A theory for flowing nematic polymers with orientational distortion

Feng, James J.; Sgalari, G.; Leal, L. Gary

doi:10.1122/1.1289278

Cited by 81 publications

(59 citation statements)

References 26 publications

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“…The inhomogeneous flows were first studied by Marrucci and Greco [17], and subsequently by many people [8,20]. Instead of using the distribution as the sole order parameter, they used a combination of the tensorial order parameter and the distribution function, and used the spatial gradients of the tensorial order parameter to describe the spatial variations.…”

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confidence: 99%

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

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Abstract. We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the EricksenLeslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional. We use x ∈ Ω ⊆ R 3 to denote the material point and f (x, m, t) to represent the number density for the number of molecules whose orientation is parallel to m at point x and time t. where De is the Deborah number, R is the rotational gradient operator(see Section 3), κ is a constant velocity gradient, and U is the mean-field interaction potential. Onsager [18]

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confidence: 99%

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

Wang

Zhang

2014

Comm Pure Appl Math

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“…The theories by Marrucci and Greco (1993), Larson and coauthors (1998), and Feng et al (2000) typically use and elaborate the molecular long rigid rod approach, proposed by Doi (1981) and extended in the text by Doi and Edwards (1986). Also, B.…”

Section: Introductionmentioning

confidence: 99%

A model of weak viscoelastic nematodynamics

Leonov

2006

Z. angew. Math. Phys.

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The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It may describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie-Ericksen-Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is simplified to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.

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“…This is exactly the Kramers expression for the polymer elastic stress tensor [64]. The same procedure can be followed for other microstructural free energies, such as the Marrucci-Greco nematic potential energy for liquid-crystalline polymers [5,65]. …”

Section: Complex Rheologymentioning

confidence: 99%

“…As a second example, we consider here the important case of a viscoelastic polymer solution modeled as a suspension of Hookean dumbbells in a Newtonian solvent [64]. Instead of the least-action principle, we follow a formally different but essentially equivalent "virtual-work principle" [5]. For a single dumbbell with a connector Q, its elastic energy is 1 2 HQ · Q, where H is the elastic constant.…”

Section: Complex Rheologymentioning

confidence: 99%

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An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

Feng

Shen

et al.

Modeling of Soft Matter

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Abstract. The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate Navier-Stokes systems for two-phase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less self-contained exposition of the method. We will briefly review the literature, present the governing equations and discuss a numerical scheme based on different numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further.

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A theory for flowing nematic polymers with orientational distortion

Cited by 81 publications

References 26 publications

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

The Small Deborah Number Limit of the Doi‐Onsager Equation to the Ericksen‐Leslie Equation

A model of weak viscoelastic nematodynamics

An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

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