An integrated smoothed particle hydrodynamics model for complex interfacial flows with large density ratios

In this paper, an integrated smoothed particle hydrodynamics (SPH) model for complex interfacial flows with large density ratios is developed. The discrete continuity equation and acceleration equation are obtained by considering the time derivative of the volume of particle and Eckart's continuum Lagrangian equation. A continuum surface force model is used to meet the fact that surface force may not be distributed uniformly on each side of the interface. An improved boundary condition is imposed to model wall free-slip and no-slip condition for interfacial flows with large density ratios. Particle shifting algorithm (PSA) is added for interfacial flows by imposing the normal correction near the interface, called as Interface-PSA. Then four representative numerical examples, including droplet deformation, Rayleigh-Taylor instability, dam breaking, and bubble rising, are presented and compared well with reference data. It is demonstrated that inherent interfacial flow physics can be well captured, including surface tension and the dynamic evolution of the complex interfaces.     

相间交界面对非饱和土应力状态的影响

非饱和土是一种三相多孔介质,不同相之间的交界面尤其是气液交界面的存在直接影响了非饱和土的宏观行为.首先对土中交界面的形式和作用进行了探讨,指出气液交界面对非饱和土的行为有重要影响,并给出了界面功和气液比表面积的表达式.在已有的非饱和土变形功表达式基础上,引入界面能影响,得到了考虑交界面影响的非饱和土自由能方程.利用所得的自由能方程,给出了考虑交界面影响的非饱和土固相和液相相应的应力变量.对考虑交界面面积的液相流动方程进行了探讨,给出了非平衡条件下的土水特征曲线表达式,指出在平衡条件下土水特征曲线中应当考虑交界面面积的影响,传统土水特征曲线是三维关系在吸力-饱和度平面上的投影.将比表面积与土水特征曲线的关系,利用已有试验数据验证了该表达式的合理性.利用界面面积的表达式计算有效应力,将其与已有试验结果进行对比,表明给出的比表面积表达式可很好地反映实际情况.不同于已有现象学研究,本文推导具有严格的理论基础,研究表明完整的有效应力表达式中应考虑土体内部作用力的影响,其不仅包含基质吸力,同时还包含其他形式的作用力,其大小与界面比表面积有关.该表达式为下一步研究界面效应对土体变形、强度和流动特性的研究提供了基础.     

The wetting problem of fluids on solid surfaces. Part 2: the contact angle hysteresis

In part 1 (Gouin, [13]), we proposed a model of dynamics of wetting for slow movements near a contact line formed at the interface of two immiscible fluids and a solid when viscous dissipation remains bounded. The contact line is not a material line and a Young-Dupré equation for the apparent dynamic contact angle taking into account the line celerity was proposed. In this paper we consider a form of the interfacial energy of a solid surface in which many small oscillations are superposed on a slowly varying function. For a capillary tube, a scaling analysis of the microscopic law associated with the Young-Dupré dynamic equation yields a macroscopic equation for the motion of the contact line. The value of the deduced apparent dynamic contact angle yields for the average response of the line motion a phenomenon akin to the stick-slip motion of the contact line on the solid wall. The contact angle hysteresis phenomenon and the modelling of experimentally well-known results expressing the dependence of the apparent dynamic contact angle on the celerity of the line are obtained. Furthermore, a qualitative explanation of the maximum speed of wetting (and dewetting) can be given.Received: 5 June 2001, Accepted: 24 May 2003, Published online: 29 July 2003PACS: 02.90, 47.50, 66.20, 68.03, 68.08     

Relative permeabilities of a near critical binary fluid

Relative permeabilities were measured at very low interfacial tensions (IFT) for two-phase mixtures of methanol and hexane flowing through Clashach sandstone. These two components pass from a two- to a single-phase system as the temperature is increased above the critical solution temperature (CST). The interfacial tension between the coexisting phases approaches zero as the solution reaches miscibility. The phase behaviour of methanol and hexane mixtures has been well characterised allowing the calculation of relative permeabilities, saturations and capillary numbers. Flow data are reported for four different temperatures in the two-phase region (i.e., four values of IFT and capillary number). The capillary desaturation curve (CDC) for the strongly wetting methanol rich phase is also presented. In addition to the novel technique presented for measurement of relative permeability, the results indicate that relative permeabilities approach straight line functions very near the critical point. Furthermore, desaturation of the wetting phase was found to be dependent on the capillary number which, in turn, depends on the location of the mixture on the fluid phase diagram and the proximity to the critical temperature.     

考虑时变效应的面板堆石坝三维湿化变形研究1）

针对堆石料浸水后的湿化变形并不是瞬时产生,而是一个渐进发展过程这一现象,建议将湿化变形进行时变计算。首先采用Prandtl-Reuss流动法则推导了湿化剪切应变分量,然后叠加湿化体积应变分量,获得三维湿化应变分量;通过分析三维湿化应变分量和单轴应力状态下的湿化应变的关系,指出有关文献推导的三维湿化应变分量计算公式不严谨;然后类比于堆石料流变变形计算公式,推导了湿化变形时变计算公式。实例分析表明,湿化引起坝顶沉降随时间逐渐增大,变形稳定的时间与湿化变形速率呈反比关系。     

THREE-DIMENSIONAL WETTING DEFORMATION FOR CONCRETE FACED ROCKFILL DAM CONSIDERING TIME-EFFECT

针对堆石料浸水后的湿化变形并不是瞬时产生,而是一个渐进发展过程这一现象,建议将湿化变形进行时变计算. 首先采用Prandtl-Reuss 流动法则推导了湿化剪切应变分量,然后叠加湿化体积应变分量,获得三维湿化应变分量;通过分析三维湿化应变分量和单轴应力状态下的湿化应变的关系,指出有关文献推导的三维湿化应变分量计算公式不严谨;然后类比于堆石料流变变形计算公式,推导了湿化变形时变计算公式. 实例分析表明,湿化引起坝顶沉降随时间逐渐增大,变形稳定的时间与湿化变形速率呈反比关系.     

Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a nonlinear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled, in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. The rates of convergence in space and time are evaluated leading to an optimal order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube, and on the wetting drop over a horizontal solid wall.     

Separation phenomena at the interface of a finitely deformed composite sphere

The problem of the finite deformation of a composite sphere subjected to a spherically symmetric dead load traction is revisited focusing on the formation of a cavity at the interface between a hyperelastic, incompressible matrix shell and a rigid inhomogeneity. Separation phenomena are assumed to be governed by a vanishingly thin interfacial cohesive zone characterized by uniform normal and tangential interface force–separation constitutive relations. Spherically symmetric cavity shapes (spheres) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation, the dead loading and the volume concentration of inhomogeneity. Spherically symmetric and non-symmetric bifurcations initiating from spherically symmetric equilibrium states are analyzed within the framework of infinitesimal strain superimposed on a given finite deformation. A simple formula for the dead load required to initiate the non-symmetrical rigid body mode is obtained and a detailed examination of a few special cases is provided. Explicit results are presented for the Mooney–Rivlin strain energy density and for an interface force–separation relation which allows for complete decohesion in normal separation.     

Constitutive modeling of complex interfaces based on a differential interfacial energy function

An important theory on the dynamics of complex interfaces is the Doi and Ohta theory where the interfacial contribution to the Cauchy stress tensor is determined from an interfacial conformation tensor. For a uniform deformation field in the Eulerian framework, Doi and Ohta adopted a decoupling approximation to reduce a fourth-order tensor into two second-order tensors and derived a differential equation governing the evolution of the interfacial conformation tensor. In this paper, a different formulation is presented for establishing the Cauchy stress tensor based on a path-independent interfacial energy function. By differentiating this interfacial energy function against a Lagrangian strain tensor, a nearly closed-form solution for the stress tensor was determined, involving only elementary algebraic and matrix operations. From this process, the stress-conformation relation proposed by Doi and Ohta is also confirmed from a thermodynamic perspective. The testing cases with uniaxial elongation and simple shear further showed improved fitting to the analytical or exact solutions.     

Interfacial localization of nanoclay particles in oil-in-water emulsions and its reflection in interfacial moduli

The localization of nanoclay particles dispersed in the oil phase of a model oil-in-water emulsion depends on the wetting property of layered nanoparticles. Investigation at a single droplet interface shows that nanoclay is located at different interfacial regions depending on the hydrophilic property of the nanoclay surface. Hydrophobic nanoclays do not present Pickering phenomena at the interface and hardly form an interfacial layer. Hydrophilic nanoclay particles quickly move to the interface and form a Pickering interface with a high interfacial shear modulus. With surfactant, poor hydrophilic nanoclays can be located at the interface due to improvement of the wetting behavior caused by the surfactants dissolved in the aqueous continuous phase. With ionic molecules changing the wetting behavior of particles, the interfacial localization of nanoclays can be controlled and improve the mechanical property of emulsion.     

Dynamics and rheology of the morphology of immiscible polymer blends – on modeling and simulation

 The material properties of heterogeneous polymer blends are crucially influenced by their morphology, i.e., by the spatial structure of the blend components and by the specific configuration of the interfaces separating the phases. Hence, in order to understand the behavior of experimentally obtained morphologies, one is interested in modeling the relevant dynamics of the morphology subject to external flow. Thus one can study, e.g., through the interfacial stress tensor the rheological properties due to the interfaces. The balance equations used for that purpose are based on a Cahn-Hilliard equation for the local concentration, the continuity equation, and a modified Navier-Stokes equation for the local velocity. The essential material and processing parameters such as surface tension, viscosity and volume fraction of both polymers, and imposed shear rate are taken into consideration as model coefficients. By regarding hydrodynamic interaction, which is proved to be important in case of immiscible blends, the interfacial relaxation is described properly. Simulations in both three and two dimensions agree at least qualitatively with experimental results concerning droplet deformation, droplet coalescence, and interfacial rheological properties of the blend. Received: 25 September 2000 Accepted: 24 April 2001     

Interfacial fracture analysis of bonded dissimilar strips with a functionally graded interlayer under antiplane deformation

This paper provides the solution to the problem of dissimilar, homogeneous semi-infinite strips bonded through a functionally graded interlayer and weakened by an embedded or edge interfacial crack. The bonded system is assumed to be under antiplane deformation, subjected to either traction-free or clamped boundary conditions along its bounding planes. Based on the Fourier integral transform, the problem is formulated in terms of a singular integral equation which has a simple Cauchy kernel for the embedded crack and a generalized Cauchy kernel for the edge crack. In the numerical results, the effects of geometric and material parameters of the bonded system on the crack-tip stress intensity factors are presented in order to quantify the interfacial fracture behavior in the presence of the graded interlayer.     

Irreversible thermodynamics of liquid crystal interfaces

A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.     

流场中聚合物共混体系液滴形变的理论模型

讨论了两相聚合物共混体系中，悬浮于另一种牛顿（或粘弹）液体中的牛顿（或粘弹）液滴的形变理论模型．影响液滴形变的主要因素有两相的组成、粘度比和弹性比、动态界面张力、临界界面张力系数，外流场形式及其强度．对于两相均为牛顿流体的体系，理论预测能够与实验相符；对于两相（或其中一相）为粘弹流体的体系，由于弹性的影响而使液滴形变的研究变得复杂，理论模型尚需完善．建立完整的液滴形变理论模型还需深入研究界面层、微观分子形变、液滴之间及液滴和连续相介质之间的相互作用对液滴形变的影响     

Stress intensity factors for an oblique edge crack in a coating/substrate system with a graded interfacial zone under antiplane shear

This paper investigates the edge crack problem for a coating/substrate system with a functionally graded interfacial zone under the condition of antiplane deformation. With the interfacial zone being modeled by a nonhomogeneous interlayer having the continuously varying shear modulus between the dissimilar, homogeneous phases of the coated medium, the coating is assumed to contain an edge crack at an arbitrary angle to the interfacial zone. The Fourier integral transform method is used in conjunction with the coordinate transformations of basic field variables. Formulation of the proposed crack problem is then reduced to solving a singular integral equation with a generalized Cauchy kernel. The mode III stress intensity factors are defined and evaluated in terms of the solution to the integral equation. In the numerical results, the values of the stress intensity factors are plotted, illustrating the effects of the crack orientation angle for various material and geometric combinations of the coating/substrate system with the graded interfacial zone.     

Interfacial excess energy, excess stress and excess strain in elastic solids: Planar interfaces

In this paper, interfacial excess energy and interfacial excess stress for coherent interfaces in an elastic solid are reformulated within the framework of continuum mechanics. It is shown that the well-known Shuttleworth relationship between the interfacial excess energy and interfacial excess stress is valid only when the interface is free of transverse stresses. To account for the transverse stress, a new relationship is derived between the interfacial excess energy and interfacial excess stress. Dually, the concept of transverse interfacial excess strain is also introduced, and the complementary Shuttleworth equation is derived that relates the interfacial excess energy to the newly introduced transverse interfacial excess strain. This new formulation of interfacial excess stress and excess strain naturally leads to the definition of an in-plane interfacial stiffness tensor, a transverse interfacial compliance tensor and a coupling tensor that accounts for the Poisson's effect of the interface. These tensors fully describe the elastic behavior of a coherent interface upon deformation.     

Interface behavior and void formation during infiltration of liquids into porous structures

A numerical approach was proposed to simulate time-dependent evolution of the liquid front during the pore-level infiltration of liquids into porous structures. It considers the multiphase problem of liquid penetration into the pore(s) initially occupied with air and the simultaneous escaping of air. The Volume-of-Fluid (VOF) method was employed using a two-dimensional model of the graphite pore structure. The proposed method is capable of tracking the evolution of liquid front and yields the infiltration criteria for wetting and non-wetting liquids. Contribution of various driving forces (resulting from pressure gradient, gravity and interfacial effects) to infiltration and interface behavior including the liquid front shape, position and velocity was investigated. Interface pinning (temporary and permanent) and wicking flow through the pore(s) were investigated during infiltration of wetting liquids, whereas pore-level fingering and void formation (entrapment of air within the pore) were observed for non-wetting liquids. The results were verified against the results of coupled VOF level-set method, known to be more accurate for interface tracking. Moreover, the results of liquid penetration length during the wicking flow through a network of pores in series were validated with good agreement against the experimental results of unidirectional horizontal infiltration of graphite foam, and a modified Washburn equation.     

Role of droplet bridging on the stability of particle-containing immiscible polymer blends

The effect of micron-sized hydrophobic calcium carbonate particles on the stabilization of polydimethylsiloxane (PDMS)/polyisobutylene (PIB) immiscible model blends is investigated in this study. The analytical splitting of bulk and liquid–liquid interface contributions from the droplet bridging one is successfully performed due to the negligible contribution of hydrophobic microparticles to the bulk rheology of phases. The presence of particles at the fluid–fluid interface is supported by wetting parameter calculation and verified by optical microscopy observations. Moreover, direct visualizations shows that particles are able to form clusters of droplets by simultaneously adsorbing at two fluid–fluid interfaces and glue-dispersed droplets together, probably due to the patchy interactions induced by heterogeneous distribution of particles along the interface. Rheological studies show that the flow-induced coalescence is slowed down upon addition of particles and almost suppressed with the addition of 4 wt% particles. The linear viscoelastic response is modeled to estimate interfacial tension by considering the contribution of particle-induced droplet aggregation in addition to bulk and droplet deformation ones. From linear and nonlinear viscoelastic responses, the improved stability of filled polymer blends is attributed to the interfacial rheology and/or the bridged structure of droplets, even though the interfacial area is not fully covered by particles. Furthermore, Doi–Ohta scaling relations are investigated by employing stress growth response upon step-up of shear flow.     

含界面脱粘压电复合材料层合板的非线性动力稳定性分析

基于Reddy提出的板高阶剪切变形简化理论,研究了含界面脱粘损伤压电复合材料层合板非线性动力稳定性问题.首先,建立了分层模型,推导了考虑几何非线性、阻尼效应、纵向惯性力和力-电耦合效应的Mathieu方程,并且给出了该方程解的解析表达式.其次,通过典型算例讨论了界面脱粘损伤以及反馈控制力对该层合板动力不稳定区域、纵向、横向共振频率和最大"牵引"深度的影响.由典型算例讨论可知:随着层合板界面脱粘损伤的扩大,其动力稳定性能逐渐减弱,其中在损伤较小时,反馈控制力对智能结构几乎没有影响;而在损伤比较大的情况下,反馈控制力将能有效地减少动力不稳定区域重合面积.     

Static wetting on flexible substrates: a finite element formulation

In static wetting on an elastic substrate, force exerted by the liquid–vapour surface tension on a solid surface deforms the substrate, producing a capillary ridge along the contact line. This paper presents a finite element formulation for predicting elastic deformation, close to the static wetting line (with angle of contact=90o and σSV=σSL).The substrate deformation is modelled with the Mooney–Rivlin constitutive law for incompressible rubber‐like solids. At the contact line, a stress singularity is known to arise, due to the surface tension acting on a line of infinitesimal thickness. To relive the stress singularity, either (i) the surface tension is applied over a finite contact region (of macroscopic thickness), or (ii) the solid crease angle is fixed. These two options suggest that normal component of Neumann's triangle law of forces, for the three surface tensions, is not applicable for elastic substrates (as for rigid ones). The vertical displacement of the contact line is a strong function of liquid/vapour surface tension and shear modulus of the solid. Copyright 2004 John Wiley & Sons, Ltd.