Linear stability of a thin non-isothermal droplet spreading on a rotating disk

Previous works show that the linear stability of the contact line of an isothermal spreading droplet depends on the base state curvature of the free interface at the position of the unperturbed contact line [1]. Here the linear stability of a thin liquid droplet on a rotating disk is investigated, where the disk has a certain radial temperature profile and the ambient passive gas a constant temperature. It is shown that two different Marangoni effects influence the spreading beside centrifugal, gravitational, and capillary effects. The stability analysis prevails, that temperature gradients in the disk amplify some and damp other modes, while cooling the entire disk (or heating the gas) damps the growth rate of all modes and therefore seems to be an appropriate way to suppress the instability. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone.     

Local well-posedness for a quasi-stationary droplet model

The moving boundary problem for the contact line evolution of a droplet is studied. Local existence and uniqueness of classical solutions is established.     

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)     

Internal Flow Analysis for Slow Moving Small Droplets in Contact with Hydrophobic Surfaces

Internal fluid flow behavior for slow moving small droplets in contact with hydrophobic surfaces is analyzed. The shape of the droplet is first computed using the Young-Laplace equation. For this purpose a Finite Element (FE) model [1], in which contact constraints are enforced through Penalty and Augmented Lagrange Multiplier methods, is used. The flow field within the droplet is then analyzed using the Stokes flow model, considering a de-coupled approach. Similar to the membrane deformation model, the formulation for the flow analysis is also expressed in the framework of FE analysis. Both, stabilized (Pressure Stabilizing/Petrov-Galerkin PSPG) and Galerkin FE formulations are considered. The motion of the fluid inside the droplet is governed by the slip condition enforced on the membrane of the droplet. Numerical examples for droplets rolling steadily are presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Contact angle effects on microdroplet deformation using CFD

In this paper we use computational fluid dynamics (CFD) to study the effect of contact angle on droplet shape as it moves through a contraction. A new non-dimensional number is proposed in order to predict situations where the deformed droplet will form a slug in the contraction and thus have the opportunity to interact with the channel wall. It is proposed that droplet flow into a contraction is a useful method to ensure that a droplet will wet a channel surface without a trapped lubrication film, and thus help ensure that a slug will remain attached to the wall downstream of the contraction. We demonstrate that when a droplet is larger than a contraction, capillary and Reynolds numbers, and fluid properties may not be sufficient to fully describe the droplet dynamics through a contraction. We show that, with everything else constant, droplet shape and breakup can be controlled simply by changing the wetting properties of the channel wall. CFD simulations with contact angles ranging from 30° to 150° show that lower contact angles can induce droplet breakup while higher contact angles can form slugs with contact angle dependent shape.     

Well‐posedness for the Navier Slip Thin‐Film equation in the case of partial wetting

Consider a viscous liquid droplet spreading on a surface. The classical slip condition at the liquid‐solid interface is the no‐slip condition. However, this condition yields infinite dissipation rate when the contact line moves (“no‐slip paradox”). For this reason other slip conditions such as the Navier slip condition have been proposed. We prove well‐posedness for a reduced 1‐D fluid model related to Navier slip. It turns out that the profile of the droplet cannot be described by a smooth function (not even for an initially smooth profile). However, existence and uniqueness can be proved in larger classes of spaces that allow for certain classes of singular expansions at the moving contact point. © 2011 Wiley Periodicals, Inc.     

Temperature effects in thin droplets

Thin droplets spreading on a solid substrate are investigated, with a special focus on temperature effects. The aim is to manipulate the fingering instability which may occur in the spreading in a spin coating process. The analysis bases on lubrication approximation, valid for flat thin droplets, which usually is the case. The dynamic of the wetting is implemented by using a generalized law of Tanner, coupling the contact angle (CA) of the droplet at the (apparent) contact line (CL) with its speed. A one-way coupling is used to investigate, whether viscous heating has to be taken into account. It can be derived that its role is negligible in the spreading process of a thin droplet, even for a relatively large viscous influence (large capillary number). Analyzing the results of a linear stability analysis of the fingering instability and taking Marangoni-stresses (MS) into account reveals, that the instability may be suppressed by cooling the ambient gas or heating the substrate during the spreading. Unfortunately an comparison with experiments for spreading droplets in a heated gas shows deviations for larger spreading radii. The influence of temperature on density is investigated and on the way a criteria, from which it may be obtained whether a simple Boussinesq-approximation (BA) is appropriate or not. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lattice Boltzmann Study on the Motion of Dual Droplets in Microchannels With Contact Angle Hysteresis北大核心CSCD

接触角滞后表现为流体在非理想固体表面上运动时前进接触角和后退接触角不同,是两相流体在润湿表面上流动的重要现象.该文采用改进的伪势格子Boltzmann(LB)多组分模型,并与几何润湿边界条件相结合,研究了两个液滴在具有接触角滞后性微通道表面上的运动行为,主要研究了通道内特征数、通道表面性质以及液滴初始参数的影响.研究结果表明：毛细数的增大有助于液滴的移动,然而并不利于液滴的排出,且毛细数的增加对上游液滴的影响大于其对下游液滴的影响;另一方面,接触角滞后性窗口越大,液滴运动和形变更迟缓,但形变程度更明显,两液滴更早地发生合并,但更晚地排出管道;液滴间距的增加使液滴的运动行为在不同阶段表现为不同的模式,但都导致通道中残留小液滴,使得液滴排出通道的时间增加.研究结果还表明：上游液滴和下游液滴的相对尺寸差距越大,越不利于液滴排出管道.     

Three-dimensional Contact of a Rigid Roller Traversing a Viscoelastic Half Space

In this paper, the authors consider a nontrivial three-dimensionalviscoelastic contact problem which has some physical significance.(Although the subject of the analysis is an elliptical roller,it is only a small step onward to the consideration of a crownedcylindrical roller.) Generally, it is the intractability of the mathematics whichhinders analytic solution of true three-dimensional problemsin (visco)elasticity. The traditional method of surmountingthis difficulty is to reduce the problem to two dimensions,either by choosing a suitable geometry, or by using an appropriateco-ordinate system. The elastic line integral theory representsanother approach; certain approximations are used to simplifythe governing equations, thus allowing the solution of the problem. After the development of a viscoelastic analogue of the Boussinesqequation valid for the solution of quasi-steady state viscoelasticcontact problems, analysis proceeds making use of near fieldand extended line integral approximations. Results are generatedshowing the velocity dependence of several physical parameters,including the size and shape of the contact zone. One additionalpoint of interest is uncovered, namely the presence of a pressurepeak near the leading edge of the contact zone.     

On the Deformation of a Viscous Droplet Caused by Variable Surface Tension

The surface tension of a small viscous droplet is made variable by the deposition of a surface active material. The resultant shear forces produce a deformation of and a fluid motion within the droplet. The process, including the nonlinear convection or spread of surfactant, is examined theoretically. Some experimental results are reported, and application to cell division is discussed.     

The transmission of a flexural-gravitational wave through a rigid end-stop in a floating plate

An elastic plate is located on the surface of a liquid, in continuous contact with it and rigidly clamped in a support along a certain straight line. The orthogonal incidence of a small amplitude flexural-gravitational wave on the support is considered. Exact expressions are obtained for the wave field in the fluid and the flexural field in the plate. The transmission coefficient of the incident flexural-gravitational wave through the support and its reflection coefficient from it are determined. The forces which arise in the support are found. The investigation is carried out for liquids of finite and infinite depth. The effect of the depth of the liquid on the wave processes is indicated. The liquid is assumed to be inviscid and its friction on the bottom and the lower surface of the plate in the neighbourhood of the support is therefore ignored.     

线张力作用下微纳米尺度液滴的非线性粘附

三相接触线上的线张力对微纳米尺度液滴的粘附行为有至关重要的影响．首次在由表面张力、线张力和液滴尺度组成的全参数空间中研究了液滴粘附的非线性行为．研究表明:液滴粘附解空间可以被归纳为4个特征区;在每个特征区内,液滴的粘附行为是相同的;在具有高度非线性的特征区中,给定材料体系,存在着多重粘附态;液滴粘附解空间中存在着两个公共不动点．这些新结果,与数值计算和实验观测相符合．     

Droplet spreading: Intermediate scaling law by PDE methods

We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity. The standard lubrication approximation leads to an evolution equation for the film height h that is ill‐posed when the spreading is limited by the no‐slip boundary condition at the liquid‐solid interface due to a singularity at the moving contact line. The most common relaxation of the no‐slip boundary condition removes this singularity but introduces a new physical length scale: the slippage length b. It is believed that this microscopic‐length scale only enters logarithmically in the effective (that is, macroscopic) spreading behavior. In this paper, we rigorously show that the naively expected spreading rate is indeed only altered by a logarithmic term involving b. More precisely, we prove a scaling law for the diameter of the apparent (that is, macroscopic) support of the droplet in time. This is an intermediate scaling law: It takes an initial layer to “forget” the initial droplet shape, whereas after a long time, the droplet is so thin that its spreading is governed by the physics on the scale b. Our proof works by deriving suitable estimates for physically relevant integral quantities: the free energy, the length of the apparent support, and their respective rates of change. As opposed to matched asymptotic methods, this PDE approach closely mimics a simple heuristic argument based on the gradient flow structure. © 2002 John Wiley & Sons, Inc.     

Computational Modeling of Liquid Droplets Moving on Rough Surfaces

We present a de-coupled approach for computational modeling of liquid droplets moving on rough substrate surfaces. The computational model comprises solving the membrane deformation problem and the fluid flow problem in a segregated manner. The droplet shape is first computed by solving the Young-Laplace equation where contact constraints, due to the droplet-substrate contact, are applied through the penalty method [1]. The resulting configuration constitutes the domain for the fluid flow problem, where the bulk fluid behavior is modeled by the unsteady Stokes' flow model expressed in Arbitrary Lagrangian-Eulerian (ALE) framework. The entire analysis is performed in the framework of Finite Element Method (FEM). Application of the approach to the case of a droplet moving on a rough surface is presented as an example. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stiffness and damping models for the oil film in line contact elastohydrodynamic lubrication and applications in the gear drive

Innovative stiffness and damping models for oil films are developed to account for the impacts in both normal and tangential directions. Given that these models are applied to a gear drive in line contact elastohydrodynamic lubrication (EHL), the combined stiffness is derived from the stiffness of both the oil film and gear tooth while the combined damping is established from the damping of these parts. The effects of three fundamental parameters (contact force, rotation speed, and tooth numbers) of the gear drive in line contact EHL on the combined stiffness and damping are then investigated. The results reveal that the small normal and tangential stiffness of the lubricant can alleviate meshing impact and shear vibration, while the impact and friction heat can be reduced by using an oil film with either a large normal damping or small tangential damping. Given that its amplitude and fluctuation are closely related to shear rate, effective viscosity, entrainment velocity, and curvature radii, the improved combined stiffness and damping can be obtained by rationally matching the geometric and operating parameters.     

On the Crenation of a Compound Liquid Droplet

The dynamic response of a small, very viscous liquid droplet composed of a core fluid surrounded by a thin fluid shell is examined as additional fluid is deposited into this incompressible shell. At early times, the shell incorporates the extra mass by ruffling its external surface, and a number of crenations form. These protuberances decrease in size and number over a longer time period, and eventually the droplet again becomes spherical, with an increased radius. This sequence of events and its dependence on the rheological properties of the fluids are studied. These effects compare well qualitatively with those obtained using a surface fluid rather than a finite thickness shell of fluid. The possible implications of this fluid model for the surface ruffling effects observed in cell biology are discussed.     

The contact problem for a piecewise-homogeneous plane with a semi-infinite inclusion

A piecewise-homogenous elastic plate, reinforced with a semi-infinite inclusion, which intersects the interface at a right angle and is loaded with shear forces is considered. The contact stresses along the contact line are determined and the behaviour of the contact stresses in the neighbourhood of singular points is established. Using methods of the theory of analytical functions and integral transformations the problem is reduced to a system of singular integro-differential equations on the semi-axis. The solution is presented in explicit form.     

锥形微通道内液滴自输运特性及力学驱动机制研究

针对液体在微通道内的自输运特性,采用数值仿真与能量解析相结合的方法研究了液滴在锥形微通道内的自输运特性及力学驱动机制,得到微通道的锥形角、液滴与微通道内壁的接触角及微通道的润湿性对液滴自输运特性的影响关系.分析表明,微通道的锥形角、液滴与微通道内壁的接触角均能影响液滴的自输运方向及驱动力大小.对于亲水性微通道,微通道的锥形角、液滴与微通道内壁的接触角其作用效果呈现整体形态;对于疏水性微通道,微通道的锥形角、液滴与微通道内壁的接触角其作用效果呈现局域形态.这可为研究液体在微通道内的自输运机理及界面内液体细观流动机制奠定理论基础.