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The quarterly journal of mechanics and applied mathematics最新文献

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Amplitude–Dependent Three–Dimensional Neutral Modes in Plane Poiseuille–Couette Flow at Large Reynolds Number 大雷诺数平面泊瓦-库埃特流动中振幅相关的三维中性模态
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-02-01 DOI: 10.1093/qjmam/hby022
R Kumar;A G Walton
{"title":"Amplitude–Dependent Three–Dimensional Neutral Modes in Plane Poiseuille–Couette Flow at Large Reynolds Number","authors":"R Kumar;A G Walton","doi":"10.1093/qjmam/hby022","DOIUrl":"https://doi.org/10.1093/qjmam/hby022","url":null,"abstract":"The non-linear stability of plane Poiseuille–Couette flow to three-dimensional disturbances is investigated asymptotically at large values of the Reynolds number \u0000<tex>$R$</tex>\u0000 based on channel half-width and the maximum velocity of the Poiseuille component. The asymptotic theory, aimed at a detailed understanding of the physical mechanisms governing the amplitude-dependent stability properties of the flow, shows that the phase shifts induced across the critical layer and a near-wall shear layer are comparable when the disturbance size \u0000<tex>$Delta=O(R^{-4/9} )$</tex>\u0000. In addition, it emerges that at this crucial size both streamwise and spanwise wavelengths of the travelling wave disturbance are comparable with the channel width, with an associated phasespeed of \u0000<tex>$O(1)$</tex>\u0000. Neutral solutions are found to exist in the range \u0000<tex>$0&lt;V&lt;2$</tex>\u0000 with \u0000<tex>$c_0 = V$</tex>\u0000 to leading order, where \u0000<tex>$c_0$</tex>\u0000 and \u0000<tex>$V$</tex>\u0000 are non-dimensional quantities representing the dominant phasespeed of the non-linear travelling waves and the wall sliding speed respectively. Moreover, these instability modes exist at sliding speeds well in excess of the linear instability cut-off. The amplitude equation governing these modes is derived analytically and we further find that this asymptotic structure breaks down in the limit \u0000<tex>$V rightarrow 2$</tex>\u0000 when the disturbance streamwise wavelength decreases to \u0000<tex>$O(R^{-1/3} )$</tex>\u0000 and the maximum of the basic flow becomes located at the upper wall.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"72 1","pages":"87-130"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hby022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49986336","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}
引用次数: 4
Inhomogeneous non-unidirectional deformations of an elastic wedge 弹性楔的非均匀非单向变形
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-02-01 DOI: 10.1093/qjmam/hby019
C Harley;K R Rajagopal
{"title":"Inhomogeneous non-unidirectional deformations of an elastic wedge","authors":"C Harley;K R Rajagopal","doi":"10.1093/qjmam/hby019","DOIUrl":"https://doi.org/10.1093/qjmam/hby019","url":null,"abstract":"We consider the deformation of a nonlinearly elastic wedge in the case of a stored energy function that is given by a power law. This model, introduced by Knowles, reduces to the classical neo-Hookean model when the power-law exponent is unity, and for certain values of the exponent the equations of anti-plane strain lose ellipticity allowing one to carry out interesting mathematical analyses of the governing equations. Reminiscent of the situation in Jeffery–Hamel flows of the Navier–Stokes fluid (see Jeffery, Phil. Mag. Series 6. 29 (1915); Hamel, Deutsch. Math. Ver.25 (1916)) we find that depending on the wedge angle, in addition to solutions that are unidirectional, that is either towards or away from the apex of the wedge, solutions that are not unidirectional, in that the displacement is towards the apex in certain regions of the wedge and away from the apex in other regions, are also obtained. Solutions that are not unidirectional have been shown to exist by McLeod and Rajagopal (Arch. Rational Mech. Anal.147 (1999)). We are able to provide numerical solutions which confirm these findings not only for the case where we consider the boundary conditions employed in their work, but also for traction boundary conditions. The case where \u0000<tex>$n&lt;0.5$</tex>\u0000 corresponding to the domain where the equation loses ellipticity, is also investigated and solutions obtained.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"72 1","pages":"1-23"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hby019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49986334","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}
引用次数: 0
Diffraction by a quarter–plane. Analytical continuation of spectral functions 四分之一平面衍射谱函数的解析延拓
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-02-01 DOI: 10.1093/qjmam/hby021
R C Assier;A V Shanin
{"title":"Diffraction by a quarter–plane. Analytical continuation of spectral functions","authors":"R C Assier;A V Shanin","doi":"10.1093/qjmam/hby021","DOIUrl":"https://doi.org/10.1093/qjmam/hby021","url":null,"abstract":"The problem of diffraction by a Dirichlet quarter-plane (a flat cone) in a 3D space is studied. The Wiener–Hopf equation for this case is derived and involves two unknown (spectral) functions depending on two complex variables. The aim of the present work is to construct an analytical continuation of these functions onto a well-described Riemann manifold and to study their behaviour and singularities on this manifold. In order to do so, integral formulae for analytical continuation of the spectral functions are derived and used. It is shown that the Wiener–Hopf problem can be reformulated using the concept of additive crossing of branch lines introduced in the article. Both the integral formulae and the additive crossing reformulation are novel and represent the main results of this work.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"72 1","pages":"51-86"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hby021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49986337","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}
引用次数: 19
Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation 薄膜方程自相似爆破曲线的匹配渐近分析
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-10-19 DOI: 10.1093/QJMAM/HBZ001
M. Dallaston
{"title":"Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation","authors":"M. Dallaston","doi":"10.1093/QJMAM/HBZ001","DOIUrl":"https://doi.org/10.1093/QJMAM/HBZ001","url":null,"abstract":"We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth order and destabilising second order term. It has previously been shown that blow up is only possible when the exponent in the second order term is above a certain critical value (dependent on the exponent in the fourth order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau--Levich `drag-out' problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"110 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79420008","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}
引用次数: 0
Method of automorphic functions for an inverse problem of antiplane elasticity 反平面弹性反问题的自同构函数方法
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-09-28 DOI: 10.1093/QJMAM/HBZ003
Y. Antipov
{"title":"Method of automorphic functions for an inverse problem of antiplane elasticity","authors":"Y. Antipov","doi":"10.1093/QJMAM/HBZ003","DOIUrl":"https://doi.org/10.1093/QJMAM/HBZ003","url":null,"abstract":"A nonlinear inverse problem of antiplane elasticity for a multiply connected domain is examined. It is required to determine the profile of $n$ uniformly stressed inclusions when the surrounding infinite body is subjected to antiplane uniform shear at infinity. A method of conformal mappings of circular multiply connected domains is employed. The conformal map is recovered by solving consequently two Riemann-Hilbert problems for piecewise analytic symmetric automorphic functions. For domains associated with the first class Schottky groups a series-form representation of a ($3n-4$) parametric family of conformal maps solving the problem is discovered. Numerical results for two and three uniformly stressed inclusions are reported and discussed.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72867290","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}
引用次数: 10
Asymptotic strategy for matching homogenized structures. Conductivity problem 均匀化结构匹配的渐近策略。导电率的问题
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-09-10 DOI: 10.1093/QJMAM/HBY017
A. Kolpakov, I. Andrianov, D. Prikazchikov
{"title":"Asymptotic strategy for matching homogenized structures. Conductivity problem","authors":"A. Kolpakov, I. Andrianov, D. Prikazchikov","doi":"10.1093/QJMAM/HBY017","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY017","url":null,"abstract":"The paper is concerned with application of the homogenization theory to bodies containing macroinhomogeneities or bodies, parts of which cannot be homogenized (partial homogenization). This situation arises, in particular, for problems of joining homogeneous and periodically inhomogeneous bodies, or combining inhomogeneous bodies of different periodic structure. The peculiarity of the problem is related to a boundary layer, possibly arising on the interface of the matched components. Moreover, this boundary layer may be either real or fictitious, with the latter occurring due to inaccurate formulation of boundary conditions along the interface, ignoring the effect of the micro-stresses. The consideration is carried out within the framework of the steadystate heat equation. The focus of current investigation is on formulation of the problem for the periodicity cell in case of discontinuous homogenized deformations, when these cannot be treated as independent of the “fast” variables. The first order correctors are constructed. The issue of consistent matching procedure, avoiding emergence of fictitious boundary layers, is discussed. It is shown that the temperature of an inhomogeneous fragment on the boundary may be determined from the solution of the homogenized problem, whereas the derivatives (temperature gradients) require fast correctors of the homogenization theory to be taken into account. The analytical consideration is confirmed by results of numerical simulations.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84438181","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}
引用次数: 4
Trapping of waves by thin floating ice floes 被薄浮冰捕获的波浪
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-09-03 DOI: 10.1093/QJMAM/HBY014
R. Porter
{"title":"Trapping of waves by thin floating ice floes","authors":"R. Porter","doi":"10.1093/QJMAM/HBY014","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY014","url":null,"abstract":"","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85087331","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}
引用次数: 9
Uniaxial Elastic-Plastic Wave Propagation 单轴弹塑性波传播
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-08-31 DOI: 10.1093/QJMAM/HBY016
H. Ockendon, J. Ockendon, P. Howell, S. J. Thomson
{"title":"Uniaxial Elastic-Plastic Wave Propagation","authors":"H. Ockendon, J. Ockendon, P. Howell, S. J. Thomson","doi":"10.1093/QJMAM/HBY016","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY016","url":null,"abstract":"This paper considers the uniaxial elastic-plastic response of a solid slab subject to a velocity applied to one boundary, over time scales on which both elastic and plastic waves can propagate. We briefly describe a simple model in which the amplitudes are small enough for these waves to satisfy linear wave equations, although the full problem is still nonlinear because of the presence of an unknown free boundary separating elastic and plastic regions. Examples are given which show that the presence of residual stress can cause the pattern of elastic and plastic regions in the ( X, t )-plane to be surprisingly complicated.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83858533","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}
引用次数: 0
Nonlinear electro-elastic bodies with residual stresses: spectral formulation 具有残余应力的非线性电弹性体:谱公式
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-08-30 DOI: 10.1093/QJMAM/HBY015
M. Shariff, R. Bustamante, J. Merodio
{"title":"Nonlinear electro-elastic bodies with residual stresses: spectral formulation","authors":"M. Shariff, R. Bustamante, J. Merodio","doi":"10.1093/QJMAM/HBY015","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY015","url":null,"abstract":"\u0000 In the present article, a spectral model is developed for residually stressed electro-elastic bodies. The model uses a total energy function that depends on the right stretch tensor, residual stress tensor and one of the electric variables. Some boundary value results with cylindrical symmetry are given. Results for the inflation of a hollow sphere, where the residual stress is assumed to depend only on the radial position, are also given. The constitutive formulation contains spectral invariants that have an immediate physical interpretation which is useful in a rigorous construction of a specific form of the total energy function via an appropriate experiment","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"267 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79812024","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}
引用次数: 10
Some three-dimensional boundary value and boundary-contact problems for an elastic mixture with double porosity 双孔隙弹性混合物的三维边值及边接触问题
The quarterly journal of mechanics and applied mathematics Pub Date : 2018-08-13 DOI: 10.1093/QJMAM/HBY011
R. Janjgava
{"title":"Some three-dimensional boundary value and boundary-contact problems for an elastic mixture with double porosity","authors":"R. Janjgava","doi":"10.1093/QJMAM/HBY011","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY011","url":null,"abstract":"","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77874945","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}
引用次数: 4
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