Showing papers by "Chun Liu published in 2007"

On a micro‐macro model for polymeric fluids near equilibrium

Abstract: In this paper, we study a micro-macro model for polymeric fluid. The system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation. We employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure. For the initial data not far from the equilibrium, we prove the global existence and uniqueness of classical solutions to the system. © 2006 Wiley Periodicals, Inc.

Global existence for a 2D incompressible viscoelastic model with small strain

Abstract: In this paper, we continue our previous study towards understanding the twodimensional hydrodynamic systems describing Oldroyd type incompressible viscoelastic fluids. We will decompose the deformation tensor into the strain and rotation components and look at their distinct contributions and structures in the small strain (with respect to viscosity) dynamics. In particular, we prove that there exist classical solutions globally in time if the strain component of the initial deformation is small enough, while we require no assumptions on smallness of the magnitude of the rotation component.

Analysis of a phase field navier-stokes vesicle-fluid interaction model

Abstract: This paper is concerned with the dynamics of vesicle membranes in incompressible viscous fluids. Some rigorous theory are presented for the phase field Navier-Stokes model proposed in [7], which is based on an energetic variation approach and incorporates the effect of bending elasticity energy for the vesicle membranes. The existence and uniqueness results of the global weak solutions are established.

Convergence of Numerical Approximations of the Incompressible Navier-Stokes Equations with Variable Density and Viscosity

Abstract: We consider numerical approximations of incompressible Newtonian fluids having variable, possibly discontinuous, density and viscosity. Since solutions of the equations with variable density and viscosity may not be unique, numerical schemes may not converge. If the solution is unique, then approximate solutions computed using the discontinuous Galerkin method to approximate the convection of the density and stable finite element approximations of the momentum equation converge to the solution. If the solution is not unique, a subsequence of these approximate solutions will converge to a solution.

Dynamics of Defect Motion in Nematic Liquid Crystal Flow: Modeling and Numerical Simulation

Abstract: The annihilation of a hedgehog-antihedgehog pair in hydrodynamics of (elastically isotropic) nematic liquid crystal materials is modeled using the Ericksen-Leslie theory which results in a nonlinear system for the flow velocity field and liquid crystal director field coupled through the transport of the directional order parameter and the induced elastic stress. An ecient and accurate numerical scheme is presented and implemented for this coupled nonlinear system in an axi-symmetric domain. Numerical simulations of annihilation of a hedgehog-antihedgehog pair with dierent types of transport are presented. In particular, it is shown that the stretching parameter in the transport equation contributes to the symmetry breaking of the pair’s moving speed during the dynamics of annihilation.

Mathematical models for the deformation of electrolyte droplets

Abstract: Using phase field methods, we introduce a penalty formulation for restricting the support of solutions of the hydrodynamic Poisson-Nernst-Plank equations to evolving subregions of the domain. The formulation is derived through variational principles from a free energy involving the phase field and electrostatic energy. We validate the model by energetic arguments and several dynamic, finite element simulations of the (linear) Navier-Stokes, Poisson-Nernst-Plank and Allen-Cahn system.

Diffuse interface energies capturing the euler number: relaxation and renomalization ∗

Abstract: We introduce a set of new interfacial energies for approximating the Euler number of level surfaces in the phase field (diffuse-interface) representation. These new formulae have simpler forms than those studied earlier in (Q. Du, C. Liu and X. Wang, Retrieving topological information for phase field models, SIAM J. Appl. Math., 65, 1913-1932, 2005), and do not contain higher order derivatives of the phase field function. Theoretical justifications are provided via formal asymptotic analysis, and practical validations are performed through numerical experiments. Relaxation and renormalization schemes are also developed to improve the robustness of the new energy functionals.

Heart-shaped bubbles rising in anisotropic liquids

Abstract: This Letter reports on numerical simulations motivated by experimental observations of an unusual inverted-heart shape for bubbles rising in an anisotropic micellar solution. We explain the bubble shape by assuming that the micelles are aligned into a nematic phase, whose anchoring energy on the bubble competes with the interfacial tension and the bulk elasticity of the nematic to modify the interfacial curvature. Numerical results show that bubbles with sufficiently strong planar anchoring rising in a vertically aligned nematic indeed assume the observed shape. The parameter values required are compared with the experimental materials and conditions.