The dynamics of a gravity-driven thin film flow with insoluble surfactant are described in the lubrication approximation by a coupled system of nonlinear PDEs. When the total quantity of surfactant is fixed, a traveling wave solution exists. For the case of constant flux of surfactant from an upstream reservoir, global traveling waves no longer exist as the surfactant accumulates at the leading edge of the thin film profile. The dynamics can be described using matched asymptotic expansions for t →∞ . The solution is constructed from quasistatically evolving traveling waves. The rate of growth of the surfactant profile is shown to be O( √ t) and is supported by numerical simulations.
{"title":"Growing Surfactant Waves in Thin Liquid Films Driven by Gravity","authors":"Thomas P. Witelski, M. Shearer, R. Levy","doi":"10.1155/AMRX/2006/15487","DOIUrl":"https://doi.org/10.1155/AMRX/2006/15487","url":null,"abstract":"The dynamics of a gravity-driven thin film flow with insoluble surfactant are described in the lubrication approximation by a coupled system of nonlinear PDEs. When the total quantity of surfactant is fixed, a traveling wave solution exists. For the case of constant flux of surfactant from an upstream reservoir, global traveling waves no longer exist as the surfactant accumulates at the leading edge of the thin film profile. The dynamics can be described using matched asymptotic expansions for t →∞ . The solution is constructed from quasistatically evolving traveling waves. The rate of growth of the surfactant profile is shown to be O( √ t) and is supported by numerical simulations.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"31 1","pages":"15487"},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81326682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
When a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behavior of a film, when there is strong slip at a liquid-substrate interface. The film can be modeled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.
{"title":"Intermediate-asymptotic structure of a dewetting rim with strong slip","authors":"P. L. Evans, J. King, A. Münch","doi":"10.1155/AMRX/2006/25262","DOIUrl":"https://doi.org/10.1155/AMRX/2006/25262","url":null,"abstract":"When a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behavior of a film, when there is strong slip at a liquid-substrate interface. The film can be modeled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"237 1","pages":"25262"},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76941464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
During the coalescence of a drop with a planar interface, a hole is generated in the microscopic film that separates the drop from the interface. A bridge forms, connecting the droptotheliquidbulk.Anexperimentalstudyhasbeenperformedtoinvestigatethetime dependence of the behavior of the bridge. A high-speed digital camera recorded the process from underneath the interface. During film rupture, the radius of the bridge demonstrated a power-law time dependence. These results agree with the power-law behavior predicted for drop/drop coalescence and we show that when using correct dimensionless forms, they fully agree with numerical simulations for variety of fluid combinations.
{"title":"Inertia-Dominated Coalescence of Drops","authors":"H. Aryafar, A. Lukyanets, H. Kavehpour","doi":"10.1155/AMRX/2006/94630","DOIUrl":"https://doi.org/10.1155/AMRX/2006/94630","url":null,"abstract":"During the coalescence of a drop with a planar interface, a hole is generated in the microscopic film that separates the drop from the interface. A bridge forms, connecting the droptotheliquidbulk.Anexperimentalstudyhasbeenperformedtoinvestigatethetime dependence of the behavior of the bridge. A high-speed digital camera recorded the process from underneath the interface. During film rupture, the radius of the bridge demonstrated a power-law time dependence. These results agree with the power-law behavior predicted for drop/drop coalescence and we show that when using correct dimensionless forms, they fully agree with numerical simulations for variety of fluid combinations.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"1 1","pages":"94630"},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83713174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Validation of a three-dimensional vortex particle method for fluid flows","authors":"A. Giovannini, Y. Gagnon","doi":"10.1155/AMRX/2006/17027","DOIUrl":"https://doi.org/10.1155/AMRX/2006/17027","url":null,"abstract":"","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"34 1","pages":"17027"},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78357137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Ω ⊂ R (N ≥ 1) be a bounded connected open set whose boundary ∂Ω is regular enough (e.g., ∂Ω ∈ C). Let ω ⊂ Ω be a (small) nonempty open subset and let T > 0. We will use the notation Q = Ω × (0, T) and Σ = ∂Ω × (0, T) and we will denote by n(x) the outward unit normal toΩ at the point x ∈ ∂Ω. In this paper we deal with the controllability properties of some nonlinear parabolic equations for which the nonlinear terms are related to time derivatives and/or second-order spatial derivatives. As usual,we will first strict ourselves to similar linear systems and then we will be concerned with the original nonlinear problems. For the controllability analysis of the linear systems, the main tool will be a new Carleman estimate that holds for very weak solutions, that is, for solutions that only belong to L(Q) = L(0, T ; L(Ω)), of appropriate linear parabolic systems. The sense we will give to these solutions comes from the formulation by transposition of the corresponding systems. More precisely, let us assume that φ ∈ H(Ω) and f, F, G, andH satisfy
设Ω∧R (N≥1)是一个边界∂Ω足够正则(例如∂Ω∈C)的有界连通开集，设Ω∧Ω是一个(小的)非空开子集，设T > 0。我们将使用符号Q = Ω x (0, T)和Σ =∂Ω x (0, T)，我们将用n(x)表示x∈∂Ω处的向外单位法线toΩ。本文研究了一类非线性项与时间导数和/或二阶空间导数有关的非线性抛物型方程的可控性。像往常一样，我们会先把自己严格限定在类似的线性系统上，然后再考虑原来的非线性问题。对于线性系统的可控性分析，主要工具将是一个新的Carleman估计，它适用于非常弱解，即只属于L(Q) = L(0, T)的解;L(Ω))，适当的线性抛物系统。我们对这些解的理解来自于相应系统的变换。更准确地说，我们假设φ∈H(Ω)和f, f, G, H满足
{"title":"Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability","authors":"E. Fernández-Cara, S. Guerrero","doi":"10.1155/AMRX/2006/75090","DOIUrl":"https://doi.org/10.1155/AMRX/2006/75090","url":null,"abstract":"Let Ω ⊂ R (N ≥ 1) be a bounded connected open set whose boundary ∂Ω is regular enough (e.g., ∂Ω ∈ C). Let ω ⊂ Ω be a (small) nonempty open subset and let T > 0. We will use the notation Q = Ω × (0, T) and Σ = ∂Ω × (0, T) and we will denote by n(x) the outward unit normal toΩ at the point x ∈ ∂Ω. In this paper we deal with the controllability properties of some nonlinear parabolic equations for which the nonlinear terms are related to time derivatives and/or second-order spatial derivatives. As usual,we will first strict ourselves to similar linear systems and then we will be concerned with the original nonlinear problems. For the controllability analysis of the linear systems, the main tool will be a new Carleman estimate that holds for very weak solutions, that is, for solutions that only belong to L(Q) = L(0, T ; L(Ω)), of appropriate linear parabolic systems. The sense we will give to these solutions comes from the formulation by transposition of the corresponding systems. More precisely, let us assume that φ ∈ H(Ω) and f, F, G, andH satisfy","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"21 1","pages":"75090"},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77987090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Author(s): Welsh, Edward; Birnir, Bjorn; Bertozzi, Andrea L. | Abstract: Analysis of an evolution model for a river channel shows how three types of shocks determine the profile of the channel. This model shows that in a young river channel, evolution is driven by white noise magnifying into a bore followed by a hydraulic jump. This mechanism produces a convex profile typical of young landscapes. A small knick-point then develops at the bottom of the unstable convex profile. This knick-point is magnified and colored into a diffusive shock which travels upslope, digging into the convex profile until the profile becomes concave, typical of mature landscapes.
{"title":"Shocks in the evolution of an eroding channel","authors":"E. Welsh, B. Birnir, A. Bertozzi","doi":"10.1155/AMRX/2006/71638","DOIUrl":"https://doi.org/10.1155/AMRX/2006/71638","url":null,"abstract":"Author(s): Welsh, Edward; Birnir, Bjorn; Bertozzi, Andrea L. | Abstract: Analysis of an evolution model for a river channel shows how three types of shocks determine the profile of the channel. This model shows that in a young river channel, evolution is driven by white noise magnifying into a bore followed by a hydraulic jump. This mechanism produces a convex profile typical of young landscapes. A small knick-point then develops at the bottom of the unstable convex profile. This knick-point is magnified and colored into a diffusive shock which travels upslope, digging into the convex profile until the profile becomes concave, typical of mature landscapes.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73435845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of minimizing a convex linear-fractional separable function over a feasible region defined by a convex inequality constraint or linear inequality constraint, and bounds on the variables (box constraints). These problems are interesting from both theoretical and practical points of view because they arise in some mathematical programming problems and in various practical problems. Polynomial algorithms for solving such problems are proposed and their convergence is proved. Some examples and results of numerical experiments are also presented.
{"title":"Minimization of a Convex Linear-Fractional Separable Function Subject to a Convex Inequality Constraint or Linear Inequality Constraint and Bounds on the Variables","authors":"S. Stefanov","doi":"10.1155/AMRX/2006/36581","DOIUrl":"https://doi.org/10.1155/AMRX/2006/36581","url":null,"abstract":"We consider the problem of minimizing a convex linear-fractional separable function over a feasible region defined by a convex inequality constraint or linear inequality constraint, and bounds on the variables (box constraints). These problems are interesting from both theoretical and practical points of view because they arise in some mathematical programming problems and in various practical problems. Polynomial algorithms for solving such problems are proposed and their convergence is proved. Some examples and results of numerical experiments are also presented.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"65 1","pages":"36581"},"PeriodicalIF":0.0,"publicationDate":"2006-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83538980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of $n$ independent copies, with good dependence in $n$.
{"title":"Concentration for independent random variables with heavy tails","authors":"F. Barthe, P. Cattiaux, C. Roberto","doi":"10.1155/AMRX.2005.39","DOIUrl":"https://doi.org/10.1155/AMRX.2005.39","url":null,"abstract":"If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of $n$ independent copies, with good dependence in $n$.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"19 1","pages":"39-60"},"PeriodicalIF":0.0,"publicationDate":"2005-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88684628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Boussinesq-Rayleigh series solutions of the unsteadyNavier-Stokes equations are computed symbolically in twodimensions. For finite Reynolds numbers, a nonlinear system ofdifferential recurrent relations admits two formal solutions: ageneral solution for flows forced by the dynamic pressure and ageneral solution for freestreams. For generating functions, whichare bounded together with their derivatives, the absoluteconvergence of the series solutions is shown symbolically byconverting the differential recurrent relations into tensorrecurrent relations and using the comparison and ratio tests. Atriangular structure of three-dimensional tensors of derivativesemployed in the tensor recurrent relations is obtained byinduction. A detailed examination of four basic forced flows andfour basic freestreams shows that the formal series solutions awayfrom boundaries are nonlinear superpositions of the Stokes flow, the Bernoulli flow, the Couette flow, and the Poiseuille flow thatare unsteady, two-dimensional continuations of the classicalsolutions at high Reynolds numbers. A tensor algorithm fornumerical evaluation and continuation of the series solutions isimplemented by parallel computing. Emergence of multi-scalecoherent structures of the Poiseuille flow at high Reynoldsnumbers is tackled by using multivalued contours of the streamfunction.
{"title":"The Boussinesq-Rayleigh series for two-dimensional flows away from boundaries","authors":"V. Miroshnikov","doi":"10.1155/AMRX.2005.183","DOIUrl":"https://doi.org/10.1155/AMRX.2005.183","url":null,"abstract":"The Boussinesq-Rayleigh series solutions of the unsteadyNavier-Stokes equations are computed symbolically in twodimensions. For finite Reynolds numbers, a nonlinear system ofdifferential recurrent relations admits two formal solutions: ageneral solution for flows forced by the dynamic pressure and ageneral solution for freestreams. For generating functions, whichare bounded together with their derivatives, the absoluteconvergence of the series solutions is shown symbolically byconverting the differential recurrent relations into tensorrecurrent relations and using the comparison and ratio tests. Atriangular structure of three-dimensional tensors of derivativesemployed in the tensor recurrent relations is obtained byinduction. A detailed examination of four basic forced flows andfour basic freestreams shows that the formal series solutions awayfrom boundaries are nonlinear superpositions of the Stokes flow, the Bernoulli flow, the Couette flow, and the Poiseuille flow thatare unsteady, two-dimensional continuations of the classicalsolutions at high Reynolds numbers. A tensor algorithm fornumerical evaluation and continuation of the series solutions isimplemented by parallel computing. Emergence of multi-scalecoherent structures of the Poiseuille flow at high Reynoldsnumbers is tackled by using multivalued contours of the streamfunction.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"1 1","pages":"183-227"},"PeriodicalIF":0.0,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89570675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pattern analysis of naturally synthesized images is crucial for a number of important fields includingimage processing, computer vision, artificial intelligence, andcomputer graphics. Benefited from several important works inexistence, the current research paper proposes a novelfree-boundary variational model for segmented imagedecomposition. As an inverse problem solver, the new modeloutputs not only the boundaries of individual objects as achievedby the Mumford-Shah model, but also a structure decompositioncomprising a smooth (or cartoonish) component, an oscillatorycomponent (or texture), and a square-integrable residue (ornoise). Motivations and justifications from vision research areemphasized, and some preliminary mathematical analysis is given.
{"title":"Piecewise H−1+H0+H1 images and the Mumford-Shah-Sobolevmodel for segmented image decomposition","authors":"Jianhong Shen","doi":"10.1155/AMRX.2005.143","DOIUrl":"https://doi.org/10.1155/AMRX.2005.143","url":null,"abstract":"Pattern analysis of naturally synthesized images is crucial for a number of important fields includingimage processing, computer vision, artificial intelligence, andcomputer graphics. Benefited from several important works inexistence, the current research paper proposes a novelfree-boundary variational model for segmented imagedecomposition. As an inverse problem solver, the new modeloutputs not only the boundaries of individual objects as achievedby the Mumford-Shah model, but also a structure decompositioncomprising a smooth (or cartoonish) component, an oscillatorycomponent (or texture), and a square-integrable residue (ornoise). Motivations and justifications from vision research areemphasized, and some preliminary mathematical analysis is given.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"15 1","pages":"143-167"},"PeriodicalIF":0.0,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79639135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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