Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time t evolves from large negative to large positive, the triangle lump reverses itself along the y -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the y -direction when time t evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.
本文研究的是 Davey-Stewartson (DS) II 方程。基于 Kadomtsev-Petviashvili (KP) 简化法和舒尔多项式理论，我们构建了一般高阶块体解。通过提取真解的舒尔多项式的前阶渐近值，得出了由基本块体及其位置组成的预测解。当选择解的索引为不同的正整数组合时，大时间的预测解反映了真解的两类块状模式。第一类是三角形的块状模式，由 Yablonskii-Vorob'ev 多项式的根结构分析描述。当时间 t 由大负值变为大正值时，三角形凸块沿 y 方向反转。第二类块状模式包括外部区域的非三角形，它由弗伦斯基-赫米特多项式的非零根结构分析描述，以及内部区域的可能三角形，它由雅布隆斯基-沃罗布夫多项式的根结构分析描述。此外，当时间 t 从大负值变为大正值时，外部区域的非三角形块状图案会旋转一个角度，而内部区域的可能三角形块状图案则会沿 y 方向反转。这些结果加深了我们对高阶凸块时间演化机制的理解。
{"title":"Prediction of general high-order lump solutions in the Davey–Stewartson II equation","authors":"Xue-Wei Yan, Haie Long, Yong Chen","doi":"10.1098/rspa.2023.0455","DOIUrl":"https://doi.org/10.1098/rspa.2023.0455","url":null,"abstract":"Under investigation in this work is the Davey–Stewartson (DS) II equation. Based on the Kadomtsev–Petviashvili (KP) reduction method and Schur polynomial theory, we construct the general high-order lump solutions. The prediction solutions consisting of fundamental lumps and their positions are derived by extracting leading-order asymptotics of the Schur polynomials of true solutions. When indexes of the solutions are chosen as different positive integer combinations, the prediction solutions at large time reflect two classes of lump patterns of the true solutions. The first class of lump pattern with triangle shape is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. When time <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive, the triangle lump reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction. The second class of lump pattern consists of non-triangle in outer region, which is analytically described by non-zero root structure of the Wronskian–Hermit polynomial, together with possible triangle in the inner region, which is analytically described by root structure of the Yablonskii–Vorob’ev polynomial. In addition, the non-triangle lump pattern in outer regions rotates at an angle while the possible triangle lump pattern in the inner region reverses itself along the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>y</mml:mi> </mml:math> </jats:inline-formula> -direction when time <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> </mml:math> </jats:inline-formula> evolves from large negative to large positive. The obtained results improve our understanding of time evolution mechanisms of high-order lumps.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":3.5,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties.
{"title":"Effective statistical control strategies for complex turbulent dynamical systems","authors":"Jeffrey Covington, Di Qi, Nan Chen","doi":"10.1098/rspa.2023.0546","DOIUrl":"https://doi.org/10.1098/rspa.2023.0546","url":null,"abstract":"Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135515145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Honeycomb materials frequently encounter hypergravity conditions in both aerospace and biological contexts during phases such as launch, reentry or under centrifugal motion. The significant body force engendered by hypergravity induces alterations in the microstructure of honeycomb materials, which in turn, influences their macroscopic mechanical behaviour. Leveraging the stiffness of the beam element as a pivotal variable, we successfully derived the equivalent moduli of the honeycomb material under hypergravity conditions. We further proposed the concept of a ‘hypergravity factor’, elucidating that the density of the base material, the dimensions of honeycomb cells and the magnitude of the hypergravity contribute to amplifying hypergravity effects. The results, numerically validated through finite-element simulations, could be reduced to the case that neglects body force. The critical buckling load of the honeycomb material under hypergravity can be assessed by setting the derived moduli to zero. In the presence of hypergravity, a honeycomb material undergoes a transition into a gradient material along the hypergravity direction, thereby exacerbating anisotropy. This phenomenon is theoretically expected to occur in virtually all porous materials. The analytical framework adopted, which employs beam stiffness as an intermediary variable, facilitates the extension of these results to honeycomb materials which encompass beam elements with functional gradients or varying cross-sectional morphologies.
{"title":"Equivalent in-plane elastic moduli of honeycomb materials under hypergravity conditions","authors":"Lei Wang, Guannan Wang, Chaofeng Lü","doi":"10.1098/rspa.2023.0638","DOIUrl":"https://doi.org/10.1098/rspa.2023.0638","url":null,"abstract":"Honeycomb materials frequently encounter hypergravity conditions in both aerospace and biological contexts during phases such as launch, reentry or under centrifugal motion. The significant body force engendered by hypergravity induces alterations in the microstructure of honeycomb materials, which in turn, influences their macroscopic mechanical behaviour. Leveraging the stiffness of the beam element as a pivotal variable, we successfully derived the equivalent moduli of the honeycomb material under hypergravity conditions. We further proposed the concept of a ‘hypergravity factor’, elucidating that the density of the base material, the dimensions of honeycomb cells and the magnitude of the hypergravity contribute to amplifying hypergravity effects. The results, numerically validated through finite-element simulations, could be reduced to the case that neglects body force. The critical buckling load of the honeycomb material under hypergravity can be assessed by setting the derived moduli to zero. In the presence of hypergravity, a honeycomb material undergoes a transition into a gradient material along the hypergravity direction, thereby exacerbating anisotropy. This phenomenon is theoretically expected to occur in virtually all porous materials. The analytical framework adopted, which employs beam stiffness as an intermediary variable, facilitates the extension of these results to honeycomb materials which encompass beam elements with functional gradients or varying cross-sectional morphologies.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some of the fundamentals of quantum information science are described, including the concepts of quantum resources, quantum states and mixedness of states. The explanations are detailed and include a combination of basic facts with fully worked examples, and some more advanced topics. The principles of quantum information are illustrated with chemical examples drawn from singlet fission, photophysics of radicals, molecular excitons, electron transfer and so on. Suggestions for prospects and challenges for the field are discussed.
{"title":"A molecular perspective on quantum information","authors":"Gregory D. Scholes","doi":"10.1098/rspa.2023.0599","DOIUrl":"https://doi.org/10.1098/rspa.2023.0599","url":null,"abstract":"Some of the fundamentals of quantum information science are described, including the concepts of quantum resources, quantum states and mixedness of states. The explanations are detailed and include a combination of basic facts with fully worked examples, and some more advanced topics. The principles of quantum information are illustrated with chemical examples drawn from singlet fission, photophysics of radicals, molecular excitons, electron transfer and so on. Suggestions for prospects and challenges for the field are discussed.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135515357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The energy transfer process for acoustic propagation and scattering in inhomogeneous, lossy media is rigorously investigated. Two problems are considered: an arbitrarily composed cluster of lossy scatterers, excited by a single point source and a layered, lossy medium excited by an internal point-source distribution. Energy conservation laws and optical theorems are established. The connection of active intensity with the scattering cross section and the relation between reactive intensity and Lagrangian density are determined. Furthermore, the influence of the losses of the involved media to the kinetic energy is unveiled. Reductions to lossless media and near-zero frequency values are derived.
{"title":"Analysis of the energy transfer process for multiple scattering problems involving lossy media","authors":"Andreas Kalogeropoulos, Nikolaos L. Tsitsas","doi":"10.1098/rspa.2023.0513","DOIUrl":"https://doi.org/10.1098/rspa.2023.0513","url":null,"abstract":"The energy transfer process for acoustic propagation and scattering in inhomogeneous, lossy media is rigorously investigated. Two problems are considered: an arbitrarily composed cluster of lossy scatterers, excited by a single point source and a layered, lossy medium excited by an internal point-source distribution. Energy conservation laws and optical theorems are established. The connection of active intensity with the scattering cross section and the relation between reactive intensity and Lagrangian density are determined. Furthermore, the influence of the losses of the involved media to the kinetic energy is unveiled. Reductions to lossless media and near-zero frequency values are derived.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135161363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive a Sturm–Liouville system of equations for the exact calculation of the survival probability in first-passage time problems. This system is the one associated with the Wiener–Hopf integral equation obtained from the theory of random walks. The derived approach is an alternative to the existing literature and we tested it against direct calculations from both discrete- and continuous-time random walks in a manageable, but meaningful, example. Within this framework, the Sparre Andersen theorem results to be a boundary condition for the system.
{"title":"Sturm–Liouville systems for the survival probability in first-passage time problems","authors":"Marcus Dahlenburg, Gianni Pagnini","doi":"10.1098/rspa.2023.0485","DOIUrl":"https://doi.org/10.1098/rspa.2023.0485","url":null,"abstract":"We derive a Sturm–Liouville system of equations for the exact calculation of the survival probability in first-passage time problems. This system is the one associated with the Wiener–Hopf integral equation obtained from the theory of random walks. The derived approach is an alternative to the existing literature and we tested it against direct calculations from both discrete- and continuous-time random walks in a manageable, but meaningful, example. Within this framework, the Sparre Andersen theorem results to be a boundary condition for the system.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The crosslinking technique (CXL) is an effective low-risk therapeutic treatment of keratoconus and other ectatic disorders of the human cornea. The effect of corneal CXL is to increase the stiffness of the stroma to prevent the progression of the cornea distortion. Several clinical and experimental studies have shown that the stiffening effects predominantly localize on the anterior portion of the stroma and that the in-depth stiffening distribution is highly dependent on the duration of treatment. Yet, how the stiffening effects distribute through the cornea thickness as a function of the treatment duration is an open question. Here, we propose an analytical model of the stiffening profile due to CXL treatment as a function of the irradiation time. We consider linear and nonlinear variations of the crosslinking effects across the thickness and implement them into a finite element model of the porcine cornea. We present a time-dependent in-depth stiffening profile that allows us to predict the post-operative cornea response to physiological intraocular pressure for different irradiation times. We anticipate that this predictive model will support the development of patient specific three-dimensional models that will allow clinicians to design customized CXL treatment, thus enhancing treatment outcomes.
{"title":"A predictive model of UV-A-riboflavin crosslinking treatment on porcine corneas","authors":"Alessandra Bonfanti, Anna Pandolfi","doi":"10.1098/rspa.2023.0323","DOIUrl":"https://doi.org/10.1098/rspa.2023.0323","url":null,"abstract":"The crosslinking technique (CXL) is an effective low-risk therapeutic treatment of keratoconus and other ectatic disorders of the human cornea. The effect of corneal CXL is to increase the stiffness of the stroma to prevent the progression of the cornea distortion. Several clinical and experimental studies have shown that the stiffening effects predominantly localize on the anterior portion of the stroma and that the in-depth stiffening distribution is highly dependent on the duration of treatment. Yet, how the stiffening effects distribute through the cornea thickness as a function of the treatment duration is an open question. Here, we propose an analytical model of the stiffening profile due to CXL treatment as a function of the irradiation time. We consider linear and nonlinear variations of the crosslinking effects across the thickness and implement them into a finite element model of the porcine cornea. We present a time-dependent in-depth stiffening profile that allows us to predict the post-operative cornea response to physiological intraocular pressure for different irradiation times. We anticipate that this predictive model will support the development of patient specific three-dimensional models that will allow clinicians to design customized CXL treatment, thus enhancing treatment outcomes.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In Tabula XCI verso of Codex Atlanticus, Leonardo da Vinci presents an ingenious masonry structure composed of segments in the shape of inverted triangles. These are assembled by contact in a chain to obtain a lintel or jack arch, where they are pressed together by the thrust of the end constraints. Drawing inspiration from Leonardo’s sketches, we show that, by connecting the segments in pair through elastic tendons, this system represents a new type of flextegrity beam. In a classical flextegrity, the contact surfaces of the segments are curved conjugate profiles, imposing a pure rolling motion along properly designed pitch lines: the consequent elongation of the tendon dictates the constitutive response as a function of the relative rotation of the segments. Here, the contact is through plane surfaces, so that the kinematics, besides the relative rotation, is characterized by segmental shearing. This system is theoretically analysed and a continuum model is derived as the length of the segments becomes small. Comparisons with experiments on three-dimensional-printed prototypes confirm the theoretical findings and highlight the possible competition between rotational and sliding types of deformation. Apart from the historical value, this type of construction can be used in innovative structures or metamaterials.
{"title":"Shear and flexural deformations in flextegrity segmental beams inspired by Leonardo’s triangular masonry construction","authors":"Claudio Boni, Gianni Royer-Carfagni","doi":"10.1098/rspa.2023.0453","DOIUrl":"https://doi.org/10.1098/rspa.2023.0453","url":null,"abstract":"In Tabula XCI verso of Codex Atlanticus, Leonardo da Vinci presents an ingenious masonry structure composed of segments in the shape of inverted triangles. These are assembled by contact in a chain to obtain a lintel or jack arch, where they are pressed together by the thrust of the end constraints. Drawing inspiration from Leonardo’s sketches, we show that, by connecting the segments in pair through elastic tendons, this system represents a new type of flextegrity beam. In a classical flextegrity, the contact surfaces of the segments are curved conjugate profiles, imposing a pure rolling motion along properly designed pitch lines: the consequent elongation of the tendon dictates the constitutive response as a function of the relative rotation of the segments. Here, the contact is through plane surfaces, so that the kinematics, besides the relative rotation, is characterized by segmental shearing. This system is theoretically analysed and a continuum model is derived as the length of the segments becomes small. Comparisons with experiments on three-dimensional-printed prototypes confirm the theoretical findings and highlight the possible competition between rotational and sliding types of deformation. Apart from the historical value, this type of construction can be used in innovative structures or metamaterials.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the release and subsequent gravity-driven spreading of a dense finite volume of fluid in an anisotropic porous medium bounded by an impermeable substrate. When the permeability in the vertical direction is much smaller than in the horizontal direction, as is the case in many real geological reservoirs, this restricts the spread of the current to a very thin layer near the impermeable base. Using a combination of asymptotic analysis and finite difference computations of Darcy flow, we show that there exist two distinct flow regimes. At early times, the bulk of the current descends slowly and uniformly, injecting fluid into thin finger-like regions near the base. At much later times, the current transitions to the classical gravity-driven solution and continues to spread with a self-similar shape. One interesting consequence is that the swept volume of the current grows differently depending on the anisotropy of the medium. This has important consequences for managing contaminant spills, where it is important to minimize the contacted volume of the aquifer, or during geological CO2 sequestration where a larger contacted volume results in more CO2 being stored.
{"title":"Anisotropy distorts the spreading of a fixed volume porous gravity current","authors":"Graham P. Benham","doi":"10.1098/rspa.2023.0271","DOIUrl":"https://doi.org/10.1098/rspa.2023.0271","url":null,"abstract":"We consider the release and subsequent gravity-driven spreading of a dense finite volume of fluid in an anisotropic porous medium bounded by an impermeable substrate. When the permeability in the vertical direction is much smaller than in the horizontal direction, as is the case in many real geological reservoirs, this restricts the spread of the current to a very thin layer near the impermeable base. Using a combination of asymptotic analysis and finite difference computations of Darcy flow, we show that there exist two distinct flow regimes. At early times, the bulk of the current descends slowly and uniformly, injecting fluid into thin finger-like regions near the base. At much later times, the current transitions to the classical gravity-driven solution and continues to spread with a self-similar shape. One interesting consequence is that the swept volume of the current grows differently depending on the anisotropy of the medium. This has important consequences for managing contaminant spills, where it is important to minimize the contacted volume of the aquifer, or during geological <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"normal\">CO</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> sequestration where a larger contacted volume results in more <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"normal\">CO</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> being stored.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Developable degree-4 (DD4) vertices have four facets and four creases and can be unfolded flat. The rigid-folding kinematics of DD4 vertices is rich in that it generally has two folding modes and can get stuck when two facets bind together. To study the full spectrum of the kinematics of DD4 vertices, parametric solutions for fold angles in terms of the cotangents of half-angles are derived from the opposite and adjacent fold angle relationships. It is shown that any two fold angles of a general DD4 vertex are related by the equation of a hyperbola. When the vertex has collinear creases or is flat-foldable, the pertinent hyperbola equations degenerate into linear relationships. Meanwhile, when DD4 vertices are classified into three categories according to Grashof’s criterion, both unique and binding folds can be readily located from the facet with the largest or smallest sector angle. The rigid-folding kinematics of typical vertices is then investigated. In addition to the flat state, the two folding modes can also be switched at the binding states if self-intersection is permitted. The results provide new formulae and clear geometric views on the rigid-folding kinematics of DD4 vertices, which are fundamental for constructing larger origami patterns.
{"title":"Parametric solutions to the kinematics of developable degree-4 rigid origami vertices","authors":"Yucai Hu, Changjun Zheng, Chuanxing Bi, Haiyi Liang","doi":"10.1098/rspa.2023.0319","DOIUrl":"https://doi.org/10.1098/rspa.2023.0319","url":null,"abstract":"Developable degree-4 (DD4) vertices have four facets and four creases and can be unfolded flat. The rigid-folding kinematics of DD4 vertices is rich in that it generally has two folding modes and can get stuck when two facets bind together. To study the full spectrum of the kinematics of DD4 vertices, parametric solutions for fold angles in terms of the cotangents of half-angles are derived from the opposite and adjacent fold angle relationships. It is shown that any two fold angles of a general DD4 vertex are related by the equation of a hyperbola. When the vertex has collinear creases or is flat-foldable, the pertinent hyperbola equations degenerate into linear relationships. Meanwhile, when DD4 vertices are classified into three categories according to Grashof’s criterion, both unique and binding folds can be readily located from the facet with the largest or smallest sector angle. The rigid-folding kinematics of typical vertices is then investigated. In addition to the flat state, the two folding modes can also be switched at the binding states if self-intersection is permitted. The results provide new formulae and clear geometric views on the rigid-folding kinematics of DD4 vertices, which are fundamental for constructing larger origami patterns.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135515334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：