# Showing papers in "Journal of Engineering Mathematics in 2005"

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Duke University

^{1}TL;DR: In this paper, a set of lubrication models for thin film flow of incompressible fluids on solid substrates is derived and studied, where the models are obtained as asymptotic limits of the Navier-Stokes equations with the slip boundary condition for different orders of magnitude for the slip-length parameter.

Abstract: A set of lubrication models for the thin film flow of incompressible fluids on solid substrates is derived and studied. The models are obtained as asymptotic limits of the Navier-Stokes equations with the Navier-slip boundary condition for different orders of magnitude for the slip-length parameter. Specifically, the influence of slip on the dewetting behavior of fluids on hydrophobic substrates is investigated here. Matched asymptotics are used to describe the dynamic profiles for dewetting films and comparison is given with computational simulations. The motion of the dewetting front shows transitions from being nearly linear in time for no-slip to t 2/3 as the slip is increased. For much larger slip lengths the front motion appears to become linear again. Correspondingly, the dewetting profiles undergo a transition from oscillatory to monotone decay into the uniform film layer for large slip. Increasing the slip further, to very large values, is associated with an increasing degree of asymmetry in the structure of the dewetting ridge profile.

148 citations

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TL;DR: In this paper, the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations and the difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions.

Abstract: Engineering materials are generally non-homogeneous, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. In the paper the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations. The difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions. This leads to a Cosserat continuum theory. A more sophisticated strategy is to homogenise the equations by means of a discrete Fourier transformation. The result is a Kunin-type non-local theory. In the following these theories are analysed by considering a model consisting of independent periodic 1D chauns of solid spheres connected by shear translational and rotational springs. It is found that the Cosserat theory offers a healthy balance between accuracy and simplicity. Kunin’s integral homogenisation theory leads to a non-local Cosserat continuum description that yields an exact solution, but does not offer any real simplification in the solution of the model equations as compared to the original discrete system. The rotational degree of freedom affects the phenomenology of wave propagation considerably. When the rotation is suppressed, only one type of wave, viz. a shear wave, exists. When the restriction on particle rotation is relaxed, the velocity of this wave decreases and another, high velocity wave arises.

99 citations

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TL;DR: In this paper, a Domain-Element Local Integro-differential Equation Method ( DELIDEM) is developed and implemented for the solution of 2D potential problems in materials with arbitrary continuous variation of the material parameters.

Abstract: A new approach (Domain-Element Local Integro-Differential-Equation Method -- DELIDEM) is developed and implemented for the solution of 2-D potential problems in materials with arbitrary continuous variation of the material parameters. The domain is discretized into conforming elements for the polynomial approximation and the local integro-differential equations (LIDE) are considered on subdomains determined by domain elements and collocated at interior nodes. At the boundary nodes, either the prescribed boundary conditions or the LIDE are collocated. The applicability and reliability of the method is tested for several numerical examples.

61 citations

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TL;DR: In this article, a macroelement condition is introduced for constructing the local stabilized formulation of the stationary Navier-Stokes equations and some numerical tests to confirm the theoretical results of the stabilized finite-element method are provided.

Abstract: A stabilized finite-element method for the two-dimensional stationary incompressible Navier-Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation of the stationary Navier-Stokes equations. By satisfying this condition, the stability of the Q1−P0 quadrilateral element and the P1−P0 triangular element are established. Moreover, the well-posedness and the optimal error estimate of the stabilized finite-element method for the stationary Navier-Stokes equations are obtained. Finally, some numerical tests to confirm the theoretical results of the stabilized finite-element method are provided.

61 citations

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TL;DR: In this article, the use of matched asymptotic expansions in option pricing has been investigated in the context of stochastic processes of diffusion type. And a tentative framework for applied matched-asymptotics expansion applied directly to stochastically processes of the diffusion type is also proposed.

Abstract: Modern financial practice depends heavily on mathematics and a correspondingly large theory has grown up to meet this demand. This paper focuses on the use of matched asymptotic expansions in option pricing; it presents illustrations of the approach in ‘plain vanilla’ option valuation, in valuation using a fast mean-reverting-stochastic volatility model, and in a model for illiquid markets. A tentative framework for matched asymptotic expansions applied directly to stochastic processes of diffusion type is also proposed.

53 citations

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TL;DR: In this paper, a uniformly valid asymptotic theory for a general "drum-shaped" electrostatically actuated device is presented and the structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented.

Abstract: The mathematical modeling and analysis of electrostatically actuated micro- and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic-field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace's equa- tion. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general "drum shaped" electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored.

45 citations

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TL;DR: In this paper, Havelock's type of expansion theorems are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities.

Abstract: Havelock’s type of expansion theorems, for an integrable function having a single discontinuity point in the domain where it is defined, are utilized to derive analytical solutions for the radiation or scattering of oblique water waves by a fully extended porous barrier in both the cases of finite and infinite depths of water in two-layer fluid with constant densities. Also, complete analytical solutions are obtained for the boundary-value problems dealing with the generation or scattering of axi-symmetric water waves by a system of permeable and impermeable co-axial cylinders. Various results concerning the generation and reflection of the axisymmetric surface or interfacial waves are derived in terms of Bessel functions. The resonance conditions within the trapped region are obtained in various cases. Further, expansions for multipole-line-source oblique-wave potentials are derived for both the cases of finite and infinite depth depending on the existence of the source point in a two-layered fluid.

44 citations

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TL;DR: In this article, the stochastic dynamics of a chemostat with three trophic levels, substrate-bacterium-worm, was analyzed and a diffusion model of the process was formulated.

Abstract: The stochastic dynamics of a chemostat with three trophic levels, substrate-bacterium-worm, is analyzed. It is assumed that the worm population is perturbed by environmental stochastic noise causing extinction in finite time. A diffusion model of the process is formulated. With singular perturbation methods applied to the corresponding Fokker-Planck equation an estimate of the expected extinction time is derived. This chemostat can be seen as an experimental sewage-treatment system in which the worm population facilitates the reduction of remaining sludge

40 citations

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TL;DR: In this paper, an implementation of the localized boundary-domain integral-equation (LBDIE) method for numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed.

Abstract: An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.

39 citations

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TL;DR: In this paper, the effects of ferrofluid on the curved squeeze film between two annular plates, when the upper plate approaches the lower one normally, are studied including the rotation of the magnetic particles and their magnetic moments.

Abstract: The effects of ferrofluid on the curved squeeze film between two annular plates, when the upper plate approaches the lower one normally, are studied including the rotation of the magnetic particles and their magnetic moments. The aim is to study the effects of rotation of the magnetic particles on the characteristics of the squeeze film. The main equation is derived in the Appendix A. Expressions for the pressure, load capacity and response time are obtained. Load capacity and response time are found to increase when the volume concentration of the solid phase, Langevin’s parameter or the curvature of the upper plate are increased.

39 citations

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TL;DR: In this article, a quasi-one-dimensional theory was proposed to predict the detonation speed and axial solution for cylindrical rate-sticks of highly non-ideal explosives with a simple model with a weakly pressure-dependent rate law and a pseudo-polytropic equation of state.

Abstract: Numerical simulations of detonations in cylindrical rate-sticks of highly non-ideal explosives are performed, using a simple model with a weakly pressure-dependent rate law and a pseudo-polytropic equation of state. Some numerical issues with such simulations are investigated, and it is shown that very high resolution (hundreds of points in the reaction zone) are required for highly accurate (converged) solutions. High-resolution simulations are then used to investigate thequalitative dependences of the detonation driving zone structure on the diameter and degree of confinement of the explosive charge. The simulation results are used to show that, given the radius of curvature of the shock at the charge axis, the steady detonation speed and the axial solution are accurately predicted by a quasi-one-dimensional theory, even for cases where the detonation propagates at speeds significantly below the Chapman-Jouguet speed. Given reaction rate and equation of state models, this quasi-one-dimensional theory offers a significant improvement to Wood-Kirkwood theories currently used in industry

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TL;DR: In this paper, the authors present a numerical study of the macro-dynamic behavior of the unsteady-state granular flow in a cylindrical hopper with flat bottom by means of a modified discrete-element method (DEM) and an averaging method.

Abstract: This paper presents a numerical study of the micro- and macro-dynamic behavior of the unsteady-state granular flow in a cylindrical hopper with flat bottom by means of a modified discrete-element method (DEM) and an averaging method. The results show that the trends of the distributions of the microscopic properties such as the velocity and forces, and the macroscopic properties such as the velocity, mass density, stress and couple stress of the unsteady-state hopper flow are similar to those of steady-state hopper flow, and do not change much with the discharge of particles. However, the magnitudes of the macroscopic properties in different regions have different rates of variation. In particular, the magnitudes of the two normal stresses vary little with time in the orifice region, but decrease in other regions. The magnitude of the shear stress decreases with time when far from the bottom wall and central axis of the hopper. The results also indicate that DEM can capture the key features of the granular flow, and facilitated with a proper averaging method, can also generate information helpful to the test and development of an appropriate continuum model for granular flow.

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TL;DR: In this paper, a plasticity model for the flow of granular materials is presented which is derived from a physically based kinematic rule and which is closely related to the double-shearing model, the double sliding free-rotating model and also to the plastic-potential model.

Abstract: A plasticity model for the flow of granular materials is presented which is derived from a physically based kinematic rule and which is closely related to the double-shearing model, the double-sliding free-rotating model and also to the plastic-potential model. All of these models incorporate various notions of the concept of rotation-rate and the crucial idea behind the model presented here is that it identifies this rotation-rate with a property associated with a Cosserat continuum, namely, the intrinsic spin. As a consequence of this identification, the stress tensor may become asymmetric. For simplicity, in the analysis presented here, the material parameters are assumed to be constant. The central results of the paper are that (a) the model is hyperbolic for two-dimensional steady-state flows in the inertial regime and (b) the model possesses a domain of linear well-posedness. Specifically, it is proved that incompressible flows are well-posed.

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TL;DR: In this paper, the authors propose to embed the oscillator in an autoparametric system by coupling to a damped oscillator, which can be used for quenching Rayleigh and van der Pol relaxation oscillations.

Abstract: Stable normal-mode vibrations in engineering can be undesirable and one of the possibilities for quenching these is by embedding the oscillator in an autoparametric system by coupling to a damped oscillator. There exists the possibility of destabilizing the undesirable vibrations by a suitable tuning and choice of nonlinear coupling parameters. An additional feature is that, to make the quenching effective in the case of relaxation oscillations, one also has to deform the slow manifold by choosing appropriate coupling; this is illustrated for Rayleigh and van der Pol relaxation.

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Lund University

^{1}TL;DR: In this paper, the process of continuous sedimentation of particles in a liquid has often been predicted by means of operating charts and mass-balance considerations, where the underlying constitutive assumption is the one by Kynch.

Abstract: The process of continuous sedimentation of particles in a liquid has often been predicted by means of operating charts and mass-balance considerations, where the underlying constitutive assumption is the one by Kynch. Much more complex operating charts (concentration-flux diagrams) can be obtained from a one-dimensional model of an ideal continuous clarifier-thickener unit. The engineering concept of ???optimal operation??? is defined generally as a special type of solution of the model equation, which is a conservation law with a source term and a space-discontinuous flux function. All qualitatively different step responses (with the unit initially in optimal operation in steady state) are presented and classified in terms of operating charts. Quantitative information relating several interesting variables are also presented concerning, for example, the time until overflow occurs as a function of the feed concentration and flux.

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TL;DR: In this paper, the shape and speed of the solid-melt interface are described at times just before complete freezing takes place, as well as the temperature field in the vicinity of the extinction point.

Abstract: The one-phase Stefan problem for the inward solidification of a three-dimensional body of liquid that is initially at its fusion temperature is considered. In particular, the shape and speed of the solid-melt interface is described at times just before complete freezing takes place, as is the temperature field in the vicinity of the extinction point. This is accomplished for general Stefan numbers by employing the Baiocchi transform. Other previous results for this problem are confirmed, for example the asymptotic analysis reveals the interface ultimately approaches an ellipsoid in shape, and furthermore, the accuracy of these results is improved. The results are arbitrary up to constants of integration that depend physically on both the Stefan number and the shape of the fixed boundary of the liquid region. In general it is not possible to determine this dependence analytically; however, the limiting case of large Stefan number provides an exception. For this limit a rather complete asymptotic picture is presented, and a recipe for the time it takes for complete freezing to occur is derived. The results presented here for fully three-dimensional domains complement and extend those given by McCue et al.[Proc. R. Soc. London A 459 (2003) 977], which are for two dimensions only, and for which a significantly different time dependence occurs.

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TL;DR: In this article, the problem of water-wave scattering by a strip of ice cover floating on the surface of deep water is investigated, where the ice cover is modelled as a thin elastic plate of very small thickness.

Abstract: The problem of water-wave scattering by a strip of ice-cover floating on the surface of deep water is investigated here. The ice-cover is modelled as a thin elastic plate of very small thickness. The problem is reduced to that of solving two singular integral equations of Carleman type over a semi-infinite range and are solved approximately by casting them into two separate Riemann-Hilbert problems by assuming the strip width to be large. The reflection and transmission coefficients are derived approximately. Numerical results for the reflection coefficient are presented graphically against the wave number and also against the ice-cover parameter. The oscillatory nature of the reflection coefficient against the wave number as well as the ice-cover parameter is found to be one of the main features of the curves. It is also seen that, in the limiting case when the ice-cover parameter tends to zero (i.e., the ice-cover is almost absent), the amount of reflection is negligible

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TL;DR: In this paper, a power-series-expansion technique was used to solve approximately the two-dimensional wave equation, where wave polynomials were used as solving functions.

Abstract: The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.

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TL;DR: In this article, the predictive capability of different classes of extended plasticity theories (bounding surface plasticity, generalized plasticity and generalized tangential plasticity) in the modeling of incremental nonlinearity is assessed.

Abstract: The objective of this paper is to assess the predictive capability of different classes of extended plasticity theories (bounding surface plasticity, generalized plasticity and generalized tangential plasticity) in the modeling of incremental nonlinearity, which is one of the most striking features of the mechanical behavior of granular soils, occurring as a natural consequence of the particular nature of grain interactions at the microscale. To this end, the predictions of the various constitutive models considered are compared to the results of a series of Distinct Element simulations performed ad hoc. In the comparison, extensive use is made of the concept of incremental strain-response envelope in order to assess the directional properties of the material response for a given initial state and stress history.

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TL;DR: In this paper, numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders, for cases in which fluid inertia can be neglected.

Abstract: Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders, for cases in which fluid inertia can be neglected The truncated power-law fluid model is used to define the relationship between the viscous stress and the rate-of-strain tensor The flow is shown to be dominated by shear effects, not extension Numerical simulation results are presented for the drag coefficient of the cylinders and the velocity variance components, and are compared against asymptotically valid analytical results Square and hexagonal arrays are considered, both for crossflow in the plane perpendicular to the alignment vector of the cylinders (flows along the axes of the array as well as off-axis flows), and for flow along the cylinders It is shown that the observed strong dependence of the drag coefficient on the power-law index (through which the stress tensor is related to the rate-of-strain tensor) can be described at all solid area fractions by scaling the drag on a cylinder with appropriate velocity and length scales The velocity variance components show only a weak dependence on the power-law index The results for steady-state drag and velocity variances are used in an approximate theory for flows accelerated from rest The numerical simulation data for unsteady flows agree very well with the approximate theory

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TL;DR: In this article, mixed boundary-value problems (BVPs) for a second-order quasi-linear elliptic partial differential equation with variable coefficients dependent on the unknown solution and its gradient are considered.

Abstract: Mixed boundary-value Problems (BVPs) for a second-order quasi-linear elliptic partial differential equation with variable coefficients dependent on the unknown solution and its gradient are considered. Localized parametrices of auxiliary linear partial differential equations along with different combinations of the Green identities for the original and auxiliary equations are used to reduce the BVPs to direct or two-operator direct quasi-linear localized boundary-domain integro-differential equations (LBDIDEs). Different parametrix localizations are discussed, and the corresponding nonlinear LBDIDEs are presented. Mesh-based and mesh-less algorithms for the LBDIDE discretization are described that reduce the LBDIDEs to sparse systems of quasi-linear algebraic equations.

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TL;DR: In this paper, an analysis of non-Newtonian effects on lubrication flows is presented based on the upper-convected Maxwell constitutive equation, which is the simplest viscoelastic model having a constant viscosity and relaxation time.

Abstract: An analysis of non-Newtonian effects on lubrication flows is presented based on the upper-convected Maxwell constitutive equation, which is the simplest viscoelastic model having a constant viscosity and relaxation time. By employing characteristic lubricant relaxation times in all order of magnitude analysis, a perturbation method is developed to analyze the flow of a non-Newtonian lubricant between two surfaces. The effect of viscoelasticity on the lubricant velocity and pressure fields is examined, and the influence of minimum film thickness on lubrication characteristics is investigated. Numerical simulations show a significant enhancement in the pressure field when the minimum film thickness is sufficiently small. This mechanism suggests that viscoelasticity does indeed produce a beneficial effect on lubrication performance, which is consistent with experimental observations.

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TL;DR: In this paper, the stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk was investigated, assuming that the magnetic field is unaffected by the motion of the fluid.

Abstract: The stability of a conducting fluid flow over a rotating disk with a uniform magnetic field applied normal to the disk, is investigated. It is assumed that the magnetic field is unaffected by the motion of the fluid. The mean flow and linear stability equations are solved for a range of magnetic field-strength parameters and the absolute/convective nature of the stability is investigated. It is found that increasing the magnetic field parameter is in general stabilizing.

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TL;DR: In this article, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillation in boilers, the last being a prob- lem of concern in engineering two-phase flows.

Abstract: Asymptotic methods for singularly perturbed delay differential equations are in many ways more chal- lenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a prob- lem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician.

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TL;DR: In this article, a micromechanical constitutive equation is developed which allows for the broad range of interparticle interactions observed in a real deforming granular assembly: microslip contact, gross slip contact, loss of contact and an evolution in these modes of contact as the deformation proceeds.

Abstract: Micromechanical constitutive equations are developed which allow for the broad range of interparticle interactions observed in a real deforming granular assembly: microslip contact, gross slip contact, loss of contact and an evolution in these modes of contact as the deformation proceeds. This was accomplished through a synergetic use of contact laws, which account for interparticle resistance to both sliding and rolling, together with strain-dependent anisotropies in contacts and the normal contact force. By applying the constitutive model to the bi-axial test it is demonstrated that the model can correctly predict the evolution of various anisotropies as well as the formation of a distinct shear band. Moreover, the predicted shear-band properties (e.g. thickness, prolonged localisation, void ratio) are an even better fit with experimental observations than were previously found by use of previously developed micromechanical models.

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TL;DR: In this paper, the weakly nonlinear, freely vibrating motion of a system of coupled spans of suspended overhead transmission lines is studied, and it is shown that the natural vibration is the gravity mode, of which the tension component vanishes in the first harmonic.

Abstract: The weakly nonlinear, freely vibrating motion of a system of coupled spans of suspended overhead transmission lines is studied. It is shown that the natural vibration is the gravity mode, of which the tension component vanishes in the first harmonic. The problem originates from a study of the phenomenon of galloping, which is a high-amplitude periodic oscillation of overhead transmission lines due to steady crosswind. Particular attention is given to an intermodal resonance, which may be interesting for galloping control.

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TL;DR: In this article, a similarity solution for strong blast waves of variable energy propagating in a dusty gas is presented, assuming that the equilibrium-flow condition is maintained and the variable energy input is supplied by a driving piston or surface according to a time-dependent power law.

Abstract: This paper presents a similarity solution for strong blast waves of variable energy propagating in a dusty gas. It is assumed that the equilibrium-flow condition is maintained and the variable energy input is supplied by a driving piston or surface according to a time-dependent power law. Three cases have been investigated: Case I corresponds to a decelerated piston, Case II to a piston of constant velocity, and Case III to a continuously accelerated piston starting from rest. Except in the case of constant front velocity, the similarity solution is valid for adiabatic flow as long as the effect of the counter-pressure is neglected. The effects of a parameter characterizing the various energy input of the blast wave on the similarity solution have been examined. The computations have been performed for various values of mass concentration of the solid particles and for the ratio of density of solid particles to the constant initial density of gas. Tables and graphs of numerical results are presented and discussed.

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TL;DR: In this paper, the Stokes flow due to the motion or presence of a rigid particle in a fluid-filled tube with arbitrary geometry is discussed with emphasis on the induced upstream to downstream pressure change.

Abstract: The computation of Stokes flow due to the motion or presence of a rigid particle in a fluid-filled tube with arbitrary geometry is discussed with emphasis on the induced upstream to downstream pressure change It is proposed that expressing the pressure change as an integral over the particle surface involving (a) the a priori unknown traction, and (b) the velocity of the pure-fluid pressure-driven flow, simplifies the numerical implementation and ameliorates the effect of domain truncation Numerical computations are performed based on the integral formulation in conjunction with a boundary-element method for a particle translating and rotating inside a cylindrical tube with a circular cross-section The numerical results are consistent with previous asymptotic solutions for small particles, and complement available numerical solutions for particular types of motion

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TL;DR: In this paper, boundary integral equations for large deformation of shear-deformable plates are presented, and two different methods are used to calculate the derivatives of the nonlinear terms in the domain integral.

Abstract: Boundary-integral equations for large deformation of shear-deformable plates are presented. Two different methods are used to calculate the derivatives of the nonlinear terms in the domain integral. The first approach requires the evaluation of a hypersingular domain integral. The second approach avoids the calculation of a hypersingular integral by utilizing radial basis functions to approximate the integrand. Quadratic isoparametric boundary-elements are used to discretise the boundary, and constant cell elements are used to discretise the domain. For the solution of a nonlinear problem four methods are presented. They include: total incremental method, cumulative-load incremental method, Euler method and nonlinear system method. Several examples are presented and comparisons with analytical results and other numerical results are made to demonstrate the accuracy of the proposed methods.

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TL;DR: In this paper, the incremental stress-strain relation of dense packings of polygons is investigated by using moleculardynamics simulations and the comparison of the simulation results to the continuous theories is performed using explicit expressions for the averaged stress and strain over a representative volume element.

Abstract: The incremental stress-strain relation of dense packings of polygons is investigated by using moleculardynamics simulations. The comparison of the simulation results to the continuous theories is performed using explicit expressions for the averaged stress and strain over a representative volume element. The discussion of the incremental response raises two important questions of soil deformation: Is the incrementally nonlinear theory appropriate to describe the soil mechanical response? Does a purely elastic regime exist in the deformation of granular materials? In both cases the answer will be “no”. The question of stability is also discussed in terms of the Hill condition of stability for non-associated materials. It is contended that the incremental response of soils should be revisited from micromechanical considerations. A micromechanical approach assisted by discrete element simulations is briefly outlined.