Abstract: Abstract In the present work, the band-pass filter characteristics of a coaxial waveguide with an impedance coated groove on the inner wall and perfectly conducting outer wall is analyzed rigorously through the Wiener–Hopf technique. By using the direct Fourier transform, the related boundary value problem is reduced to the Wiener–Hopf equation whose solution contains infinitely many constants satisfying an infinite system of linear algebraic equations. These equations are solved numerically and the field terms, which depends on the solution obtained numerically, are derived explicitly. The problem is also analyzed by applying a mode-matching technique and the results are compared numerically. Besides, some computational results illustrating the effects of parameters such as depth of the groove, surface impedances and radii of the walls are also presented. PubDate: 2020-09-15
Abstract: Abstract Investigated in this paper is the coupled fluid–body motion of a thin solid body undergoing a skimming impact on a shallow-water layer. The underbody shape (the region that makes contact with the liquid layer) is described by a smooth polynomic curve for which the magnitude of underbody thickness is represented by the scale parameter C. The body undergoes an oblique impact (where the horizontal speed of the body is much greater than its vertical speed) onto a liquid layer with the underbody’s trailing edge making the initial contact. This downstream contact point of the wetted region is modelled as fixed (relative to the body) throughout the skimming motion with the liquid layer assumed to detach smoothly from this sharp trailing edge. There are two geometrical scenarios of interest: the concave case ( \(C<0\) producing a hooked underbody) and the convex case ( \(C > 0\) producing a rounded underbody). As C is varied the rebound dynamics of the motion are predicted. Analyses of small-time water entry and of water exit are presented and are shown to be broadly in agreement with the computational results of the shallow-water model. Reduced analysis and physical insights are also presented in each case alongside numerical investigations and comparisons as C is varied, indicating qualitative analytical/numerical agreement. Increased body thickness substantially changes the interaction structure and accentuates inertial forces in the fluid flow. PubDate: 2020-09-08
Abstract: Abstract Because of geometric or material discontinuities, stress singularities can occur around the vertex of a V-notch. The singular order is an important parameter for characterizing the degree of the stress singularity. The present paper focuses on the calculation of the singular order for the anti-plane propagating V-notch in a composite material structure. Starting from the governing equation of elastodynamics and the displacement asymptotic expansion, an ordinary differential eigen equation with respect to the singular order is proposed. The interpolating matrix method is then employed to solve the established eigen equation to conduct the singular orders. The effects of the material principal axis direction, shear modulus, and propagation velocity and acceleration on the singular orders of the V-notch are respectively investigated, and some conclusions are drawn. The singular orders of the V-notches decrease with an increase in the material principal axis direction angle, except for the crack, whose singular orders do not change with the principal axis direction. The singular orders increase with an increase in the shear modulus \(G_{13} \) , while they decrease with an increase in the shear modulus \(G_{23} \) . The singular orders increase as the magnitude of the propagation velocity increases, while they decrease as the direction of the propagation velocity increases. The singular orders increase with an increase in the value of the propagation acceleration, while they decrease with an increase in the direction of the propagation acceleration. The singular orders become larger when the ratio of \(G_{13}^{(2)} /G_{13}^{(1)} \) increases, while they become smaller when the ratio of \(G_{23}^{(2)} /G_{23}^{(1)} \) increases, and they increase with an increase in the mass density ratio for the bi-material V-notch. PubDate: 2020-09-03
Abstract: Abstract A ferrofluid saturated porous layer convection problem is studied in the variable gravitational field for a local thermal nonequilibrium (LTNE) model. Internal heating and variation (increasing or decreasing) in gravity with distance through the layer affected the stability of the convective system. The Darcy model is employed for the momentum equation and the LTNE model for the energy equation. The boundaries are considered to be rigid-isothermal and paramagnetic. For the linear stability analysis of the three-dimensional problem, the normal mode has been applied and the eigenvalue problem is solved numerically using Chebyshev pseudospectral method, while weakly nonlinear analysis is carried out with a truncated Fourier series. The effect of different dimensionless parameters on the Rayleigh number has also been studied. We found that the system becomes unstable on the increasing value of nonlinearity index of magnetization ( \(M_3\) ), porosity-modified conductivity ratio ( \(\beta \) ), and internal heat parameter ( \(\xi \) ). It is observed that the system is stabilized by increasing the value of the Langevin parameter ( \(\alpha _\mathrm{{L}}\) ), variable gravity coefficient ( \(\delta \) ), and effective heat transfer parameter ( \(H_1\) ). Runge–Kutta–Gill method has been used for solving the finite-amplitude equations to study the transient behavior of the Nusselt number. Streamlines and isotherms patterns are determined for the steady case and are presented graphically. PubDate: 2020-09-01
Abstract: Abstract A third-order highly non-linear ODE that arises in applications of non-Newtonian boundary-layer fluid flow, governed by a power-law Ostwald–de Waele rheology, is considered. The model appears in many disciplines related to applied and engineering mathematics, in addition to engineering and industrial applications. The aim is to use a new set of variables, defined via the first and second-order derivatives of the dependent variable, to transform the problem to a bounded domain, where we study properties of solutions, discuss existence and uniqueness of solutions, and investigate some physical parameter values and limitations leading to non-existence of solutions. PubDate: 2020-08-13
Abstract: Abstract For a pulsating free surface source in a three-dimensional finite depth fluid domain, the Green function of the source presented by John [Communs. Pure Appl. Math. 3:45–101, 1950] is superposed as the Rankine source potential, an image source potential and a wave integral in the infinite domain \((0, \infty )\) . When the source point together with a field point is on the free surface, John’s integral and its gradient are not convergent since the integration \(\int ^\infty _\kappa \) of the corresponding integrands does not tend to zero in a uniform manner as \(\kappa \) tends to \(\infty \) . Thus evaluation of the Green function is not based on direct integration of the wave integral but is obtained by approximation expansions in earlier investigations. In the present study, five images of the source with respect to the free surface mirror and the water bed mirror in relation to the image method are employed to reformulate the wave integral. Therefore the free surface Green function of the source is decomposed into the Rankine potential, the five image source potentials and a new wave integral, of which the integrand is approximated by a smooth and rapidly decaying function. The gradient of the Green function is further formulated so that the same integration stability with the wave integral is demonstrated. The significance of the present research is that the improved wave integration of the Green function and its gradient becomes convergent. Therefore evaluation of the Green function is obtained through the integration of the integrand in a straightforward manner. The application of the scheme to a floating body or a submerged body motion in regular waves shows that the approximation is sufficiently accurate to compute linear wave loads in practice. PubDate: 2020-08-12
Abstract: Abstract An effective-medium theory for the electrical conductivity of Ohmic dispersions taking explicit account of particle shape and spatial distribution independently is available from the work of Ponte Castañeda and Willis [J Mech Phys Solids 43:1919–1951, 1996]. When both shape and distribution take particular “ellipsoidal” forms, the theory provides analytically explicit estimates. The purpose of the present work is to evaluate the predictive capabilities of these estimates when dispersions exhibit dissimilar particle shape and distribution. To this end, comparisons are made with numerical bounds for coated ellipsoid assemblages computed via the finite element method. It is found that estimates and bounds exhibit good agreement for the entire range of volume fractions, aspect ratios, and conductivity contrasts considered, including those limiting values corresponding to an isotropic distribution of circular cracks. The fact that the explicit estimates lie systematically within the numerical bounds hints at their possible realizability beyond the class of isotropic dispersions. PubDate: 2020-08-01
Abstract: Abstract A recent asymptotic model for the operation of a vanadium redox flow battery (VRFB) is extended to include the dissociation of sulphuric acid—a bulk chemical reaction that occurs in the battery’s porous flow-through electrodes, but which is often omitted from VRFB models. Using asymptotic methods and time-dependent two-dimensional numerical simulations, we show that the charge–discharge curve for the model with the dissociation reaction is almost identical to that for the model without, even though the concentrations of the ionic species in the recirculating tanks, although not the state of charge, are considerably different in the two models. The ability of the asymptotic model to extract both the qualitative and quantitative behaviour of the considerably more time-consuming numerical simulations correctly indicates that it should be possible to add further physical phenomena to the model without incurring significant computational expense. PubDate: 2020-08-01
Abstract: Abstract As an extension of the discrete Sommerfeld problems on lattices, the scattering of a time-harmonic wave is considered on an infinite square lattice when there exists a pair of semi-infinite cracks or rigid constraints. Due to the presence of stagger, also called offset, in the alignment of the defect edges the asymmetry in the problem leads to a matrix Wiener–Hopf kernel that cannot be reduced to scalar Wiener–Hopf in any known way. In the corresponding continuum model the same problem is a well-known formidable one which possesses certain special structure with exponentially growing elements on the diagonal of kernel. From this viewpoint the present paper tackles a discrete analog of the same by reformulating the Wiener–Hopf problem and reducing it to a finite set of linear algebraic equations; the coefficients of which can be found by an application of the scalar Wiener–Hopf factorization. The considered discrete paradigm involving lattice waves is relevant for modern applications of mechanics and physics at small length scales. PubDate: 2020-07-31
Abstract: Abstract The algebraic and semi-algebraic formulations of a new multiscale elliptic solver based on a non-overlapping domain decomposition method are presented. By algebraic we mean that all information needed to implement the proposed procedure can be extracted from the underlying fine grid finite volume linear system. In addition to the entries of this linear system, the proposed semi-algebraic procedure will also use the coefficients of the elliptic equation at subdomain boundaries. Initially we construct multiscale basis functions (or local solutions) subject to Robin boundary conditions. Although the implementation of Robin conditions in the formulation of the multiscale method uses the coefficients of the elliptic equation of interest, we modify it such that only entries of the finite volume linear system appear in the calculation of local solutions. A linear combination of local solutions gives a (discontinuous at subdomain boundaries) solution that is refined in a smoothing step. To this end we propose an algebraic scheme to remove discontinuities that needs a staggered (or dual) coarse grid. By iterating with the defect correction scheme, we construct an effective two-stage preconditioner that combines both local solutions and the smoothing step; the final result is obtained in a small number of iterations. Our focus in this work is the presentation of the algebraic and semi-algebraic formulations of the multiscale elliptic solver, which are carefully explained, and in verifying their accuracy and efficiency in the solution of challenging problems. We consider two- and three-dimensional problems with coefficients exhibiting high-contrast, anisotropic, and channelized structures that are solved to reveal the good properties of the proposed schemes. PubDate: 2020-07-31
Abstract: Abstract Hyperbolic towers are towers in the shape of a single-sheet hyperboloid, and they are interesting in architecture. In this paper, we deal with the infinitesimal bending of a curve on a hyperboloid of one sheet; that is, we study the flexibility of the net-like structures used to make a hyperbolic tower. Visualization of infinitesimal bending has been carried out using Mathematica, and some examples are presented and discussed. PubDate: 2020-07-31
Abstract: Abstract We find a general method to obtain the radially symmetric solutions of Dirichlet problem for Pennes bioheat equation in the exterior domain of a circle through the computation of Green’s function of a naturally related operator. We apply this technique to solve a problem in radio-frequency ablation. PubDate: 2020-07-30
Abstract: Abstract The radiative–conductive transfer equation in the \(S_N\) approximation for spherical geometry is solved using a modified decomposition method. The focus of this work is to show how to distribute the source terms in the recursive equation system in order to guarantee arithmetic stability and thus numerical convergence of the obtained solution, guided by a condition number criterion. Some examples are compared with results from literature and parameter combinations are analyzed, for which the condition number analysis indicates convergence of the solutions obtained by the recursive scheme. PubDate: 2020-07-30
Abstract: Abstract In geo-engineering, mechanical problems of surcharge loads acting on ground surface and shallow tunnel excavation are often encountered. When the complex variable method is applied, such problems turn to non-zero resultant issues and generally result in infinite displacement singularity in geomaterial at infinity, which violates the fact that displacement components at infinity should be zero in field observations. The displacement singularity at infinity caused by non-zero resultants corresponds non-zero coefficients of logarithmic items in complex potentials. To eliminate the singularity at infinity, the symmetrical method is adopted so that the far-field displacement obtained from the modified displacement solution conducted by the complex variable method would be logically identical to the reality. Two fundamental cases of single surcharge load acting on ground surface and single tunnel excavation are presented to verify the modified displacement solution, and good agreements are observed. The main advantage that the proposed solution eliminates displacement singularity at infinity and makes far-field displacement logically accord with field observation is presented by comparing to the original solution. The drawback that the proposed solution is dependent on modified depth is discussed as well. PubDate: 2020-06-06
Abstract: Abstract The problem of oblique scattering of surface waves by a thick partially immersed rectangular barrier or a thick submerged rectangular barrier extending infinitely downwards in deep water is studied here to obtain the reflection and transmission coefficients semi-analytically. Use of Havelock’s expansion of water wave potential function reduces each problem to an integral equation of first kind on the horizontal component of velocity across the gap above or below the barrier. Multi-term Galerkin approximations involving polynomials as basis functions multiplied by appropriate weight functions are used to solve these equations numerically. Evaluated numerical results for the reflection coefficients are plotted graphically for both the barriers. The study reveals that the reflection coefficient depends significantly on the thickness of the barrier. The accuracy of the numerical results is checked by using energy identity and by obtaining results available in the literature as special cases. PubDate: 2020-06-04
Abstract: Abstract Localized buckling modes in the axially compressed strut on a distributed-spring elastic foundation with a softening quadratic nonlinearity are numerically calculated based on a modified Petviashvili method in the spatial frequency domain. As the load decreases (increases), the maximum displacement of the corresponding localized buckling mode increases (decreases) and its width decreases (increases). Then, under the influence of longitudinal and transverse perturbations, stabilities of these localized buckling modes are numerically investigated. The adopted numerical method is the spatial Fourier transform in space and the finite difference method in time. For initial positive longitudinal perturbations, localized buckling modes are unstable showing focusing-type finite-time blowup singularities. For initial negative perturbations, localized buckling modes are unstable and become dispersed showing small-amplitude oscillations. For initial transverse perturbations, localized buckling modes are unstable and are transformed into three-dimensional locally confined modes, finally showing finite-time blowup singularities. PubDate: 2020-06-02
Abstract: Abstract The paper presents analytical and numerical results of radiation phenomena at the far field and solution of the acoustic wave equation with boundary conditions imposed by the pipe wall. A semi-infinite pipe with partial lining and interior perforated screen is considered. The study is important because of its applications in noise reduction in exhausts of automobile engines, in modern aircraft jet and turbofan engines. This problem is described by a Wiener–Hopf equation and then solved numerically. The solution involves three infinite sets of coefficients satisfying three infinite systems of linear algebraic equations. Numerical solutions of these systems are obtained for various values of the parameters of the problem and their effects on the radiation phenomenon are shown graphically. PubDate: 2020-04-25
Abstract: Abstract In the hot end of a 3-D printer, polymer feedstock flows through a heated cylinder in order to become pliable. This setup determines a natural upper limit to the speed at which the polymer may be extruded. The case of polymers which undergo the crystalline-melt transition is considered; the resulting mathematical model is a Stefan-like moving boundary-value problem for the polymer temperature. Using the heat balance integral method provides an analytical approximation for the temperature. Several different conditions which use this temperature to establish the maximum velocity are considered; using a pointwise polymer exit temperature in the hot end matches well with experimental data. PubDate: 2020-04-13
Abstract: Abstract Multi-parameter sensitivity algorithms can be used to construct a Hessian matrix and second-degree Taylor expansion. In terms of an asymmetric dynamic system, two multi-parameter sensitivity algorithms are proposed in this paper. The modal method with its consistence proof is firstly derived to compute the first- and second-order sensitivities of the eigenpair, and the algebraic method with its stability proof is also proposed. One significant difference between the algebraic method and the modal method is that the algebraic method uses the derivative of the normalization condition as the bordered equation to remove the singularity of the coefficient matrix in the sensitivity dominant equation, whereas the modal method uses the derivative of the normalization condition as the supplementary equation to determine the special coefficient in the modal superposition for a normalized undamped mode shape. As both the proposed methods adopt the same normalization condition, the resulting sensitivities are consistent with each other. Three numerical example are used to determine the correctness, accuracy and validity of the proposed methods in three cases: a single-parameter system, a two-parameter system, and a special system which has complex eigenpairs caused by the asymmetric property of the dynamic system. PubDate: 2020-04-13
Abstract: Abstract Thermal electrochemical models for porous electrode batteries (such as lithium ion batteries) are widely used. Due to the multiple scales involved, solving the model accounting for the porous microstructure is computationally expensive; therefore, effective models at the macroscale are preferable. However, these effective models are usually postulated ad hoc rather than systematically upscaled from the microscale equations. We present an effective thermal electrochemical model obtained using asymptotic homogenisation, which includes the electrochemical model at the cell level coupled with a thermal model that can be defined at either the cell or the battery level. The main aspects of the model are the consideration of thermal effects, the diffusion effects in the electrode particles, and the anisotropy of the material based on the microstructure, all of them incorporated in a systematic manner. We also compare the homogenised model with the standard electrochemical Doyle, Fuller & Newman model. PubDate: 2020-04-11
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