We consider the short-time transient of the diffusion of mass (or heat) to a body whose surface is held at zero concentration, with the space outside initially at unit concentration. The problem is expressed in terms of the probability that a brownian path of short duration intersects the body. A three-term asymptotic series for the absorption rate is derived for an arbitrary smooth body, together with the leading corrections due to edges and lines of contact with insulating surfaces. A three-term series is also derived for a plane laminar conductor with a smooth boundary, or equivalently a conducting film mounted on an insulating plane. These results are used to derive short-time absorption rates for shapes such as discs, rings and cylinders, commonly used for microelectrodes and hot-film devices.
{"title":"The short-time transient of diffusion outside a conducting body","authors":"C. G. Phillips, Kalvis M. Jansons","doi":"10.1098/rspa.1990.0042","DOIUrl":"https://doi.org/10.1098/rspa.1990.0042","url":null,"abstract":"We consider the short-time transient of the diffusion of mass (or heat) to a body whose surface is held at zero concentration, with the space outside initially at unit concentration. The problem is expressed in terms of the probability that a brownian path of short duration intersects the body. A three-term asymptotic series for the absorption rate is derived for an arbitrary smooth body, together with the leading corrections due to edges and lines of contact with insulating surfaces. A three-term series is also derived for a plane laminar conductor with a smooth boundary, or equivalently a conducting film mounted on an insulating plane. These results are used to derive short-time absorption rates for shapes such as discs, rings and cylinders, commonly used for microelectrodes and hot-film devices.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"11 1","pages":"431 - 449"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75874534","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 paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P, enthalpy h, dryness fraction x, velocity w, and length coordinate z. The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π, which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity af and the equilibrium velocity ae. The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω < af to w > af can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.
{"title":"Physical aspects of the relaxation model in two-phase flow","authors":"Z. Bilicki, J. Kestin","doi":"10.1098/rspa.1990.0040","DOIUrl":"https://doi.org/10.1098/rspa.1990.0040","url":null,"abstract":"The paper explores the potential of the homogeneous relaxation model (HRM) as a basis for the description of adiabatic, one-dimensional, two-phase flows. To this end, a rigorous mathematical analysis highlights the similarities and differences between this and the homogeneous equilibrium model (HEM) emphasizing the physical and qualitative aspects of the problem. Special attention is placed on a study of dispersion, characteristics, choking and shock waves. The most essential features are discovered with reference to the appropriate and convenient phase space Ω for HRM, which consists of pressure P, enthalpy h, dryness fraction x, velocity w, and length coordinate z. The geometric properties of the phase space Ω enable us to sketch the topological pattern of all solutions of the model. The study of choking is intimately connected with the occurrence of singular points of the set of simultaneous first-order differential equations of the model. The very powerful centre manifold theorem allows us to reduce the study of singular points to a two-dimensional plane Π, which is tangent to the solutions at a singular point, and so to demonstrate that only three singular-point patterns can appear (excepting degenerate cases), namely saddle points, nodal points and spiral points. The analysis reveals the existence of two limiting velocities of wave propagation, the frozen velocity af and the equilibrium velocity ae. The critical velocity of choking is the frozen speed of sound. The analysis proves unequivocally that transition from ω < af to w > af can take place only via a singular point. Such a condition can also be attained at the end of a channel. The paper concludes with a short discussion of normal, fully dispersed and partly dispersed shock waves.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"39 1","pages":"379 - 397"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73990740","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 investigate the ray paths from a point-source S on a slightly oblate ellipsoidal shell. The caustics are found to form a 4-star, i.e. a regular, 4-cusped hypocycloid, centred on the point antipodal to S. The length-scale of the 4-star varies as ϵ cos2λ, where ϵ is the eccentricity and λ is the latitude of the antipodal point.
{"title":"Ray paths and caustics on a slightly oblate ellipsoid","authors":"M. Longuet-Higgins","doi":"10.1098/rspa.1990.0035","DOIUrl":"https://doi.org/10.1098/rspa.1990.0035","url":null,"abstract":"We investigate the ray paths from a point-source S on a slightly oblate ellipsoidal shell. The caustics are found to form a 4-star, i.e. a regular, 4-cusped hypocycloid, centred on the point antipodal to S. The length-scale of the 4-star varies as ϵ cos2λ, where ϵ is the eccentricity and λ is the latitude of the antipodal point.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"32 1","pages":"283 - 290"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88643051","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 exclusion of steady downward unsaturated seepage from underground cavities is reducible to a linear convection—diffusion equation with a no normal-flux condition at the cavity surface. Various exact solutions indicate that a roof boundary-layer analysis centred on the upstream stagnation point, and neglecting peripheral variation, gives to remarkable accuracy the quantity θmax, the crucial dimensionless potential determining whether or not water enters the cavity. The great accuracy of this analysis is attributed to the use of curvilinear coordinates natural to the cavity configuration. Global information (such as up to three separate characteristic lengthscales) is injected into the localized boundary-layer formulation via the metric coefficient of the natural coordinates. These are essential to the boundary-layer analysis. Cartesian coordinates, on the other hand, invariably suggest that no boundary layer exists! Definition of the natural coordinates is discussed and means of constructing them about arbitrary cavities are developed. Results for smooth cavities support the conjecture that roof geometry near the upstream stagnation point largely determines θmax, with downstream details unimportant. Comparison of solutions for flat-roofed rectangular and cylindrical cavities with those for strips and discs indicate, however, that the conjecture applies only in weak form to cavities of polygonal cross-section.
{"title":"Conjectures on certain boundary-layer equations and natural coordinates","authors":"J. Philip","doi":"10.1098/rspa.1990.0037","DOIUrl":"https://doi.org/10.1098/rspa.1990.0037","url":null,"abstract":"The problem of exclusion of steady downward unsaturated seepage from underground cavities is reducible to a linear convection—diffusion equation with a no normal-flux condition at the cavity surface. Various exact solutions indicate that a roof boundary-layer analysis centred on the upstream stagnation point, and neglecting peripheral variation, gives to remarkable accuracy the quantity θmax, the crucial dimensionless potential determining whether or not water enters the cavity. The great accuracy of this analysis is attributed to the use of curvilinear coordinates natural to the cavity configuration. Global information (such as up to three separate characteristic lengthscales) is injected into the localized boundary-layer formulation via the metric coefficient of the natural coordinates. These are essential to the boundary-layer analysis. Cartesian coordinates, on the other hand, invariably suggest that no boundary layer exists! Definition of the natural coordinates is discussed and means of constructing them about arbitrary cavities are developed. Results for smooth cavities support the conjecture that roof geometry near the upstream stagnation point largely determines θmax, with downstream details unimportant. Comparison of solutions for flat-roofed rectangular and cylindrical cavities with those for strips and discs indicate, however, that the conjecture applies only in weak form to cavities of polygonal cross-section.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"29 1","pages":"307 - 324"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80063959","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}
This theoretical study is motivated by the experimental observations (a) on the thickening of a turbulent boundary layer compared with its laminar counterpart, (b) on the erupting tongue of fluid that forms the leading edge of a turbulent spot in a boundary layer, (c) on the wall-layer and mid-flow scales, and (d) on the slugs of vorticity that occur in the middle of turbulent channel and pipe flows. It appears that no previous rational explanation has been put forward for these experimental observations. The present tentative suggestions for (a), (b) and (d) centre on the existence of small-deficit fast-travelling zones of concentrated vorticity governed by the nonlinear Euler equations to leading order at high Reynolds numbers Re but crucially influenced by viscosity nevertheless. In the boundary-layer case these zones travel outside the original boundary layer and hence act to increase the effective boundary-layer thickness. The structure of such zones and their scales, governing equations and amplitude dependence are discussed for assumed planar boundary layers and channel flows and for three-dimensional pipe flows in turn. Allied with this, the theory addresses the closure of the amplitude-dependent neutral curve at high Reynolds numbers, the connection with other Euler-type flows and the possibility of delay in sublayer bursting, as well as aiming to give some guidance on nonlinear aspects of unsteady two- and three-dimensional computations for Euler and related flows. The aspects in (c) above, concerning the turbulent scales both of the thin wall layer (O(Re-1 In Re), from a renormalizing and scale-cascade argument) and of the thicker mid-flow zone (containing the Kolmogorov microscale O(Re-3/4)) which lies between that layer and the extensive small-deficit outer zone, are also discussed tentatively in terms of their dynamics, leading to apparently good agreement with turbulent-flow experiments and empirical models, for those scales. Other qualitative comparisons are presented.
本理论研究的动机是实验观察(a)湍流边界层与层流边界层相比增厚，(b)形成边界层湍流点前缘的流体喷发舌，(c)壁面层和中流尺度，以及(d)湍流通道和管道流动中间出现的涡量段塞。似乎以前没有人对这些实验观察提出合理的解释。目前对(a)、(b)和(d)的尝试性建议集中在高雷诺数Re下存在由非线性欧拉方程主导的小赤字快行区集中涡度，但仍然受到粘度的重要影响。在边界层的情况下，这些区域在原始边界层之外移动，从而增加了有效边界层厚度。在假定的平面边界层和通道流动以及三维管道流动的情况下，依次讨论了这些区域的结构及其规模、控制方程和振幅依赖性。与此相结合，该理论解决了高雷诺数下振幅相关中性曲线的闭合，与其他欧拉型流动的联系以及亚层破裂延迟的可能性，并旨在对非定常二维和三维欧拉及相关流动计算的非线性方面提供一些指导。上面(c)中关于薄壁层(O(Re-1 in Re)，从重整化和尺度级联的角度来看)和位于该层和广泛的小缺陷外区之间的较厚的中流区(包含Kolmogorov微尺度O(Re-3/4))的湍流尺度的各方面，也在其动力学方面进行了初步讨论，导致与这些尺度的湍流实验和经验模型明显吻合。还提出了其他质的比较。
{"title":"On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers, and slugs of vorticity in channel and pipe flows","authors":"F. Smith, D. Doorly, A. Rothmayer","doi":"10.1098/rspa.1990.0034","DOIUrl":"https://doi.org/10.1098/rspa.1990.0034","url":null,"abstract":"This theoretical study is motivated by the experimental observations (a) on the thickening of a turbulent boundary layer compared with its laminar counterpart, (b) on the erupting tongue of fluid that forms the leading edge of a turbulent spot in a boundary layer, (c) on the wall-layer and mid-flow scales, and (d) on the slugs of vorticity that occur in the middle of turbulent channel and pipe flows. It appears that no previous rational explanation has been put forward for these experimental observations. The present tentative suggestions for (a), (b) and (d) centre on the existence of small-deficit fast-travelling zones of concentrated vorticity governed by the nonlinear Euler equations to leading order at high Reynolds numbers Re but crucially influenced by viscosity nevertheless. In the boundary-layer case these zones travel outside the original boundary layer and hence act to increase the effective boundary-layer thickness. The structure of such zones and their scales, governing equations and amplitude dependence are discussed for assumed planar boundary layers and channel flows and for three-dimensional pipe flows in turn. Allied with this, the theory addresses the closure of the amplitude-dependent neutral curve at high Reynolds numbers, the connection with other Euler-type flows and the possibility of delay in sublayer bursting, as well as aiming to give some guidance on nonlinear aspects of unsteady two- and three-dimensional computations for Euler and related flows. The aspects in (c) above, concerning the turbulent scales both of the thin wall layer (O(Re-1 In Re), from a renormalizing and scale-cascade argument) and of the thicker mid-flow zone (containing the Kolmogorov microscale O(Re-3/4)) which lies between that layer and the extensive small-deficit outer zone, are also discussed tentatively in terms of their dynamics, leading to apparently good agreement with turbulent-flow experiments and empirical models, for those scales. Other qualitative comparisons are presented.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"25 1","pages":"255 - 281"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74225231","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}
Understanding the rate dependencies of the tensile strength of reinforcing fibres is a key for the understanding of the rate dependencies of the properties of the corresponding composite materials. Hence, in this study it is attempted to clarify the mechanical responses of aramid and carbon fibres at different rates of strain in the light of our previous observations of strain rate dependence of the corresponding hybrid composites under both static and fatigue flexural conditions. In addition, it is attempted to correlate the rate sensitivity with the degree of structural order in the fibres. The study is carried out with low-, medium- and high-modulus pitch based carbon fibres and with Kevlar 29, 49 and 149 para-aramid fibres, whose strengths were tested at strain rates ranging between 0.004 to 2.0% s-1. It is shown that the strength results of the two fibre families follow the Weibull distribution at all strain rates studied. In the case of the carbon fibres two different régimes are observed for the scale parameter as a function of strain rate. At low strain rates the scale parameter increases slowly with the rate, whereas a strong decrease is observed at higher strain rates. This trend becomes more evident as the crystallinity of the fibre increases. The low strain rate behaviour is governed by the power law breakdown rule model, whereas the high strain rate behaviour is accounted for by the rate of growth of a sharp inter-crystallite flaw. In the case of the aramid fibres the scale parameter is insensitive to the strain rate, which supposedly results from a situation where fracture in these fibres does not necessarily involve an activation volume controlled mechanism.
{"title":"Dependence of the tensile strength of pitch-based carbon and para-aramid fibres on the rate of strain","authors":"H. Wagner, J. Aronhime, G. Marom","doi":"10.1098/rspa.1990.0045","DOIUrl":"https://doi.org/10.1098/rspa.1990.0045","url":null,"abstract":"Understanding the rate dependencies of the tensile strength of reinforcing fibres is a key for the understanding of the rate dependencies of the properties of the corresponding composite materials. Hence, in this study it is attempted to clarify the mechanical responses of aramid and carbon fibres at different rates of strain in the light of our previous observations of strain rate dependence of the corresponding hybrid composites under both static and fatigue flexural conditions. In addition, it is attempted to correlate the rate sensitivity with the degree of structural order in the fibres. The study is carried out with low-, medium- and high-modulus pitch based carbon fibres and with Kevlar 29, 49 and 149 para-aramid fibres, whose strengths were tested at strain rates ranging between 0.004 to 2.0% s-1. It is shown that the strength results of the two fibre families follow the Weibull distribution at all strain rates studied. In the case of the carbon fibres two different régimes are observed for the scale parameter as a function of strain rate. At low strain rates the scale parameter increases slowly with the rate, whereas a strong decrease is observed at higher strain rates. This trend becomes more evident as the crystallinity of the fibre increases. The low strain rate behaviour is governed by the power law breakdown rule model, whereas the high strain rate behaviour is accounted for by the rate of growth of a sharp inter-crystallite flaw. In the case of the aramid fibres the scale parameter is insensitive to the strain rate, which supposedly results from a situation where fracture in these fibres does not necessarily involve an activation volume controlled mechanism.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"17 1","pages":"493 - 510"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74253813","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 axisymmetric perturbations of static space-times with prevailing sources (a Maxwell field or a perfect fluid) are considered; and it is shown how a flux integral can be derived directly from the relevant linearized equations. The flux integral ensures the conservation of energy in the attendant scattering of radiation and the sometimes accompanying transformation of one kind of radiation into another. The flux integral derived for perturbed Einstein-Maxwell space-times will be particularly useful in this latter context (as in the scattering of radiation by two extreme Reissner-Nordström black-holes) and in the setting up of a scattering matrix. And the flux integral derived for a space-time with a perfect-fluid source will be directly applicable to the problem of the non-radial oscillations of a star with accompanying emission of gravitational radiation and enable its reformulation as a problem in scattering theory.
{"title":"The flux integral for axisymmetric perturbations of static space-times","authors":"S. Chandrasekhar, V. Ferrari","doi":"10.1098/rspa.1990.0038","DOIUrl":"https://doi.org/10.1098/rspa.1990.0038","url":null,"abstract":"The axisymmetric perturbations of static space-times with prevailing sources (a Maxwell field or a perfect fluid) are considered; and it is shown how a flux integral can be derived directly from the relevant linearized equations. The flux integral ensures the conservation of energy in the attendant scattering of radiation and the sometimes accompanying transformation of one kind of radiation into another. The flux integral derived for perturbed Einstein-Maxwell space-times will be particularly useful in this latter context (as in the scattering of radiation by two extreme Reissner-Nordström black-holes) and in the setting up of a scattering matrix. And the flux integral derived for a space-time with a perfect-fluid source will be directly applicable to the problem of the non-radial oscillations of a star with accompanying emission of gravitational radiation and enable its reformulation as a problem in scattering theory.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"28 1","pages":"325 - 349"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86672528","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}
Asymptotic evolution laws for plane dilatational shock waves travelling in simple materials with memory are derived in this paper by using two approximation methods. The first method is a combination of singular surface theory and perturbation methods. A system of two coupled first-order ordinary differential equations is derived for the shock amplitude and the amplitude of the accompanying second-order discontinuity. The shock amplitude is assumed to be small, but the accompanying second-order discontinuity may be taken either to be finite or to be small with the shock amplitude. The first case corresponds to the situation in which the duration time of the applied load is small compared with the viscous relaxation time and we show that the evolutionary behaviour of the two discontinuities is strongly affected by material nonlinearity. The second case, however, corresponds to the situation in which the duration time is comparable with the viscous relaxation time and we are able to show that the evolutionary behaviour is as predicted by the linear theory of viscoelasticity. In both cases the corresponding elastic results are obtained on allowing the viscous relaxation time to tend to infinity. The second approximation method is the shock-fitting method applied to a modulated simple wave theory, which is itself an approximation based on a small-amplitude finite-rate assumption equivalent to the first case discussed above. The two approximation methods are shown to yield the same evolution laws within their common range of validity.
{"title":"One-dimensional shock waves in simple materials with memory","authors":"Y. B. Fu, N. H. Scott","doi":"10.1098/rspa.1990.0047","DOIUrl":"https://doi.org/10.1098/rspa.1990.0047","url":null,"abstract":"Asymptotic evolution laws for plane dilatational shock waves travelling in simple materials with memory are derived in this paper by using two approximation methods. The first method is a combination of singular surface theory and perturbation methods. A system of two coupled first-order ordinary differential equations is derived for the shock amplitude and the amplitude of the accompanying second-order discontinuity. The shock amplitude is assumed to be small, but the accompanying second-order discontinuity may be taken either to be finite or to be small with the shock amplitude. The first case corresponds to the situation in which the duration time of the applied load is small compared with the viscous relaxation time and we show that the evolutionary behaviour of the two discontinuities is strongly affected by material nonlinearity. The second case, however, corresponds to the situation in which the duration time is comparable with the viscous relaxation time and we are able to show that the evolutionary behaviour is as predicted by the linear theory of viscoelasticity. In both cases the corresponding elastic results are obtained on allowing the viscous relaxation time to tend to infinity. The second approximation method is the shock-fitting method applied to a modulated simple wave theory, which is itself an approximation based on a small-amplitude finite-rate assumption equivalent to the first case discussed above. The two approximation methods are shown to yield the same evolution laws within their common range of validity.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"56 1","pages":"547 - 571"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83096155","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}
An analysis of compound rotations, such as occur in eulerian cradles, is presented in terms of a calculus of rotation axes, without reference to the associated coordinate transformations. The general case of three rotation shafts mounted on one another, with any relation between them at datum zero, is presented. The problem and its solution may be represented entirely in terms of a plane octagon in which four sides have directions that are instrumental constants and the other four sides have lengths that are instrumental constants. When the first four sides are given lengths that express both the rotation angle and the axial direction of the required rotation, then the remaining four sides have directions that directly express the rotations in the drive shafts, that will generate the required rotation. Analytic expressions are given for the shaft setting angles in the general case. If the first and third axes are parallel and the intermediate one perpendicular to these at datum zero (as in the four-circle diffractometer) then these reduce to θ1 = arctan (μ, σ) + [arctan (λ, v) - ψ -½8π], θ2 = 2s arcsin (λ2 + v2)½, θ3 = (μ, σ) - [arctan (λ, v) - ψ - ½8π], s = ± 1, 0 ≤ arcsin (λ2 + v2)½ ≤ ½π, in which λ, μ, v and σ are the four components of a rotation vector constructed such that λ, μ and v are the direction cosines of the rotation axis multiplied by sin½θ for a rotation angle θ and σ is cos½θ. ψ is a constant determined by the choice of directions to which λ and v are measured. The results for the general case are also expressed in terms of more conventional variables.
{"title":"On the factorization of rotations with examples in diffractometry","authors":"R. Diamond","doi":"10.1098/rspa.1990.0043","DOIUrl":"https://doi.org/10.1098/rspa.1990.0043","url":null,"abstract":"An analysis of compound rotations, such as occur in eulerian cradles, is presented in terms of a calculus of rotation axes, without reference to the associated coordinate transformations. The general case of three rotation shafts mounted on one another, with any relation between them at datum zero, is presented. The problem and its solution may be represented entirely in terms of a plane octagon in which four sides have directions that are instrumental constants and the other four sides have lengths that are instrumental constants. When the first four sides are given lengths that express both the rotation angle and the axial direction of the required rotation, then the remaining four sides have directions that directly express the rotations in the drive shafts, that will generate the required rotation. Analytic expressions are given for the shaft setting angles in the general case. If the first and third axes are parallel and the intermediate one perpendicular to these at datum zero (as in the four-circle diffractometer) then these reduce to θ1 = arctan (μ, σ) + [arctan (λ, v) - ψ -½8π], θ2 = 2s arcsin (λ2 + v2)½, θ3 = (μ, σ) - [arctan (λ, v) - ψ - ½8π], s = ± 1, 0 ≤ arcsin (λ2 + v2)½ ≤ ½π, in which λ, μ, v and σ are the four components of a rotation vector constructed such that λ, μ and v are the direction cosines of the rotation axis multiplied by sin½θ for a rotation angle θ and σ is cos½θ. ψ is a constant determined by the choice of directions to which λ and v are measured. The results for the general case are also expressed in terms of more conventional variables.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"14 1","pages":"451 - 472"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76593648","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 describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.
{"title":"Mesh duality and Legendre duality","authors":"S. Chynoweth, M. J. Sewell, D. Jones","doi":"10.1098/rspa.1990.0039","DOIUrl":"https://doi.org/10.1098/rspa.1990.0039","url":null,"abstract":"We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"13 1","pages":"351 - 377"},"PeriodicalIF":0.0,"publicationDate":"1990-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75725366","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}
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：