Showing papers by "Chun Liu published in 2009"
TL;DR: In this article, the authors discuss the general energetic variational approaches for hydrodynamic systems of complex fluids, and discuss the important roles of MDP in designing numerical method for computations of hydrodynamics in complex fluids.
Abstract: We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.
137 citations
TL;DR: In this paper, a hydrodynamic system was established to study vesicle deformations under external flow fields, in the Eulerian formulation, involving the coupling of the incompressible flow system and a phase field equation.
84 citations
TL;DR: In this paper, the convergence of global strong solutions to single steady states as time tends to infinity was studied, both in 2D for arbitrary regular initial data and in 3D for certain particular cases.
Abstract: We study the asymptotic behavior of global solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes The coupling system consists of Navier-Stokes equations and kinematic transport equations for the molecular orientations We prove the convergence of global strong solutions to single steady states as time tends to infinity as well as estimates on the convergence rate both in 2D for arbitrary regular initial data and in 3D for certain particular cases
57 citations
01 Jun 2009
40 citations
TL;DR: In this paper, the authors study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport.
Abstract: In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton’s method are applied to this strongly nonlinear phase fieldmodel on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve. AMS subject classifications: 65B99, 65K05, 65K10, 65N12, 65N22, 65N30, 65N55, 65Z05
33 citations
01 Jun 2009
TL;DR: In this article, the authors introduce the theory of Incompressible Inviscid Flows (IIVF) and conservation laws of IIVF and the Compressible Euler System in Two Space Dimensions.
Abstract: Introduction to the Theory of Incompressible Inviscid Flows (T Y Hou & X Yu) Systems of Conservation Laws. Theory, Numerical Approximation and Discrete Shock Profiles (D Serre) Kinetic Theory and Conservation Laws: An Introduction (S Ukai & T Yang) Elementary Statistical Theories with Applications to Fluid Systems (X Wang) The Compressible Euler System in Two Space Dimensions (Y Zheng).
5 citations
01 Jan 2009
Abstract: In this paper we will discuss several issues related to the moment-closure approximation of multiscale models for viscoelastic polymeric fluids. These moment-closure approaches are based on special ansatz for the probability density function (PDF) in the finite extensible nonlinear elastic (FENE) dumbbell micro-macro models which consists of the coupled incompressible Navier-Stokes equations and the Fokker-Planck equations. We present the exact energy law of the resulting closure systems and introduce a post-modification scheme to preserve the positivity of PDF. The scheme not only reduces the region of negative PDF values but also preserves the structure of the induced stress tensor resulting from the molecular behaviors such as stretching and rotation. Numerical verifications are provided for the moment-closure system with some standard external flows. We also explore the relation of the maximum entropy principle (MEP) and the moment-closure approach. 1. Introduction. In this paper, we consider the hydrodynamical systems of dilute polymeric fluids. The viscoelastic flow of rheological complex fluids can be described by a multiscale (micro-macro) model. The multiscale-multiphysics model includes the coupling between the continuum mechanic theory [5] in macroscopic level and the kinetic theory in microscopic level. This micro-macro model also reflects the interaction between two different scales as the macroscopic flow/deformation will affect the microscopic structure through kinematic transport/deformation relation; while the averaging (coarsening) effects of the microscopic molecular configurations such as stretching and orientation will affect the macroscopic flow field through the induced elastic stresses. In many applications, we are more interested in the macroscopic quantity, such as the induced elastic stresses, rather than the detail behavior of molecular/microscopic variables. These stresses, resulting from the average of molecular behaviors, can be described in many situations by the moments of distribution function of molecular configurations. Notice that the PDF of the molecular configurations carries all the microscopic information of the system. Among different molecular models, the two most used well-known models are the Hookean dumbbell model, which is related to the Oldroyd-B viscoelasticity [1, 12], and the finite-extensible-nonlinear-elastic (FENE) dumbbell model [1, 2]. While the Hookean models are the best understood one and form the basis for most analytic studies, in this paper we focus on the FENE spring dumbbell model
3 citations
Posted Content•
13 Jan 2009
TL;DR: In this paper, the long time behavior of the classical solutions to a hydrodynamical system modeling the flow of nematic liquid crystals was studied, and the convergence of global solutions to single steady states as time tends to infinity was proved.
Abstract: In this paper we study the long time behavior of the classical solutions to a hydrodynamical system modeling the flow of nematic liquid crystals. This system consists of a coupled system of Navier-Stokes equations and kinematic transport equations for the molecular orientations. By using a suitable Lojasiewicz-Simon type inequality, we prove the convergence of global solutions to single steady states as time tends to infinity. Moreover, we provide estimates for the convergence rate.
2 citations
01 Jun 2009
TL;DR: In this article, the long-time behavior of the classical solutions to a hydrodynamical system modeling the flow of nematic liquid crystals with rod-like molecules was studied and the convergence of global solutions to single steady states as time tends to infinity was proved.
Abstract: In this paper we study the long-time behavior of the classical solutions to a hydrodynamical system modeling the flow of nematic liquid crystals with rod-like molecules. This system consists of a coupled system of Navier–Stokes equations and kinematic transport equations for the molecular orientations. By using a suitable Łojasiewicz–Simon type inequality, we prove the convergence of global solutions to single steady states as time tends to infinity. Moreover, we provide estimates for the convergence rate.
2 citations
TL;DR: In this article, the existence of both local and global smooth solutions to the Cauchy problem 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 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.