Simon M Finney, Matthew G Hennessy, Andreas Münch, Sarah L Waters
We study an elastic particle translating axially along the centre-line of a rigid cylindrical tube filled with a Newtonian viscous fluid. The flow is pressure-driven and an axial body force is applied to the particle. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small elastic strains in the particle. In this case, there is a one-way decoupling of the fluid-structure interaction problem. The leading-order fluid problem is shown to be pressure-driven Stokes flow past a rigid sphere, and is solved using the semi-analytical method of reflections. The traction exerted by the fluid on the particle can be computed and used to formulate a pure solid-mechanics problem for the deformation of the particle, which can be solved analytically. This framework is used to investigate the role of the background flow, an axial body force, and the tube wall on the particle’s leading-order translational velocity, resulting deformation, and induced solid stress. By considering the first-order fluid problem the next-order correction to the translational velocity of the particle is shown to be zero. Depending on the magnitude of the ratio of applied body force to viscous forces, the particle can either have a bullet-like shape, an anti-bullet shape, or retain its original spherical shape. A non-linear arbitrary Lagrangian-Eulerian finite element implementation is used, in conjunction with various existing results from the literature, to validate the method of reflections solutions and interrogate their range of validity.
{"title":"The impact of confinement on the deformation of an elastic particle under axisymmetric tube flow","authors":"Simon M Finney, Matthew G Hennessy, Andreas Münch, Sarah L Waters","doi":"10.1093/imamat/hxae022","DOIUrl":"https://doi.org/10.1093/imamat/hxae022","url":null,"abstract":"We study an elastic particle translating axially along the centre-line of a rigid cylindrical tube filled with a Newtonian viscous fluid. The flow is pressure-driven and an axial body force is applied to the particle. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small elastic strains in the particle. In this case, there is a one-way decoupling of the fluid-structure interaction problem. The leading-order fluid problem is shown to be pressure-driven Stokes flow past a rigid sphere, and is solved using the semi-analytical method of reflections. The traction exerted by the fluid on the particle can be computed and used to formulate a pure solid-mechanics problem for the deformation of the particle, which can be solved analytically. This framework is used to investigate the role of the background flow, an axial body force, and the tube wall on the particle’s leading-order translational velocity, resulting deformation, and induced solid stress. By considering the first-order fluid problem the next-order correction to the translational velocity of the particle is shown to be zero. Depending on the magnitude of the ratio of applied body force to viscous forces, the particle can either have a bullet-like shape, an anti-bullet shape, or retain its original spherical shape. A non-linear arbitrary Lagrangian-Eulerian finite element implementation is used, in conjunction with various existing results from the literature, to validate the method of reflections solutions and interrogate their range of validity.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"4 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Takeo Kamizawa, Andrzej Jamiołkowski, Takashi Matsuoka, Noboru Watanabe
In this paper we will discuss the p-reducibility/irreducibility of positive polynomials, and we will give some sufficient conditions for quintic polynomials to be p-reducible/irreducible. This research is closely related to some problems in bio-chemistry, especially to the cooperativity in bio-systems. We will study some applications of our results to these problems.
{"title":"On the P-Irreducibility of Quintic Positive Polynomials","authors":"Takeo Kamizawa, Andrzej Jamiołkowski, Takashi Matsuoka, Noboru Watanabe","doi":"10.1093/imamat/hxae021","DOIUrl":"https://doi.org/10.1093/imamat/hxae021","url":null,"abstract":"In this paper we will discuss the p-reducibility/irreducibility of positive polynomials, and we will give some sufficient conditions for quintic polynomials to be p-reducible/irreducible. This research is closely related to some problems in bio-chemistry, especially to the cooperativity in bio-systems. We will study some applications of our results to these problems.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"60 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new Omega Calculus (a.k.a MacMahon’s Partition Analysis) based integral-free representation for the solution of a non-autonomous and non-homogeneous evolution equation. Our new representation generalizes some of the main results of the recent work of Francisco Neto (2024, A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus. Ann. Inst. Henri Poincaré D, 11, 383) and Bassom et al. (2023, An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation. Appl. Math. Lett., 139, 108537) and show that we can indeed compute the coefficients of the Maclaurin series solution associated with the evolution equation starting with the Peano-Baker series. Furthermore, we discuss in the context of our framework the inverse problem for homogeneous evolution equations in a Hilbert space answering a question left open by Bassom et al. in this case; that is, assuming the solution of the homogeneous evolution equation is a known analytic function the problem concerns the determination of the associated generator of the dynamics. Finally, in order to illustrate the versatility of our approach we explicitly determine the Maclaurin series solution related to the power series method in the context of the vibration problems for the non-uniform (tapered) Euler-Bernoulli beam and thus we explicitly solve the recursion relations considered by Adair and Jaeger (2018, A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams. J. Vib. Control, 24, 3855-3864).
我们给出了一种新的基于欧米茄微积分（又称麦克马洪分区分析法）的无积分表示法，用于求解非自治和非均质演化方程。我们的新表示法概括了弗朗西斯科-内托（Francisco Neto，2024，A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus.Ann.Henri Poincaré D, 11, 383）和 Bassom 等人（2023，经典非自治抽象演化方程的显式麦克劳林级数解。Appl.Lett., 139, 108537），并证明我们确实可以从皮诺-贝克级数开始计算与演化方程相关的麦克劳林级数解的系数。此外，我们还讨论了希尔伯特空间中同质演化方程的逆问题，回答了巴索姆等人在这种情况下提出的一个问题，即假设同质演化方程的解是一个已知的解析函数，那么问题就涉及如何确定相关的动力学发电机。最后，为了说明我们方法的多功能性，我们在非均匀（锥形）欧拉-伯努利梁的振动问题中明确确定了与幂级数方法相关的 Maclaurin 级数解，从而明确解决了 Adair 和 Jaeger（2018，A power series solution for rotating nonuniform Euler-Bernoulli cantilever beams.J. Vib.Control, 24, 3855-3864）。
{"title":"An explicit Maclaurin series solution to non-autonomous and non-homogeneous evolution equation, Omega Calculus, and associated applications","authors":"Antônio Francisco Neto","doi":"10.1093/imamat/hxae020","DOIUrl":"https://doi.org/10.1093/imamat/hxae020","url":null,"abstract":"We give a new Omega Calculus (a.k.a MacMahon’s Partition Analysis) based integral-free representation for the solution of a non-autonomous and non-homogeneous evolution equation. Our new representation generalizes some of the main results of the recent work of Francisco Neto (2024, A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus. Ann. Inst. Henri Poincaré D, 11, 383) and Bassom et al. (2023, An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation. Appl. Math. Lett., 139, 108537) and show that we can indeed compute the coefficients of the Maclaurin series solution associated with the evolution equation starting with the Peano-Baker series. Furthermore, we discuss in the context of our framework the inverse problem for homogeneous evolution equations in a Hilbert space answering a question left open by Bassom et al. in this case; that is, assuming the solution of the homogeneous evolution equation is a known analytic function the problem concerns the determination of the associated generator of the dynamics. Finally, in order to illustrate the versatility of our approach we explicitly determine the Maclaurin series solution related to the power series method in the context of the vibration problems for the non-uniform (tapered) Euler-Bernoulli beam and thus we explicitly solve the recursion relations considered by Adair and Jaeger (2018, A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams. J. Vib. Control, 24, 3855-3864).","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"39 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tamara G Grossmann, Urszula Julia Komorowska, Jonas Latz, Carola-Bibane Schönlieb
Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen–Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.
{"title":"Can physics-informed neural networks beat the finite element method?","authors":"Tamara G Grossmann, Urszula Julia Komorowska, Jonas Latz, Carola-Bibane Schönlieb","doi":"10.1093/imamat/hxae011","DOIUrl":"https://doi.org/10.1093/imamat/hxae011","url":null,"abstract":"Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen–Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"20 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141145861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Martin Benning, Tatiana A Bubba, Luca Ratti, Danilo Riccio
Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.
{"title":"Trust your source: quantifying source condition elements for variational regularisation methods","authors":"Martin Benning, Tatiana A Bubba, Luca Ratti, Danilo Riccio","doi":"10.1093/imamat/hxae008","DOIUrl":"https://doi.org/10.1093/imamat/hxae008","url":null,"abstract":"Source conditions are a key tool in regularisation theory that are needed to derive error estimates and convergence rates for ill-posed inverse problems. In this paper, we provide a recipe to practically compute source condition elements as the solution of convex minimisation problems that can be solved with first-order algorithms. We demonstrate the validity of our approach by testing it on two inverse problem case studies in machine learning and image processing: sparse coefficient estimation of a polynomial via LASSO regression and recovering an image from a subset of the coefficients of its discrete Fourier transform. We further demonstrate that the proposed approach can easily be modified to solve the machine learning task of identifying the optimal sampling pattern in the Fourier domain for a given image and variational regularisation method, which has applications in the context of sparsity promoting reconstruction from magnetic resonance imaging data.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"76 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140117449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates a two-species chemotaxis-fluid system with indirect pursuit-evasion interaction in a bounded domain with smooth boundary. Under suitably regular initial data and no-flux/no-flux/no-flux/no-flux/Dirichlet boundary conditions, we prove that the system possesses a global bounded classical solution in the two-dimensional and three-dimensional cases. Our results extend the result obtained in previously known ones and partly result is new.
{"title":"Global existence and boundedness in a two-species chemotaxis-fluid system with indirect pursuit-evasion interaction","authors":"Chao Liu, Bin Liu","doi":"10.1093/imamat/hxae009","DOIUrl":"https://doi.org/10.1093/imamat/hxae009","url":null,"abstract":"This paper investigates a two-species chemotaxis-fluid system with indirect pursuit-evasion interaction in a bounded domain with smooth boundary. Under suitably regular initial data and no-flux/no-flux/no-flux/no-flux/Dirichlet boundary conditions, we prove that the system possesses a global bounded classical solution in the two-dimensional and three-dimensional cases. Our results extend the result obtained in previously known ones and partly result is new.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140026269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A comparative analysis is performed on the effects of heterogeneity in both linear and nonlinear material characteristics of half-space on the propagation of bright and dark solitary Love waves in a nonlinear layered half-space. The layer is assumed to be homogeneous, nonlinear, elastic while the half-space is vertically heterogeneous. The problem is formulated for two types of elastic materials, incompressible and generalized neo-Hookean materials, and the differences caused by the two materials in the formulation are revealed. For the media composed of generalized neo-Hookean materials, a nonlinear Schrödinger (NLS) equation describing the self interaction of Love waves is obtained by utilizing multiple scales method. The differences in the effects of linear and nonlinear material properties of the half-space on both the existence and nonlinear evolution of bright and dark solitary Love waves are compared graphically.
{"title":"The comparison between effects of heterogeneous and homogeneous half-spaces underlying homogeneous layer on solitary Love waves","authors":"Ekin Deliktas-Ozdemir","doi":"10.1093/imamat/hxae007","DOIUrl":"https://doi.org/10.1093/imamat/hxae007","url":null,"abstract":"A comparative analysis is performed on the effects of heterogeneity in both linear and nonlinear material characteristics of half-space on the propagation of bright and dark solitary Love waves in a nonlinear layered half-space. The layer is assumed to be homogeneous, nonlinear, elastic while the half-space is vertically heterogeneous. The problem is formulated for two types of elastic materials, incompressible and generalized neo-Hookean materials, and the differences caused by the two materials in the formulation are revealed. For the media composed of generalized neo-Hookean materials, a nonlinear Schrödinger (NLS) equation describing the self interaction of Love waves is obtained by utilizing multiple scales method. The differences in the effects of linear and nonlinear material properties of the half-space on both the existence and nonlinear evolution of bright and dark solitary Love waves are compared graphically.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"12 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139954311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.
{"title":"Tangential Tensor Fields on Deformable Surfaces – How to Derive Consistent L2-Gradient Flows","authors":"Ingo Nitschke, Souhayl Sadik, Axel Voigt","doi":"10.1093/imamat/hxae006","DOIUrl":"https://doi.org/10.1093/imamat/hxae006","url":null,"abstract":"We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"4 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139926822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Siiri Rautio, Rashmi Murthy, Tatiana A Bubba, Matti Lassas, Samuli Siltanen
Limited-angle tomography is a highly ill-posed linear inverse problem. It arises in many applications, such as digital breast tomosynthesis. Reconstructions from limited-angle data typically suffer from severe stretching of features along the central direction of projections, leading to poor separation between slices perpendicular to the central direction. In this paper, a new method is introduced, based on machine learning and geometry, producing an estimate for interfaces between regions of different X-ray attenuation. The estimate can be presented on top of the reconstruction, indicating more reliably the separation between features. The method uses directional edge detection, implemented using complex wavelets and enhanced with morphological operations. By using convolutional neural networks, the visible part of the singular support is first extracted and then extended to the full domain, filling in the parts of the singular support that would otherwise be hidden due to the lack of measurement directions.
限角断层扫描是一个高难度线性逆问题。它出现在许多应用中，如数字乳腺断层合成。有限角度数据的重建通常会受到沿投影中心方向特征严重拉伸的影响，导致垂直于中心方向的切片之间分离不佳。本文介绍了一种基于机器学习和几何学的新方法，可对不同 X 射线衰减区域之间的界面进行估计。该估计值可以在重建的基础上显示，从而更可靠地显示特征之间的分离情况。该方法使用复杂小波实现定向边缘检测，并通过形态学操作进行增强。通过使用卷积神经网络，首先提取奇异支撑的可见部分，然后扩展到全域，填补奇异支撑中由于缺乏测量方向而被隐藏的部分。
{"title":"Learning a microlocal prior for limited-angle tomography","authors":"Siiri Rautio, Rashmi Murthy, Tatiana A Bubba, Matti Lassas, Samuli Siltanen","doi":"10.1093/imamat/hxae005","DOIUrl":"https://doi.org/10.1093/imamat/hxae005","url":null,"abstract":"Limited-angle tomography is a highly ill-posed linear inverse problem. It arises in many applications, such as digital breast tomosynthesis. Reconstructions from limited-angle data typically suffer from severe stretching of features along the central direction of projections, leading to poor separation between slices perpendicular to the central direction. In this paper, a new method is introduced, based on machine learning and geometry, producing an estimate for interfaces between regions of different X-ray attenuation. The estimate can be presented on top of the reconstruction, indicating more reliably the separation between features. The method uses directional edge detection, implemented using complex wavelets and enhanced with morphological operations. By using convolutional neural networks, the visible part of the singular support is first extracted and then extended to the full domain, filling in the parts of the singular support that would otherwise be hidden due to the lack of measurement directions.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"283 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139753193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lyndon Koens, Rohan Vernekar, Timm Krüger, Maciej Lisicki, David W Inglis
The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media and microfluidic arrays, this solution is important for many real-world systems. We asymptotically determine the flow around a general rectangular doubly-periodic array of infinite slender cylinders, extending the existing asymptotic solution for square arrays. The flow in the cell is represented by a collection of doubly-periodic, rapidly-convergent two-dimensional singularity solutions, and the boundary condition on the surface of the cylinder is solved asymptotically in powers of the cylinder radius. The asymptotic solution provides an easily computed closed-form estimate for the flow and forces as a function of the radius and the dimensions of the cell. The force is compared to results from lattice-Boltzmann simulations of low-Reynolds-number flows in the same geometry, and the accuracy of the no-slip condition on the surface of the cylinder, predicted by the asymptotic theory, is assessed. Finally, the behaviour of the flow, flux, force and effective permeability of the cell is investigated as a function of the geometric parameters. The structure of the asymptotic permeability is consistent with previous single-geometry predictions but provides a closed-form estimate for how the aspect ratio of the cell changes the leading-order behaviour. These models could be used to help understand the flows within porous systems composed of fibres and systems involving periodic arrays such as systems based on deterministic lateral displacement.
{"title":"The slow viscous flow around a general rectangular doubly-periodic arrays of infinite slender cylinders","authors":"Lyndon Koens, Rohan Vernekar, Timm Krüger, Maciej Lisicki, David W Inglis","doi":"10.1093/imamat/hxae003","DOIUrl":"https://doi.org/10.1093/imamat/hxae003","url":null,"abstract":"The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media and microfluidic arrays, this solution is important for many real-world systems. We asymptotically determine the flow around a general rectangular doubly-periodic array of infinite slender cylinders, extending the existing asymptotic solution for square arrays. The flow in the cell is represented by a collection of doubly-periodic, rapidly-convergent two-dimensional singularity solutions, and the boundary condition on the surface of the cylinder is solved asymptotically in powers of the cylinder radius. The asymptotic solution provides an easily computed closed-form estimate for the flow and forces as a function of the radius and the dimensions of the cell. The force is compared to results from lattice-Boltzmann simulations of low-Reynolds-number flows in the same geometry, and the accuracy of the no-slip condition on the surface of the cylinder, predicted by the asymptotic theory, is assessed. Finally, the behaviour of the flow, flux, force and effective permeability of the cell is investigated as a function of the geometric parameters. The structure of the asymptotic permeability is consistent with previous single-geometry predictions but provides a closed-form estimate for how the aspect ratio of the cell changes the leading-order behaviour. These models could be used to help understand the flows within porous systems composed of fibres and systems involving periodic arrays such as systems based on deterministic lateral displacement.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"29 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139661342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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