The Flow of Ice, Treated as a Newtonian Viscous Liquid, around a Cylindrical Obstacle Near the Bed of a Glacier

AbstractThis paper describes an analytical solution of the equations of motion and heat conduction for ice flowing around a cylindrical solid inclusion and over a solid plane boundary. This is intended to be a simplified representation of the flow of clean glacier ice around a stone and over a rigid rock bed. The ice is treated as a Newtonian viscous liquid and the equations are solved in two dimensions. Regelation boundary conditions are applied at both ice–rock interfaces. It is found that finite solutions for the temperature and stream function only exist for the special cases in which two dimensionless critical wavelengths are zero. That is, unless the stone is very far from the glacial bed, the classical regelation boundary conditions cannot be obeyed over the whole of its surface.

Download Full-text

Model of vortex flow of charge in a vessel with turbine impeller

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19871888 ◽

1987 ◽

Vol 52 (8) ◽

pp. 1888-1904

Author(s):

Miloslav Hošťálek ◽

Ivan Fořt

Keyword(s):

Boundary Conditions ◽

Equations Of Motion ◽

Theoretical Data ◽

Eddy Diffusion ◽

Quantitative Agreement ◽

Creeping Flow ◽

Model Parameters ◽

Mean Flow ◽

Two Dimensional Flow ◽

Vorticity Transport

A theoretical model is described of the mean two-dimensional flow of homogeneous charge in a flat-bottomed cylindrical tank with radial baffles and six-blade turbine disc impeller. The model starts from the concept of vorticity transport in the bulk of vortex liquid flow through the mechanism of eddy diffusion characterized by a constant value of turbulent (eddy) viscosity. The result of solution of the equation which is analogous to the Stokes simplification of equations of motion for creeping flow is the description of field of the stream function and of the axial and radial velocity components of mean flow in the whole charge. The results of modelling are compared with the experimental and theoretical data published by different authors, a good qualitative and quantitative agreement being stated. Advantage of the model proposed is a very simple schematization of the system volume necessary to introduce the boundary conditions (only the parts above the impeller plane of symmetry and below it are distinguished), the explicit character of the model with respect to the model parameters (model lucidity, low demands on the capacity of computer), and, in the end, the possibility to modify the given model by changing boundary conditions even for another agitating set-up with radially-axial character of flow.

Download Full-text

Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

International Journal of Fracture ◽

10.1007/s10704-021-00541-y ◽

2021 ◽

Author(s):

Javier Bonet ◽

Antonio J. Gil

Keyword(s):

Kinetic Energy ◽

Crack Propagation ◽

Mathematical Models ◽

Linear Elasticity ◽

Strain Energy ◽

Equations Of Motion ◽

Two Dimensions ◽

Discontinuous Solutions ◽

Linear Elasticity Theory ◽

Intersonic Crack Propagation

AbstractThis paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

Download Full-text

Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

Boundary Value Problems ◽

10.1186/s13661-019-01316-0 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Maozhu Zhang ◽

Kun Li ◽

Hongxiang Song

Keyword(s):

Boundary Conditions ◽

Operator Theory ◽

Resolvent Convergence ◽

Spectral Inclusion ◽

Regular Approximation ◽

Special Cases ◽

Abstract Operator ◽

Norm Resolvent Convergence ◽

Sturm Liouville ◽

Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

Download Full-text

Impurity induced scale-free localization

Communications Physics ◽

10.1038/s42005-021-00547-x ◽

2021 ◽

Vol 4 (1) ◽

Author(s):

Linhu Li ◽

Ching Hua Lee ◽

Jiangbin Gong

Keyword(s):

Boundary Conditions ◽

Skin Effect ◽

Reciprocal Lattice ◽

Periodic Boundary Conditions ◽

Open Boundary ◽

Open Boundary Conditions ◽

Translational Invariance ◽

Scale Free ◽

Special Cases ◽

Simulation Results

AbstractNon-Hermitian systems have been shown to have a dramatic sensitivity to their boundary conditions. In particular, the non-Hermitian skin effect induces collective boundary localization upon turning off boundary coupling, a feature very distinct from that under periodic boundary conditions. Here we develop a full framework for non-Hermitian impurity physics in a non-reciprocal lattice, with periodic/open boundary conditions and even their interpolations being special cases across a whole range of boundary impurity strengths. We uncover steady states with scale-free localization along or even against the direction of non-reciprocity in various impurity strength regimes. Also present are Bloch-like states that survive albeit broken translational invariance. We further explore the co-existence of non-Hermitian skin effect and scale-free localization, where even qualitative aspects of the system’s spectrum can be extremely sensitive to impurity strength. Specific circuit setups are also proposed for experimentally detecting the scale-free accumulation, with simulation results confirming our main findings.

Download Full-text

Low Buckling Loads of Axially Compressed Conical Shells

Journal of Applied Mechanics ◽

10.1115/1.3408517 ◽

1970 ◽

Vol 37 (2) ◽

pp. 384-392 ◽

Cited By ~ 68

Author(s):

M. Baruch ◽

O. Harari ◽

J. Singer

Keyword(s):

Boundary Conditions ◽

Axial Compression ◽

Plane Boundary ◽

Essential Difference ◽

Physical Reason ◽

Buckling Loads ◽

Conical Shells ◽

Simple Supports ◽

Interaction Curves ◽

The Stability

The stability of simply supported conical shells under axial compression is investigated for 4 different sets of in-plane boundary conditions with a linear Donnell-type theory. The first two stability equations are solved by the assumed displacement, while the third is solved by a Galerkin procedure. The boundary conditions are satisfied with 4 unknown coefficients in the expression for u and v. Both circumferential and axial restraints are found to be of primary importance. Buckling loads about half the “classical” ones are obtained for all but the stiffest simple supports SS4 (v = u = 0). Except for short shells, the effects do not depend on the length of the shell. The physical reason for the low buckling loads in the SS3 case is explained and the essential difference between cylinder and cone in this case is discussed. Buckling under combined axial compression and external or internal pressure is studied and interaction curves have been calculated for the 4 sets of in-plane boundary conditions.

Download Full-text

Actions, topological terms and boundaries in first-order gravity: A review

International Journal of Modern Physics D ◽

10.1142/s0218271816300111 ◽

2016 ◽

Vol 25 (04) ◽

pp. 1630011 ◽

Cited By ~ 19

Author(s):

Alejandro Corichi ◽

Irais Rubalcava-García ◽

Tatjana Vukašinac

Keyword(s):

Boundary Conditions ◽

Symplectic Structure ◽

Equations Of Motion ◽

Hamiltonian Formalism ◽

Action Principle ◽

Internal Boundary ◽

Four Dimensions ◽

First Order ◽

Conserved Charges ◽

Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

Download Full-text

Surface Wave Speed of Functionally Graded Magneto-Electro-Elastic Materials with Initial Stresses

Journal of Theoretical and Applied Mechanics ◽

10.2478/jtam-2014-0016 ◽

2014 ◽

Vol 44 (3) ◽

pp. 49-64 ◽

Cited By ~ 5

Author(s):

Li Li ◽

P. J. Wei

Keyword(s):

Surface Wave ◽

Boundary Conditions ◽

Elastic Material ◽

Short Circuit ◽

Equations Of Motion ◽

Open Circuit ◽

Functionally Graded ◽

Initial Stresses ◽

Shear Surface ◽

Traction Boundary Conditions

Abstract The shear surface wave at the free traction surface of half- infinite functionally graded magneto-electro-elastic material with initial stress is investigated. The material parameters are assumed to vary ex- ponentially along the thickness direction, only. The velocity equations of shear surface wave are derived on the electrically or magnetically open circuit and short circuit boundary conditions, based on the equations of motion of the graded magneto-electro-elastic material with the initial stresses and the free traction boundary conditions. The dispersive curves are obtained numerically and the influences of the initial stresses and the material gradient index on the dispersive curves are discussed. The investigation provides a basis for the development of new functionally graded magneto-electro-elastic surface wave devices.

Download Full-text

Relativistic equations for elementary particles

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1948.0005 ◽

1948 ◽

Vol 192 (1029) ◽

pp. 195-218 ◽

Cited By ~ 19

Keyword(s):

Elementary Particle ◽

Wave Equation ◽

Equations Of Motion ◽

Rest Mass ◽

Wave Form ◽

Total Charge ◽

Field Equations ◽

Relativistic Approximation ◽

Special Cases ◽

Linear Differential

From the general principles of quantum mechanics it is deduced that the wave equation of a particle can always be written as a linear differential equation of the first order with matrix coefficients. The principle of relativity and the elementary nature of the particle then impose certain restrictions on these coefficient matrices. A general theory for an elementary particle is set up under certain assumptions regarding these matrices. Besides, two physical assumptions concerning the particle are made, namely, (i) that it satisfies the usual second-order wave equation with a fixed value of the rest mass, and (ii) either the total charge or the total energy for the particle-field is positive definite. It is shown that in consequence of (ii) the theory can be quantized in the interaction free case. On introducing electromagnetic interaction it is found that the particle exhibits a pure magnetic moment in the non-relativistic approximation. The well-known equations for the electron and the meson are included as special cases in the present scheme. As a further illustration of the theory the coefficient matrices corresponding to a new elementary particle are constructed. This particle is shown to have states of spin both 3/2 and 1/2. In a certain sense it exhibits an inner structure in addition to the spin. In the non-relativistic approximation the behaviour of this particle in an electromagnetic field is the same as that of the Dirac electron. Finally, the transition from the particle to the wave form of the equations of motion is effected and the field equations are given in terms of tensors and spinors.

Download Full-text

Boundary states for chiral symmetries in two dimensions

Journal of High Energy Physics ◽

10.1007/jhep09(2020)018 ◽

2020 ◽

Vol 2020 (9) ◽

Author(s):

Philip Boyle Smith ◽

David Tong

Keyword(s):

Boundary Conditions ◽

Ground State ◽

Zero Mode ◽

Central Charge ◽

Two Dimensions ◽

Dirac Fermions ◽

Chiral Symmetries ◽

Boundary States ◽

Ground State Degeneracy ◽

Left And Right

Abstract We study boundary states for Dirac fermions in d = 1 + 1 dimensions that preserve Abelian chiral symmetries, meaning that the left- and right-moving fermions carry different charges. We derive simple expressions, in terms of the fermion charge assignments, for the boundary central charge and for the ground state degeneracy of the system when two different boundary conditions are imposed at either end of an interval. We show that all such boundary states fall into one of two classes, related to SPT phases supported by (−1)F , which are characterised by the existence of an unpaired Majorana zero mode.

Download Full-text