New solutions for surface tension driven spreading of a thin film

The standard fourth-order non-linear PDE modelling the flow of thin fluid film subject to surface tension is studied. The Lie group method is used to reduce the model equation from a fourth-order PDE to a fourth-order ODE. Analytical solutions are obtained for certain cases. Where analytical progress cannot be made, we determine numerical solutions.     

An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk

The axisymmetric spreading of a thin liquid drop under the influence of gravity and rotation is investigated. The effects of the Coriolis force and surface tension are ignored. The Lie group method is used to analyse the non-linear diffusion-convection equation modelling the spreading of the liquid drop under gravity and rotation. A stationary group invariant solution is obtained. The case when rotation is small is considered next. A straightforward perturbation approach is used to determine the effects of the small rotation on the solution given for spreading under gravity only. Over a short period of time no real difference is observed between the approximate solution and the solution for spreading under gravity only. After a long period of time, the approximate solution tends toward a dewetting solution. We find that the approximate solution is valid only in the interval t∈[0,t∗), where t∗ is the time when dewetting takes place. An approximation to t∗ is obtained.     

A new method for manufacture of thin film heat flux gauges

The manufacture of thin film gauges for measuring transient temperatures and heat fluxes is described. A new method of using ceramic substrates (ZrO₂) with two sintered platinum wires is described. Examples of static and dynamic calibrations are given. Sample measurements in a shock tunnel are presented. The gauges show good mechanical strength and sensitivity.This article was processed using Springer- Verlag T_EX Shock Waves macro package 1990.     

The numerical solution of the thin film flow surrounding a horizontal cylinder resulting from a vertical cylindrical jet

The numerical solution of the thin film flow surrounding a horizontal cylinder resulting from a single vertical cylindrical jet is obtained. This is effected by transforming the domain of the flow, which contains a free surface, onto a rectangular parallelepiped and using a marching strategy to solve the ensuing parabolic equations. The flow terminates at a finite distance along the cylinder, its position depending on the velocity and mass flux of the jet. A comparison with the usual two-dimensional model in which the jet is replaced by a vertical sheet shows that such a representation is valid provided the overall width of the flow is not too large. In particular, the differences in heat transfer characteristics amount to a few per cent, thus validating the use of the two-dimensional model when applied to heat exchanger tubes. A comparison with the more usual multicolumn case is also considered.     

A computational model for the indentation and phase transformation of a martensitic thin film

We propose a computational model for a stress-induced martensitic phase transformation of a single-crystal thin film by indentation and its reverse transformation to austenite by heating. Our model utilizes a surface energy that allows sharp interfaces with finite energy and a penalty that forces the film to lie above the indenter and undergo a stress-induced austenite-to-martensite phase transformation. We introduce a method to nucleate the martensite-to-austenite phase transformation since in our model the film would otherwise remain in the martensitic phase in a local minimum of the energy.     

Axisymmetric spreading of a thin power-law fluid under gravity on a horizontal plane

Separable solutions admitted by a nonlinear partial differential equation modeling the axisymmetric spreading under gravity of a thin power-law fluid on a horizontal plane are investigated. The model equation is reduced to a highly nonlinear second-order ordinary differential equation for the spatial variable. Using the techniques of Lie group analysis, the nonlinear ordinary differential equation is linearized and solved. As a consequence of this linearization, new results are obtained.     

Unraveling surfactant transport on a thin liquid film

When a drop of insoluble surfactant is deposited on the surface of a thin liquid film, a radial flow is induced by the resulting surface tension gradient. It is difficult in practice to measure or visualize the evolution of the surfactant concentration and the corresponding surface tension field. In this contribution, we propose a numerical technique which allows, in theory, the reconstruction of the surfactant concentration and surface tension fields from the knowledge of the free surface velocity. The method also requires the knowledge of the equation of state relating the surfactant concentration to the surface tension. The proposed method is based on a reformulation of the lubrication approximation which then takes as an input the free surface velocity field. As a by-product, the film thickness is also reconstructed. We also show in this contribution, that the surface diffusion coefficient can also be estimated, in principle. The methodologies are successfully tested on ideal, synthetic data-sets but also on under-resolved, noisy, data-sets more representative of true experimental conditions. This contribution may help shed some light on the phenomena involved in surfactant transport.     

Simulation and measurement of 3D shear-driven thin liquid film flow in a duct

Three-dimensional flow behavior of thin liquid film that is shear-driven by turbulent air flow in a duct is measured and simulated. Its film thickness and width are reported as a function of air velocity, liquid flow rate, surface tension coefficient, and wall contact angle. The numerical component of this study is aimed at exploring and assessing the suitability of utilizing the FLUENT-CFD code and its existing components, i.e. Volume of Fluid model (VOF) along with selected turbulence model, for simulating the behavior of 3D shear-driven liquid film flow, through a comparison with measured results. The thickness and width of the shear-driven liquid film are measured using an interferometric technique that makes use of the phase shift between the reflections of incident light from the top and bottom surfaces of the thin liquid film. Such measurements are quite challenging due to the dynamic interfacial instabilities that develop in this flow. The results reveal that higher air flow velocity decreases the liquid film thickness but increases its width, while higher liquid flow rate increases both its thickness and width. Simulated results provide good estimates of the measured values, and reveal the need for considering a dynamic rather than a static wall contact angle in the model for improving the comparison with measured values.     

The Mexican hat effect on the delamination buckling of a compressed thin film

Because of the interaction between film and substrate,the film buckling stress can vary significantly,depending on the delamination geometry,the film and substrate mechanical properties.The Mexican hat effect indicates such interaction.An analytical method is presented,and related dimensional analysis shows that a single dimensionless parameter can effectively evaluate the effect.     

Wave regimes on power-law fluid film flowing down a porous plane

Non-linear waves on the surface of a falling film of power-law fluid on a vertical porous plane are investigated. The waves are described by evolution equations generalising equations previously derived in the case of solid plane. It is shown that the slip condition on the interface between pure liquid and the porous substrate drastically changes structure of the steady waves travelling in the film.     

Studying thin film damping in a micro-beam resonator based on non-classical theories

In this paper, a mathematical model is presented for studying thin film damping of the surrounding fluid in an in-plane oscillating micro-beam resonator. The proposed model for this study is made up of a clamped-clamped micro-beam bound between two fixed layers. The micro-gap between the micro-beam and fixed layers is filled with air. As classical theories are not properly capable of pre-dicting the size dependence behaviors of the micro-beam, and also behavior of micro-scale fluid media, hence in the presented model, equation of motion governing longitudinal displacement of the micro-beam has been extracted based on non-local elasticity theory. Furthermore, the fluid field has been modeled based on micro-polar theory. These coupled equations have been simplified using Newton-Laplace and continuity equations. After transforming to non-dimensional form and linearizing, the equations have been discretized and solved simultaneously using a Galerkin-based reduced order model. Considering slip boundary conditions and applying a complex frequency approach, the equivalent damping ratio and quality factor of the micro-beam resonator have been obtained. The obtained values for the quality factor have been compared to those based on classical theories. We have shown that applying non-classical theories underestimate the values of the quality factor obtained based on classical theo-ries. The effects of geometrical parameters of the micro-beam and micro-scale fluid field on the quality factor of the res-onator have also been investigated.     

Surface concentration measurement of a spreading monolayer film using a displacement current technique

There are numerous techniques for measuring surface concentration of monolayer films. While most techniques are suitable only for stationary films, a novel technique is examined for its feasibility in measuring the concentration of a polar monolayer during the spreading of the film. Measurement of the concentration of an octadecanol monolayer while spreading over a water surface is demonstrated with a displacement current measurement technique in conjunction with Wilhelmy plate measurements of the surface pressure. It was found that the displacement current technique can easily measure changes in the film concentration well below those detectable by the Wilhelmy plate. A gaseous film region was observed ahead of the leading edge detectable by the Wilhelmy plate. This film region has not been previously reported in the literature. Furthermore, the film front appears to diffuse during the spreading process. Surfactant concentration behavior during spreading is useful for examining and developing models of surface tension driven flows.     

A film flow model for analysing gravity-driven,thin wavy fluid films

To analyse the physics underlying gravity-driven runoff of thin wavy films, a film flow model is developed, and is solved with computational fluid dynamics. This model is based on the lubrication theory, and takes into account the gravitational, wall shear and surface tension forces. A key characteristic of the model is that it assumes only one computational cell over the film height, which enables studying film flow on larger computational domains. A main aim of this study is to perform a detailed validation of the numerical model. The film flow model is validated against several experiments of gravity-driven, thin fluid films on smooth surfaces. The time-averaged film thickness and the fluid speed profiles predicted by the model show very good agreement with experimental results. Similarly, the film flow model is able to predict the wave speeds with sufficient accuracy. The energy spectra of the waves, where higher frequency waves are present in film flows at higher Reynolds numbers, show an exponentially decaying trend at these high frequencies. The model performs better than the Nusselt equation for film flows, which under-predicts the time-averaged film thickness and over-predicts the time-averaged fluid speeds, even for flows at low Reynolds numbers. The film flow model is compared qualitatively for fingering behaviour. This model also allows to investigate film flows on large surfaces, which can be rough, curved and of complex geometrical shape.     

Effects of air-flow and evaporation on the development of thin liquid film over a rotating annular disk

Axisymmetric flow of thin pure liquid film on a spinning horizontal annular disk is studied under the action of air shear at the liquid–air interface and evaporation. The non-linear evolution equation that is obtained by singular perturbation method is solved analytically, for small Reynolds number, by using the method of characteristic and numerically by the use of Newton–Kantorovich method for any Reynolds number. Font breakdown time and its location from the center of the disk is predicted both by analytically and numerically. The result shows that the thinning of the initial film increases as air stress increase, same result is also escalated in presence of evaporation.     

Mixed-mode interfacial adhesive strength of a thin film on an anisotropic substrate

The mixed-mode interfacial adhesion strength between a gold (Au) thin film and an anisotropic passivated silicon (Si) substrate is measured using laser-induced stress wave loading. Test specimens are prepared by bonding a fused silica (FS) prism to the back side of a 〈1 0 0〉 Si substrate with a thin silicon nitride (Si_xN_y) passivation layer deposited on the top surface. A high-amplitude stress wave is developed by pulsed laser ablation of a sacrificial absorbing layer on one of the lateral surfaces of the FS prism. Due to the negative non-linear elastic properties of the FS, the compressive stress wave evolves into a decompression shock with fast fall time. Careful selection of the incident angle between the pulse and the FS/Si interface generates a mode-converted shear wave in refraction, subjecting the Si_xN_y/Au thin film interface to dynamic mixed-mode loading, sufficient to cause interfacial fracture. A detailed analysis of the anisotropic wave propagation combined with interferometric measurements of surface displacements enables calculation of the interfacial stresses developed under mixed-mode loading. The mixed-mode interfacial strength is compared to the interfacial strength measured under purely tensile loading.     

Modeling of progressive delamination in a thin film driven by diffusion-induced stresses

A semi-analytical method based on the cohesive model has been developed to investigate the progressive growth of interface delamination in an axisymmetric thin film electrode driven by diffusion-induced stresses under the assumption that the electrode remains elastic during the Li-ion diffusion process. The evolutions of the cohesive zone and debonding zone with respect to charging time have been predicted. The cohesive zone propagates in an accelerating manner and the debonding zone advances in a slowing down manner. The key parameters that control the interfacial stresses and delamination have been identified from the obtained governing equations. And according to the discussions on the key parameters, design insights into the geometry, charging velocity and material properties of the electrode have been provided.     

Quantitative characterization of the interfacial adhesion of Ni thin film on steel substrate: A compression-induced buckling delamination test

A compression-induced buckling delamination test is employed to quantitatively characterize the interfacial adhesion of Ni thin film on steel substrate. It is shown that buckles initiate from edge flaws and surface morphologies exhibit symmetric, half-penny shapes. Taking the elastoplasticity of film and substrate into account, a three-dimensional finite element model for an edge flaw with the finite size is established to simulate the evolution of energy release rates and phase angles in the process of interfacial buckling-driven delamination. The results show that delamination propagates along both the straight side and curved front. The mode II delamination plays a dominant role in the process with a straight side whilst the curved front experiences almost the pure mode I. Based on the results of finite element analysis, a numerical model is developed to evaluate the interfacial energy release rate, which is in the range of 250–315 J/m² with the corresponding phase angle from −41° to −66°. These results are in agreement with the available values determined by other testing methods, which confirms the effectiveness of the numerical model.     

An investigation of an Emden-Fowler equation from thin film flow

A third-order ordinary differential equation (ODE) for thin film flow with both Neumann and Dirichlet boundary conditions is transformed into a second-order nonlinear ODE with Dirichlet boundary conditions.Numerical solutions of the nonlinear second-order ODE are investigated using finite difference schemes.A finite difference formulation to an Emden-Fowler representation of the second-order nonlinear ODE is shown to converge faster than a finite difference formulation of the standard form of the second-order nonlinear ODE.Both finite difference schemes satisfy the von Neumann stability criteria.When mapping the numerical solution of the second-order ODE back to the variables of the original third-order ODE we recover the position of the contact line.A nonlinear relationship between the position of the contact line and physical parameters is obtained.     

Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt

In this paper, we prove in general that the homotopy perturbation method (HPM) proposed in 1998 is only a special case of the homotopy analysis method (HAM) profound in 1992 when ħ = −1. Besides, by using the thin film flows of Sisko and Oldroyd 6-constant fluids on a moving belt as examples, we show that the solutions given by HPM (Siddiqui, A.M., Ahmed, M., Ghori, Q.K.: Chaos Solitons and Fractals (2006) in press) are divergent, and thus useless. However, by choosing a proper value of the auxiliary parameter ħ, we give convergent series solution by means of the HAM. These two examples also show that, different from the HPM and other traditional analytic techniques, the HAM indeed provides us with a simple way to ensure the convergence of the solution.     

Waves on the surface of a falling power-law fluid film

Waves that occur at the surface of a falling film of thin power-law fluid on a vertical plane are investigated. Using the method of integral relations an evolution equation is derived for two types of waves equation which are possible under long wave approximation. This equation reveals the presence of both kinematic and dynamic wave processes which may either act together or singularly dominate the wave field depending on the order of different parameters. It is shown that, at a small flow rate, kinematic waves dominate the flow field and the energy is acquired from the mean flow during the interaction of the waves, while, for high flow rate, inertial waves dominate and the energy comes from the kinematic waves. It is also found that this exchange of energy between kinematic and inertial waves strongly depends on the power-law index n. Linear stability analysis predicts the contribution of different terms in the wave mechanism. Further, it is found that the surface tension plays a double role: for a kinematic wave process, it exerts dissipative effects so that a finite amplitude case may be established, but for a dynamic wave process it yields dispersion. Further, it is shown that the non-Newtonian character n plays a vital role in controlling the role of the term that contains surface tension in the above processes.