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)     

Theoretical analysis of Rayleigh–Taylor instability on a spherical droplet in a gas stream

A linear analysis of the Rayleigh–Taylor (R–T) instability on a spherical viscous liquid droplet in a gas stream is presented. Different from the most previous studies in which the external acceleration is usually assumed to be radial, the present study considers a unidirectional acceleration acting on a spherical droplet with arbitrary initial disturbances and therefore can provide insights into the influence of R–T instability on the atomization of spherical droplets. A general recursion relation coupling different spherical modes is derived and two physically prevalent limiting cases are discussed. In the limiting case of inviscid droplet, the critical Bond numbers to excite the instability and the growth rates for a given Bond number are obtained by solving two eigenvalue problems. In the limiting case of large droplet acceleration, different spherical modes are asymptotically decoupled and an explicit dispersion relation is derived. For given Bond number and Ohnesorge numbers, the critical size of stable droplet, the most-unstable mode and its corresponding growth rate are determined theoretically.     

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)     

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 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)     

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.     

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)     

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.     

液滴对弹性梯度薄基变形的影响

液滴润湿现象在细胞的变形和软器件的设计和制作中具有潜在的指导意义.该文在考虑三相接触线处线张力的情况下,研究了液滴引起的梯度薄基变形问题.首先,利用积分变换法求解了基底变形的本构方程,给出了变形的法向位移表达式.其次,讨论了基底弹性模量非梯度、指数型梯度和幂型梯度变化时基底的变形情况.最后,给出了液滴大小、弹性模量、线张力及梯度指数变化时位移的变化情况.数值结果表明弹性模量逐渐减小和梯度指数逐渐增大时,湿润脊逐渐变高,变形也越大;线张力和特征深度越小,位移的峰值越高,变形也越大;液滴半径较小时,湿润脊的对称性会变得更好.     

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.     

A fully meshless method for ‘gas – evaporating droplet’ flow modelling

A meshless method for modelling two-phase flows with phase transition is described. The method is based on consideration of three systems: viscous-vortex blobs, thermal-blobs and droplets; and can be applied for numerical simulation of 2D non-isothermal flows of ‘gas-evaporating droplets’ in the framework of the one-way coupled two-fluid approach. The carrier phase is viscous incompressible gas. The dispersed phase is presented by a cloud of identical spherical droplets, and, due to evaporation, the radius and mass of droplets are time dependent. The carrier phase parameters are calculated using the viscous-vortex and thermal-blob method; the dispersed phase parameters are calculated using the Lagrangian approach. Two applications have been considered: (i) a standard benchmark – Lamb vortex; (ii) a cold spray injected into a hot quiescent gas. In the latter problem three cases corresponding to three droplet sizes were investigated. The smallest droplets (of the three cases considered) are more readily entrained by the carrier phase and form ring-like structures; the flow shows better mixing. Larger droplets evaporate less intensively. The medium sized droplets collect into two narrow bands stretched along the jet axis. The largest droplets form a two-phase jet, which remains close to the jet axis. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Octopus-shaped instabilities of evaporating drops

We report on instabilities during spreading of volatile liquids, with emphasis on the novel instability observed when isopropyl alcohol (IPA) is deposited on a monocrystaline Si wafer. This instability is characterized by emission of drops ahead of the expanding front, with each drop followed by smaller, satellite droplets, forming the structures which we nickname ‘octopi’ due to their appearance. A less volatile liquid, or a substrate of larger heat conductivity, suppress this instability. We have formulated a theoretical model that reproduces the main features of the experiment. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The trajectory and stability of a spiralling liquid jet: Viscous theory

We examine a spiralling slender viscous jet emerging from a rapidly rotating orifice, extending Wallwork et al. [I.M. Wallwork, S.P. Decent, A.C. King, R.M.S.M. Schulkes, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory, J. Fluid Mech. 459 (2002) 43–65] by incorporating viscosity. The effects of viscosity on the trajectory of the jet and its linear instability are determined using a mixture of computational and asymptotic methods, and verified using experiments. A non-monotonic relationship between break-up length and rotation rate is demonstrated with the trend varying with viscosity. The sizes of the droplets produced by this instability are determined by considering the most unstable wave mode. It is also found that there is a non-monotonic relationship between droplet size and viscosity. Satellite droplet formation is also considered by analysing very short wavelength modes. The effects of long wavelength modes are examined, and a wave which propagates down the trajectory of the jet is identified for the highly viscous case. A comparison between theoretical and experimental results is made, with favourable agreement. In particular, a quantitative comparison is made between droplet sizes predicted from the theory with experimental observations, with encouraging agreement obtained. Four different types of break-up are identified in our experiments. The experimentally observed break-up mechanisms are discussed in light of our theory.     

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)     

化学驱粘性指进微观渗流模拟研究

化学驱是提高原油采收率的重要方法,化学剂的加入使得地下流体粘度界面张力发生变化,粘性指进演变特征更为复杂.基于格子玻尔兹曼伪势多相流模型,综合考虑油水两相间相互作用、聚合物增粘与非牛顿特性、表面活性剂传质扩散作用及降低界面张力机理,建立了化学驱微观渗流格子玻尔兹曼模拟方法.基于数值模拟方法了研究二维单通道内粘性指进发展...     

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 Wetting of a Surface by a Very Viscous Fluid

The movement of a small viscous droplet on a flat surface is considered in self-spreading, in roundup, and as a result of an adhesion gradient. Edge conditions based on the dynamic contact angle are discussed within the context of lubrication theory. Tangential slip at the contact line is shown to be small compared to the normal velocity. The shapes of the droplet and its contact line are calculated, and nonlinear and inertial effects assessed.     

Modeling of aerosol spray characteristics for synthesis of sensor thin film from solution

A model is developed of aerosol spray for synthesis of sensor film from solution. The synthesis technique considered involves atomization of a solution of mixed salts in methanol, spraying of solution droplets, droplet deposition on a heated substrate, evaporation and chemical reaction to produce mixed oxides, and subsequent film growth. The precise control of oxide nanoparticle size distribution and inter-particle spacing in the film is crucial to achieving high sensitivity. These in turn largely depend on the droplet characteristics prior to impingement on the substrate. This paper focuses on the development of a model to describe the atomization and spray processes prior to the film growth. Specifically, a mathematical model is developed utilizing computational fluid dynamics solution of the equations governing the transport of atomized droplets from the nozzle to the substrate in order to predict droplet characteristics in flight. The predictions include spatial distribution of droplet size and concentration, and the effect on these characteristics of swirling inlet flow at the spray nozzle.     

乙醇液滴撞击高温壁面蒸发过程的模拟预测研究北大核心CSCD

采用CLSVOF方法,引入描述壁面润湿特性的动态接触角,建立了乙醇液滴撞击高温壁面的数值模型,对乙醇液滴撞击高温壁面后的沸腾蒸发过程展开了研究,并与实验数据进行了对比验证.研究表明:在相同液滴温度下,壁面温度越高,亲水性越强,乙醇液滴的撞击速度越快,液滴的沸腾时间越早,蒸发完成所用时间也越短.在此研究基础上,基于机器学习算法,建立了液滴蒸发预测模型,对乙醇液滴撞击高温壁面后蒸发剩余量随时间的变化进行了预测研究,并通过将不同机器学习算法的预测结果与模拟结果对比,选出最优预测模型.     

Heat flux effect on the droplet entrainment and deposition in annular flow dryout

This paper discusses the internal mass transfer process in annular flow dryout. The emphasis is put on the order of magnitude estimation of respective hydrodynamic and thermal mechanisms and the analysis of the heat flux effect on droplet entrainment and deposition. A simple interfacial turbulence model is developed to characterize the turbulence intensity suppression due to interface tension. The heat flux effect on droplet entrainment depends on the competition between the shear force decrease due to vapor effusion and the bubble emission: in low flow condition, the bubble emission outweighs the shear force decrease, thus the net effect is to increase the droplet entrainment; in high flow condition, the situation may reverse. The heat flux impact on droplet deposition is significant only for very fine droplets (less than 1 μm) because of the coupled effect of interface turbulence damping and the radial vapor effusion due to evaporation, but for droplets of medium and large sizes the heat flux effect is negligible. The analysis is then used to develop constitutive equations for droplet entrainment and deposition rates to take into account the interaction between thermal and hydrodynamic mechanisms, which gives improved CHF prediction for limiting quality regime (LQR) CHF experimental data.