Journal articles: 'Thin Film Equation'

1

Huang, F. Y., and C. D. Mote. "Derivation of a Thin Film Equation by a Direct Approach." Journal of Applied Mechanics 63, no. 2 (June 1, 1996): 467–73. http://dx.doi.org/10.1115/1.2788891.

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A new model of the thin viscous fluid film, constrained between two translating, flexible surfaces, is presented in this paper: The unsteady inertia of the film is included in the model. The derivation starts with the reduced three-dimensional Navier-Stokes equations for an incompressible viscous fluid with a small Reynolds number. By introduction of an approximate velocity field, which satisfies the continuity equation and the no-slip boundary conditions exactly, into weighted integrals of the three-dimensional equations over the film thickness, a two-dimensional thin film equation is obtained explicitly in a closed form. The 1th thin film equation is obtained when the velocity field is approximated by 21th order polynominals, and the three-dimensional viscous film is described with increasing accuracy by thin film equations of increasing order. Two cases are used to illustrate the coupling of the film to the vibration of the structure and to show that the second thin film equation can be applied successfully to the prediction of a coupled film-structure response in the range of most applications. A reduced thin film equation is derived through approximation of the second thin film equation that relates the film pressure to transverse accelerations and velocities, and to slopes and slope rates of the two translating surfaces.

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2

Segatti, Antonio, and Juan Luis Vázquez. "On a fractional thin film equation." Advances in Nonlinear Analysis 9, no. 1 (March 26, 2020): 1516–58. http://dx.doi.org/10.1515/anona-2020-0065.

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Abstract This paper deals with a nonlinear degenerate parabolic equation of order α between 2 and 4 which is a kind of fractional version of the Thin Film Equation. Actually, this one corresponds to the limit value α = 4 while the Porous Medium Equation is the limit α = 2. We prove existence of a nonnegative weak solution for a general class of initial data, and establish its main properties. We also construct the special solutions in self-similar form which turn out to be explicit and compactly supported. As in the porous medium case, they are supposed to give the long time behaviour or the wide class of solutions. This last result is proved to be true under some assumptions. Lastly, we consider nonlocal equations with the same nonlinear structure but with order from 4 to 6. For these equations we construct self-similar solutions that are positive and compactly supported, thus contributing to the higher order theory.

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LIU, CHANGCHUN, JINGXUE YIN, and HONGJUN GAO. "A GENERALIZED THIN FILM EQUATION." Chinese Annals of Mathematics 25, no. 03 (July 2004): 347–58. http://dx.doi.org/10.1142/s0252959904000329.

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4

van Odyck, D. E. A., and C. H. Venner. "Stokes Flow in Thin Films." Journal of Tribology 125, no. 1 (December 31, 2002): 121–34. http://dx.doi.org/10.1115/1.1506317.

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Present understanding of the mechanisms of lubrication and the load carrying capacity of lubricant films mainly relies on models in which the Reynolds equation is used to describe the flow. The narrow gap assumption is a key element in its derivation from the Navier Stokes equations. However, the tendency in applications is that lubricated contacts have to operate at smaller film thickness levels, and because engineering surfaces are never perfectly smooth, locally in the film this narrow gap assumption may violated. In addition to this geometric limitation of the validity of the Reynolds equation may come a piezoviscous and compressibility related limitation. In this paper the accuracy of the predictions of the Reynolds model in relation to the local geometry of the gap is investigated. A numerical solution algorithm for the flow in a narrow gap has been developed based on the Stokes equations. For a model problem the differences between the pressure and velocity fields according to the Stokes model and the Reynolds equation have been investigated. The configuration entails a lower flat surface together with an upper surface (flat or parabolic) in which a local defect (single asperity) of known geometry has been embedded. It is investigated how the magnitude of the differences develops as a function of the geometric parameters of the film and the feature. Finally, it is discussed to what extend for these problems a perturbation approach can provide accurate corrections to be applied to the Reynolds solution.

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5

Chen, Kuen Tsann, Jui Hsing Chang, and Jiun Yu Wu. "Modified Stoney's Equation for Evaluation of Residual Stresses on Thin Film." Applied Mechanics and Materials 789-790 (September 2015): 25–32. http://dx.doi.org/10.4028/www.scientific.net/amm.789-790.25.

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In the article, a simple method for the modification of the Stoney's equation was presented. The Stoney's equation is proposed from the assumption of equi-biaxial residual stresses in thin films. In this present method, biaxial stresses are different in x-axis and y-axis on thin film. The location of neutral axis depends on the material parameters and the film thickness. The finite element method (FEM) was used to simulate the thermal stress on the thin film. The results of the modified methods are compared with the results of FEM and other literatures. The present method is more accurate than the Stoney's equation in the evaluation of such films.

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6

Ruschak, Kenneth J., and Steven J. Weinstein. "Viscous Thin-Film Flow Over a Round-Crested Weir." Journal of Fluids Engineering 121, no. 3 (September 1, 1999): 673–77. http://dx.doi.org/10.1115/1.2823522.

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Gravity-driven flow over a round-crested weir is analyzed for viscous flow. An equation for the entire flow profile is obtained by simplifying the equations for slowly varying film thickness, assuming a velocity profile, and integrating across the film. Solution of the resulting first order, ordinary differential equation requires a boundary condition generated at a critical point of the flow, beyond which waves cannot propagate upstream. Results for the relationship between head and flow rate are consolidated on a dimensionless master curve represented by an empirical equation.

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7

Ruschak, Kenneth J., and Steven J. Weinstein. "Laminar, Gravitationally Driven Flow of a Thin Film on a Curved Wall." Journal of Fluids Engineering 125, no. 1 (January 1, 2003): 10–17. http://dx.doi.org/10.1115/1.1522412.

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Gravitationally driven flow of a thin film down an arbitrarily curved wall is analyzed for moderate Reynolds number by generalizing equations previously developed for flow on a planar wall. In the analysis, the ratio of the characteristic film thickness to the characteristic dimension of the wall is presumed small, and terms estimated to be first order in this parameter are retained. Partial differential equations are reduced to ordinary differential equations by the method of von Ka´rma´n and Pohlhausen; namely, an expression for the velocity profile is assumed, and the equation for conservation of linear momentum is averaged across the film. The assumed velocity profile changes shape in the flow direction because a self-similar profile, one of fixed shape but variable magnitude, leads to an equation that typically fails under critical conditions. The resulting equations for film thickness routinely accommodate subcritical-to-supercritical transitions and supercritical-to-subcritical transitions as classified by the underlying wave propagation. The more severe supercritical-to-subcritical transition is manifested by a standing wave where the film noticeably thickens; this standing wave is a simple analogue of a hydraulic jump. Predictions of the film-thickness profile and variations in the velocity profile compare favorably with those from the Navier-Stokes equation obtained by the finite element method.

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8

Abd Alsamieh, M. F. "Non-Newtonian Film Thickness Formation in Ultra-thin Film." International Journal of Automotive and Mechanical Engineering 16, no. 1 (March 21, 2019): 6230–44. http://dx.doi.org/10.15282/ijame.16.1.2019.11.0473.

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This paper aims to show the characteristics of ultra-thin films for non-Newtonian fluid using Ree-Eyring model where intermolecular forces of solvation and Van der Waal's are considered in addition to the hydrodynamic action to fulfill an identified need for such a conjunction. In this case, the film thickness and pressure distribution are obtained by simultaneous solution of the modified Reynolds’ equation incorporating the effect of non-Newtonian fluid, film thickness equation including elastic deformation caused by all contributing pressures and the load balance equation using Newton-Raphson method with Gauss-Seidel iterations. Effect of changing the operating conditions of speed, load, Eyring shear stress and slide-roll ratio on the characteristic of the contact has been studied. The results show that, for the case where the hydrodynamic action is the only pressure acting to support the applied load capacity, the film thickness and the pressure gradient at the exit of the contact obtained using non-Newtonian model is different than that formed using the Newtonian model especially for the increased value of slide-roll ratio. The main results of this study are that for ultra-thin film, the film thickness formed using non-Newtonian model is smaller compared to that obtained using Newtonian case and the discretization of the film thickness as the gap is reduced occurs similar to the results obtained using Newtonian model. The pressure shape shows no difference compared to that formed using the Newtonian case in which an oscillation around the Hertizan contact pressure shape due to the solvation effect appears. The results also show that for ultra-thin film, changing the Eyring shear stress does not affect the film thickness formation.

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9

van Odyck, D. E. A., and C. H. Venner. "Compressible Stokes Flow in Thin Films." Journal of Tribology 125, no. 3 (June 19, 2003): 543–51. http://dx.doi.org/10.1115/1.1539058.

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A multigrid numerical solution algorithm has been developed for the laminar (Stokes) flow of a compressible medium in a thin film. The solver has been applied to two model problems each representative of lubrication problems in a specific way. For both problems the solutions of the Stokes equations are compared with the solutions of the Reynolds equation. The configurations of both model problems were chosen such that based on the ratio film thickness to contact length (H/L) the difference between the Reynolds and the Stokes solutions will be very small, so the geometry of the gap itself does not lead to a significant cross film dependence of the pressure. It is shown that in this situation the compressibility can still lead to a cross-film pressure dependence which is predicted by the Stokes solution and not by the Reynolds solution. The results demonstrate that limitations exist to the validity of the Reynolds equation related to the compressibility of the medium.

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10

Fabes, B. D., W. C. Oliver, R. A. McKee, and F. J. Walker. "The determination of film hardness from the composite response of film and substrate to nanometer scale indentations." Journal of Materials Research 7, no. 11 (November 1992): 3056–64. http://dx.doi.org/10.1557/jmr.1992.3056.

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Two equations for determining the hardness of thin films from depth-sensing indentation data are examined. The first equation is based on an empirical fit of hardness versus indenter displacement data obtained from finite element calculations on a variety of hypothetical films. The second equation is based on a model which assumes that measured hardness is determined by the weighted average of the volume of plastically deformed material in the coating and that in the substrate. The equations are evaluated by fitting the predicted hardness versus contact depth to data obtained from titanium coatings on a sapphire substrate. Only the volume fractions model allows the data to be fitted with a single adjustable parameter, the film hardness; the finite element equation requires two thickness-dependent parameters to obtain acceptable fits. It is argued that the difficulty in applying the finite element model lies in the use of an unrealistic area function for the indenter. For real indenters, which have finite radii, the area function must appear explicitly in the final equation. This is difficult to do with the finite element approach, but is naturally incorporated into the volume fractions equation. Finally, using the volume fractions approach the hardnesses of the titanium films are found to be relatively insensitive to film thickness. Thus, the apparent increase in hardness with decreasing film thickness for the titanium films is most likely due to increased interactions between the film and substrate for the thinner films rather than to a change in the basic structure of the titanium films.

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11

Yekta, Ahmad, Zahra Masoumi, and Mitchell A. Winnik. "Luminescence measurements of oxygen permeation and oxygen diffusion in thin polymer films." Canadian Journal of Chemistry 73, no. 11 (November 1, 1995): 2021–29. http://dx.doi.org/10.1139/v95-250.

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A new method is developed for analyzing the diffusion of oxygen in thin polymer films via fluorescence quenching measurements. We begin by reviewing previous methods, all of which involve approximations, for the measurement of permeability and diffusion coefficient by luminescence quenching; their shortcomings are clarified. An exact analytic theory is developed that successfully couples Fick's laws of diffusion to the Stern–Volmer equation of intensity quenching. Various modes of experimentation with polymeric films are considered. The equations we derive make the unexpected prediction that the rate of emission intensity decay when O2 diffuses into a polymer film is much faster than the rate of emission intensity enhancement when O2 diffuses out of the same film, even when the molecular diffusivity remains unchanged. Experiments show that this is indeed the observed behaviour. Keywords: diffusion coefficient, oxygen permeability, polymers, quenching, luminescence quenching, Fick's laws. Stern–Volmer equation.

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12

Chu, Li-Ming, Jaw-Ren Lin, and Jiann-Lin Chen. "Effects of Surface Roughness and Surface Force on the Thin Film Elastohydrodynamic Lubrication of Circular Contacts." Zeitschrift für Naturforschung A 67, no. 6-7 (July 1, 2012): 412–18. http://dx.doi.org/10.5560/zna.2012-0035.

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The effects of surface roughness and surface force on thin film elastohydrodynamic lubrication (TFEHL) circular contact problems are analyzed and discussed under constant load condition. The multi-level multi-integration (MLMI) algorithm and the Gauss-Seidel iterative method are used to simultaneously solve the average Reynolds type equation, surface force equations, the load balance equation, the rheology equations, and the elastic deformation equation. The simulation results reveal that the difference between the TFEHL model and the traditional EHL model increase with decreasing film thickness. The effects of surface forces become significant as the film thickness becomes thinner. The surface forces have obvious effects in the Hertzian contact region. The oscillation phenomena in pressure and film thickness come mainly from the action of solvation forces

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13

Raible, Martin, Stefan J. Linz, and Peter Hänggi. "Amorphous thin film growth: Minimal deposition equation." Physical Review E 62, no. 2 (August 1, 2000): 1691–705. http://dx.doi.org/10.1103/physreve.62.1691.

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14

Giacomelli, Lorenzo, Andrey Shishkov, and Roberta Passo. "THE THIN FILM EQUATION WITH NONLINEAR DIFFUSION." Communications in Partial Differential Equations 26, no. 9-10 (September 1, 2001): 1509–57. http://dx.doi.org/10.1081/pde-100107451.

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15

King, J. R. "Two generalisations of the thin film equation." Mathematical and Computer Modelling 34, no. 7-8 (October 2001): 737–56. http://dx.doi.org/10.1016/s0895-7177(01)00095-4.

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16

Gandarias, M. L., and M. S. BruzÃ³n. "Symmetry analysis for a thin film equation." PAMM 7, no. 1 (December 2007): 2040021–22. http://dx.doi.org/10.1002/pamm.200700408.

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17

SHKLYAEV, OLEG E., and ELIOT FRIED. "Stability of an evaporating thin liquid film." Journal of Fluid Mechanics 584 (July 25, 2007): 157–83. http://dx.doi.org/10.1017/s0022112007006350.

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We use a newly developed set of interface conditions to revisit the problem of an evaporating thin liquid film. In particular, instead of the conventional Hertz–Knudsen–Langmuir equation for the evaporation mass flux, we impose a more general equation expressing the balance of configurational momentum. This balance, which supplements the conventional conditions enforcing the balances of mass, momentum and energy on the film surface, arises from a consideration of configurational forces within a thermodynamical framework. We study the influence of two newly introduced terms on the evolution of the liquid film. One of these terms accounts for the transport of energy within the liquid–vapour interface. The other term, which we refer to as the effective pressure, accounts for vapour recoil. Both new terms are found to be stabilizing. Furthermore, the effective pressure is found to affect a time-dependent base state of the evaporating film and to be an important factor in applications involving liquid films with thicknesses of one or two monolayers. Specifically, we demonstrate that consideration of the effective pressure makes it possible to observe the influence of the van der Waals interactions on film evolution close to the instant of rupture. Dimensional considerations indicate that one of the most significant influences of these effects occurs for molten metals.

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18

BERTSCH, MICHIEL, ROBERTA DAL PASSO, STEPHEN H. DAVIS, and LORENZO GIACOMELLI. "Effective and microscopic contact angles in thin film dynamics." European Journal of Applied Mathematics 11, no. 2 (April 2000): 181–201. http://dx.doi.org/10.1017/s0956792599004015.

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We introduce and analyse a class of quasi-self-similar solutions of the thin film equation to describe the dynamics of expanding liquid films on a solid surface. Using these solutions as intermediate asymptotics profiles, we obtain a quantitative expression for the shape of the film and a relation between the speed of the contact line and the macroscopic and microscopic contact angles.

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19

Hadda, Mohammed, and Mouhcine Tilioua. "Thin Film Limits in Magnetoelastic Interactions." Mathematical Problems in Engineering 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/165962.

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This paper deals with classical dimensional reductions 3D-2D and 3D-1D in magnetoelastic interactions. We adopt a model described by the Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We identify the limit problem both for flat and slender media by using the so-called energy method.

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20

Li, Wang-Long. "Ultra-Thin Gas Squeeze Film Characteristics for Infinitely Large Squeeze Number." Journal of Tribology 120, no. 4 (October 1, 1998): 750–57. http://dx.doi.org/10.1115/1.2833775.

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In this study, the characteristics of ultra-thin gas films are analyzed asymptotically for infinite squeeze number using the molecular gas film lubrication equation with coupled roughness and rarefaction effects taken into consideration. The governing equation of the internal region was obtained by a time averaged technique, and the boundary conditions were obtained numerically from the matching conditions near the boundaries. Two new functions, H and H−1 were proposed for deriving the matching equation near the boundary. Finally, the characteristics of squeeze film bearings with infinite width were analyzed for various roughness parameters (Peklenik number, standard deviations of the composite roughness, and roughness orientation angles), rarefaction parameter (Knudsen number), and operation conditions (excursion ratio).

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21

YILBAS, B. S., and S. BIN MANSOOR. "FREQUENCY DEPENDENT PHONON TRANSPORT IN TWO-DIMENSIONAL SILICON AND DIAMOND THIN FILMS." Modern Physics Letters B 26, no. 17 (June 3, 2012): 1250104. http://dx.doi.org/10.1142/s0217984912501047.

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Phonon transport in two-dimensional silicon and aluminum films is investigated. The frequency dependent solution of Boltzmann transport equation is obtained numerically to account for the acoustic and optical phonon branches. The influence of film size on equivalent equilibrium temperature distribution in silicon and aluminum films is presented. It is found that increasing film width influences phonon transport in the film; in which case, the difference between the equivalent equilibrium temperature due to silicon and diamond films becomes smaller for wider films than that of the thinner films.

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22

Pelusi, F., M. Sega, and J. Harting. "Liquid film rupture beyond the thin-film equation: A multi-component lattice Boltzmann study." Physics of Fluids 34, no. 6 (June 2022): 062109. http://dx.doi.org/10.1063/5.0093043.

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Under the condition of partial surface wettability, thin liquid films can be destabilized by small perturbations and rupture into droplets. As successfully predicted by the thin film equation (TFE), the rupture dynamics are dictated by the liquid–solid interaction. The theory describes the latter using the disjoining pressure or, equivalently, the contact angle. The introduction of a secondary fluid can lead to a richer phenomenology, thanks to the presence of different fluid/surface interaction energies but has so far not been investigated. In this work, we study the rupture of liquid films with different heights immersed in a secondary fluid using a multi-component lattice Boltzmann (LB) approach. We investigate a wide range of surface interaction energies, equilibrium contact angles, and film thicknesses. We found that the rupture time can differ by about one order of magnitude for identical equilibrium contact angles but different surface free energies. Interestingly, the TFE describes the observed breakup dynamics qualitatively well, up to equilibrium contact angles as large as 130°. A small film thickness is a much stricter requirement for the validity of the TFE, and agreement with LB results is found only for ratios [Formula: see text] of the film height h and lateral system size L, such as [Formula: see text].

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23

Chapman, Navid, Mingyu Chapman, and William B. Euler. "Modeling of Poly(methylmethacrylate) Viscous Thin Films by Spin-Coating." Coatings 11, no. 2 (February 9, 2021): 198. http://dx.doi.org/10.3390/coatings11020198.

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A predictive film thickness model based on an accepted equation of state is applied to the spin-coating of sub-micron poly(methylmethacrylate) viscous thin films from toluene. Concentration effects on density and dynamic viscosity of the spin-coating solution are closely examined. The film thickness model is calibrated with a system-specific film drying rate and was observed to scale with the square root of spin speed. Process mapping is used to generate a three-dimensional design space for the control of film thickness.

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24

Falope, Federico Oyedeji, and Enrico Radi. "Finite Thin Cover on an Orthotropic Elastic Half Plane." Modelling and Simulation in Engineering 2016 (2016): 1–11. http://dx.doi.org/10.1155/2016/5393621.

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The present work deals with the mechanical behaviour of thin films bonded to a homogeneous elastic orthotropic half plane under plain strain condition and infinitesimal strain. Both the film and semi-infinite substrate display linear elastic orthotropic behaviour. By assuming perfect adhesion between film and half plane together with membrane behaviour of the film, the compatibility condition between the coating and substrate leads to a singular integral equation with Cauchy kernel. Such an equation is straightforwardly solved by expanding the unknown interfacial stress in series of Chebyshev polynomials displaying square-root singularity at the film edges. This approach allows handling the singular behaviour of the shear stress and, in turn, reducing the problem to a linear algebraic system of infinite terms. Results are found for two loading cases, with particular reference to concentrated axial forces acting at the edges of the film. The corresponding mode II stress intensity factor has been assessed, thus providing the stress concentrations at both ends of the covering. Possible applications of the results here obtained range from MEMS, NEMS, and solar Silicon cell for energy harvesting to welded joint and building foundation.

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25

Mansoor, Saad Bin, and Bekir Sami Yilbas. "Phonon transport in aluminum and silicon film pair: laser short-pulse irradiation at aluminum film surface." Canadian Journal of Physics 92, no. 12 (December 2014): 1614–22. http://dx.doi.org/10.1139/cjp-2013-0710.

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Phonon transport in paired aluminum and silicon thin films is considered under laser short-pulse irradiation at the aluminum film surface. The Boltzmann equation is incorporated to formulate energy transport in the films. To include a volumetric source resembling laser irradiation in the aluminum film, the Boltzmann equation is modified. Thermal boundary resistance is located at the interface of the film pair. An equivalent equilibrium temperature is introduced to assess the thermal resistance of the film during the laser heating process. The phonon temperature obtained from solution of the Boltzmann equation is compared with the findings of the two-temperature model. It is found that phonon temperature obtained from the solution of the Boltzmann equation is lower than that corresponding to the two-temperature model, which is particularly true in the surface region of the aluminum film. Phonon temperature increases gradually while, early on, the electron temperature rises and decays sharply in the surface region of the aluminum film.

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Shao, Hong Yang, Kan Zhang, Yi Dan Zhang, Mao Wen, and Wei Tao Zheng. "NbN Thin Film of Alternating Textures." Materials Science Forum 898 (June 2017): 1431–37. http://dx.doi.org/10.4028/www.scientific.net/msf.898.1431.

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The δ-NbN thin films with different thickness have been prepared by reactive magnetron sputtering at different deposition time and exhibited alternating textures between (111) and (200) orientations as a function of thickness. In addition, the grain size, peak position, morphology, residual stress and orientation distributions of the deposited films were explored by X-ray diffraction, low-angel X-ray reflectivity, scanning electron microscopy and surface profiler. The film deposited at 300 s showed a (111) preferred orientation, changing to (200) preferred orientation at 600 s, and exhibited alternating textures between (111) and (200) preferred orientations. With further increasing deposition time, in which (200) peak position and the full width at half maximum of (111) peak also displayed a trend of alternating variation with varying deposition time. The intrinsic stress for δ-NbN films calculated by Stoney equation alternately changed with alternating textures, in which (111) orientation always takes place at relatively high intrinsic stress state and vice versa. Meanwhile, the film with (111) preferred orientation showed higher density than (200) preferred orientation. The film deposited at 4800 s owned a mixed texture of (111) and (200), showing an anisotropy distribution of (111)-oriented and (200)-oriented grains, while film deposited at 7200 s owned a strong (200) texture, displaying an isotropy distribution of (200)-oriented grains. The competitive growth between (111)-oriented and (200)-oriented grains was responsibility for alternating texture.

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GOTTLIEB, O., and ALEXANDER ORON. "STABILITY AND BIFURCATIONS OF PARAMETRICALLY EXCITED THIN LIQUID FILMS." International Journal of Bifurcation and Chaos 14, no. 12 (December 2004): 4117–41. http://dx.doi.org/10.1142/s0218127404011958.

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We investigate the stability and bifurcations of parametrically excited thin liquid films. A recently derived nonlinear evolution equation for the two-dimensional spatio-temporal dynamics of falling liquid films on an oscillating vertical wall is expanded to low order Fourier modes. A fourth-order modal dynamical system is validated to yield the primary bifurcation structure of the fundamental falling film dynamics described by the Benney equation, and accurately predicts the quasi-periodic structure of the temporally modulated Benney equation (TMBE). The stability of fundamental steady and periodic solutions is analytically and numerically investigated so as to reveal the threshold for nonstationary and chaotic solutions corresponding to aperiodic modulated traveling waves. The reduced modal dynamical system enables construction of a comprehensive bifurcation structure, which is verified by numerical simulation of the evolution equation.

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28

Seis, Christian. "The thin-film equation close to self-similarity." Analysis & PDE 11, no. 5 (April 11, 2018): 1303–42. http://dx.doi.org/10.2140/apde.2018.11.1303.

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Boutat, M., S. Hilout, J. E. Rakotoson, and J. M. Rakotoson. "A generalized thin-film equation in multidimensional space." Nonlinear Analysis: Theory, Methods & Applications 69, no. 4 (August 2008): 1268–86. http://dx.doi.org/10.1016/j.na.2007.06.028.

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BOWEN, M., and T. P. WITELSKI. "Pressure-dipole solutions of the thin-film equation." European Journal of Applied Mathematics 30, no. 2 (April 2, 2018): 358–99. http://dx.doi.org/10.1017/s095679251800013x.

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We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

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31

He, Ji-Huan, and Chang Sun. "A variational principle for a thin film equation." Journal of Mathematical Chemistry 57, no. 9 (August 31, 2019): 2075–81. http://dx.doi.org/10.1007/s10910-019-01063-8.

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32

Meleshko, S. V., N. F. Samatova, and A. V. Melechko. "Group analysis of the thin film dewetting equation." International Journal of Non-Linear Mechanics 47, no. 1 (January 2012): 9–13. http://dx.doi.org/10.1016/j.ijnonlinmec.2011.08.005.

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33

King, John R., and Roman M. Taranets. "Asymmetric travelling waves for the thin film equation." Journal of Mathematical Analysis and Applications 404, no. 2 (August 2013): 399–419. http://dx.doi.org/10.1016/j.jmaa.2013.03.047.

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34

Liang, Bo, Meishan Wang, Yang Cao, and Huiying Shen. "A thin film equation with a singular diffusion." Applied Mathematics and Computation 227 (January 2014): 1–10. http://dx.doi.org/10.1016/j.amc.2013.10.087.

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35

Li, Dong, Zhonghua Qiao, and Tao Tang. "Gradient bounds for a thin film epitaxy equation." Journal of Differential Equations 262, no. 3 (February 2017): 1720–46. http://dx.doi.org/10.1016/j.jde.2016.10.025.

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36

Charalambous, Kyriakos, and Christodoulos Sophocleous. "Symmetry properties for a generalised thin film equation." Journal of Engineering Mathematics 82, no. 1 (November 7, 2012): 109–24. http://dx.doi.org/10.1007/s10665-012-9577-6.

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37

Kitavtsev, G., L. Recke, and B. Wagner. "Spectrum asymptotics for the linearized thin film equation." PAMM 8, no. 1 (December 2008): 10727–28. http://dx.doi.org/10.1002/pamm.200810727.

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Chu, H. M., R. T. Lee, S. Y. Hu, and Y. P. Chang. "Rheological Characteristics for Thin Film Elastohydrodynamic Lubrication." Journal of Mechanics 21, no. 2 (June 2005): 77–84. http://dx.doi.org/10.1017/s172771910000455x.

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ABSTRACTThis paper uses three lubrication models to explore the differential phenomenon in the status of thin film lubrication (TFL). According to the viscous adsorption theory, the modified Reynolds equation for thin film elastohydrodynamic lubrication (TFEHL) is derived. In this theory, the film thickness between lubricated surfaces is simplified as three fixed layers across the film, and the viscosity and density of the lubricant vary with pressure in each layer. Under certain conditions, such as a rough or concentrated contact of a nominally flat surface, films may be of nanometer scale. The thin film elastohydrodynamic lubrication (EHL) analysis is performed on a surface forces (SF) model which includes van der waals and solvation forces. The results show that the proposed TFEHL model can reasonably calculate the film thickness and the average relative viscosity under thin film EHL. The adsorption layer thickness and the viscosity influence significantly the lubrication characteristics of the contact conjunction. The differences in pressure distribution and film shape between surface forces model and classical EHL model were obvious, especially in the Hertzian contact area. The solvation force has the greatest influence on pressure distribution.

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39

Kuznetsova, I. A., O. V. Savenko, and D. N. Romanov. "The influence of Fermi surface anisotropy and the charge carrier surface scattering kinetics on the electrical conductivity of a thin metal film in the view of the quantum size effect." Journal of Physics: Conference Series 2056, no. 1 (October 1, 2021): 012018. http://dx.doi.org/10.1088/1742-6596/2056/1/012018.

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Abstract The electrical conductivity of a thin metal film in an alternating electric field is calculated considering the quantum size effect. The Fermi surface of the metal has the shape of an ellipsoid of rotation, the main axis of which is parallel to the plane of the film. The quantum kinetic equation obtained from the von Neumann equation (the Liouville quantum equation) is solved. The Soffer model is used as the boundary conditions for the distribution function. The dependence of the electrical conductivity on the film thickness is analyzed. A comparison is made with experimental data on the electrical conductivity of bismuth thin films.

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Flik, M. I., and C. L. Tien. "Size Effect on the Thermal Conductivity of High-Tc Thin-Film Superconductors." Journal of Heat Transfer 112, no. 4 (November 1, 1990): 872–81. http://dx.doi.org/10.1115/1.2910494.

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Using the kinetic theory approximation and reported data, this study shows that at low temperatures, the phonon mean free path in polycrystalline ceramic YBa2Cu3O7 can be of the order of the thickness of thin-film superconductors. In this case, boundary scattering reduces the thermal conductivity with decreasing film thickness. A simple method accounts for the size effect on conduction in thin films. This analysis rests solely on geometric arguments and does not consider the effect of grain boundaries. For conduction along the film, this model approximates well an analytical solution of the Boltzmann transport equation, and is in good agreement with experimental data for thin lead films. The model is also employed to analyze the size effect on conduction across the film and the influence of anisotropy.

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Liu, Yunyan, Hongsheng Song, Junshan Xiu, Meiling Sun, Dong Zhao, Zisheng Su, Gongxiang Wei, and Fangming Jin. "Surface Dynamics Transition of Vacuum Vapor Deposited CH3NH3PbI3 Perovskite Thin Films." Advances in Condensed Matter Physics 2018 (2018): 1–7. http://dx.doi.org/10.1155/2018/8297918.

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The growth dynamics of CH3NH3PbI3 perovskite thin films on ITO covered glass substrate were investigated. The evolution of the film could be divided into two stages. A mound-like surface was obvious at the first stage. Stable dynamic scaling was observed for thicker films at the second stage. Through analyzing the scaling exponent, growth exponent β, and 2D fast Fourier transform, it is concluded that, at the second stage, the growth mechanism of mound formation does not play a major role, and the film growth mechanism can be described by Mullins diffusion equation.

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Duruk, Selin, Edouard Boujo, and Mathieu Sellier. "Thin Liquid Film Dynamics on a Spinning Spheroid." Fluids 6, no. 9 (September 6, 2021): 318. http://dx.doi.org/10.3390/fluids6090318.

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The present work explores the impact of rotation on the dynamics of a thin liquid layer deposited on a spheroid (bi-axial ellipsoid) rotating around its vertical axis. An evolution equation based on the lubrication approximation was derived, which takes into account the combined effects of the non-uniform curvature, capillarity, gravity, and rotation. This approximate model was solved numerically, and the results were compared favorably with solutions of the full Navier–Stokes equations. A key advantage of the lubrication approximation is the solution time, which was shown to be at least one order of magnitude shorter than for the full Navier–Stokes equations, revealing the prospect of controlling film dynamics for coating applications. The thin film dynamics were investigated for a wide range of geometric, kinematic, and material parameters. The model showed that, in contrast to the purely gravity-driven case, in which the fluid drains downwards and accumulates at the south pole, rotation leads to a migration of the maximum film thickness towards the equator, where the centrifugal force is the strongest.

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Ruschak, Kenneth J., and Steven J. Weinstein. "Thin-Film Flow at Moderate Reynolds Number." Journal of Fluids Engineering 122, no. 4 (July 5, 2000): 774–78. http://dx.doi.org/10.1115/1.1319499.

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Viscous, laminar, gravitationally-driven flow of a thin film over a round-crested weir is analyzed for moderate Reynolds numbers. A previous analysis of this flow utilized a momentum integral approach with a semiparabolic velocity profile to obtain an equation for the film thickness (Ruschak, K. J., and Weinstein, S. J., 1999, “Viscous Thin-Film Flow Over a Round-Crested Weir,” ASME J. Fluids Eng., 121, pp. 673–677). In this work, a viscous boundary layer is introduced in the manner of Haugen (Haugen, R., 1968, “Laminar Flow Around a Vertical Wall,” ASME J. Appl. Mech. 35, pp. 631–633). As in the previous analysis of Ruschak and Weinstein, the approximate equations have a critical point that provides an internal boundary condition for a bounded solution. The complication of a boundary layer is found to have little effect on the thickness profile while introducing a weak singularity at its beginning. The thickness of the boundary layer grows rapidly, and there is little cumulative effect of the increased wall friction. Regardless of whether a boundary layer is incorporated, the approximate free-surface profiles are close to profiles from finite-element solutions of the Navier-Stokes equation. Similar results are obtained for the related problem of developing flow on a vertical wall (Cerro, R. L., and Whitaker, S., 1971, “Entrance Region Flows With a Free Surface: the Falling Liquid Film,” Chem. Eng. Sci., 26, pp. 785–798). Less accurate results are obtained for decelerating flow on a horizontal wall (Watson, E. J., 1964, “The Radial Spread of a Liquid Jet Over a Horizontal Plane,” J. Fluid Mech. 20, pp. 481–499) where the flow is not gravitationally driven. [S0098-2202(00)01904-0]

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Morozov, Matvey, and Ofer Manor. "An extended Landau–Levich model for the dragging of a thin liquid film with a propagating surface acoustic wave." Journal of Fluid Mechanics 810 (November 25, 2016): 307–22. http://dx.doi.org/10.1017/jfm.2016.728.

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In this paper we revisit the Landau and Levich analysis of a coating flow in the case where the flow in the thin liquid film is supported by a Rayleigh surface acoustic wave (SAW), propagating in the solid substrate. Our theoretical analysis reveals that the geometry of the film evolves under the action of the propagating SAW in a manner that is similar to the evolution of films that are being deposited using the dip coating technique. We show that in a steady state the thin-film evolution equation reduces to a generalized Landau–Levich equation with the dragging velocity, imposed by the SAW, depending on the local film thickness. We demonstrate that the generalized Landau–Levich equation has a branch of stable steady state solutions and a branch of unstable solutions. The branches meet at a saddle-node bifurcation point corresponding to the threshold value of the SAW intensity. Below the threshold value no steady states were found and our numerical computations suggest a gradual thinning of the liquid film from its initial geometry.

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Gorman, G. L., M. M. Chen, G. Castillo, and R. C. C. Perera. "Density Measurement of Thin Sputtered Carbon Films." Advances in X-ray Analysis 32 (1988): 323–30. http://dx.doi.org/10.1154/s0376030800020632.

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AbstractThe densities of sputtered thin carbon films have been determined using a novel X-ray technique. This nondestructive method involves the measurement of the transmitivity of a characteristic soft (low energy) X-ray line through the carbon film, and using the established equation I1 = I0eμpt where I1/I0 is the transmitivity, fi the photo absorption cross section, t the independently measured thickness, the density p can be easily solved for. This paper demonstrates the feasibility of using this simple technique to measure densities of carbon films as thin as 300 Å, which is of tremendous practical interest as carbon films on this order of thickness are used extensively as abrasive and corrosive barriers (overcoats) for metallic recording media disks. The dependence of the density upon film thickness for a fixed processing condition is presented, as also its dependence (for a fixed thickness) upon different processing parameters (e.g., sputtering gas pressure and target power). The trends noted in this study indicate that the sputtering gas pressure plays the most important role, changing the film density from 2.4gm/cm3 at 1 mTorr to 1.5gm/cm3 at 30 mTorr for 1000 Å thick films.

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Gorla, Rama Subba Reddy. "Rupture of Thin Power-Law Liquid Film on a Cylinder." Journal of Applied Mechanics 68, no. 2 (November 2, 2000): 294–97. http://dx.doi.org/10.1115/1.1355033.

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The dynamic rupture process of a thin power-law type non-Newtonian liquid film on a cylinder has been analyzed by investigating the stability to finite amplitude disturbances. The dynamics of the liquid film is formulated using the balance equations including a body force term due to van der Waals attractions. The governing equation for the film thickness was solved by finite difference method as part of an initial value problem for spatial periodic boundary conditions. A decrease in the cylinder radius will induce a stronger lateral capillary force and thus will accelerate the rupture process. The influence of the power-law exponent on rupture is discussed.

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van der Poel, C. J. "Rapid crystallization of thin solid films." Journal of Materials Research 3, no. 1 (February 1988): 126–32. http://dx.doi.org/10.1557/jmr.1988.0126.

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Laser-beam-controlled heating appears to be an excellent technique for driving isothermal transformations in a thin solid film on a thick substrate. The transformation is detected by the change in optical properties of the film as it evolves from the initial to the final state. Results are presented for the amorphous-to-crystalline transition in 100 nm thick films of InSb and of some Te alloys on thick glass substrates. Discrimination between interface growth and homogeneous crystallization can be made from the data. The crystallization of InSb films can be desribed by an Avrami equation with a single activation energy of 154 ± 6 kJ/mol over the full range of measured crystallization times between 100μs and 1000s. For low temperatures the results are consistent with differential scanning calorimetry (DSC) measurements. For Te-alloy films, the large temperature interval covered by the experimental method enables clear observation of the curvature of the temperature-time-transmission (T–t–t) plot.

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ARMENDÁRIZ, J., and M. MATALON. "Evaporation and combustion of thin films of liquid fuels." Journal of Fluid Mechanics 435 (May 25, 2001): 351–76. http://dx.doi.org/10.1017/s002211200100413x.

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We consider the evaporation and subsequent burning of thin films of liquid fuels. Previous studies on liquid films, with and without evaporation, have primarily considered the gas phase to be passive. The new element in this study is the introduction of combustion and the examination of both the liquid and gas phases and their effect on the film's behaviour. For the case of a liquid film burning in quiescent air we show that the problem can be simplified to a single nonlinear evolution equation for the film thickness. All remaining variables, which are simply expressed in terms of the function describing the instantaneous position of the liquid–vapour interface, are subsequently determined. This equation is then solved in order to understand the dynamics of the film in the presence of evaporation and combustion.The planar configuration is discussed first. Predictions for the total evaporation time are obtained, along with the time history of the film thickness, the interfacial surface temperature, the flame standoff distance and its temperature, and the mass burning rate. The dependence of the burning characteristics on the fuel and oxidizer Lewis numbers, which measure the relative importance of thermal and molecular diffusivities, is also determined. Second, we analyse the case of a non-planar interface, where temperature variations along the film's surface cause fluid motion in the liquid that could either dampen or amplify spatial non-uniformities. We show that, while thermocapillarity has the tendency to destabilize the planar interface, combustion acts to reduce this effect. In particular, when the heat release by combustion is substantial, all disturbances are obliterated, the film remains nearly planar and the burning occurs along nearly horizontal surfaces.

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Chu, Li-Ming, Jaw-Ren Lin, and Cai-Wan Chang-Jian. "Effects of adsorption layers and elastic deformation on thin-film lubrication of circular contacts with non-Newtonian lubricants." Industrial Lubrication and Tribology 70, no. 2 (March 12, 2018): 363–70. http://dx.doi.org/10.1108/ilt-02-2017-0029.

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Purpose The modified Reynolds equation for non-Newtonian lubricant is derived using the viscous adsorption theory for thin-film elastohydrodynamic lubrication (TFEHL) of circular contacts. The proposed model can reasonably calculate the phenomenon in the thin-film lubrication (TFL) unexplained by the conventional EHL model. The differences between classical EHL and TFEHL with the non-Newtonian lubricants are discussed. Design/methodology/approach The power-law lubricating film between the elastic surfaces is modeled in the form of three layers: two adsorption layers on each surface and one middle layer. The modified Reynolds equation with power-law fluid is derived for TFEHL of circular contacts using the viscous adsorption theory. The finite difference method and the Gauss–Seidel iteration method are used to solve the modified Reynolds equation, elasticity deformation, lubricant rheology equations and load balance equations simultaneously. Findings The simulation results reveal that the present model can reasonably calculate the pressure distribution, the film thickness, the velocity distribution and the average viscosity in TFL with non-Newtonian lubricants. The thickness and viscosity of the adsorption layer and the flow index significantly influence the lubrication characteristics of the contact conjunction. Originality/value The present model can reasonably predict the average viscosity, the turning point and the derivation (log film thickness vs log speed) phenomena in the TFEHL under constant load conditions.

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Gorla, Madhu Sudan Reddy, and Rama Subba Reddy Gorla. "Nonlinear Theory of Tear Film Rupture." Journal of Biomechanical Engineering 122, no. 5 (March 22, 2000): 498–503. http://dx.doi.org/10.1115/1.1289997.

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Nonlinear thin film rupture has been analyzed by investigating the stability of tear films to finite amplitude disturbances. The dynamics of the liquid film is formulated using the Navier–Stokes equations, including a body force term due to van der Waals attractions. The governing equation was solved by the finite difference method as part of an initial value problem for spatial periodic boundary conditions. The rupture of the tear film covering the cornea and the formation of dry spots is an important phenomenon in various pathological states associated with a dry eye. [S0148-0731(00)00605-1]

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Journal articles on the topic 'Thin Film Equation'

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