A scaling approach to the derivation of hydrodynamic boundary conditions

Qian, Tiezheng; Qiu, Chunyin; Sheng, Ping

doi:10.1017/s0022112008002863

Cited by 12 publications

(12 citation statements)

References 37 publications

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“…This can also be formally justified by the surface layer scaling approach within a thin layer next to the solid wall (cf. [75]). For an isothermal closed system, evolution of the binary mixtures is characterized by the following energy dissipation law (cf.…”

Section: The Energetic Variational Approachmentioning

confidence: 99%

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

2019

Arch Rational Mech Anal

101

117

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The Cahn-Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn-Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints such as conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as t → +∞ and characterize the stability of local energy minimizers for the system.

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Section: The Energetic Variational Approachmentioning

confidence: 99%

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

2019

Arch Rational Mech Anal

101

117

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“…It has been shown 22,23 that the continuum hydrodynamic model for contact line motion in two-component fluids can be derived in a variational approach based on the Onsager principle of minimum energy dissipation (entropy production). 24,25 This principle, as formulated by Onsager for small perturbations away from equilibrium, 24,25 can also be applied to onecomponent liquid-gas systems in the linear response regime.…”

Section: A Two Coexisting Mechanisms For Contact Line Motionmentioning

confidence: 99%

“…20 Its numerical implementation has produced continuum solutions in quantitative agreement with MD simulation results. 20,21 Recently, it has been shown 22,23 that the GNBC can be derived in a variational approach based on the Onsager principle of minimum energy dissipation. 24,25 We would like to point out that in all the other studies based on diffuse-interface modeling for binary mixtures, 14,15,17 the no-slip boundary condition is kept and the nonintegrable stress singularity is removed by introducing diffusive transport through the fluid-fluid interface.…”

Section: Introductionmentioning

confidence: 99%

Contact line motion in confined liquid–gas systems: Slip versus phase transition

Qian

2010

The Journal of Chemical Physics

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In two-phase flows, the interface intervening between the two fluid phases intersects the solid wall at the contact line. A classical problem in continuum fluid mechanics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a nonintegrable stress singularity. Recently, various diffuse-interface models have been proposed to explain the contact line motion using mechanisms missing from the sharp-interface treatments in fluid mechanics. In one-component two-phase (liquid-gas) systems, the contact line can move through the mass transport across the interface while in two-component (binary) fluids, the contact line can move through diffusive transport across the interface. While these mechanisms alone suffice to remove the stress singularity, the role of fluid slip at solid surface needs to be taken into account as well. In this paper, we apply the diffuse-interface modeling to the study of contact line motion in one-component liquid-gas systems, with the fluid slip fully taken into account. The dynamic van der Waals theory has been presented for one-component fluids, capable of describing the two-phase hydrodynamics involving the liquid-gas transition [A. Onuki, Phys. Rev. E 75, 036304 (2007)]. This theory assumes the local equilibrium condition at the solid surface for density and also the no-slip boundary condition for velocity. We use its hydrodynamic equations to describe the continuum hydrodynamics in the bulk region and derive the more general boundary conditions by introducing additional dissipative processes at the fluid-solid interface. The positive definiteness of entropy production rate is the guiding principle of our derivation. Numerical simulations based on a finite-difference algorithm have been carried out to investigate the dynamic effects of the newly derived boundary conditions, showing that the contact line can move through both phase transition and slip, with their relative contributions determined by a competition between the two coexisting mechanisms in terms of entropy production. At temperatures very close to the critical temperature, the phase transition is the dominant mechanism, for the liquid-gas interface is wide and the density ratio is close to 1. At low temperatures, the slip effect shows up as the slip length is gradually increased. The observed competition can be interpreted by the Onsager principle of minimum entropy production.

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“…By the use of the Cahn-Hilliard (CH) hydrodynamic formulation for two-phase flows [13,14,17], the implementation of the GNBC leads to continuum solutions in quantitative agreement with MD simulation results [16,18,19]. Recently, it has been shown [20,21] that the GNBC can be derived in a variational approach from the Onsager principle of minimum energy dissipation [22,23].…”

Section: Introductionmentioning

confidence: 91%

Stick-Slip Motion of Moving Contact Line on Chemically Patterned Surfaces

Lei

Qian³

et al. 2010

CICP

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Abstract. Based on our continuum hydrodynamic model for immiscible two-phase flows at solid surfaces, the stick-slip motion has been predicted for moving contact line at chemically patterned surfaces [Wang et al., J. Fluid Mech., 605 (2008), pp. 59-78]. In this paper we show that the continuum predictions can be quantitatively verified by molecular dynamics (MD) simulations. Our MD simulations are carried out for two immiscible Lennard-Jones fluids confined by two planar solid walls in Poiseuille flow geometry. In particular, one solid surface is chemically patterned with alternating stripes. For comparison, the continuum model is numerically solved using material parameters directly measured in MD simulations. From oscillatory fluid-fluid interface to intermittent stick-slip motion of moving contact line, we have quantitative agreement between the continuum and MD results. This agreement is attributed to the accurate description down to molecular scale by the generalized Navier boundary condition in our continuum model. Numerical results are also presented for the relaxational dynamics of fluid-fluid interface, in agreement with a theoretical analysis based on the Onsager principle of minimum energy dissipation.

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A scaling approach to the derivation of hydrodynamic boundary conditions

Cited by 12 publications

References 37 publications

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

Contact line motion in confined liquid–gas systems: Slip versus phase transition

Stick-Slip Motion of Moving Contact Line on Chemically Patterned Surfaces

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