On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam

This paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one “wall” is flexible and flat. The model is new in the sense that the composite dynamics of the three-dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the Reissner-Mindlin plate equations on the two-dimensional flat wall of the chamber, while, if a two-dimensional acoustic chamber is considered, the Timoshenko beam equations describe the deflections of the one-dimensional “wall.” With a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model, viz. in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium. We present the uniform stability result for the case of a two-dimensional chamber and outline the method for the three-dimensional model which shows strong resemblance with the system of dynamic plane elasticity.     

On muons of energy ≥150 Gev in association with EAS

Preliminary results of an investigation on energetic penetrating particles (muons) of energy ≥150 Gev are reported. These particles are recorded at a depth of 580 m.w.e. underground, in association with extensive air showers (EAS) of size 10⁴–10⁵ particles. Neon flash tube technique is being used to study detailed characteristics of these energetic particles. The probability of a muon being accompanied by an electron shower, produced in rock and lead absorber, is compared with the results of other workers. Two cases of large double bursts have been recorded. Some of the possible interpretations have been given. About 16% of the total events observed have more than one particle.     

Effect of variable density on hydromagnetic mixed convection flow of a non-Newtonian fluid past a moving vertical plate

The effect of temperature-dependent density on MHD mixed convection flow of power-law fluid past a moving semi-infinite vertical plate for high temperature differences between the plate and the ambient fluid is studied. The fluid density is assumed to decrease exponentially with temperature. The usual Boussinesq approximations are not considered due to the large temperature differences. The surface temperature of the moving plate was assumed to vary according to a power-law form, that is, T_w(x) = T_∞ + Ax^γ. The fluid is permeated by a uniform magnetic field imposed perpendicularly to the plate on the assumption of small magnetic Reynolds number. A numerical shooting algorithm for two unknown initial conditions with fourth-order Runge–Kutta integration scheme has been used to solve the coupled non-linear boundary value problem. The effects of various parameters on the velocity and temperature profiles as well as the local skin-friction coefficient and the local Nusselt number are presented graphically and in the tabular form. The results show that application of Boussinesq approximations in a non-Newtonian fluid subjected to high temperature differences gives a significant error in the values of the skin-friction coefficient and the application of an external magnetic field reduces this error markedly in the case of shear-thickening fluid.     

Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers CP¹→CP¹ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.     

On the vibrations of a plate with a concentrated mass and very small thickness

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^–m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. The thickness plate h being fixed, we describe the asymptotic behaviour, as ε→O, of the eigenvalues and eigenfunctions of the corresponding spectral problem, depending on the value of m: Low‐ and high‐frequency vibrations are studied for m>2. We also consider the case where the thickness plate h depends on ε; then, different values of m are singled out. Copyright © 2003 John Wiley & Sons, Ltd.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Primary cosmic ray nucleon spectrum from sea-level muon spectrum and scaling hypothesis parameters

Based on the measured sea-level muon energy spectrum and the fitted parameters of Feynman’s scaling hypothesis that describe high energy collisions, we have deduced the primary cosmic ray nucleon spectrum in the energy range 50 GeV < E < 5000 GeV. It is shown that the statistical errors in the resulting nucleon spectrum are among the lowest that exist now and any residual systematic uncertainties in the evaluated flux can be reduced further by using accurate high energy collision data of nucleons and pions in light nuclei at current accelerator energies.     

Exact solutions of accelerated flows for a Burgers’ fluid II. The cases γ = λ2/4 and γ > λ2/4

The purpose of this study is to provide the exact analytic solutions of accelerated flows for a Burgers’ fluid when the relaxation times satisfy the conditions γ = λ²/4 and γ > λ²/4. The velocity field and the adequate tangential stress that is induced by the flow due to constantly accelerating plate and flow due to variable accelerating plate are determined by means of Laplace transform. All the solutions that have been obtained are presented in the form of simple or multiple integrals in terms of Bessel functions. A comparison between Burgers’ and Newtonian fluids for the velocity and the shear stress is also made through several graphs.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

A highly accurate numerical solution of a biharmonic equation

The coefficients for a nine–point high–order accuracy discretization scheme for a biharmonic equation ∇ ⁴u = f(x, y) (∇² is the two–dimensional Laplacian operator) are derived. The biharmonic problem is defined on a rectangular domain with two types of boundary conditions: (1) u and ∂²u/∂n² or (2) u and ∂u/part;n (where ∂/part;n is the normal to the boundary derivative) are specified at the boundary. For both considered cases, the truncation error for the suggested scheme is of the sixth-order O(h⁶) on a square mesh (h_x = h_y = h) and of the fourth-order O(h⁴_xh²_xh²_y h⁴_y) on an unequally spaced mesh. The biharmonic equation describes the deflection of loaded plates. The advantage of the suggested scheme is demonstrated for solving problems of the deflection of rectangular plates for cases of different boundary conditions: (1) a simply supported plate and (2) a plate with built-in edges. In order to demonstrate the high–order accuracy of the method, the numerical results are compared with exact solutions. © John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 375–391, 1997     

Natural frequencies of thin rectangular plates using homotopy-perturbation method

The homotopy perturbation method (HPM) was developed to search for asymptotic solutions of nonlinear problems involving parabolic partial differential equations with variable coefficients. This paper illustrates that HPM be easily adapted to solve parabolic partial differential equations with constant coefficients. Natural frequencies of a rectangular plate of uniform thickness, simply-supported on all sides, are obtained with minimum amount of computation. The solution is shown to converge rapidly to a combination of sine and cosine functions. Truncating the series solution by using only the first three terms of the sine and cosine functions as compared to the exact solution results in an absolute error not exceeding 2 × 10⁻⁴ and 9×10⁻⁴ for the trigonometric functions respectively. HPM is then applied to solve the nonlinear problem of a rectangular plate of variable thickness. A direct expression for the eigenvalues (natural frequencies) of the rectangular plate is obtained as compared to determining its eigenvalues by solving the characteristic equation using the conventional method. Comparison of results for the frequency parameter with existing literature show that HPM is highly efficient and accurate. Natural frequencies of a simply-supported guitar soundboard were obtained using an equivalent rectangular plate with the same boundary condition.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

Rheological and dynamic thermomechanical propert ies of epoxy composites reinforced with single- and multiwalled carbon nanotubes

An epoxy resin (Epon 828) was filled with single- and multiwalled carbon nanotubes (SWCNT and MWCNT) in two steps by using the high shear mixing and ultrasonication techniques. The melt flow of the composites was characterized in a plate/plate rheometer. The thermomechanical properties of the composites were determined in dynamic mechanical analysis tests performed at various frequencies and temperatures. It was found that the incorporation of SWCNT or MWCNT increased the viscosity and stiffness of epoxy above its glass-transition temperature. The time-temperature superposition principle was employed to estimate the storage modulus of the composites as a function of frequency (f = 10^–33–10³ Hz) in the form of master curves.     

Ein supraleitender Gleichspannungsverstärker für Helium-Temperaturen

Summary A simpleDC Voltage Amplifier using the resistive transition curve of a superconducting wire of 8% Pb in In is described. This wire is held in the intermediate state by a constant magnetic field. The small current from the measuring voltage is amplified by a superconducting transformer and used to build up a small additional field over the wire which in turn changes its resistance. This change in resistance is measured by a conventional potential drop technique with an ordinary galvanometer. The sensitivity of that device is 10^–9 Volt. It is limited by zero point fluctuations caused by instabilities of the intermediate state.     

Alpha effect in helically forced turbulence

The effect on large-scale dynamics of small-scale helicity injection in three-dimensional resistive magnetohydrodynamic turbulence is investigated for a weak initial magnetic field. The magnetic configuration of α² dynamo prototype flows is such that, generally, the energy concentrates on one large-scale mode. However, we obtain that alpha effect is not limited to the dominant mode and that a non-local equation (in Fourier space) is more appropriate to describe it. It gives some insights into the non-local theory of Pouquet et al. [J. Fluid Mech. 77 (1976) 321] where the inverse cascade results from a competition between the helicity and Alfvén effects.     

Effect of Second Order Chemical Reaction on MHD Free Convective Radiating Flow over an Impulsively Started Vertical Plate

An attempt has been made to study laminar convective heat and mass transfer flow of an incompressible, viscous and electrically conducting fluid over an impulsively started vertical plate with conduction-radiation embedded in a porous medium in presence of transverse magnetic field. The influence of both second order chemical reaction and heat generation are taken into account. The governing coupled partial differential equations are solved by Crank-Nicolson method. The effects of important physical parameters on the velocity, temperature and concentration have been analyzed through graphs. The results of the present study agree well with the previous solutions. Applications of the present study are shown in material processing systems and different industries. The important findings of present study are: chemical reaction parameter acts as resistive force to reduce the velocity whereas heat source parameter enhances the velocity.     

A compact C0 discontinuous Galerkin method for Kirchhoff plates

A compact C⁰ discontinuous Galerkin (CCDG) method is developed for solving the Kirchhoff plate bending problems. Based on the CDG (LCDG) method for Kirchhoff plate bending problems, the CCDG method is obtained by canceling the term of global lifting operator and enhancing the term of local lifting operator. The resulted CCDG method possesses the compact stencil, that is only the degrees of freedom belonging to neighboring elements are connected. The advantages of CCDG method are: (1) CCDG method just requires C⁰ finite element spaces; (2) the stiffness matrix is sparser than CDG (LCDG) method; and (3) it does not contain any parameter which can not be quantified a priori compared to C⁰ interior penalty (IP) method. The optimal order error estimates in certain broken energy norm and H¹‐norm for the CCDG method are derived under minimal regularity assumptions on the exact solution with the help of some local lower bound estimates of a posteriori error analysis. Some numerical results are included to verify the theoretical convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1265–1287, 2015     

Port-Hamiltonian formulation and symplectic discretization of plate models Part II: Kirchhoff model for thin plates

The mechanical model of a thin plate with boundary control and observation is presented as a port-Hamiltonian system (PHs¹), both in vectorial and tensorial forms: the Kirchhoff-Love model of a plate is described by using a Stokes-Dirac structure and this represents a novelty with respect to the existing literature. This formulation is carried out both in vectorial and tensorial forms. Thanks to tensorial calculus, this model is found to mimic the interconnection structure of its one-dimensional counterpart, i.e. the Euler-Bernoulli beam.The Partitioned Finite Element Method (PFEM²) is then extended to obtain a suitable, i.e. structure-preserving, weak form. The discretization procedure, performed on the vectorial formulation, leads to a finite-dimensional port-Hamiltonian system. This part II of the companion paper extends part I, dedicated to the Mindlin model for thick plates. The thin plate model comes along with additional difficulties, because of the higher order of the differential operator under consideration.     

Casimir force between metal plate and dielectric plate

The Casimir effect between metal plate and dielectric plate is discussed with 1 + 1-dimensional potential model without using cut-off method. Calculation shows that the Casimir force between metal plate and dielectric plate is determined not only by the potentialV ₀, the dielectric thickness and the distancea between the metal plate and dielectric plate, but also by the dimension of the vessel. Whena is far less than the dimension of the vessel, the Casimir forceF _c ∝a ^-1; converselyF _c ∝a ^-2 This result is significant for Casimir force experiment.