Arbitrary Smooth Cross Section Research Articles

Diffraction of flexural-gravity waves by a vertical cylinder of non-circular cross section

The linear three-dimensional problem of flexural-gravity wave (hydro-elastic wave) diffraction by a vertical cylinder of an arbitrary smooth cross section is studied using an asymptotic approach combined with the vertical mode method for water of finite depth. The surface of the water is covered by an infinite, continuous elastic ice plate. The rigid cylinder extends from the sea bottom to the ice surface. The ice plate is frozen to the cylinder. The ice deflection is described by the equation of a thin elastic plate of constant thickness with clamped edge conditions at the cylinder. The flow under the ice is described by the linear theory of potential flows. The coupled problem of wave diffraction is solved in two steps. First, the problem is solved without evanescent waves similar to the problem of water waves diffracted by a vertical cylinder. This solution does not satisfy the edge conditions. Second, a radiation problem with a prescribed motion of the ice plate edge is solved by the vertical mode method. The sum of these two solutions solve the original problem. Both solutions are obtained by an asymptotic method with a small parameter quantifying a small deviation of the cylinder cross section from a circular one. Third-order asymptotic solutions are obtained by solving a set of two-dimensional boundary problems for Helmholtz equations in the exterior of a circle. Strains along the edge, where the ice plate is frozen to the cylinder, are investigated for nearly square and elliptic cross sections of the vertical cylinders depending on the characteristics of ice and incident wave. The strains are shown to be highest in the places of high curvatures of the cross sections. The derived asymptotic formulae can be used in design of vertical columns in ice. They directly relate the strains in ice plate to the shape of the column.

Modeling the gravitational flows of a viscoplastic fluid on a conical surface

рідка плівка,The problem of a stationary waveless gravitational flow of a viscoplastic fluid over the surface of a cone with an arbitrary smooth cross section is considered. It is assumed that the axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. A curvilinear orthogonal coordinate system (ξ, η, ζ) associated with the body surface is introduced: ξ is the coordinate along the generatrix of the body, η is the polar angle in the plane perpendicular to the axis of the body of revolution, ζ is the distance along the normal to the surface. To describe the flow of a liquid film, a viscous incompressible fluid model is used, which is based on partial differential equations - the equations of motion and continuity. The following boundary conditions are used: sticking conditions on the solid surface; on the surface separating liquid and gas, the conditions for continuity of stresses and normal component of the velocity vector. To close the system of differential equations, the Shvedov-Bingham rheological model is used. To simplify the system of differential equations, the small parameter method is used. The small parameter is the relative film thickness. It is assumed that the generalized Reynolds number has an order equal to one. The solution of the equations of continuity and motion (taking into account the principal terms of the expansion) was obtained in an analytical form. The obtained formulas for the components of the velocity and pressure vector generalize the known relations for flat surfaces. To determine the unknown film thickness, an initial-boundary value problem was formulated for a first-order partial differential equation. The solution to this problem is found with the help of the finite difference method. The results of calculations by the proposed method for cones with a cross section in the form of a circle and a square with rounded corners are presented. Calculations show that the plasticity parameter and the cross-sectional shape significantly affect the velocity and distribution profiles of the thickness of the viscous layer over the surface of the body.

Wave action by arrays of vertical cylinders with arbitrary smooth cross-section

This paper presents an analytical model to solve the linear wave diffraction problem by arrays of bottom-mounted cylinders with arbitrary smooth cross-section. Based on the assumption of ideal fluid and potential theory, the unknown coefficients of total velocity potential can be solved by system of linear equations, which are obtained from the boundary conditions. The accuracy of the present method is verified by comparing it with the numerical tool in terms of the wave force and wave run-up. Multiple cylinders with different configurations are tested, the cross-section of which is circular with cosine perturbation. The results show that the proposed method could obtain an accurate prediction of the wave action with multiple cylinder problems. Finally, the diffraction wave is investigated on arrays of bottom-mounted cylinders with different cross-section and layout. The near-trapping problem with effects of the multi-body interaction are also investigated.

Investigation of linear wave action around a truncated cylinder with non-circular cross section

This paper presents an analytical model to solve the linear wave diffraction problem using a truncated cylinder with an arbitrary smooth cross section. Using the method of separation of variables, the boundary condition equations for the surface of a truncated cylinder are derived. Based on the condition that the radius function of a cylinder surface can be expanded into a Fourier series, the unknown coefficients of the diffraction potential can be obtained from a linear system equation. The surface elevation and wave force are calculated and analyzed in terms of the total velocity potential for different cases. The truncated error introduced in the solving process is discussed. Then, the accuracy of this method is verified by comparison with the results of the boundary element method. Finally, this method is further extended to a practical engineering application with a quasi-ellipse caisson for different drafts and exposures.

On the quantum mechanics of a particle constrained to generalized cylinders with application to Möbius strips

The intrinsic geometric approach to quantum mechanics has been applied to study nanostructures, as well as aromaticity and other electronic properties of molecules. Usually strong hypotheses on the geometry are made in order to allow for analytical solutions. We undertake the problem of stability, with respect to smooth deformations, of the spectra obtained under such simplifying hypotheses. In this article we show that symmetries can sometimes be removed by solving the problem of a particle constrained to a right cylinder with an arbitrary smooth cross section considering the effects of intrinsic geometry. The topology of the cross section, rather than its geometry, is shown to play a key role in this context. Our solution is applied to a generalized Mobius strip when the median curve is nearly a circle of diameter sufficiently large compared to the width of the strip. We also provide a systematic method for ordering the quantum states of generalized right cylinders according to their energies and apply this method to order the spectra of Mobius strips whose diameters are much larger than their widths. Finally, for the sake of comparison, our results are used to calculate the \(\uppi \) electron energy spectra of five aromatic molecules (benzene, pyrazine, pyridine, 1,3 diazyne and 1,3,5 triazyne), showing a fair agreement with experimental data and quantum chemistry calculations.

Torsion of an arbitrarily shaped nanosized bar

We investigate the Saint-Venant torsion problem of a nanosized bar with arbitrary smooth cross section by enhancing the classical continuum-based model with a theory of surface elasticity based on a version of the Gurtin–Murdoch surface/interface model. An elegant analytical solution is derived by means of conformal mapping, Faber series, Fourier series, and simple matrix algebra. Our solution leads to the establishment of a complex torsion function which, in turn, allows for a concise expression for the torsional rigidity. Our results clearly demonstrate that both the stress field and the normalized torsional rigidity are size dependent. An approximate formula for the size-dependent torsional rigidity is also proposed.

Exponentially Converging Nyström Methods in Oblique Diffraction of Arbitrarily Polarized Waves by Bianisotropic/Chiral Cylinders With Arbitrary Smooth Cross Section

We study oblique diffraction of arbitrarily polarized plane waves by a uniform homogeneous bianisotropic cylinder with arbitrarily shaped smooth boundary. Following a boundary integral equations formulation we develop a 4 × 4 system of combined field integral equations and another 4 × 4 system of Muller-type Fredholm integral equations. Both these systems, having the tangential to the surface of the cylinder components of the electric and magnetic fields as the unknowns, are of the second kind and uniquely solvable. Efficient discretization via simple and rapidly converging Nystrom algorithms enables one to obtain highly accurate results even for electrically large objects. The proposed solution technique: a) fully accounts for the singular nature of the kernels; b) yields simple closed form expressions for all matrix elements; and c) appears to converge exponentially versus matrix size. The analogous diffraction problem for an arbitrarily shaped smooth chiral cylinder may be treated as a simple special case of the present analysis.

Transonic Shocks in Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity(non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Frechet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.

Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

We prove that the spherically symmetric subsonic flows in an infinitely long straight divergent nozzle with arbitrary smooth cross-section are unique for the three-dimensional steady potential flow equation. The proof depends on an extreme principle for elliptic equations in an unbounded conical domain, under the assumption that the gradient of the solution is of order \({O\left(\frac{1}{|x|}\right)}\) as \({|x|\rightarrow\infty}\) . Similar result holds for steady subsonic Euler flows in two-dimensional infinitely long straight divergent nozzles.

Existence and uniqueness of solutions for acoustic scattering over infinite obstacles

The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂ D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.

Electromagnetic scattering by a smooth convex impedance cone

The problem of the diffraction of an electromagnetic plane wave by a convex cone of arbitrary smooth cross-section with impedance (Leontovich) boundary conditions is studied. The vector problem is reduced to that for the Debye potentials. By means of Kontorovich-Lebedev integrals, two spectral functions are introduced and the corresponding boundary value problem is formulated. The spectral functions for the potentials are found to satisfy the Helmholtz equations on the unit sphere and to be coupled through non-traditional boundary conditions of the impedance type with shifts on the spectral variable. The use of the Green theorem permits us to establish an integral formulation of the boundary value problem for the spectral functions. The formal asymptotic solution of the problem is then given for the case of a narrow cone. For this, two different methods are given: a method of perturbation applied to the spectral integral equations and an adaptation of the method of matching the asymptotic series in spectral domain. Both methods lead to the same closed-form result for the leading term of the scattering diagram asymptotics.

Analysis of Electromagnetic Scattering By Infinite Conducting Cylinders of Arbitrary Smooth Cross Section Using a Genetically Optimised Technique (Ga/Mas)

The utilisation of elemental auxiliary sources located inside the scatterer in order to approximately describe the scattered field has proved to be an efficient numerical method for the analysis of electromagnetic (EM) scattering by metallic bodies. In this paper the problem of TM scattering by perfectly conducting cylinders of arbitrary but smooth cross section is treated with the integration of the above technique with the strategy of Genetic Algorithm (GA) optimisation. The GA is employed in order to allocate the auxiliary sources inside the scatterer and minimise the boundary condition error (BCE), which is an endogenous effect of the point matching procedure involved in the numerical solution. For such a case, the trial and error stage usually required for the selection of the source locations is omitted and at the same time GA stochastic search can be potentially conducted in a similar way in complex space. For a predefined accuracy of the BCE, the resulting pleiotropic, GA-optimised allocations of the sources are non-intuitive but notably affected by the geometry of the scatterer. Results are presented for the cases of infinite, perfectly conducting circular, elliptic, and periodically corrugated cylinders, demonstrating sufficient agreement with available solutions.

Radiation and radiation impedance on fluid-loaded elastic shells of arbitrary shape

A ray method is systematically used to derive a relation between radiated acoustic waves and elastic waves traveling over a smooth elastic shell of arbitrary shape. The radiated acoustic field is found to be intimately connected with the geometries of the shell's surface and the elastic wavefronts. This connection leads to an asymptotic expression for the local radiation impedance associated with each surface ray under the condition k f R ⪢ 1, where k f is the wavenumber in fluid and R the smallest radius of curvature of the shell. The first term in this formula is the result for an infinite flat plate, which is homogeneous and isotropic. The second term introduces inhomogeneity and anisotropy into the radiation impedance because it explicitly depends on the surface ray direction and the local curvatures of the shell's surface and the elastic wavefront. The general result is further simplified for a cylindrical shell of arbitrary smooth cross-section. The comparisons with the exact solutions for circular cylindrical and spherical shells illustrate that the present results can reproduce the first two terms in the asymptotic expansions of the exact solutions.

Approximate Solution of Second Kind Integral Equations on Infinite Cylindrical Surfaces

The paper considers second kind integral equations of the form $\phi (x) = g(x) + \int_S {k(x,y)} \phi (y)ds(y)$ (abbreviated $\phi = g + K\phi $), in which S is an infinite cylindrical surface of arbitrary smooth cross section. The “truncated equation” (abbreviated $\phi _a = E_a g + K_a \phi _a $), obtained by replacing S by $S_a $, a closed bounded surface of class $C^2 $, the boundary of a section of the interior of S of length $2a$, is also discussed. Conditions on k are obtained (in particular, implying that K commutes with the operation of translation in the direction of the cylinder axis) which ensure that $I - K$ is invertible, that $I - K_a $ is invertible and $(I - K_a )^{ - 1} $ is uniformly bounded for all sufficiently large a, and that $\phi _a $ converges to $\phi $ in an appropriate sense as $a \to \infty $. Uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution (of the integral equation and the corresponding boundary value problem) and convergence of a particular collocation method.

Analysis of two-dimensional acoustic scattering from periodic structures using a hybrid source model

A method of moments solution is presented for the problem of two-dimensional acoustic scattering of a plane wave from a general piecewise homogeneous periodic structure. The reduction of the general problem to a consideration of the fields over a suitably selected period, referred to as the unit cell, is facilitated by the Floquet theorem. The solution then uses fictitious spatially periodic and properly modulated strip sources to simulate the fields in homogeneous regions crossed by the unit cell boundaries and fictitious filamentary sources to simulate the fields in homogeneous regions completely enclosed within this cell. The fields radiated by the strip sources are represented in terms of Floquet modes. Finally, the simulated fields are forced to obey the continuity conditions for the pressure and the normal component of the velocity at selected sets of points on the boundaries among the regions within the unit cell. The procedure is simple to implement, rapidly converging, and is applicable to structures of arbitrary smooth cross section. Acoustically rigid and soft regions are also treated within the general procedure. Results are given and compared with available data and with an analytic approximation.

Analysis of two‐dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model

A method of moments solution is presented for the problem of two‐dimensional electromagnetic scattering from a periodic grating composed of an infinite set of penetrable cylinders illuminated by a TM (transverse magnetic relative to the cylinder axis) plane wave. The reduction of the general problem to a consideration of the fields over a suitably selected period, referred to as the unit cell, is facilitated by the Floquet theorem. The solution then uses fictitious spatially periodic and properly modulated electric current strips to simulate the scattered field in the exterior region within the unit cell and fictitious filamentary currents to simulate the field penetrated into the cylinder enclosed within this cell. The fields radiated by the current strips are represented in terms of Floquet modes. Finally, the simulated fields are forced to obey the continuity conditions for the tangential components of the electric and magnetic fields at a selected set of points on the cylinder boundary within the unit cell. The procedure is simple to implement, rapidly converging, and is applicable to cylinders of arbitrary smooth cross section. Perfectly conducting cylinders are treated as reduced cases of the general procedure for penetrable cylinders. Results are given and compared with available data. The efficiency of the suggested method is demonstrated.