Showing papers by "Chun Liu published in 2008"
TL;DR: In this paper, the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near-equilibrium initial data was proved.
Abstract: We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near- equilibrium initial data. The results hold in both two- and three-dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals, and mixture problems.
255 citations
TL;DR: In this paper, the authors present results for the existence of classical solutions of a hydrodynamical system modeling the flow of liquid crystals, which consists of a coupled system of NN equations and various kinematic transport equations for the molecular orientations.
Abstract: In this paper we present results for the existence of classical
solutions of a hydrodynamical system modeling the flow of nematic
liquid crystals. The system consists of a coupled system of
Navier-Stokes equations and various kinematic transport equations
for the molecular orientations. A formal physical derivation of the
induced elastic stress using least action principle reflects the
special coupling between the transport and the induced stress terms.
The derivation and the analysis of the system falls into a general
energetic variational framework for complex fluids with elastic
effects due to the presence of nontrivial microstructures.
134 citations
TL;DR: In this paper, the authors used the Onsager principle to derive a two-phase continuum formulation for the hydrodynamics of the electrorheological fluid, consisting of dielectric microspheres dispersed in an insulating liquid.
Abstract: We use the Onsager principle to derive a two-phase continuum formulation for the hydrodynamics of the electrorheological (ER) fluid, consisting of dielectric microspheres dispersed in an insulating liquid. Predictions of the theory are in excellent agreement with the experiments. In particular, it is shown that whereas the usual configuration of applied electric field being perpendicular to the shearing direction can lead to shear thinning at high shear rates and thus the loss of ER effect, the interdigitated, alternating electrodes configuration can eliminate the shear-thinning effect.
40 citations
TL;DR: It is shown that for the underlying Fokker–Planck type of equations, any preassigned distribution on the boundary will become redundant once the nondimensional number $\text{{\it Li\/}} := \frac{Hb}{k_BT} \geq 2$.
Abstract: We consider the microscopic equation of finite extensible nonlinear elasticity (FENE) models for polymeric fluids under a steady flow field. It is shown that for the underlying Fokker–Planck type of equations, any preassigned distribution on the boundary will become redundant once the nondimensional number $\text{{\it Li\/}} := \frac{Hb}{k_BT} \geq 2$, where H is the elasticity constant, $\sqrt{b}$ is the maximum dumbbell extension, T is the temperature, and $k_B$ is the usual Boltzmann constant. Moreover, if the probability density function is regular enough for its trace to be defined on the sphere $|m| = \sqrt{b}$, then the trace is necessarily zero when $\text{{\it Li\/}} > 2$. These results are consistent with our numerical simulations as well as some recent well-posedness results by preassuming a zero boundary distribution.
33 citations
TL;DR: In this article, the authors considered the finite extensible nonlinear elasticity (FENE) model in viscoelastic polymeric fluids and employed the maximum entropy principle to obtain the solution which maximizes the entropy of FENE model in stationary situations.
Abstract: We consider the finite extensible nonlinear elasticity (FENE) dumbbell
model in viscoelastic polymeric fluids. We employ the maximum
entropy principle for FENE model to obtain the solution which
maximizes the entropy of FENE model in stationary situations. Then
we approximate the maximum entropy solution using the second order
terms in microscopic configuration field to get an probability
density function (PDF). The approximated PDF gives a solution to
avoid the difficulties caused by the nonlinearity of FENE model.
We perform the moment-closure approximation procedure with the PDF
approximated from the maximum entropy solution, and compute the
induced macroscopic stresses. We also show that the moment-closure
system satisfies the energy dissipation law. Finally, we show some
numerical simulations to verify the PDF and moment-closure system.
22 citations
TL;DR: In this article, a two-phase, electrical-hydrodynamic model for the description of electrorheological fluid dynamics, based on the Onsager principle of minimum energy dissipation, is presented.
Abstract: We present the formulation of a two-phase, electrical-hydrodynamic model for the description of electrorheological fluid dynamics, based on the Onsager principle of minimum energy dissipation. By considering the energetics of (induced) dipole-dipole interaction between the solid particles in terms of a field variable n(� x ), we employ the Onsager principle to derive the relevant coupled hydrodynamic equations, together with a continuity equation for n(� x ). Numerical solution of the relevant equations yields predictions that display very realistic behaviors as seen experimentally. In particular, we show that while the predicted results have features that resemble Bingham fluids, there can be important differences. For example, the yield stress obtained by extrapolating the shear rate to zero is 30-40% lower than that obtained from the maximum of the stress-strain relation. Moreover, for the conventional electrode configuration where the field is perpendicular to the shearing direction, there is very clear shearing-thinning effect that has been seen experimentally.
21 citations
TL;DR: An enhanced moment-closure approximation to the finite extensible nonlinear elastic (FENE) models of polymeric fluids, which takes into account the drastic split into two spikes and centralized behavior under the large macroscopic flow effects.
Abstract: We present an enhanced moment-closure approximation to the finite extensible nonlinear elastic (FENE) models of polymeric fluids. This new moment-closure method involves the perturbation of the equilibrium probability distribution function (PDF), which takes into account of the drastic split into two spikes and centralized behavior under the large macroscopic flow effects. The resulting macroscopic system includes the moment-closure equations, the momentum (force balance) equations, as well as an auxiliary equation representing implicitly the dynamics of the spikes for the microscopic configurations. It also inherits the energy dissipation law from the original micro-macro models. Through numerical experiments, we demonstrate the accuracy and robustness of the moment-closure system for some special external flow with a wide range of flow rates.
20 citations
TL;DR: It is shown that whereas the usual configuration of applied electric field being perpendicular to the shearing direction can lead to shear thinning at high shear rates and thus the loss of ER effect, the interdigitated, alternating electrodes configuration can eliminate the shear-thinning effect.
Abstract: We present the formulation of a two‐phase, electrical‐hydrodynamic model for the description of electrorheological fluid dynamics. By considering the energetics of (induced) dipole‐dipole interaction between the solid particles in terms of a field variable n(x), we employ the Onsager principle to derive the relevant coupled hydrodynamic equations, together with a continuity equation for n(x). Numerical solution of the relevant equations yields predictions that display very realistic behaviors as seen experimentally.
19 citations
01 Jan 2008
TL;DR: In this article, energy law preserving finite element schemes were developed to simulate motions of two-phase incompressible flows governed by hydrodynamical phase field models of Allen-Cahn and Cahn-Hilliard.
Abstract: We develop energy law preserving finite element schemes to simulate motions of two-phase incompressible flows governed by hydrodynamical phase field models of Allen-Cahn and Cahn-Hilliard. We reformulate the original hydrodynamical phase field models in the weak form consistent with the continuous energy law. Then, convenient conformal C0 finite elements are adopted in the Galerkin discretization. The resulting temporal discrete schemes are carefully crafted to either respect the energy law or control possible oscillations near the fluid-interface. A fixed iterative method is used to solve the discrete nonlinear system so that a matrix free time evolution can be achieved in which the velocity and phase variable are solved separately. A few two-fluid flow examples are computed to demonstrate the performance of the method.