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Discrete & Continuous Dynamical Systems - B

March 2011 , Volume 15 , Issue 2

Special Issue on Computational and Theoretical Modeling of Complex Fluids/Soft Matter

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Preface
Qi Wang and Xiaofeng Yang
2011, 15(2): i-i doi: 10.3934/dcdsb.2011.15.2i +[Abstract](613) +[PDF](76.6KB)
Abstract:
Complex fluids or soft matter are fluids in which elementary molecules are constrained so that their degrees of freedom are reduced. When their degrees of freedom are reduced, the motion of elementary molecules is confined through intermolecular as well as intramolecular interaction. The extent to which the elementary molecules are constrained dictates their collective responses to the ambient and their mutual interaction, thereby leading to a wide spectrum of materials properties exhibited in complex fluids. It's been shown that the mesoscopic morphology and dynamics in complex fluids, given the fact that they are normally consisted of macromolecules, can impact their macroscopic property significantly.

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An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems
Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang and Hong-Ying Man
2011, 15(2): 325-341 doi: 10.3934/dcdsb.2011.15.325 +[Abstract](815) +[PDF](800.4KB)
Abstract:
In this paper we prove an optimal-order error estimate for a family of characteristic mixed method with arbitrary degree of mixed finite element approximations for the numerical solution of transient convection diffusion equations. This paper generalizes the results in [1, 61]. The proof of the main results is carried out via three lemmas, which are utilized to overcome the difficulties arising from the combination of MMOC and mixed finite element methods. Numerical experiments are presented to justify the theoretical analysis.
Using the immersed boundary method to model complex fluids-structure interaction in sperm motility
Robert H. Dillon and Jingxuan Zhuo
2011, 15(2): 343-355 doi: 10.3934/dcdsb.2011.15.343 +[Abstract](837) +[PDF](1690.5KB)
Abstract:
We describe work on the development of immersed boundary methods for sperm motility in complex fluids. This includes an Oldroyd-B formulation and a Lagrangian mesh method. We also describe the development of an immersed boundary rheometer for the studying the properties of viscoelastic fluids. We present preliminary simulation results for the Oldroyd-B and Lagrangian mesh rheometers and compare sperm motility in Newtonian, Oldroyd-B and Lagrangian mesh fluids using an existing immersed boundary model for sperm motility.
Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one
Shijin Ding, Changyou Wang and Huanyao Wen
2011, 15(2): 357-371 doi: 10.3934/dcdsb.2011.15.357 +[Abstract](765) +[PDF](399.0KB)
Abstract:
We consider the equation modeling the compressible hydrodynamic flow of liquid crystals in one dimension. In this paper, we establish the existence of a weak solution $(\rho, u,n)$ of such a system when the initial density function $0\le \rho_0 \in L^\gamma$ for $\gamma>1$, $u_0\in L^2$, and $n_0\in H^1$. This extends a previous result by [12], where the existence of a weak solution was obtained under the stronger assumption that the initial density function $0$<$c\le \rho_0\in H^1$, $u_0\in L^2$, and $n_0\in H^1$.
On the stochastic immersed boundary method with an implicit interface formulation
Qiang Du and Manlin Li
2011, 15(2): 373-389 doi: 10.3934/dcdsb.2011.15.373 +[Abstract](763) +[PDF](367.1KB)
Abstract:
In this paper, we present a consistent and rigorous derivation of some stochastic fluid-structure interaction models based on an implicit interface formulation of the stochastic immersed boundary method. Based on the fluctuation-dissipation theorem, a proper form can be derived for the noise term to be incorporated into the deterministic hydrodynamic fluid-structure interaction models in either the phase field or level-set framework. The resulting stochastic systems not only capture the fluctuation effect near equilibrium but also provide an effective tool to model the complex interfacial morphology in a fluctuating fluid.
Bound on the yield set of fiber reinforced composites subjected to antiplane shear
Guillermo H. Goldsztein
2011, 15(2): 391-400 doi: 10.3934/dcdsb.2011.15.391 +[Abstract](730) +[PDF](340.7KB)
Abstract:
We consider fiber reinforced composites where both the matrix and the fibers are made of ideally plastic materials with the fibers being much stronger than the matrix. We restrict our attention to microstructures and applied stresses that lead to both microscopic and macroscopic antiplane shear deformations. We obtain a bound on the yield set of the composite in terms of the shapes of the fibers, their volume fraction and the yield set of the matrix.
Accurate simulations of 2-D phase shift masks with a generalized discontinuous Galerkin (GDG) method
Xia Ji and Wei Cai
2011, 15(2): 401-415 doi: 10.3934/dcdsb.2011.15.401 +[Abstract](817) +[PDF](1319.8KB)
Abstract:
In this paper, we apply a newly developed generalized discontinuous Galerkin (GDG) method for rigorous simulations of 2-D phase shift masks (PSM). The main advantage of the GDG method is its accurate treatment of jumps in solutions using the Dirac $\delta$ generalized functions as source terms of partial differential equations. The scattering problem of the PSM is cast with a total field/scattering field formulation while the GDG method is used to handle the inhomogeneous jump conditions between the total and scattering fields along the physical and perfectly matched layer (PML) interfaces. Numerical results demonstrate the high order accuracy of the GDG method and its capability of handling the non-periodic structures such as optical images near mask edges.
A multicomponent model for biofilm-drug interaction
Brandon Lindley, Qi Wang and Tianyu Zhang
2011, 15(2): 417-456 doi: 10.3934/dcdsb.2011.15.417 +[Abstract](745) +[PDF](1892.2KB)
Abstract:
We develop a tri-component model for the biofilm and solvent mixture, in which the extracellular polymeric substance (EPS) network, bacteria and effective solvent consisting of the solvent, nutrient, drugs etc. are modeled explicitly. The tri-component mixture is assumed incompressible as a whole while inter-component mixing, dissipation, and conversion are allowed. A linear stability analysis is conducted on constant equilibria revealing up to two unstable modes corresponding to possible bacterial growth induced by the bacterial and EPS production and dependent upon the regime of the model parameters. A 1-D transient simulation is carried out to investigate the nonlinear dynamics of the EPS network, bacteria distribution, drug and nutrient distribution in a channel with and without shear. Finally, the transient biofilm dynamics are studied with respect to a host of diffusive properties of the drug and nutrient present in the biofilm.
Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data
Ke Xu, M. Gregory Forest and Xiaofeng Yang
2011, 15(2): 457-473 doi: 10.3934/dcdsb.2011.15.457 +[Abstract](660) +[PDF](4838.4KB)
Abstract:
Liquid crystalline polymers have been extensively studied in shear starting from an equilibrium nematic phase. In this study, we explore the transient and long-time behavior as a steady shear cell experiment commences during an isotropic-nematic (I-N) phase transition. We initialize a localized Gaussian nematic droplet within an unstable isotropic phase with nematic, vorticity-aligned equilibrium at the walls. In the absence of flow, the simulation converges to a homogeneous nematic phase, but not before passing through quite intricate defect arrays and patterns due to physical anchoring, the dimensions of the shear cell, and transient backflow generated around the defect arrays during the I-N transition. Snapshots of this numerical experiment are then used as initial data for shear cell experiments at controlled shear rates. For homogeneous stable nematic equilibrium initial data, the Leal group [4, 5, 6] and the authors [12] confirm the Larson-Mead experimental observations [7, 8]: stationary 2-D roll cells and defect-free 2-D orientational structure at low shear rates, followed at higher shear rates by an unstable transition to an unsteady 2-D cellular flow and defect-laden attractor. We show at low shear rates that the memory of defect-laden data lasts forever; 2-D steady attractors of [4, 5, 12] emerge for defect free initial data, whereas 1-D unsteady attractors arise for defect-laden initial data.
One order parameter tensor mean field theory for biaxial liquid crystals
Xiaoyu Zheng and Peter Palffy-Muhoray
2011, 15(2): 475-490 doi: 10.3934/dcdsb.2011.15.475 +[Abstract](595) +[PDF](461.4KB)
Abstract:
In this paper, we present a simple one tensor mean field model of biaxial nematic liquid crystals. The salient feature of our approach is that material parameters appear explicitly in the order parameter tensor. We construct the free energy from a mean field potential based on anisotropic dispersion interactions, identify the order parameter tensor and its elements, and obtain self-consistent equations, which are then solved numerically. The results are illustrated in a 3D ternary phase diagram. The phase behavior can be simply related to molecular parameters. The results may be useful for designing molecules that show a thermotropic biaxial phase.

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