Mathematical Analysis of Multi-Scale Models of Complex Fluids

Li, Tiejun; Zhang, Pingwen

doi:10.4310/cms.2007.v5.n1.a1

Cited by 30 publications

(29 citation statements)

References 73 publications

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“…We note that the Cartesian coordinate and the spherical coordinate are related by the following relations: So we have 25) i.e., r 1 M 12 = 0.…”

Section: Extended Nematics With Permanent Dipoles (µ = 0)mentioning

confidence: 99%

“…Given the rising interest in kinetic theory in the mathematics community these days, various attempts have been made to analyze the properties of the partial differential equations in the kinetic theories and obtain their solutions semianalytically and numerically [23,11,12,13,14,15,16,17,18,5,6,7,10,23,34,35,26,40,38,20]. A recent review of the state of the art in the mathematical and numerical analysis of multi-scale models of complex fluids is given by Li and Zhang [25]. …”

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confidence: 99%

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Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

Wang²,

Zhang³

et al. 2007

Communications in Mathematical Sciences

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Abstract. We study the steady state phase behavior of homogeneous, rigid, extended (polar) nematic polymers or nematic dispersions under imposed magnetic (or electric fields), in which the intermolecular dipole-dipole and excluded volume interaction as well as external field contribution are accounted for. We completely characterize the phase diagram for polar nematics with and without permanent magnetic moments (or dipoles). For nematics without magnetic moments, the steady state is either purely nematic or polar depending on the strength of the excluded volume and the dipole-dipole interaction, in which the nonzero polarity vector (the first moment vector) is coaxial with the second moment tensor; thereby the steady state pdf is determined essentially by up to three scalar order parameters and a rotational group of SO(2) transverse to the imposed field direction. For nematics with permanent magnetic moments (or dipoles), the steady states are polar and the polarity vector is parallel to the external field direction when a necessary condition for parameters is met. When the condition is violated, only stable steady states have their polarity vector parallel to the external field direction and there are thermodynamically unstable nonparallel states. The stability of the steady states is inferred from the minimum of the free energy density.Key words. nematic liquid crystal, kinetic theory, polymers, magnetic field, electric field, bifurcation, steady state, phase transition, polarity, stability AMS subject classifications. 45K05, 70K50, 82B26, 82B27 IntroductionKinetic theory is a useful tool in modeling soft matter and complex fluids [2,4,8,24,9]. In the past, it has been perceived as colossal and complicated-hardly accessible to theoretical analysis. Given the rising interest in kinetic theory in the mathematics community these days, various attempts have been made to analyze the properties of the partial differential equations in the kinetic theories and obtain their solutions semianalytically and numerically [23,11,12,13,14,15,16,17,18,5,6,7,10,23,34,35,26,40,38,20]. A recent review of the state of the art in the mathematical and numerical analysis of multi-scale models of complex fluids is given by Li and Zhang [25].

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“…We note that the Cartesian coordinate and the spherical coordinate are related by the following relations: So we have 25) i.e., r 1 M 12 = 0.…”

Section: Extended Nematics With Permanent Dipoles (µ = 0)mentioning

confidence: 99%

mentioning

confidence: 99%

Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

Wang²,

Zhang³

et al. 2007

Communications in Mathematical Sciences

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“…For a review of numerical algorithms for the approximation of kinetic models of dilute polymers see, for example, Section 4 of the survey article of Li and Zhang [32]; for recent progress on deterministic algorithms for the approximation of Fokker-Planck and coupled Navier-Stokes-Fokker-Planck systems, see, for example, Lozinski et al [37,38], and Knezevic and Süli [26,27].…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of kinetic dilute polymer models with microscopic cut-off

Barrett¹,

Süli²

2010

ESAIM: M2AN

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Abstract. We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω ⊂ R d , d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function β L (·) := min(·, L) in the drag and convective terms, where L 1. We focus on finitelyextensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-StokesFokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker-Planck-Navier-Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellianweighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions. We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms, and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

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“…A delicate analysis for the nonexplosive behavior of Q is deduced. In a high dimensional case, the local well-posedness is proved in [6] for the coupled system under the polynomial growth condition of Q |∇ m F (Q)| ≤ 1 + |Q| p (m = 0, 1, 2, 3, 4) (1.6) for some positive real p. The most recent progress for the mathematical analysis of complex fluids is reviewed in [12]. To the best knowledge of the authors, there are no results concerning the convergence of the BCF methods for the dumbbell model in the nonshear flow case, even for the Hookean spring, which is the main contribution of this paper.…”

Section: Introductionmentioning

confidence: 99%

Convergence Analysis of BCF Method for Hookean Dumbbell Model with Finite Difference Scheme

Zhang²

2006

Multiscale Model. Simul.

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Abstract.A convergence analysis of the Brownian configuration fields (BCF) method [M. A. Hulsen, A. P. G. van Heel, and B. H. A. A. van den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79-101] for the Hookean dumbbell model with finite difference scheme in dimension 2 or 3 is given in this paper under the assumption that the continuous solution is smooth enough. An explicit solution of the Hookean dumbbell model is obtained via deformation tensor. A large deviation-type estimate for the error of polymeric stress E(QQ) is given, which is a key step in the proof. It is shown that if the number of configuration fields N , the space stepsize h, and the time stepsize δt are chosen appropriately, the convergence of second order in space and first order in time may be proved after excluding a set of small probability. Simultaneous discretization of Monte Carlo and space and the inverse inequality trick are essential for the proof. 1. Introduction. The dumbbell model is the simplest model of polymeric fluids that takes into account the microscopic behavior of the solute polymers [2,4,16]. It models the polymers in dilute solution by dumbbells, each with two beads connected by a spring. The configuration of the spring then specifies the conformation of the polymer. Denote by u and p the velocity and pressure of the fluid and by Q the configuration of the spring; then one has the following coupled macroscopic-kinetic equations:

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Mathematical Analysis of Multi-Scale Models of Complex Fluids

Cited by 30 publications

References 73 publications

Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

Steady states and their stability of homogeneous, rigid, extended nematic polymers under imposed magnetic fields

Finite element approximation of kinetic dilute polymer models with microscopic cut-off

Convergence Analysis of BCF Method for Hookean Dumbbell Model with Finite Difference Scheme

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