Pub Date : 2019-11-11DOI: 10.1525/abt.2016.78.1.80
R. Lord
Today I must concentrate on just one of the many possible aspects of plankton research, one that 1 believe has a real future given the opportunity and the backing. This is: how the movement and mixing of water masses affect the plankton and, in turn, the effect this has on the other dependent communities and so on the fish themselves. We know that the mixing of different water masses produces conditions that are usually more productive than in either, and this can happen on a large scale, as in the convergences, or on a small scale in quite local areas. Sometimes the cause of the greater productivity is obvious—for example, off the coast of Peru where off-shore winds tend to drive away the depleted surface waters. These are replaced by the upwelling of nutrient-rich waters from below; some mixing takes place seeding the rich water with phytoplankton and resulting in one of the richest areas of production in the oceans and incidentally leading to an export of cheap fish meal that is having repercussions in the European markets.
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This paper studies the resonances and points of spectral concentration of the block operator matrix ( − d2 d x 2 +q tw tw ) in the space L2(0,1)⊕L2(0,1). In particular we study the dynamics of the resonance/eigenvalue λ(t), showing that an embedded eigenvalue can evolve into a resonance and that eigenvalues which are absorbed by the essential spectrum give rise to resonance points. A connection is also established between resonances and points of spectral concentration. Finally, some numerical examples are given which show that each of the above theoretical possibilities can be realized.
本文研究了L2(0,1)⊕L2(0,1)空间中块算子矩阵(- d2 d x 2 +q tw tw)的谱集中共振点和谱集中点。特别地，我们研究了共振/特征值λ(t)的动力学，表明嵌入的特征值可以演变成共振，并且被本质谱吸收的特征值产生共振点。在共振和谱集中点之间也建立了联系。最后给出了一些数值算例，表明上述每种理论可能性都是可以实现的。
{"title":"Spectral concentrations and resonances of a second–order block operator matrix and an associated λ–rational Sturm-Liouville problem","authors":"B. M. Brown, M. Langer, M. Marletta","doi":"10.1098/rspa.2003.1272","DOIUrl":"https://doi.org/10.1098/rspa.2003.1272","url":null,"abstract":"This paper studies the resonances and points of spectral concentration of the block operator matrix ( − d2 d x 2 +q tw tw ) in the space L2(0,1)⊕L2(0,1). In particular we study the dynamics of the resonance/eigenvalue λ(t), showing that an embedded eigenvalue can evolve into a resonance and that eigenvalues which are absorbed by the essential spectrum give rise to resonance points. A connection is also established between resonances and points of spectral concentration. Finally, some numerical examples are given which show that each of the above theoretical possibilities can be realized.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75390778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We report the discovery of a new mathematical identity for the Navier–Stokes equations, which model the flow of a viscous compressible gas. The identity relates the sum of the squares of the viscous flux Jacobians in rotated coordinate systems, and takes the form of a remarkably compact similarity transformation: ∑ i=1 3 A * i d =G{ ∑ i=1 3 A i d A i d } G T . The proof of an analogous relationship for the sum of the squares of the Euler flux Jacobians appeared in Venkatasubban (Proc. R. Soc. Lond. (2001) A457, 1111–1114). At that time, the extension to the Navier–Stokes equations was not evident, given that this requires a separate and detailed derivation. It may be remarked that these relationships possess an elegant simplicity that the flux Jacobians themselves do not have. In developing numerical algorithms for flow solvers based on the Navier–Stokes equations, it is often necessary or advantageous to work in a rotated coordinate frame. This relation thus has potential applications in the development of new and improved algorithms for Navier–Stokes computer codes used to solve industrial problems in aerodynamics and fluid dynamics.
我们报告了对模拟粘性可压缩气体流动的Navier-Stokes方程的一个新的数学恒等式的发现。恒等式涉及旋转坐标系中粘性通量雅可比矩阵的平方和，并采用非常紧凑的相似变换形式:∑i=1 3a * id =G{∑i=1 3a * id a id} G T。欧拉通量雅可比矩阵平方和的一个类似关系的证明出现在Venkatasubban (Proc. R. Soc)中。Lond。(2001) 457, 1111-1114)。当时，对Navier-Stokes方程的扩展还不明显，因为这需要一个单独而详细的推导。可以注意到，这些关系具有通量雅可比矩阵本身所没有的优雅的简单性。在开发基于Navier-Stokes方程的流动求解数值算法时，在旋转坐标系中工作往往是必要的或有利的。因此，这种关系在开发用于解决空气动力学和流体动力学工业问题的纳维-斯托克斯计算机代码的新的和改进的算法方面具有潜在的应用。
{"title":"Rotational transformations of the sum of the squares of the flux Jacobians of the Navier–Stokes equations","authors":"Chittur S. Venkatasubban","doi":"10.1098/rspa.2004.1365","DOIUrl":"https://doi.org/10.1098/rspa.2004.1365","url":null,"abstract":"We report the discovery of a new mathematical identity for the Navier–Stokes equations, which model the flow of a viscous compressible gas. The identity relates the sum of the squares of the viscous flux Jacobians in rotated coordinate systems, and takes the form of a remarkably compact similarity transformation: ∑ i=1 3 A * i d =G{ ∑ i=1 3 A i d A i d } G T . The proof of an analogous relationship for the sum of the squares of the Euler flux Jacobians appeared in Venkatasubban (Proc. R. Soc. Lond. (2001) A457, 1111–1114). At that time, the extension to the Navier–Stokes equations was not evident, given that this requires a separate and detailed derivation. It may be remarked that these relationships possess an elegant simplicity that the flux Jacobians themselves do not have. In developing numerical algorithms for flow solvers based on the Navier–Stokes equations, it is often necessary or advantageous to work in a rotated coordinate frame. This relation thus has potential applications in the development of new and improved algorithms for Navier–Stokes computer codes used to solve industrial problems in aerodynamics and fluid dynamics.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87106749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The general spinning motion of an axisymmetric rigid body on a horizontal table is analysed, allowing for slip and friction at the point of contact P. Attention is focused on the case of spheroids (prolate or oblate), and particularly on spheroids whose density distribution is such that the centre–of–mass and centre–of–volume coincide. Four classes of fixed points (i.e. steady states) are identified, and the linear stability properties in each case are determined, assuming viscous friction at P. The governing dynamical system is six–dimensional. Trajectories of the system are computed, and are shown in projection in a three–dimensional subspace; these start near unstable fixed points and (in the case of viscous friction) end at stable fixed points. It is shown inter alia that a uniform prolate spheroid set in sufficiently rapid spinning motion with its axis horizontal is unstable, and its axis rises to a stable steady state, at either an intermediate angle or the vertical, depending on the initial angular velocity. These computations allow an assessment of the circumstances under which the condition described as ‘gyroscopic balance’ is realized. Under this condition, the evolution from an unstable to a stable state is greatly simplified, being described by a first–order differential equation. Oscillatory modes which are stable on linear analysis may be destabilized during this evolution, with consequential oscillations in the normal reaction R at the point of support. The computations presented here are restricted to circumstances in which R remains positive.
{"title":"Dynamics of an axisymmetric body spinning on a horizontal surface. I. Stability and the gyroscopic approximation","authors":"H. K. Moffatt, Y. Shimomura, M. Branicki","doi":"10.1098/rspa.2004.1329","DOIUrl":"https://doi.org/10.1098/rspa.2004.1329","url":null,"abstract":"The general spinning motion of an axisymmetric rigid body on a horizontal table is analysed, allowing for slip and friction at the point of contact P. Attention is focused on the case of spheroids (prolate or oblate), and particularly on spheroids whose density distribution is such that the centre–of–mass and centre–of–volume coincide. Four classes of fixed points (i.e. steady states) are identified, and the linear stability properties in each case are determined, assuming viscous friction at P. The governing dynamical system is six–dimensional. Trajectories of the system are computed, and are shown in projection in a three–dimensional subspace; these start near unstable fixed points and (in the case of viscous friction) end at stable fixed points. It is shown inter alia that a uniform prolate spheroid set in sufficiently rapid spinning motion with its axis horizontal is unstable, and its axis rises to a stable steady state, at either an intermediate angle or the vertical, depending on the initial angular velocity. These computations allow an assessment of the circumstances under which the condition described as ‘gyroscopic balance’ is realized. Under this condition, the evolution from an unstable to a stable state is greatly simplified, being described by a first–order differential equation. Oscillatory modes which are stable on linear analysis may be destabilized during this evolution, with consequential oscillations in the normal reaction R at the point of support. The computations presented here are restricted to circumstances in which R remains positive.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87400887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The fluctuation of mechanical fields arising in polycrystals is investigated. These materials are viewed as composites of the Hashin–Shtrikman type with a large number of anisotropic phases and a ‘granular’ topology. We show that the estimation of the intra–phase stress and strain (rate) second moments comes down to the resolution of a linear system of equations. Applied to a linear viscous face–centred cubic (FCC) polycrystal, it is observed that significant local slip rates are estimated even when the corresponding Schmid factor vanishes, due to the intergranular interactions. For the application to viscoplastic polycrystals, the secant and affine nonlinear extensions of the self–consistent scheme are compared. At large stress sensitivity (n = 30), it is observed that the secant linearization leads to almost uniform slip rates for all slip systems in every phase, whereas the affine approach predicts a much larger spread. Furthermore, there is no one–to–one relation between the phase–average stress (or strain rate) and the corresponding second moment. It is emphasized that intra–phase strain–rate heterogeneities should be accounted for when dealing with microstructure evolution.
{"title":"Mechanical field fluctuations in polycrystals estimated by homogenization techniques","authors":"R. Brenner, O. Castelnau, L. Badea","doi":"10.1098/rspa.2004.1278","DOIUrl":"https://doi.org/10.1098/rspa.2004.1278","url":null,"abstract":"The fluctuation of mechanical fields arising in polycrystals is investigated. These materials are viewed as composites of the Hashin–Shtrikman type with a large number of anisotropic phases and a ‘granular’ topology. We show that the estimation of the intra–phase stress and strain (rate) second moments comes down to the resolution of a linear system of equations. Applied to a linear viscous face–centred cubic (FCC) polycrystal, it is observed that significant local slip rates are estimated even when the corresponding Schmid factor vanishes, due to the intergranular interactions. For the application to viscoplastic polycrystals, the secant and affine nonlinear extensions of the self–consistent scheme are compared. At large stress sensitivity (n = 30), it is observed that the secant linearization leads to almost uniform slip rates for all slip systems in every phase, whereas the affine approach predicts a much larger spread. Furthermore, there is no one–to–one relation between the phase–average stress (or strain rate) and the corresponding second moment. It is emphasized that intra–phase strain–rate heterogeneities should be accounted for when dealing with microstructure evolution.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76087719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Correction for ‘Efficient computation of Schlömilch-type series’ by C. M. Linton and D. C. Barnett (Proc. R. Soc. Lond. A 460, 1177–1191. (doi: 10.1098/rspa.2003.1202)). Figure 1 appears in its correct form below.
{"title":"Correction for Linton and Barnett, Efficient computation of Schlömilch-type series","authors":"C. Linton, D. C. Barnett","doi":"10.1098/RSPA.2004.2000","DOIUrl":"https://doi.org/10.1098/RSPA.2004.2000","url":null,"abstract":"Correction for ‘Efficient computation of Schlömilch-type series’ by C. M. Linton and D. C. Barnett (Proc. R. Soc. Lond. A 460, 1177–1191. (doi: 10.1098/rspa.2003.1202)). Figure 1 appears in its correct form below.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86223059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of diffraction by a perfectly reflecting screen occupying an infinite sector of the equatorial plane is addressed by the random–walk method. The solution is represented as a superposition of the wave field completely determined by an elementary ray analysis and of the field formed by the waves diffracted by the tip of the screen. The diffracted field is explicitly represented as the mathematical expectation of a specified functional on trajectories of the random motion, the radial component of which runs in a complex space while the two–dimensional angular component remains real valued. The numerical results confirm the efficiency of the random–walk approach to the analysis of diffraction by wedge–shaped screens of arbitrary angles.
{"title":"Diffraction by a plane sector","authors":"B. Budaev, D. Bogy","doi":"10.1098/rspa.2004.1322","DOIUrl":"https://doi.org/10.1098/rspa.2004.1322","url":null,"abstract":"The problem of diffraction by a perfectly reflecting screen occupying an infinite sector of the equatorial plane is addressed by the random–walk method. The solution is represented as a superposition of the wave field completely determined by an elementary ray analysis and of the field formed by the waves diffracted by the tip of the screen. The diffracted field is explicitly represented as the mathematical expectation of a specified functional on trajectories of the random motion, the radial component of which runs in a complex space while the two–dimensional angular component remains real valued. The numerical results confirm the efficiency of the random–walk approach to the analysis of diffraction by wedge–shaped screens of arbitrary angles.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89184086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A rigorous way of obtaining sharp bounds for Stokes constants is introduced and illustrated on a concrete problem arising in applications.
本文介绍了一种求得Stokes常数尖锐边界的严格方法，并举例说明了应用中出现的一个具体问题。
{"title":"Rigorous bounds of Stokes constants for some nonlinear ordinary differential equations at rank-one irregular singularities","authors":"O. Costin, R. Costin, M. Kohut","doi":"10.1098/rspa.2004.1310","DOIUrl":"https://doi.org/10.1098/rspa.2004.1310","url":null,"abstract":"A rigorous way of obtaining sharp bounds for Stokes constants is introduced and illustrated on a concrete problem arising in applications.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77234181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A computational framework is developed which couples a series of models, each describing vastly different physical events, in order to characterize particle growth (agglomeration) in thermochemically reacting granular flows. The modelling is purposely simplified to expose the dominant mechanisms which control agglomeration. The overall system is comprised of relatively simple coupled submodels describing impact, heat production, bonding and fragmentation, each of which can be replaced by more elaborate descriptions, if and when they are available. Inverse problems, solved with a genetic algorithm, are then constructed to ascertain system parameters which maximize agglomeration likelihood within a range of admissible data.
{"title":"A computational framework for agglomeration in thermochemically reacting granular flows","authors":"T. Zohdi","doi":"10.1098/rspa.2004.1277","DOIUrl":"https://doi.org/10.1098/rspa.2004.1277","url":null,"abstract":"A computational framework is developed which couples a series of models, each describing vastly different physical events, in order to characterize particle growth (agglomeration) in thermochemically reacting granular flows. The modelling is purposely simplified to expose the dominant mechanisms which control agglomeration. The overall system is comprised of relatively simple coupled submodels describing impact, heat production, bonding and fragmentation, each of which can be replaced by more elaborate descriptions, if and when they are available. Inverse problems, solved with a genetic algorithm, are then constructed to ascertain system parameters which maximize agglomeration likelihood within a range of admissible data.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82944882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many of the kinematic and dynamic concepts relating to rotational motion have been generalized for N–dimensional rigid bodies. In this paper a new derivation of the generalized Euler equations is presented. The development herein of the N–dimensional rotational equations of motion requires the introduction of a new symbol, which is a numerical relative tensor, to relate the elements of an N Ö N skew–symmetric matrix to a vector form. This symbol allows the Hamel coefficients associated with general N–dimensional rotations to be computed. Using these coefficients, Lagrange's equations are written in terms of the angular–velocity components of an N–dimensional rigid body. The new derivation provides a convenient vector form of the equations, allows the study of systems with forcing functions, and allows for the sensitivity of the kinetic energy to the generalized coordinates.
{"title":"Hamel coefficients for the rotational motion of an N–dimensional rigid body","authors":"J. Hurtado, A. Sinclair","doi":"10.1098/rspa.2004.1320","DOIUrl":"https://doi.org/10.1098/rspa.2004.1320","url":null,"abstract":"Many of the kinematic and dynamic concepts relating to rotational motion have been generalized for N–dimensional rigid bodies. In this paper a new derivation of the generalized Euler equations is presented. The development herein of the N–dimensional rotational equations of motion requires the introduction of a new symbol, which is a numerical relative tensor, to relate the elements of an N Ö N skew–symmetric matrix to a vector form. This symbol allows the Hamel coefficients associated with general N–dimensional rotations to be computed. Using these coefficients, Lagrange's equations are written in terms of the angular–velocity components of an N–dimensional rigid body. The new derivation provides a convenient vector form of the equations, allows the study of systems with forcing functions, and allows for the sensitivity of the kinetic energy to the generalized coordinates.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81306793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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