Showing papers by "Chun Liu published in 2005"

On hydrodynamics of viscoelastic fluids

Abstract: In this paper, we study a hydrodynamic system describing fluids with viscoelastic properties After a brief examination of the relations between several models, we shall concentrate on a few analytical issues concerning them In particular, we establish local existence and global existence (with small initial data) of classical solutions for an Oldroyd system without an artificially postulated damping mechanism

A phase field formulation of the Willmore problem

Abstract: In this paper, we demonstrate, through asymptotic expansions, the convergence of a phase field formulation to model surfaces minimizing the mean curvature energy with volume and surface area constraints. Under the assumption of the existence of a smooth limiting surface, it is shown that the interface of a phase field, which is a critical point of the elastic bending energy, converges to a critical point of the surface energy. Further, the elastic bending energy of the phase field converges to the surface energy and the Lagrange multipliers associated with the volume and surface area constraints remain uniformly bounded. This paper is a first step to analytically justify the numerical simulations performed by Du, Liu and Wang in 2004 to model equilibrium configurations of vesicle membranes.

Diffuse-interface simulations of drop coalescence and retraction in viscoelastic fluids

Abstract: Drop dynamics plays a central role in defining the interfacial morphology in two-phase complex fluids such as emulsions and polymer blends. In such materials, the components are often microstructured complex fluids themselves. To model and simulate drop behavior in such systems, one has to deal with the dual complexity of non-Newtonian rheology and evolving interfaces. Recently, we developed a diffuse-interface formulation which incorporates complex rheology and interfacial dynamics in a unified framework. This paper uses a twodimensional implementation of the method to simulate drop coalescence after head-on collision and drop retraction from an elongated initial shape in a quiescent matrix. One of the two phases is a viscoelastic fluid modeled by an Oldroyd-B equation and the other is Newtonian. For the parameter values examined here, numerical results show that after drop collision, film drainage is enhanced when either phase is viscoelastic and drop coalescence happens more readily than in a comparable Newtonian system. The last stage of coalescence is dominated by a short-range molecular force in the model that is comparable to van der Waals force. The retraction of drops from an initial state of zero-velocity and zero-stress is hastened at first, but later resisted by viscoelasticity in either component. When retracting from an initial state with pre-existing stress, produced by cessation of steady shearing, viscoelasticity in the matrix hinders retraction from the beginning while that in the drop initially enhances retraction but later resists it. These results and the physical mechanisms that they reveal are consistent with prior experimental observations. © 2005 Elsevier B.V. All rights reserved.

Viscoelastic effects on drop deformation in steady shear

Abstract: This paper applies a diffuse-interface model to simulate the deformation of single drops in steady shear flows when one of the components is viscoelastic, represented by an Oldroyd-B model. In Newtonian fluids, drop deformation is dominated by the competition between interfacial tension and viscous forces due to flow. A fundamental question is how viscoelasticity in the drop or matrix phase influences drop deformation in shear. To answer this question, one has to deal with the dual complexity of nonNewtonian rheology and interfacial dynamics. Recently, we developed a diffuse-interface formulation that incorporates complex rheology and interfacial dynamics in a unified framework. Using a two-dimensional spectral implementation, our simulations show that, in agreement with observations, a viscoelastic drop deforms less than a comparable Newtonian drop. When the matrix is viscoelastic, however, the drop deformation is suppressed when the Deborah number De is small, but increases with De for larger De. This non-monotonic dependence on matrix viscoelasticity resolves an apparent contradiction in previous experiments. By analysing the flow and stress fields near the interface, we trace the effects to the normal stress in the viscoelastic phase and its modification of the flow field. These results, along with prior experimental observations, form a coherent picture of viscoelastic effects on steady-state drop deformation in shear.

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

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 suitable 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.

FENE Dumbbell Model and Its Several Linear and Nonlinear Closure Approximations

Abstract: We present some analytical and numerical studies on the finite extendible nonlinear elasticity (FENE) model of polymeric fluids and its several moment-closure approximations. The well-posedness of the FENE model is established under the influence of a steady flow field. We further infer existence of long-time and steady-state solutions for purely symmetric or antisymmetric velocity gradients. The stability of the steady-state solution for a general velocity gradient is illuminated by the analysis of the FENE-P closure approximation. We also propose a new linear closure approximation utilizing higher moments, which is shown to generate more accurate approximations than other existing closure models for moderate shear or extension rates. An instability phenomenon under a large strain is also investigated. This paper is a sequel to our earlier work [P. Yu, Q. Du, and C. Liu, Multiscale Model. Simul., 3 (2005), pp. 895--917].

Retrieving Topological Information for Phase Field Models

Abstract: The phase field approach has become a popular tool in modeling interface motion, microstructure evolution, and more recently the shape transformation of vesicle membranes under elastic bending energy. While it is advantageous to employ phase field models in numerical simu- lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes, detecting topological events may also become important for many applications such as those in the simulation of blood cells. Motivated by such considerations, a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events. It can also be used as a control method to avoid unphys- ical changes of topology due to the numerical methods, should it become necessary for particular practical applications. Through numerical experiments, we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes.

Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation

Abstract: In this paper, we study the effects of the spontaneous curvature on the static deformation of a vesicle membrane under the elastic bending energy, with prescribed bulk volume and surface area. Generalizing the phase field models developed in our previous works, we deduce a new energy formula involving the spontaneous curvature effects. Several axis-symmetric configurations are obtained through numerical simulations. Some analysis on the effects of the spontaneous curvature on the vesicle membrane shapes are also provided.

From micro to macro dynamics via a new closure approximation to the fene model of polymeric fluids

Abstract: We present a new closure approximation needed for deriving effective macroscopic moment equations from the microscopic finite-extensible-nonlinear-elastic kinetic theory modeling viscoelastic polymeric fluids. The closure is based on restricting the otherwise general probability distribution functions (PDFs) to a class of smooth distributions motivated by perturbing the equilibrium PDF\@. The simplified system coupling the moment equations and the Navier--Stokes equations still possesses an energy law analogous to the original micro-macro system. Some theoretical analysis and numerical experiments are presented to ensure the validity of the moment-closure system, and to illustrate the excellent agreement of the simplified model with the original system solved using a Monte Carlo approach, for a certain regime of physical parameters.

Transient drop deformation upon startup of shear in viscoelastic fluids

Abstract: Recent experiments show that upon abrupt start of a shear flow, a suspended drop undergoes an overshoot in deformation if either the drop or the matrix is a polymeric fluid Using a diffuse-interface formulation, we carry out two-dimensional numerical simulations that trace the origin of the transient to the mismatch of two time scales: a capillary time for drop deformation and a relaxation time for the polymers in the viscoelastic component The results are in qualitative agreement with experiments

On electro-kinetic fluids: one dimensional configurations

Abstract: Electro-kinetic fluids can be modeled by hydrodynamic systems describing the coupling between fluids and electric charges. The system consists of a momentum equation together with transport equations of charges. In the dynamics, the special coupling between the Lorentz force in the velocity equation and the material transport in the charge equation gives an energy dissipation law. In stationary situations, the system reduces to a Poisson-Boltzmann type of equation. In particular, under the no flux boundary conditions, the conservation of the total charge densities gives nonlocal integral terms in the equation. In this paper, we analyze the qualitative properties of solutions to such an equation, especially when the Debye constant $\epsilon$ approaches zero. Explicit properties can be derived for the one dimensional case while some may be generalized to higher dimensions. We also present some numerical simulation results of the system.