Stability of a thin liquid layer,spreading on a quasi–rotating disk

In many industrial processes solids are coated to obtain specific surface properties, as e.g. corrosion resistance, mechanical (wear) resistance, optical, or electrical properties. Even today many coating processes are not fully understood and the choice of parameters is largely based on experience. Hence, a prediction of the complete hydrodynamic process and the appearance of instabilities in its dependency on the parameters appears highly desirable. This would serve to optimize the quality of the coating. A common coating technique is the so-called spin coating. The coating agent is dissolved or suspended in a liquid, brought onto the solid, spread by rotation, and the carrier liquid is finally removed by evaporation or by chemical reactions. In this article an evolution equation is derived from lubrication theory, valid for thin liquid layers. The model involves a dynamic contact angle, centrifugal, capillary, and gravitational forces. The evolution equation can be solved analytically, provided the capillary number is small. Then a coupled linear stability analysis of the contact line and the free interface is performed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experiments on the spreading of silicone oil droplets on horizontal rotating plates

The spreading of liquid films is involved in many coating processes, e.g. in the spin-coating process. To achieve a high quality of the coating, the spreading liquid layer should be thin and homogeneous. Instabilities at the wetting front may lead to an inhomogeneous thickness of the coating layer and to uncoated areas. In this article the spreading of perfectly-wetting silicone oil droplets with viscosity of 100 mPa s on rotating glass plates is discussed. A Schlieren system is set up to observe the wetted area and a traversed chromatic confocal distance sensor is used to measure the contour of the droplet. The experimental data are presented and compared to an analytical model which is derived from lubrication theory and valid for thin liquid layers. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupled wetting meniscus model for the mechanism of spontaneous capillary action

Capillarity plays a significant role in many natural and artificial processes, but the mechanism responsible for its dynamics is not completely understood. In this study, we consider capillary flow characteristics and propose a coupled wetting meniscus model for the mechanism of spontaneous capillary action. In this model, capillary action is considered as the dynamic coupling of two interfacial forces, i.e., the wall wetting force at the contact line and the meniscus restoring force on the free interface. The wetting force promotes the motion of the contact line directed toward an equilibrium contact angle, whereas the meniscus restoring force promotes a reduction in the interface curvature, which is more consistent with a 90° contact angle. The competing interaction between these two forces is coupled together via the evolution of the interface shape. The model is then incorporated into a finite volume method for a two-fluid flow with an interface. Capillary flow experiments were performed, including vertical and horizontal flows. Phenomena analysis and data comparisons were conducted to verify the proposed model. According to the results of our study, the model can explain the capillary flow process well and it can be also used to accurately guide capillary flow calculations.     

充液弹性毛细管低温相变的力学分析

充液弹性毛细管广泛存在于生物体（如毛细血管、植物导管等）和工程领域（如微流控冰阀门、制冷系统热管、MEMS微通道谐振器等）.低温工作环境中,充液弹性毛细管内部的液柱会发生相变并引发冻胀效应,从而导致管壁的变形、损伤乃至断裂.该文建立并求解了考虑温度梯度、界面张力及液体冻胀作用的弹性毛细管平衡方程,分析了液柱低温相变过程中毛细管壁的径向和环向应力,发现管壁应力分布受热毛细弹性数和冻毛细弹性数的影响,且影响大小跟壁厚相关.该研究不仅有助于理解生物体内充液弹性毛细管冻胀失效机制,还可为MEMS微流控芯片的抗冻胀失效设计提供理论指导.     

高温作用下混凝土热-水-力耦合损伤分析模型

以混合物理论为基础建立了高温作用下混凝土的热-水-力耦合损伤分析模型．将混凝土视为由固体骨架、液态水、水蒸气、干燥气体和溶解气体共5种组分构成的混合物,模型的宏观平衡方程包括各组分的质量守恒方程、整体的能量守恒方程及动量守恒方程,模型所需的状态方程及本构关系全部给出,最后给出基于4个主要参数（固体骨架位移、气压力、毛细压力和温度）的控制方程．模型考虑了混凝土在高温作用下,水分的蒸发与冷凝、胶结材料的水化及脱水、溶解气的溶解与挥发等相变过程;从材料变形破坏过程中能量耗散特征入手,基于Lemaitre应变等价性假说和能量守恒原理得到力学损伤演化方程,并考虑了高温引起的热损伤对材料力学性能及力学损伤演化规律的影响,建立了热-力耦合损伤本构模型．     

Numerical simulation of non-viscous liquid pinch off using a coupled level set boundary integral method

The pinch off of an inviscid fluid column is described using a potential flow model with capillary forces. The interface velocity is obtained via a Galerkin boundary integral method for the 3D axisymmetric Laplace equation, whereas the interface location and the velocity potential on the free boundary are both approximated using level set techniques on a fixed domain. The algorithm is validated computing the Raleigh-Taylor instability for liquid columns which provides an analytical solution for short times. The simulations show the time evolution of the fluid tube and the algorithm is capable of handling pinch-off and after pinch-off events. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

原子力显微镜中液桥生成机理探讨

液桥是引起大气环境下原子力显微镜（AFM）图像失真的重要原因,同时也是大气环境下黏着力的主要成分.研究液桥对于成像机理和样品特性的理解有重要意义.提出了AFM液桥生成的物理机理,由3个不同的物理过程组成,即:挤出过程、毛细凝聚和液膜流动.这3种过程的特征平衡时间对认识液桥生成的动力学过程非常重要,挤出过程的平衡时间与接触方式有关,毛细凝聚的平衡时间在微秒量级,而液膜流动的平衡时间随液膜黏度不同变化较大.在此基础上分析了这3种形成机理在AFM不同的操作模式下对液桥体积、毛细力和耗散能的贡献.     

A phase field approach to solidification and solute separation in water solutions

We propose a phase field model for the solid–liquid phase transition in a water-salt (sodium chloride) solution in the absence of macroscopic motion, under possibly non-isothermal conditions. A thermodynamic approach based on a free energy functional is assumed. The model consists of three evolution equations: a time-dependent Ginzburg–Landau equation for the solid–liquid phase change, a diffusion equation of the Cahn–Hilliard kind for the solute dynamics and the heat equation for the temperature change. The proposed system is aimed to contribute to the modelling of the brine channels formation in the ice of the polar seas.     

Film flow over strongly undulated bottom profiles at low Reynolds numbers – Analytical results

Film flows occur in many technical processes (e.g. coating techniques) as well as in nature. In this paper we establish a new analytical method in order to calculate both the velocity field and the surface shape of a two–dimensional gravity driven film flow of a Newtonian fluid down a periodically varying bottom topology. We discuss the special case of a sinusoidal bottom shape.     

Thin liquid droplet on top of a rotating non-isothermal liquid layer

Wafers are usually coated by using spin-coating, where centrifugal forces are used to spread a droplet on the rotating wafer. This flow is unstable to the fingering instability, where several segments of the wetting front spread faster than the average, resulting in several fingers. The liquid flows via the existing fingers while the area in between does not get coated. A precise experimental investigation is problematic, as the droplet has to be placed quite exactly in the center of the rotation. Replacing the wafer by a second liquid should lead to a parabolic-shaped free interface and the droplet should center itself due to gravity. Here, we derive a model for the free interfaces of a thin droplet on top of a rotating liquid by taking gravity, centrifugal forces, friction, (thermo-)capillary and line forces into account. Additionally, this setup is the simplest example of multiple coating, where several free interfaces and contact lines influence each other. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and analysis of a two-phase thin film model with insoluble surfactant

In this paper we consider a two-phase thin film consisting of two immiscible viscous fluids endowed with a layer of insoluble surfactant on the surface of the upper fluid. The governing equations for the two film heights and the surfactant concentration are derived using a lubrication approximation. Taking gravitational forces into account but neglecting capillary effects, the resulting system of evolution equations is parabolic, strongly coupled, of second order and degenerated in the equations for the two film heights. Incorporating on the contrary capillary forces and neglecting the effects of gravitation, the system of evolution equations is parabolic, degenerated and of fourth-order for the film heights, strongly coupled to a second-order transport equation for the surfactant concentration. Local well-posedness and asymptotic stability are shown for both systems.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

On the Solutions of a $$$$-Dimensional Model for Epitaxial Growth with Axial Symmetry

In this paper, we study the evolution equation derived by Xu and Xiang (SIAM J Appl Math 69(5):1393–1414, 2009) to describe heteroepitaxial growth in \(2+1\) dimensions with elastic forces on vicinal surfaces is in the radial case and uniform mobility. This equation is strongly nonlinear and contains two elliptic integrals and defined via Cauchy principal value. We will first derive a formally equivalent parabolic evolution equation (i.e., full equivalence when sufficient regularity is assumed), and the main aim is to prove existence, uniqueness and regularity of strong solutions. We will extensively use techniques from the theory of evolution equations governed by maximal monotone operators in Banach spaces.     

On the predictive capabilities of a multiphase SPH model for hydrodynamic spreading dynamics

We assess the predictive capabilities of a multiphase Smoothed-Particle Hydrodynamics (SPH) model to simulate two-phase flow processes that are decisively governed by the dynamics of hydrodynamically moving contact lines. The spreading of a liquid droplet on a completely wettable, smooth and flat substrate serves as prototype validation problem. Our simulation results reproduce the main features of hydrodynamic spreading in the quasi-static limit, i. e. when the balance of viscous and capillary forces near the contact line governs the dynamics. The exponential spreading rate is consistent with Tanner's law. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydromagnetic instability of two coaxial streaming cylinders with double perturbed interfaces

The magnetohydrodynamic (MHD) stability of a double interface perturbed streaming liquid cylinder coaxial with a streaming fluid mantle acting upon capillary, inertial, pressure gradient and electromagnetic forces has been developed. The problem is formulated, solved and the stability criterion of the model is estabilished. The latter is discussed analytically and the results are confirmed numerically and interpreted physically. Some reported works are recovered as limiting cases. The capillary force is stabilizing or not according to restrictions. The magnetic field has a strong stabilizing influence. The radii (liquid–fluid) ratio plays an important role in increasing the MHD stabilizing domains. The density of liquid–fluid ratio has a little stabilizing effect. The streaming has a destabilizing influence for all kinds of (non-) axisymmetric perturbation modes. However, if the magnetic field strength is so strong such that the Alfvén wave velocity is greater than the streaming velocity, then the destabilizing character due to capillary force or/and streaming is completely suppressed and stability sets in. In the absence of the magnetic field and we neglect the fluid inertial force, the present results are in good agreement with the experimental results of (Kendall J.M. Phys Fluids 1986;29:2086).     

Radial spreading and stability of a thin rotating liquid droplet

Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and homogeneous fluid layers. In the present investigations, we treat droplets, spreading on rotating solid substrates. Micro-scale effects appear, firstly, at the wetting front, where the film height tends to zero. Secondly, micro-scale effects may appear at other locations, where the free liquid/gas-interface approaches the solid substrate, as e.g. at film rupture. For such situations, molecular effects need to be considered, e.g. in form of the disjoining pressure (DJP), to get physically-correct solutions. Otherwise, the spreading can be modeled within the frame of continuum mechanics, augmented by the (empirical) law of Tanner to capture the contact-line dynamics. We present, on the one hand, an overview of several interesting issues, as (i) spreading with and without considering the DJP, (ii) spreading after central rupture, including hysteresis effects, and (iii) non-isothermal spreading, including temperature-dependent surface tension (Marangoni effect) and temperature-dependent density (Rayleigh-Bénard effect). On the other hand, we present results for the instability of the contact line, for which the contact line gets corrugated (under isothermal conditions). This instability goes along with a transition from (rotationally-symmetric) two-dimensional to three-dimensional behavior. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the construction of a set of stochastic differential equations on the basis of a given integral manifold independent of velocities

We construct the Lagrange equation, Hamilton equation, and Birkhoff equation on the basis of given properties of motion under random perturbations. It is assumed that random perturbation forces belong to the class of Wiener processes and that given properties of motion are independent of velocities. The obtained results are illustrated by an example of motion of an Earth satellite under the action of gravitational and aerodynamic forces.     

Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

Stability and Asymptotic Analysis of a Fluid-Particle Interaction Model

We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov–Fokker–Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models.