Radiation of sound waves by a coaxial rigid duct with perforated screen is investigated by using the Mode Matching technique in conjunction with the Jones’ Method. The geometry of the problem consists semi-infinite outer duct and infinite inner duct. It is assumed that the duct walls are fully rigid. The solution of current study contains an infinite sets of coefficients satisfying an infinite systems of linear algebraic equations. These systems are truncated at a certain number and then solved numerically. The effects of open and perforated case, frequency and porosity on the radiation phenomenon are shown graphically. In the present study, perforated screen makes the problem more interesting when it is compared with the unperforated screen. In this context, solution of the problem is compered numerically with similar studies, which are used different method to obtain Wiener–Hopf equation, existing in the literature. As a result, it is observed that in the absence of a perforated screen, there is a perfect agreement between the two results.
{"title":"Radiation of sound waves from a coaxial duct with perforated screen","authors":"Burhan Tiryakioglu;Ayse Tiryakioglu","doi":"10.1093/imamat/hxab016","DOIUrl":"10.1093/imamat/hxab016","url":null,"abstract":"Radiation of sound waves by a coaxial rigid duct with perforated screen is investigated by using the Mode Matching technique in conjunction with the Jones’ Method. The geometry of the problem consists semi-infinite outer duct and infinite inner duct. It is assumed that the duct walls are fully rigid. The solution of current study contains an infinite sets of coefficients satisfying an infinite systems of linear algebraic equations. These systems are truncated at a certain number and then solved numerically. The effects of open and perforated case, frequency and porosity on the radiation phenomenon are shown graphically. In the present study, perforated screen makes the problem more interesting when it is compared with the unperforated screen. In this context, solution of the problem is compered numerically with similar studies, which are used different method to obtain Wiener–Hopf equation, existing in the literature. As a result, it is observed that in the absence of a perforated screen, there is a perfect agreement between the two results.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"828-844"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45606456","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 use the classical factorization method proposed firstly by Kirsch to reconstruct the support of the mixed inhomogeneous medium associated with complex valued refractive indexes and different transmission boundary conditions. We will show that for well-chosen inhomogeneous backgrounds, one obtains a necessary and sufficient condition characterizing the support of the medium via the eigensystem of a self-adjoint operator, which is related to the far field operator. Moreover, for completeness of our problem, the variational method is applied to solve the direct scattering problem. And, we present a variant of numerical examples in 2D to verify the effectiveness and robustness of the proposed inverse algorithms.
{"title":"The factorization method for the scattering by a mixed inhomogeneous medium","authors":"Jianli Xiang;Guozheng Yan","doi":"10.1093/imamat/hxab017","DOIUrl":"10.1093/imamat/hxab017","url":null,"abstract":"We use the classical factorization method proposed firstly by Kirsch to reconstruct the support of the mixed inhomogeneous medium associated with complex valued refractive indexes and different transmission boundary conditions. We will show that for well-chosen inhomogeneous backgrounds, one obtains a necessary and sufficient condition characterizing the support of the medium via the eigensystem of a self-adjoint operator, which is related to the far field operator. Moreover, for completeness of our problem, the variational method is applied to solve the direct scattering problem. And, we present a variant of numerical examples in 2D to verify the effectiveness and robustness of the proposed inverse algorithms.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"662-687"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43048568","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 optimize the spring constants $k^{i,j}$ (stiffness) of circular spring-mass systems with nearest-neighbour (NN) and next-nearest-neighbour (NNN) springs only. In this optimization problem, such systems are also subjected to damping and periodic external forces. The function to be minimized is the average ratio of the square norm of the on-site internal forces (response) to the square norm of the external on-site forces (input). Under the average of this response/input ratio is meant the average over time and over all configurations of external forces. As main result, it is established that the optimum stiffness matrix converges to the discrete and periodic bi-Laplacian operator as the size $n$ of the system increases. Such a result is obtained under the following assumptions: (a) the system has the natural mode shape (eigenvector) of alternating $1$ s and $-1$ s; and (b) the (external) forcing frequency is at least $1.095$ times higher than the highest natural frequency. It is remarkable that this optimum stiffness matrix exhibits negative stiffness for the springs linking NNN point masses. More specifically, as $n$ increases, $0> k^{i,i+2} , , = , , - tfrac{1}{4} , k^{i,i+1}$ is the relation between the optimum NNN spring constant and the optimum NN spring constant. Such systems illustrate that the introduction of negative stiffness springs in some specific positions does in fact reduce the average response/input ratio. Numerical tables illustrating the main result are given.
{"title":"Optimizing the spring constants of forced, damped and circular spring-mass systems—characterization of the discrete and periodic bi-Laplacian operator","authors":"L L A de Oliveira;M V Travaglia","doi":"10.1093/imamat/hxab021","DOIUrl":"10.1093/imamat/hxab021","url":null,"abstract":"We optimize the spring constants \u0000<tex>$k^{i,j}$</tex>\u0000 (stiffness) of circular spring-mass systems with nearest-neighbour (NN) and next-nearest-neighbour (NNN) springs only. In this optimization problem, such systems are also subjected to damping and periodic external forces. The function to be minimized is the average ratio of the square norm of the on-site internal forces (response) to the square norm of the external on-site forces (input). Under the average of this response/input ratio is meant the average over time and over all configurations of external forces. As main result, it is established that the optimum stiffness matrix converges to the discrete and periodic bi-Laplacian operator as the size \u0000<tex>$n$</tex>\u0000 of the system increases. Such a result is obtained under the following assumptions: (a) the system has the natural mode shape (eigenvector) of alternating \u0000<tex>$1$</tex>\u0000s and \u0000<tex>$-1$</tex>\u0000s; and (b) the (external) forcing frequency is at least \u0000<tex>$1.095$</tex>\u0000 times higher than the highest natural frequency. It is remarkable that this optimum stiffness matrix exhibits negative stiffness for the springs linking NNN point masses. More specifically, as \u0000<tex>$n$</tex>\u0000 increases, \u0000<tex>$0> k^{i,i+2} , , = , , - tfrac{1}{4} , k^{i,i+1}$</tex>\u0000 is the relation between the optimum NNN spring constant and the optimum NN spring constant. Such systems illustrate that the introduction of negative stiffness springs in some specific positions does in fact reduce the average response/input ratio. Numerical tables illustrating the main result are given.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"785-807"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45366354","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 study a hybrid impulsive reaction-diffusion equation composed with a discrete-time map in bounded domain $varOmega $ in space dimension $nin mathbb N$ . We assume that the exterior of domain is not lethal (not completely hostile) but hostile. We consider Robin boundary conditions which are used for mixed or reactive or semipermeable boundaries. Given geometry of the domain $varOmega $ , we establish critical domain sizes for the persistence and extinction of a species. Specifically, for habitats with the shape of $n$ -hypercube and ball of fixed radius, we formulate the critical domain sizes depending on parameters of the model, including $h$ , i.e. a measure of the hostility of the external (to $varOmega $ ) environment. For a general habitat, called Lipschitz domains, we apply isoperimetric inequalities and variational methods to find the associated critical domain sizes. We also provide applications of the main results in marine reserve, terrestrial reserve and insect pest outbreaks.
研究了在空间维数为$n mathbb n $的有界域$varOmega $上由一个离散时间映射组成的混合脉冲反应扩散方程。我们假设领域的外部不是致命的(不是完全敌对的)，而是敌对的。我们考虑Robin边界条件，它用于混合边界或反应边界或半渗透边界。给定域$varOmega $的几何形状，我们建立了物种持续和灭绝的临界域大小。具体来说，对于形状为$n$-超立方体和固定半径球的栖息地，我们根据模型参数(包括$h$)制定了关键域尺寸，即外部(对$varOmega $)环境敌意的度量。对于称为Lipschitz域的一般生境，我们应用等周不等式和变分方法来找到相关的临界域大小。我们还提供了主要结果在海洋保护区、陆地保护区和虫害暴发中的应用。
{"title":"Critical domain sizes of a discrete-map hybrid and reaction-diffusion model on hostile exterior domains","authors":"Mostafa Fazly","doi":"10.1093/imamat/hxab019","DOIUrl":"10.1093/imamat/hxab019","url":null,"abstract":"We study a hybrid impulsive reaction-diffusion equation composed with a discrete-time map in bounded domain \u0000<tex>$varOmega $</tex>\u0000 in space dimension \u0000<tex>$nin mathbb N$</tex>\u0000. We assume that the exterior of domain is not lethal (not completely hostile) but hostile. We consider Robin boundary conditions which are used for mixed or reactive or semipermeable boundaries. Given geometry of the domain \u0000<tex>$varOmega $</tex>\u0000, we establish critical domain sizes for the persistence and extinction of a species. Specifically, for habitats with the shape of \u0000<tex>$n$</tex>\u0000-hypercube and ball of fixed radius, we formulate the critical domain sizes depending on parameters of the model, including \u0000<tex>$h$</tex>\u0000, i.e. a measure of the hostility of the external (to \u0000<tex>$varOmega $</tex>\u0000) environment. For a general habitat, called Lipschitz domains, we apply isoperimetric inequalities and variational methods to find the associated critical domain sizes. We also provide applications of the main results in marine reserve, terrestrial reserve and insect pest outbreaks.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"739-760"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43174103","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}
Marvin Fritz;Christina Kuttler;Mabel L Rajendran;Barbara Wohlmuth;Laura Scarabosio
In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo–Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.
{"title":"On a subdiffusive tumour growth model with fractional time derivative","authors":"Marvin Fritz;Christina Kuttler;Mabel L Rajendran;Barbara Wohlmuth;Laura Scarabosio","doi":"10.1093/imamat/hxab009","DOIUrl":"https://doi.org/10.1093/imamat/hxab009","url":null,"abstract":"In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo–Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"688-729"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50350067","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}
The numerical solution of partial differential equations that govern fluid dynamics with turbulence and combustion is challenging due to the multiscale nature of the dynamical system and the need to resolve small-scale physical features. In addition, the uncertainties in the dynamical system, including those in the physical models and parameters, initial and boundary conditions and numerical methods, impact the computational fluid dynamics (CFD) prediction of turbulence and chemical reactions. To improve the CFD prediction, this study focuses on the development and application of a maximum likelihood ensemble filter (MLEF), an ensemble-based data assimilation (DA), for flows featuring combustion and/or turbulence. MLEF finds the optimal analysis and its uncertainty by maximizing the posterior probability density function. The novelty of the study lies in the combination of advanced DA and CFD methods for a new comprehensive application to predict engineering fluid dynamics. The study combines important aspects, including an ensemble-based DA with analysis and uncertainty estimation, an augmented control vector that simultaneously adjusts initial conditions and model empirical parameters and an application of DA to CFD modeling of combustion and flows with complex geometry. The DA performance is validated by a turbulent Couette flow. The new CFD–DA system is then applied to solve the time-evolving shear-layer mixing with methane-air combustion and the turbulent flow over a bluff-body geometry. Results demonstrate the improvement of estimates of model parameters and the uncertainty reduction in initial conditions (ICs) for CFD modeling of flames and flows by the MLEF method.
{"title":"The maximum likelihood ensemble filter for computational flame and fluid dynamics","authors":"Yijun Wang;Stephen Guzik;Milija Zupanski;Xinfeng Gao","doi":"10.1093/imamat/hxab010","DOIUrl":"10.1093/imamat/hxab010","url":null,"abstract":"The numerical solution of partial differential equations that govern fluid dynamics with turbulence and combustion is challenging due to the multiscale nature of the dynamical system and the need to resolve small-scale physical features. In addition, the uncertainties in the dynamical system, including those in the physical models and parameters, initial and boundary conditions and numerical methods, impact the computational fluid dynamics (CFD) prediction of turbulence and chemical reactions. To improve the CFD prediction, this study focuses on the development and application of a maximum likelihood ensemble filter (MLEF), an ensemble-based data assimilation (DA), for flows featuring combustion and/or turbulence. MLEF finds the optimal analysis and its uncertainty by maximizing the posterior probability density function. The novelty of the study lies in the combination of advanced DA and CFD methods for a new comprehensive application to predict engineering fluid dynamics. The study combines important aspects, including an ensemble-based DA with analysis and uncertainty estimation, an augmented control vector that simultaneously adjusts initial conditions and model empirical parameters and an application of DA to CFD modeling of combustion and flows with complex geometry. The DA performance is validated by a turbulent Couette flow. The new CFD–DA system is then applied to solve the time-evolving shear-layer mixing with methane-air combustion and the turbulent flow over a bluff-body geometry. Results demonstrate the improvement of estimates of model parameters and the uncertainty reduction in initial conditions (ICs) for CFD modeling of flames and flows by the MLEF method.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"631-661"},"PeriodicalIF":1.2,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44645899","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}
Pub Date : 2021-05-15DOI: 10.21203/RS.3.RS-563354/V1
Zofia Wr'oblewska, P. Kowalczyk, Lukasz Plociniczak
We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.
{"title":"Stability of Fixed Points in an Approximate Solution of the Spring-mass Running Model","authors":"Zofia Wr'oblewska, P. Kowalczyk, Lukasz Plociniczak","doi":"10.21203/RS.3.RS-563354/V1","DOIUrl":"https://doi.org/10.21203/RS.3.RS-563354/V1","url":null,"abstract":"\u0000 We consider a classical spring-mass model of human running which is built upon an inverted elastic pendulum. Based on our previous results concerning asymptotic solutions for large spring constant (or small angle of attack), we construct analytical approximations of solutions in the considered model. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). By appropriately concatenating asymptotic solutions for the two phases we are able to reduce the dynamics to a one-dimensional apex to apex return map. We find sufficient conditions for this map to have a unique stable fixed point. By numerical continuation of fixed points with respect to energy, we find a transcritical bifurcation in our model system.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41588949","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}
In this note we consider 1D problems within the context of a new class of elastic bodies. Under suitable conditions on the constitutive equations we prove instability and nonexistence of solutions similar to those in place for the linearized theory. The last section is devoted to describing the spatial behavior of the solutions.
{"title":"On the instability, nonexistence and spatial behaviour of the one-dimensional response of a new class of elastic bodies","authors":"R Quintanilla;K R Rajagopal","doi":"10.1093/imamat/hxab014","DOIUrl":"10.1093/imamat/hxab014","url":null,"abstract":"In this note we consider 1D problems within the context of a new class of elastic bodies. Under suitable conditions on the constitutive equations we prove instability and nonexistence of solutions similar to those in place for the linearized theory. The last section is devoted to describing the spatial behavior of the solutions.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"565-576"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41910739","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}
The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a $4times 4$ matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a $4times 4$ matrix Riemann–Hilbert (RH) problem constructed in the complex $k$ -plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel'fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou's nonlinear steepest descent method and some numerical methods.
{"title":"An initial-boundary value problem for the general three-component nonlinear Schrödinger equations on a finite interval","authors":"Zhenya Yan","doi":"10.1093/imamat/hxab007","DOIUrl":"10.1093/imamat/hxab007","url":null,"abstract":"The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a \u0000<tex>$4times 4$</tex>\u0000 matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a \u0000<tex>$4times 4$</tex>\u0000 matrix Riemann–Hilbert (RH) problem constructed in the complex \u0000<tex>$k$</tex>\u0000-plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel'fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou's nonlinear steepest descent method and some numerical methods.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"427-489"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45172944","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}
The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is $-1$ and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by $beta $ . The integral of the solution over the interior, denoted by $Q$ , is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of $Q$ on $beta $ and the polygon's geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate $R(beta)$ is a rational function which approximates $Q(beta)$ closely.
{"title":"Steady slip flow of Newtonian fluids through tangential polygonal microchannels","authors":"Grant Keady","doi":"10.1093/imamat/hxab008","DOIUrl":"10.1093/imamat/hxab008","url":null,"abstract":"The concern in this paper is the problem of finding—or, at least, approximating—functions, defined within and on the boundary of a tangential polygon, functions whose Laplacian is \u0000<tex>$-1$</tex>\u0000 and which satisfy a homogeneous Robin boundary condition on the boundary. The parameter in the Robin condition is denoted by \u0000<tex>$beta $</tex>\u0000. The integral of the solution over the interior, denoted by \u0000<tex>$Q$</tex>\u0000, is, in the context of flows in a microchannel, the volume flow rate. A variational estimate of the dependence of \u0000<tex>$Q$</tex>\u0000 on \u0000<tex>$beta $</tex>\u0000 and the polygon's geometry is studied. Classes of tangential polygons treated include regular polygons and triangles, especially isosceles: the variational estimate \u0000<tex>$R(beta)$</tex>\u0000 is a rational function which approximates \u0000<tex>$Q(beta)$</tex>\u0000 closely.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":"86 1","pages":"547-564"},"PeriodicalIF":1.2,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/imamat/hxab008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47756270","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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