The nonlinear evolution of stability waves in mixing layers of a perfect gas with a symmetric mean temperature profile is studied for subsonic Mach numbers in the high Reynolds number limit where viscous and thermal diffusion effects enter first and dominate in the critical layer. The linear, neutral eigensolution of the inviscid theory for temperature profiles having either an excess or deficit of mean temperature in the shear layer is used as a basis for the weakly nonlinear, slightly viscous analysis. The coefficients of viscosity and thermal conductivity are assumed to have a power-law dependence on the temperature and the effect of viscous dissipation is included. An analytical expression for the Landau constant, and other constants appearing in the nonlinear evolution equation for the amplitude of the eigenmode, have been obtained. It is found that the temperature excess or deficit at the critical level and the Mach number have a strong nonlinear effect, even to the extent of changing the sign of the Landau constant.
{"title":"Nonlinear stability of subsonic mixing layers with symmetric temperature variations","authors":"V. Djordjevic, L. Redekopp","doi":"10.1098/rspa.1989.0127","DOIUrl":"https://doi.org/10.1098/rspa.1989.0127","url":null,"abstract":"The nonlinear evolution of stability waves in mixing layers of a perfect gas with a symmetric mean temperature profile is studied for subsonic Mach numbers in the high Reynolds number limit where viscous and thermal diffusion effects enter first and dominate in the critical layer. The linear, neutral eigensolution of the inviscid theory for temperature profiles having either an excess or deficit of mean temperature in the shear layer is used as a basis for the weakly nonlinear, slightly viscous analysis. The coefficients of viscosity and thermal conductivity are assumed to have a power-law dependence on the temperature and the effect of viscous dissipation is included. An analytical expression for the Landau constant, and other constants appearing in the nonlinear evolution equation for the amplitude of the eigenmode, have been obtained. It is found that the temperature excess or deficit at the critical level and the Mach number have a strong nonlinear effect, even to the extent of changing the sign of the Landau constant.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"78 3 1","pages":"287 - 330"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86964077","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 is concerned with an asymptotic behaviour of a contour integral with three critical points: a saddle point, a pole and a branch point. Two leading terms of a uniform asymptotic expansion of the integral have been effectively constructed. The expansion remains valid as its critical points approach one another or coalesce. In the special case that the saddle point is bounded away from one of the remaining critical points, or from both, the expansion reduces to simpler in form quasi-uniform and non-uniform expansions, respectively. With the aid of the non-uniform expansion the integral has been interpreted as describing three different waves, each one associated with a corresponding critical point.
{"title":"Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point","authors":"A. Ciarkowski","doi":"10.1098/rspa.1989.0126","DOIUrl":"https://doi.org/10.1098/rspa.1989.0126","url":null,"abstract":"The paper is concerned with an asymptotic behaviour of a contour integral with three critical points: a saddle point, a pole and a branch point. Two leading terms of a uniform asymptotic expansion of the integral have been effectively constructed. The expansion remains valid as its critical points approach one another or coalesce. In the special case that the saddle point is bounded away from one of the remaining critical points, or from both, the expansion reduces to simpler in form quasi-uniform and non-uniform expansions, respectively. With the aid of the non-uniform expansion the integral has been interpreted as describing three different waves, each one associated with a corresponding critical point.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"24 1","pages":"273 - 286"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75324081","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 one-dimensional inverse electromagnetic scattering problem for the inversion of amplitude data of either linear polarization state is investigated. The method exploits the complex structure of the field scattered from a class of inhomogeneous dielectrics and enables the analytic signal to be reconstructed from measurements of the amplitude alone. The method is demonstrated and exemplified with experimental data in both transverse electric and transverse magnetic polarization states. The implications of the method as a means for regularization of scattered data are briefly discussed.
{"title":"On the inversion and phase recovery of scattered electromagnetic amplitude data","authors":"K. Hopcraft, B. Carroll, P. Smith","doi":"10.1098/rspa.1989.0130","DOIUrl":"https://doi.org/10.1098/rspa.1989.0130","url":null,"abstract":"The one-dimensional inverse electromagnetic scattering problem for the inversion of amplitude data of either linear polarization state is investigated. The method exploits the complex structure of the field scattered from a class of inhomogeneous dielectrics and enables the analytic signal to be reconstructed from measurements of the amplitude alone. The method is demonstrated and exemplified with experimental data in both transverse electric and transverse magnetic polarization states. The implications of the method as a means for regularization of scattered data are briefly discussed.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"6 1","pages":"361 - 375"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89085680","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}
By extending the theory of homogenization, we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales. Because for geological and engineering applications the driving force (e. g. the pressure gradient) and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity, we derive the phenomenological equations of motion by using the perturbation technique of multiple scales. All the effective coefficients are defined by boundary-value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix. In Part I the porous matrix is assumed to be rigid; the case of three scales is treated first. Symmetry and positiveness of the effective Darcy permeability tensor are proven. Extensions to four and more scales are then discussed. In Part II we allow the solid matrix to be deformable and deduce the equations of consolidation of a two-phase medium. The coupling of fluid flow and the quasi-static elastic deformation of the matrix is considered when there are three disparate length scales. Effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores. Several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed.
{"title":"Mechanics of heterogeneous porous media with several spatial scales","authors":"C. Mei, J. Auriault","doi":"10.1098/rspa.1989.0132","DOIUrl":"https://doi.org/10.1098/rspa.1989.0132","url":null,"abstract":"By extending the theory of homogenization, we consider a heterogeneous porous medium whose material structure is characterized by multiple periodicity over several disparate length scales. Because for geological and engineering applications the driving force (e. g. the pressure gradient) and the global motion of primary interest is usually at a scale much larger than the largest of structural periodicity, we derive the phenomenological equations of motion by using the perturbation technique of multiple scales. All the effective coefficients are defined by boundary-value problems on unit cells in smaller scales and no constitutive assumptions are added aside from the basic equations governing the mechanics of the pore fluid and the solid matrix. In Part I the porous matrix is assumed to be rigid; the case of three scales is treated first. Symmetry and positiveness of the effective Darcy permeability tensor are proven. Extensions to four and more scales are then discussed. In Part II we allow the solid matrix to be deformable and deduce the equations of consolidation of a two-phase medium. The coupling of fluid flow and the quasi-static elastic deformation of the matrix is considered when there are three disparate length scales. Effective coefficients of various kinds are deduced in terms of cell problems on the scale of the pores. Several general symmetry relations as well as specific properties of a medium composed of layers of porous matrix are discussed.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"149 1","pages":"391 - 423"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72682001","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 say that an n-dimensional (classically) integral lattice ⋀ is s-integrable, for an integer s, if it can be described by vectors s-½(x1,...,xk), with all xi ∊ Z, in a euclidean space of dimension k ≽ n. Equivalently, ⋀ is s-integrable if and only if any quadratic form f(x) corresponding to ⋀ can be written as s-1 times a sum of k squares of linear forms with integral coefficients, or again, if and only if the dual lattice ⋀* contains a eutactic star of scale s. This paper gives many techniques for s-integrating low-dimensional lattices (such as Es and the Leech lattice). A particular result is that any one-dimensional lattice can be 1-integrated with k = 4: this is Lagrange’s four-squares theorem. Let ϕ(s) be the smallest dimension n in which there is an integral lattice that is not s-integrable. In 1937 Ko and Mordell showed that ϕ(1) = 6. We prove that ϕ(2) = 12, ϕ(3) = 14, 21 ≼ ϕ(4) ≼ 25, 16 ≼ ϕ(5) ≼ 22, ϕ(s) ≼ 4s + 2 (s odd), ϕ(s) ≼ 2πes(1 + o(1)) (s even) and ϕ(s) ≽ 2In In s/ln In In s(1 + o(1)).
我们说，对于整数s，一个n维(经典)积分晶格，如果它可以用向量s-1 / 2 (x1，…，xk)在维数为k的欧几里得空间中用所有的xi Z来描述，那么它是s-可积的。等价地，当且仅当与之对应的任何二次形式f(x)可以写成s-1乘以具有积分系数的线性形式的k平方的和，或者，当且仅当对偶格*包含一个尺度为s的等规星。本文给出了许多低维格(如Es和Leech格)的s积分技术。一个特殊的结果是，任何一维晶格都可以被k = 4的1积分:这就是拉格朗日的四平方定理。设φ (s)为存在不可s积的整格的最小维n。1937年Ko和Mordell证明了ϕ(1) = 6。我们证明ϕ(2)= 12,ϕ(3)= 14日21≼ϕ(4)≼25日16≼ϕ(5)≼22日ϕ≼4 s (s) + 2(奇怪),ϕ(s)≼2πes (1 + o (1)) (s)和ϕ(s)≽在s / ln在年代2 (1 + o(1))。
{"title":"Low-dimensional lattices V. Integral coordinates for integral lattices","authors":"J. Conway, N. Sloane","doi":"10.1098/rspa.1989.0124","DOIUrl":"https://doi.org/10.1098/rspa.1989.0124","url":null,"abstract":"We say that an n-dimensional (classically) integral lattice ⋀ is s-integrable, for an integer s, if it can be described by vectors s-½(x1,...,xk), with all xi ∊ Z, in a euclidean space of dimension k ≽ n. Equivalently, ⋀ is s-integrable if and only if any quadratic form f(x) corresponding to ⋀ can be written as s-1 times a sum of k squares of linear forms with integral coefficients, or again, if and only if the dual lattice ⋀* contains a eutactic star of scale s. This paper gives many techniques for s-integrating low-dimensional lattices (such as Es and the Leech lattice). A particular result is that any one-dimensional lattice can be 1-integrated with k = 4: this is Lagrange’s four-squares theorem. Let ϕ(s) be the smallest dimension n in which there is an integral lattice that is not s-integrable. In 1937 Ko and Mordell showed that ϕ(1) = 6. We prove that ϕ(2) = 12, ϕ(3) = 14, 21 ≼ ϕ(4) ≼ 25, 16 ≼ ϕ(5) ≼ 22, ϕ(s) ≼ 4s + 2 (s odd), ϕ(s) ≼ 2πes(1 + o(1)) (s even) and ϕ(s) ≽ 2In In s/ln In In s(1 + o(1)).","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"19 1","pages":"211 - 232"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84424538","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 formulae are derived for the effective dielectric constants of composites containing densely packed spherical inclusions with dielectric constants differing greatly from that of the background. The asymptotic results are obtained by treating the influence of nearest-neighbouring inclusions analytically. All the results obtained are in agreement with results obtained using other methods and in some cases provide an additional term not given before. Uniform asymptotic results are obtained as a function of both the spacing of the spheres and their dielectric constants simultaneously, whereas previous analyses treat these separately. Results are obtained for a pair of spheres, for an infinite chain of spheres and for a lattice of spheres having the sodium chloride structure. Results are also presented for some Bravais lattices and a large class of structures having specified nearest-neighbour configurations. The results may also be of use in studying percolating, structures.
{"title":"Asymptotic behaviour of the effective dielectric constants of composite materials","authors":"L. Poladian","doi":"10.1098/rspa.1989.0129","DOIUrl":"https://doi.org/10.1098/rspa.1989.0129","url":null,"abstract":"Asymptotic formulae are derived for the effective dielectric constants of composites containing densely packed spherical inclusions with dielectric constants differing greatly from that of the background. The asymptotic results are obtained by treating the influence of nearest-neighbouring inclusions analytically. All the results obtained are in agreement with results obtained using other methods and in some cases provide an additional term not given before. Uniform asymptotic results are obtained as a function of both the spacing of the spheres and their dielectric constants simultaneously, whereas previous analyses treat these separately. Results are obtained for a pair of spheres, for an infinite chain of spheres and for a lattice of spheres having the sodium chloride structure. Results are also presented for some Bravais lattices and a large class of structures having specified nearest-neighbour configurations. The results may also be of use in studying percolating, structures.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"11 1","pages":"343 - 359"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82552493","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 the transient motion of ledged interphase boundaries is presented where volume diffusion of solute in the matrix to the riser of the step is assumed to control the growth rate. The analysis leads to a nonlinear integral equation governing step motion and can be applied in principle to situations involving many interacting steps. Detailed attention is given to the case of a single step in an infinite medium where comparisons are made with numerical results of Enomoto (obtained by finite difference solution of the diffusion equation). Attention is also given to transient motion in a medium of finite extent. The treatment here thus generalizes the steady-state theory of Atkinson.
{"title":"The growth kinetics of ledged interphase boundaries: transient motion, an analytical treatment","authors":"C. Atkinson, P. Wilmott","doi":"10.1098/rspa.1989.0131","DOIUrl":"https://doi.org/10.1098/rspa.1989.0131","url":null,"abstract":"An analysis of the transient motion of ledged interphase boundaries is presented where volume diffusion of solute in the matrix to the riser of the step is assumed to control the growth rate. The analysis leads to a nonlinear integral equation governing step motion and can be applied in principle to situations involving many interacting steps. Detailed attention is given to the case of a single step in an infinite medium where comparisons are made with numerical results of Enomoto (obtained by finite difference solution of the diffusion equation). Attention is also given to transient motion in a medium of finite extent. The treatment here thus generalizes the steady-state theory of Atkinson.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"34 1","pages":"377 - 389"},"PeriodicalIF":0.0,"publicationDate":"1989-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82058844","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 paper is concerned with numerical methods for finding m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation -y" + qy = λy on [0, ∞) and the results are applied to the problem of finding best constants for Everitt’s extension to the Hardy-Little-wood-Pόlya (HELP) integral inequality.
本文研究了奇异特征值方程-y”+ qy = λy在[0，∞]上求Titchmarsh-Weyl m系数m(λ)的数值方法，并将所得结果应用于求Everitt对hardy - little -wood- p lya (HELP)积分不等式的推广的最佳常数问题。
{"title":"Numerical determination of the Titchmarsh-Weyl m-coefficient and its applications to HELP inequalities","authors":"B. M. Brown, V. Kirby, J. Pryce","doi":"10.1098/rspa.1989.0122","DOIUrl":"https://doi.org/10.1098/rspa.1989.0122","url":null,"abstract":"This paper is concerned with numerical methods for finding m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation -y\" + qy = λy on [0, ∞) and the results are applied to the problem of finding best constants for Everitt’s extension to the Hardy-Little-wood-Pόlya (HELP) integral inequality.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"10 1","pages":"167 - 188"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79793274","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 paper gives asymptotic solutions to generalized Burgers equations governing the propagation of weakly nonlinear acoustic waves under the influence of geometrical spreading and thermoviscous diffusion. Geometrical effects are included through a general ray tube area function, A(r), and solutions are obtained up to arbitrarily large ranges for initially sinusoidal waves by the use of rational asymptotic techniques in the two cases of weak diffusivity and of high initial wave amplitude. These solutions use results obtained earlier by Nimmo & Crighton. Simpler approximate techniques to obtain similar solutions are then discussed. The two approximate methods considered, proposed by Shooter et al. and Rudnick, are based on physical considerations, rather than asymptotic theory. The validity of such methods is demonstrated for a broad, though restricted, range of physical situations.
{"title":"Approximate solution methods for nonlinear acoustic propagation over long ranges","authors":"P. Hammerton, D. Crighton","doi":"10.1098/rspa.1989.0120","DOIUrl":"https://doi.org/10.1098/rspa.1989.0120","url":null,"abstract":"This paper gives asymptotic solutions to generalized Burgers equations governing the propagation of weakly nonlinear acoustic waves under the influence of geometrical spreading and thermoviscous diffusion. Geometrical effects are included through a general ray tube area function, A(r), and solutions are obtained up to arbitrarily large ranges for initially sinusoidal waves by the use of rational asymptotic techniques in the two cases of weak diffusivity and of high initial wave amplitude. These solutions use results obtained earlier by Nimmo & Crighton. Simpler approximate techniques to obtain similar solutions are then discussed. The two approximate methods considered, proposed by Shooter et al. and Rudnick, are based on physical considerations, rather than asymptotic theory. The validity of such methods is demonstrated for a broad, though restricted, range of physical situations.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"42 1","pages":"125 - 152"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78782342","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}
Solutions of the small disturbance equations for shock-induced thermal explosions at high activation energies are discussed in the limit when the adiabatic and isothermal sound speeds lie close together. Temperature and pressure perturbations are shown to possess logarithmic singularities at ignition. Although the density perturbation appears to have a similar behaviour, a local adjustment occurs exponentially close to the ignition point and the density at ignition is bounded. Excellent agreement with numerical calculations is found.
{"title":"Shock-generated ignition: the induction zone","authors":"P. Blythe, D. Crighton","doi":"10.1098/rspa.1989.0123","DOIUrl":"https://doi.org/10.1098/rspa.1989.0123","url":null,"abstract":"Solutions of the small disturbance equations for shock-induced thermal explosions at high activation energies are discussed in the limit when the adiabatic and isothermal sound speeds lie close together. Temperature and pressure perturbations are shown to possess logarithmic singularities at ignition. Although the density perturbation appears to have a similar behaviour, a local adjustment occurs exponentially close to the ignition point and the density at ignition is bounded. Excellent agreement with numerical calculations is found.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":"63 1","pages":"189 - 209"},"PeriodicalIF":0.0,"publicationDate":"1989-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74320026","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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