic competition and choose technologies to maximize profits. With capital as fixed costs of production, increasing returns in the manufacturing sector exist. In the unique steady state, first, when individuals become more patient, the saving rate increases while the level of income of an individual decreases. Second, an increase in population or percentage of income spent on the manufactured good does not change steady-state technology while decreases the level of income of an individual. Third, an increase in the wage rate leads a manufacturing firm to choose a more advanced technology and the steady-state capital stock increases. Finally, an increase in the level of subsidy to technology adoption does not change steady-state technology

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

Rheological Signatures in Limit Cycle Behaviour of Dilute, Active, Polar Liquid Crystalline Polymers in Steady Shear

We consider the dilute regime of active suspensions of liquid crystalline polymers (LCPs), addressing issues motivated by our kinetic model and simulations in Forest et al. (Forest et al. 2013 Soft Matter 9, 5207-5222 (doi:10.1039/c3sm27736d)). In particular, we report unsteady two-dimensional heterogeneous flow-orientation attractors for pusher nanorod swimmers at dilute concentrations where passive LCP equilibria are isotropic. These numerical limit cycles are analogous to longwave (homogeneous) tumbling and kayaking limit cycles and two-dimensional heterogeneous unsteady attractors of passive LCPs in weak imposed shear, yet these states arise exclusively at semi-dilute concentrations where stable equilibria are nematic. The results in Forest et al. mentioned above compel two studies in the dilute regime that complement recent work of Saintillan & Shelley (Saintillan & Shelley 2013 C. R. Physique 14, 497-517 (doi: 10.1016/j.crhy.2013.04.001)): linearized stability analysis of the isotropic state for nanorod pushers and pullers; and an analytical-numerical study of weakly and strongly sheared active polar nanorod suspensions to capture how particle-scale activation affects shear rheology. We find that weakly sheared dilute puller versus pusher suspensions exhibit steady versus unsteady responses, shear thickening versus thinning and positive versus negative first normal stress differences. These results further establish how sheared dilute nanorod pusher suspensions exhibit many of the characteristic features of sheared semi-dilute passive nanorod suspensions

Chaotic Boundaries of Nematic Polymers in Mixed Shear and ExtensionalFlows

Chaotic orientational dynamics of sheared nematic polymers is documented in laboratory experiments and predicted by Doi-Hess kinetic theory for infinitely thin rods. We address robustness of rheochaos when simple shear is modified by a planar straining flow, and the macromolecules have finite aspect ratio. We predict persistence of sheared chaotic response up to a threshold straining flow strength and minimum aspect ratio, beyond which chaotic behavior is arrested. More intriguing, a straining component can induce chaos from periodic shear responses