Simulation of mass transfer at free liquid/liquid interfaces

A numerical scheme for the simulation of mass transfer processes at free liquid/liquid interfaces using the interface tracking method is presented. Due to comparable diffusion coefficients in liquid/liquid systems, the mass transfer resistance in both phases is relevant for the entire transient mass transfer process. Exemplary, the extraction process from a free rising spherical droplet of constant shape is used. The presented approach can be used in general for any multiphase steady-state mass transfer system. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Surface-tension-driven flow in a slender cone

After a droplet has broken away from a slender thread or jetof liquid, the tip of the thread or jet recoils rapidly. Atthe moment of break-off, the tip of the thread/jet is observedto have the shape of a cone close to the bifurcation point.In this paper, we study the evolution of an ideal fluid whichis initially conical, where the only force acting on the fluidis due to surface tension. We find an asymptotic solution tothe problem in terms of the aspect ratio of the cone which isassumed to be small. Using a similarity transformation, whichis valid for small times after the bifurcation, we identifya rapidly oscillating non-linear wave which propagates awayfrom the tip, as observed in experiments.     

The numerical analysis of the trapezoidal thread rolling process

The thread rolling is difficult technological process. Improve quality and contemporary reduce manufacture cost of the trapezoidal thread requires acquaintances of physical phenomena observed in the contact zone between rolls and deform work-pieces. Therefore, in this paper the physical and mathematical models of deformations (displacements and strains) and stress in the cold process of trapezoidal thread rolling, were developed. The process is considered as a geometrical and physical non-linear, initial as well as boundary value problem. The phenomena on a typical incremental step were described using a step-by-step incremental procedure, with an updated Lagrangian formulation. The state of strains was described by Green-Lagrange's tensor, while the state of stress by the second symmetrical Pioli-Kirchhoff's tensor. The object was treated as an elastic (in the reversible zone) and visco-plastic body (in non-reversible zone) with mixed hardening. The variational equation of motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. The application developed for in the ANSYS programme, which provides a complex time analysis for displacement, strains and stresses occurring in the object. The recommendations concern modeling the trapezoidal thread rolling process, where reduce degrees of freedom in numerical model is very important and provide convergence calculated results for maximum stress and strain values in the thread surface layer, were elaborated. The influence a various process conditions on the states deformation and stress for examples calculations, were presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Capillary breakup of liquid threads: a singularity-free solution

The process of capillary breakup of a thread of Newtonian liquidis considered theoretically in the simplest case where the threadis surrounded by an inviscid, dynamically passive gas. The goalis to remove the singularities inherent in the known solutionsto the problem obtained in the framework of the standard modeland explain some puzzling qualitative features of the processobserved in experiments. The analysis is based on the idea that,since the known solutions indicate that the rate at which freshfree-surface area is created tends to infinity as breakup isapproached, one has that the surface tension, whose relaxationto equilibrium is always associated with a small but finiterelaxation time, is bound to deviate from its equilibrium valuein the process of breakup. This gives rise to a surface-tensiongradient which starts to pull the liquid thread apart (the flow-inducedMarangoni effect), whilst the role of the capillary pressure-drivensqueezing of the liquid out of the neck diminishes as the surfacetension in the minimal cross-section decreases. An earlier developedtheory incorporating the interface formation process is appliedwithout any ad hoc alterations and analysed in the frameworkof the slender-jet approximation. The resulting solution issingularity-free and allows one to describe some previouslyunexplained features of experiment by Kowalewski (1996, FluidDyn. Res., 17, 121).     

The Numerical Analysis of the External Round Thread Rolling

The papers [1] describe modeling of the contact problem in the external thread rolling process. This paper shown an application of obtained model for the thread with round outline. The mathematical model of process, were presented. The variational and finite element method, were used. The algorithm of numerical analysis in ANSYS system, were elaborated. Numerical computions of displacements, strains and stresses have been conducted without the necessity to introduce boundary condition in the contact zone – a proper definition of the contact zone by single surface auto 2D. Exemplary results of numerical analysis for various condition of the process realization in the discretized model has been presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Microstructural changes and material parameter development

At previous GAMM-meetings the first two authors reported on the spinodal decomposition and coarsening in lead-containing as well as lead-free solders. They derived an extended diffusion equation for the corresponding computer modelling on the basis of a multi-component theory of mixtures. Consequently, they performed numerical studies that are capable of predicting quantitatively the development of SnPb and AgCu microstructures. Knowing this development is quite important from a technological point-of-view in order to quantify the current joining capability of the solder material (“Resteigenschaften”). On the basis of these results this paper now attempts to predict the corresponding change of the materials properties, in particular of the elastic constants. This is done by means of a homogenization technique developed by the last two authors. In this paper we will briefly summarize the results related to the extended diffusion equation, present numerical simulations of the coarsening process, outline the theory underlying the homogenization and, finally, show how the material properties behave over time. Wherever possible, alternative analytical solutions for the homogenized constants are also presented (e.g., by laminate theory). (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of van der Waals force with smoothed particle hydrodynamics: Application to the rupture of thin liquid films

The rupture of thin liquid films driven by the van der Waals force is of significance in many engineering processes, and most previous studies have relied on the lubrication approximation. In this paper, we develop a smoothed particle hydrodynamics (SPH) representation for the van der Waals force and simulate the rupture of thin liquid films without resort to lubrication theory. The van der Waals force in SPH is only imposed on one layer, i.e., the outermost layer of fluid particles, where a weighting function is deployed to evaluate the contributions of particles on or near the interface. However, to obtain an accurate hydrostatic pressure in reaction to the van der Waals force, a smaller smoothing length is used for the calculation of the weighting function than that used for SPH discretizations of the bulk fluid. The same surface particles are also used to model the surface tension. To deal with the rupture of a thin liquid film with a very small aspect ratio ε (ε = thickness/length), a coordinate transformation is introduced to shrink the length of the liquid film to achieve accurate numerical resolution with a manageable number of particles. As verifications of our physical model and numerical algorithm, we simulate the hydrostatic pressure in a stationary film and the relaxation of an initially square droplet and compare the SPH results with the analytical solutions. The method is then applied to simulate the rupture of thin liquid films with moderate and small aspect ratios (ε = 0.5 and 0.005). The convergence of the method is verified by refining particle spacing to four different levels. The effect of the capillary number on the rupture process is analyzed.     

Numerical Investigation of Dense Turbulent Sprays with an Eulerian Approach and DQMOM

The spray characteristics of the injected liquid fuel predominantly influence the combustion and emissions in IC-engines and gas turbines. This is predetermined by a dense spray region in which droplet-droplet interaction processes play a significant role. In order to accurately describe and control the dense spray behavior in modern engines, an appropriate numerical modeling tool is needed. This contribution aims at including droplet-droplet interactions into an Eulerian approach coupled to the Direct Quadrature Method of Moments (DQMOM) in order to describe evaporating droplet polydispersity and analyze dense turbulent sprays phenomena. Among the advantages of the Eulerian approach are a lower computational cost through optimal parallel computing and a straightforward liquid-gas phase coupling. To assess the designed tool, numerical results are compared to Phase Doppler Anemometry (PDA) measurements of a hollow-cone water spray. The experiment provides comprehensive validation data that include gas velocities, droplet size distribution, droplet mass fluxes and droplet velocities. Turbulence is captured by two different k-ε based models. Preliminary results show that the designed tool is able to capture the process under study in a satisfactory way. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and numerical analysis of the thread rolling process

The paper presents the physical and mathematical models of deformations (displacements and strains) and the stress in the cold process of the thread rolling. The process is considered as a geometrical and physical nonlinear, initial as well as a boundary value problem. The phenomena of a typical incremental step were described using a step-by-step incremental procedure, in the updated Lagrangian formulation. The state of strains was described by Green-Lagrange's tensor, while the state of stress was described by the second symmetrical Pioli-Kirchhoff's tensor. The object was treated as an elastic (in the reversible zone) and visco-plastic body (in the non-reversible zone) with mixed hardening. The variational equation of the motion in three dimensions for this case was proposed. Then, the finite elements methods (FEM) and dynamic explicit method (DEM) were used to obtain the solution. In a numerical analysis, boundary condition for a displacement increment, was determined in the model investigation. The results of a numerical analysis were compared and verified in an experimental investigation. Examples of calculations of the influence of a friction coefficient on the state of the deformation and stress, and an example application for this method of thread rolling were presented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

An analytical solution for the unidirectional solidification problem

The extraction of heat from a molten casting is resisted by an imperfect thermal contact at the mold-casting interface. The nature of the contact varies throughout the casting process and has the effect of increasing the thermal resistance at the interface. This can be modelled by incorporating a gaseous gap at the mold-casting interface that grows with increasing time.

This paper is concerned with an analytical solution of the unidirectional solidification problem, which incorporates movement of the casting at the interface. The derivation of the analytical solution requires the simultaneous solution of the transient heat equations, for the mold, gaseous gap, and solid and liquid parts of the melt. The analytical solution is extended so that contamination layers on the mold and casting can be incorporated as well as an initial gap. This is achieved by introducing virtual layers of mold, gas, and casting. Using the extended solution, the effects of interfacial resistance, air conductivity, and gap variation on solidification rates are examined.     

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.     

Numerical Simulation and analysis of molten flow in the shot sleeve of die casting process

This work presents a numerical study of the flow of molten metal in the shot sleeve of horizontal high-pressure die casting machine during the injection process. The analysis of the wave of the molten metal created by plunger motion can be of help in reducing porosity in manufactured parts caused by air entrapment. The used numerical model, which considers the problem as two-dimensional, is based on the conservation equations of mass and momentum. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于格子Boltzmann方法的液滴撞击具有不同润湿性孔板的研究

基于格子Boltzmann方法,对液滴撞击不同湿润性节流孔板表面进行了数值模拟.主要研究了在液滴撞击过程中,Weber数(We)、孔板表面湿润性和孔板尺寸对液滴通过孔板时不同状态的影响.数值模拟结果表明:孔板为亲水特性时,在较低We下,液滴不会与孔板表面脱离,而是附着在孔板下表面,并且在毛细作用下液滴会在孔道中上升一段...     

A composite Level Set and Extended-Domain-Eigenfunction Method for simulating 2D Stokes flow involving a free surface

In this paper, the Extended-Domain-Eigenfunction-Method (EDEM) is combined with the Level Set Method in a composite numerical scheme for simulating a moving boundary problem. The liquid velocity is obtained by formulating the problem in terms of the EDEM methodology and solved using a least square approach. The propagation of the free surface is effected by a narrow band Level Set Method. The two methods both pass information to each other via a bridging process, which allows the position of the interface to be updated. The numerical scheme is applied to a series of problems involving a gas bubble submerged in a viscous liquid moving subject to both an externally generated flow and the influence of surface tension.     

A Two-Dimensional Computational Droplet Contact Model

The interaction between capillary fluid films and micro-structural rough surfaces is one of the main challenges in studying self-cleaning mechanisms. The surface behavior of the deformable fluid film is governed by the Young-Laplace equation, which is highly non-linear. Therefore, a numerical solution is introduced using the finite element method, based on a continuum mechanical formulation. Surface and line contact at the fluid-structure interface are modeled by enforcing a contact constraint, and a contact angle, respectively. The numerical solution is validated against the analytical solution of a test case. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Numerical study of the primary breakup of a plane liquid sheet with co-flowing air streams

The objective of this study is to model the primary breakup of a plane liquid sheet emerging from an air-blast nozzle. In the present work the interface compression scheme proposed by OpenCFD Ltd. [1] has been used to capture the interface between the liquid and gas. A One-equation subgrid scale (sgs) turbulent energy transport model attributed to Yoshizawa [2] is used for modeling the effects of turbulence. The set up case selected for this study is based on the experiments carried out by Mitra [3]. The 2D simulations performed in this study predict the breakup length of the plane liquid sheet in good agreement with the experimental data. Future work will involve, performing 3D simulations of the plane liquid sheet generated by the air-blast nozzle and performing comparisons of the resulting droplet characteristics with the experimental data. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)