We construct asymptotic expansions as ε → +0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three dimensional channel with a thin screen of thickness O(ε). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Λ† of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, λ = Λ† − O(ε) and λ = Λ† −O(ε) in the cases I > 0 and I = 0, respectively. We prove that in the case I < 0 there are no embedded eigenvalues in the interval [0,Λ†], while this interval contains exactly one eigenvalue, if I ≥ 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min-principle.
{"title":"Embedded Eigenvalues for Water-Waves in a Three-Dimensional Channel with a Thin Screen","authors":"V. C. Piat, S. Nazarov, J. Taskinen","doi":"10.1093/QJMAM/HBY002","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY002","url":null,"abstract":"We construct asymptotic expansions as ε → +0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three dimensional channel with a thin screen of thickness O(ε). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Λ† of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, λ = Λ† − O(ε) and λ = Λ† −O(ε) in the cases I > 0 and I = 0, respectively. We prove that in the case I < 0 there are no embedded eigenvalues in the interval [0,Λ†], while this interval contains exactly one eigenvalue, if I ≥ 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min-principle.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"187-220"},"PeriodicalIF":0.9,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBY002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42214677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper focuses on the problem of inviscid, irrotational, incompressible fluid sloshing in a rectangular vessel with rigid, impermeable side-wall baffles, and investigates the feasibility of using time-dependent conformal mappings to numerically simulate the evolution of the unknown free-surface in fully-dynamic simulations. An algorithm which uses conformal mappings of a multiply-connected domain to relate the conjugate harmonic functions along the free-surface is documented, and kinematic results presented for a prescribed free-surface motion. The results show that the specific mapping for an infinite depth fluid has one free, within specific bounds, mapping parameter, while the mapping for finite depth fluids has two free mapping parameters. It is shown that having two free parameters gives a wider range of situations under which the conformal mapping can be computed, and it is concluded that the finite depth mapping should be used (in the appropriate limit) even for infinite depth simulations. Overall it is found that a computationally efficient algorithm can be devised to relate the conjugate harmonic functions along the free-surface of the flow domain.
{"title":"Fluid sloshing in rectangular vessels with side-wall baffles using conformal mappings of multiply-connected domains","authors":"M. Turner","doi":"10.1093/QJMAM/HBY004","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY004","url":null,"abstract":"This paper focuses on the problem of inviscid, irrotational, incompressible fluid \u0000sloshing in a rectangular vessel with rigid, impermeable side-wall baffles, and \u0000investigates the feasibility of using time-dependent conformal mappings to numerically \u0000simulate the evolution of the unknown free-surface in fully-dynamic simulations. An \u0000algorithm which uses conformal mappings of a multiply-connected domain to relate \u0000the conjugate harmonic functions along the free-surface is documented, and kinematic \u0000results presented for a prescribed free-surface motion. The results show that the \u0000specific mapping for an infinite depth fluid has one free, within specific bounds, \u0000mapping parameter, while the mapping for finite depth fluids has two free mapping \u0000parameters. It is shown that having two free parameters gives a wider range of \u0000situations under which the conformal mapping can be computed, and it is concluded \u0000that the finite depth mapping should be used (in the appropriate limit) even for infinite \u0000depth simulations. Overall it is found that a computationally efficient algorithm can \u0000be devised to relate the conjugate harmonic functions along the free-surface of the \u0000flow domain.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"245-272"},"PeriodicalIF":0.9,"publicationDate":"2018-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBY004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46567244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Carta, M. Nieves, I. Jones, N. Movchan, A. Movchan
This article presents a novel chiral structure, consisting of Euler–Bernoulli beams connected to gyroscopic spinners.Anew type of boundary condition is introduced, which is referred to as a gyrohinge. In this system, flexural waves are coupled with rotational motion.Time-harmonic conditions are derived by assuming small nutation angles of the spinners. It is shown that the eigenfrequencies of a finite beam with gyro-hinges at one or both ends change dramatically with the moments of inertia and the spin and precession rates of the spinners. The formulation is then extended to elastic beams with periodically-spaced gyro-hinges, whose dispersion properties are investigated in detail. In particular, it is shown how stop-bands and standing modes are affected by the introduction of gyroscopic spinners at the junctions. It is also demonstrated that a periodic system composed of beams connected by gyro-hinges represents a good approximation of a gyrobeam, a theoretical structural element consisting of an elastic beam possessing a continuous distribution of stored angular momentum. The gyricity coefficient of a gyrobeam is then interpreted in terms of the physical parameters of the system of beams with gyroscopic spinners. This article opens a new perspective on the design and practical implementation of chiral mechanical systems.
{"title":"Elastic Chiral Waveguides with Gyro-Hinges","authors":"G. Carta, M. Nieves, I. Jones, N. Movchan, A. Movchan","doi":"10.1093/QJMAM/HBY001","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY001","url":null,"abstract":"This article presents a novel chiral structure, consisting of Euler–Bernoulli beams connected to gyroscopic spinners.Anew type of boundary condition is introduced, which is referred to as a gyrohinge. In this system, flexural waves are coupled with rotational motion.Time-harmonic conditions are derived by assuming small nutation angles of the spinners. It is shown that the eigenfrequencies of a finite beam with gyro-hinges at one or both ends change dramatically with the moments of inertia and the spin and precession rates of the spinners. The formulation is then extended to elastic beams with periodically-spaced gyro-hinges, whose dispersion properties are investigated in detail. In particular, it is shown how stop-bands and standing modes are affected by the introduction of gyroscopic spinners at the junctions. It is also demonstrated that a periodic system composed of beams connected by gyro-hinges represents a good approximation of a gyrobeam, a theoretical structural element consisting of an elastic beam possessing a continuous distribution of stored angular momentum. The gyricity coefficient of a gyrobeam is then interpreted in terms of the physical parameters of the system of beams with gyroscopic spinners. This article opens a new perspective on the design and practical implementation of chiral mechanical systems.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"157-185"},"PeriodicalIF":0.9,"publicationDate":"2018-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBY001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42106369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new method for determining the reflection and transmission properties of lattices is developed. The method uses multipole expansions, and certain transformations of the algebraic equation systems that appear when boundary conditions are applied. It is more direct, and much simpler, than earlier approaches based on integral transforms and the Wiener–Hopf technique. The method is demonstrated for the case of a semi-infinite lattice of sound soft acoustic point scatterers, but can easily be generalised to account for finite size effects, and more general boundary conditions.
{"title":"A Direct Method for Bloch Wave Excitation by Scattering at the Edge of a Lattice. Part I: Point Scatterer Problem","authors":"I. Thompson, R. I. Brougham","doi":"10.1093/QJMAM/HBX022","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX022","url":null,"abstract":"A new method for determining the reflection and transmission properties of lattices is developed. The method uses multipole expansions, and certain transformations of the algebraic equation systems that appear when boundary conditions are applied. It is more direct, and much simpler, than earlier approaches based on integral transforms and the Wiener–Hopf technique. The method is demonstrated for the case of a semi-infinite lattice of sound soft acoustic point scatterers, but can easily be generalised to account for finite size effects, and more general boundary conditions.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"1-24"},"PeriodicalIF":0.9,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43484456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scaling in Cavity—Expansion Equations using the Isovector Method","authors":"E. Albright, S. Ramsey, Joseph Schmidt, R. Baty","doi":"10.1093/QJMAM/HBX023","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX023","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"25-45"},"PeriodicalIF":0.9,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42917391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine the effect of wall inertia on the onset of high-frequency self-excited oscillations in flow through an elastic-walled tube. The previous asymptotic model of Whittaker et al. (Proc. Roy. Soc. A466, 2010), for a long-wavelength high-frequency instability in a Starling-resistor set-up, neglected inertia in the tube wall. Here, we extend this model by modifying the ‘tube-law’ for the wall mechanics to include inertial effects. The resulting coupled model for the fluid and solid mechanics is solved to find the normal modes of oscillation for the system, together with their frequencies and growth rates. In the system and parameter regime considered, the addition of wall inertia reduces the oscillation frequency of each mode, however its effect on the stability of the system is not as straightforward. Increasing wall inertia lowers the mean flow rate required for the onset of instability, and is therefore destabilising. However, at higher flow rates the instability growth rate is decreased, and so wall inertia is stabilising here. Overall, the addition of wall inertia decreases the sensitivity of the system to the mean axial flow rate. The theoretical results show good qualitative and reasonable quantitative agreement with direct numerical simulations performed using the oomph-lib framework.
{"title":"THE EFFECT OF WALL INERTIA ON HIGH-FREQUENCY INSTABILITIES OF FLOW THROUGH AN ELASTIC-WALLED TUBE","authors":"M. Walters, M. Heil, R. Whittaker","doi":"10.1093/QJMAM/HBX024","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX024","url":null,"abstract":"We examine the effect of wall inertia on the onset of high-frequency self-excited oscillations in flow through an elastic-walled tube. The previous asymptotic model of Whittaker et al. (Proc. Roy. Soc. A466, 2010), for a long-wavelength high-frequency instability in a Starling-resistor set-up, neglected inertia in the tube wall. Here, we extend this model by modifying the ‘tube-law’ for the wall mechanics to include inertial effects. The resulting coupled model for the fluid and solid mechanics is solved to find the normal modes of oscillation for the system, together with their frequencies and growth rates. In the system and parameter regime considered, the addition of wall inertia reduces the oscillation frequency of each mode, however its effect on the stability of the system is not as straightforward. Increasing wall inertia lowers the mean flow rate required for the onset of instability, and is therefore destabilising. However, at higher flow rates the instability growth rate is decreased, and so wall inertia is stabilising here. Overall, the addition of wall inertia decreases the sensitivity of the system to the mean axial flow rate. The theoretical results show good qualitative and reasonable quantitative agreement with direct numerical simulations performed using the oomph-lib framework.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"47-77"},"PeriodicalIF":0.9,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX024","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42047279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The flow of a system of two viscous fluids between two concentric counter-rotating cylinders is discussed. A simple theory is presented that describes the evolution of shape of the interface between the fluids when they have near equal densities and identical viscosities. This suggests that the interface is neutrally stable, but that after sufficient time there are nevertheless points on the profile at which the curvature becomes very large. As a consequence, the interface develops cusp-like portions in its profile. A novel spectral method is developed for this problem in which the interface is represented as a region of finite width and over which the density changes rapidly but smoothly. The results confirm the general predictions of the asymptotic theory for rotation in a horizontal plane but when the rotation occurs vertically additional features develop in the flow.
{"title":"Interfacial behaviour in two-fluid taylor-couette flow","authors":"L. Forbes, A. Bassom","doi":"10.1093/QJMAM/HBX025","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX025","url":null,"abstract":"The flow of a system of two viscous fluids between two concentric counter-rotating cylinders is discussed. A simple theory is presented that describes the evolution of shape of the interface between the fluids when they have near equal densities and identical viscosities. This suggests that the interface is neutrally stable, but that after sufficient time there are nevertheless points on the profile at which the curvature becomes very large. As a consequence, the interface develops cusp-like portions in its profile. A novel spectral method is developed for this problem in which the interface is represented as a region of finite width and over which the density changes rapidly but smoothly. The results confirm the general predictions of the asymptotic theory for rotation in a horizontal plane but when the rotation occurs vertically additional features develop in the flow.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"79-97"},"PeriodicalIF":0.9,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX025","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43521123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Does the Weaving and Knitting Pattern of a Fabric Determine its Relaxation Time","authors":"J. Orlik, H. Andrä, I. Argatov, S. Staub","doi":"10.1093/QJMAM/HBX011","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX011","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"70 1","pages":"337-361"},"PeriodicalIF":0.9,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48326057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Mixed Boundary-Value Problems in a Wedge","authors":"P. A. Martin","doi":"10.1093/QJMAM/HBX013","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX013","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"70 1","pages":"373-386"},"PeriodicalIF":0.9,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48396509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Suction Energy for Double-Stranded DNA Inside Single-Walled Carbon Nanotubes","authors":"Mansoor H. Alshehri, J. M. Hill","doi":"10.1093/QJMAM/HBX016","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX016","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"70 1","pages":"387-400"},"PeriodicalIF":0.9,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42099768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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