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Cylindrical and prismatic nano-structures (nano-wires and pores) with circular and hexagonal cross-section are studied using mirror boundary conditions to solve the Schrodinger equation in effective mass approximation. In comparison with “quantum billiard” problem, the solution using mirror boundary conditions allows to obtain the results in a much simpler way. It is possible to use even and odd mirror boundary conditions depending on the sign of the wave-function equated in the original and the image points, respectively. The even mirror boundary conditions provides non-vanishing wave-function at the boundary, corresponding to weak confinement allowing quantum tunneling. The odd mirror boundary conditions set wave-function to zero at the boundary, corresponding to a strong confinement. We report on spatial distributions of probability density in cross-section of a cylindrical and prismatic nano-structures, presenting the formulas for the energy of the corresponding quantum states.

The method of images (MI) is an elementary concept in electromagnetism, which is a fundamental course in electrical engineering. In this work, the surface charge density of a conductive sphere in the inductive electrostatic field of a uniformly charged ring is analysed with MI. Meanwhile the surface charge distribution of this sphere is discussed when the sphere is at different positions of the axis of this ring. This work is aimed to help students understand MI and lay a solid foundation of electrical engineering learning.

We consider a thin flat inclusion of arbitrary shape located inside a transversely isotropic elastic half-space in the plane parallel to its boundary z = 0. An arbitrary tangential displacement is prescribed on the inclusion. The boundary of the half-space is stress-free. We need to find the complete field of stresses and displacements in this half-space. A governing integral equation is derived by the generalized method of images, introduced by the author. The case of circular inclusion is considered as an example. Two methods of solution of the governing integral equation are derived. A detailed solution is presented for the particular cases of radial expansion, torsion and lateral displacement of the inclusion. The solution is also valid for the case of isotropy. The governing integral equation for the case of isotropy is derived.

In order to use a linear vibration motor for the actuator of a haptic interface, the motor must provide a higher reaction rate and longer service life than typical rotational motors with an eccentric mass. In this paper, we propose a linear vibration motor that is equipped with a voice-coil actuator and permanent-magnet springs. To concentrate the magnetic flux in the actuator, a Halbach-style magnetization pattern is used. Permanent-magnet springs replace mechanical springs to help increase the service life. We use the method of equivalent current sheets and the method of images to analyze and model the proposed vibration motor. These methods are validated using finite element analyses and experiments. A prototype motor is designed and fabricated. Tests with the prototype show the feasibility of the proposed linear vibration motor.

Conductivity of n non-overlapping disks embedded in a two-dimensional background can be investigated by the method of images which is based on the successive application of the inversions with respect to circles. For closely placed disks, the classical method of images yields slowly convergent series. The method of images can be treated as an application of successive approximations to a system of functional equations. In this paper, modified functional equations are deduced to essentially accelerate the convergence. Highlights? The method of images is treated as iterations applied to functional equations. ? Modified functional equations are deduced to essentially accelerate the convergence. ? A fast method of images is constructed.

An analysis method for slug tests performed in a partially penetrating well within a vertical cutoff wall is presented. A steady-state shape factor for evaluating hydraulic conductivity of the material within the wall was derived by applying the method of images to the previously developed analytical solution of Zlotnik et al. (2010) for an infinite aquifer. Two distinct boundary conditions were considered: constant-head boundary for the case of direct contact between the wall and the aquifer, and no-flux boundary representing an impermeable filter cake on the sides of the wall. The constant-head and no-flux boundary conditions yield significantly higher and lower shape factors, respectively, than those for the infinite aquifer. Consequently the conventional line-fitting method for slug test analysis would yield an inaccurate estimate of the hydraulic conductivity of a vertical cutoff wall.

Based on the method of images, the three-dimensional soil additive load formula caused by pipe jacking soil loss was derived by using non-uniform soil displacement mode. Combining the formulas of additive load induced both by bulkhead additive thrust, force of friction between shield and soil, force of friction between follow-up pipes, which are based on the Mindlin solution, the formulas of total soil additive load were obtained. The distribution regularity of total additive load on adjacent pile during pipe jacking construction, which are caused by all of the factors, were analyzed by using an analytical calculation case. The analytical results show that, the distribution regularity of additive load is closely related with the relative position of pile and shield, and it is a three-dimensional problem. In the thrust direction, the additive stress changes from tension behind into pressure ahead of the excavation face. The additive stress in thrust direction to the pipe wall has larger extension over the additive stress in perpendicular direction. In addition, the value of additive load in thrust direction is bigger than that of perpendicular to the pipe wall. The pile foundation is subject to a small vertical additive stress, and in the vicinity of the tunnel axis, there is an additive stress in the opposite direction to both ends showing bow shaped distribution curve.

We apply both numerical modelling, by means of single-phase and two-phase models, and analytical calculation for estimation of far-field pressure buildup, considering a generic scenario of dipping aquifer and a site-specific scenario for the Scania site, southwest Sweden. We examine the effect of depth-dependent fluid and material properties as well as formation geometries on far-field pressure buildup. The use of simple analytical calculations, including the Theis equation and superposition based on the method of images, for the estimation of pressure buildup is evaluated. The results are also discussed in terms of comparison between the numerical simulations and simple analytical calculations.

In this paper, a new method for the calculation of leakage inductances between short-circuited windings is proposed. The main objective is to find an analytical alternative to the finite element method (FEM) for the computation of short-circuit inductances in electromagnetic transients type (EMT-type) programs. Typical EMT-type tools do not provide complex FEM based computation engines for this sole purpose. The new approach is derived from the method of images and uses analytical formulations for magnetic vector potential and energy developed in this work. It is shown that the difference between the inductance calculated with the FEM in 2-D and that computed with the method of images decreases as the number of layers of images increases. Also, it is demonstrated that the classical approach, based on an axial flux distribution, can lead to considerable error if the windings have unequal heights and if they are located far from the core yokes.

In our work, we investigate magnetic properties of metamaterial–metal interface with the help of the local field theory combined with the method of images. Proceeding from microscopic description of the substance and calculating its macroscopic properties, for the first time the modified Clausius–Mossotti relation has been obtained for permeability of the metamaterial bordering a metal.

The radiation arising under uniform motion of non-relativistic charged particle by (or through) perfectly conducting sphere is considered. The rigorous results are obtained using the method of images known from electrostatics., Presented on the conference "Channeling-2016", Sirmione - Desenzano del Garda, Italy, September 25 - 30, 2016 v2: Figure 5 corrected, Journal reference added

Realistic 3D human model reconstruction is an important component in computer graphics and computer vision. In particular, texturing on the surface of models is a key stage of reconstruction. In this paper, we dispose the texture mapping on the model’s surface as an optimization of image stitching, and present an effective method to generate a seamless, integrated and smooth texture on the surface of 3D human model. First, we build a corresponding Markov Random Field model with respect to color images and triangular meshes of the surface. On the basis of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α-expansion optimization for this Markov Random Field model, a 2D translation coordinate of color image, as an adaptive iterative factor, is introduced into the optimization to match the color content at the boundary of adjacent meshes. That compensates for the misalignment of adjacent color images, which caused by the inaccuracy of depth data and multi-view misregistration. Then we apply Poisson blending to a composite vector field in gradient domain, to resolve the small but noticeable illumination variations between different color images. To repair the blank regions, we parameterize the model’s surface and project it onto a 2D plane. Then the K-Nearest Neighbor algorithm is applied to fill up the blank regions with texture contents. Finally, we evaluate our method by comparison with another three advanced methods on some human models, and the results demonstrate that our method of images stitching creates a best texture on the surface of 3D human model both in visual effect and quantitative analysis.

In the paper, mathematical models for determination of resistivity of thin antistatic coatings are presented. The models are combination of the classic four-point method and the method of images. Models 2D and 3D are taken into considerations. In both cases are analyzed the parameters of measuring system and coating layer for example: the relations between the sample size and the distance between the electrode set and the edge of the sample; thickness of the coating; type of the substrate layer. Optimization of the parameters of the system has to allow using much simpler formulas for infinite flat object.

COMPUMAG 2015, Conference on the Computation of Electromagnetic Fields, Montreal, Canada, 28 June-2 July 2015.; International audience; An electromagnetic model is proposed to compute translational motion eddy current in a conductive plate. The eddy currents are due to the movement of the plate in a dc magnetic field created by a PM inductor. Firstly, the magnetic field due to the PMs is computed in 3D where the iron yokes influence is considered thanks to the method of images. Then, the motional eddy currents are computed such that the edge effects are correctly taken into account through an iterative procedure which uses magnetic images. The computations are very fast and the obtained results are close to those issued from 3D FE method and from experiments.

The Green function (GF) method is used to analyze the boundary effects produced by a Chern Simons (CS) extension to electrodynamics. We consider the electromagnetic field coupled to a $\theta$ term that is piecewise constant in different regions of space, separated by a common interface $\Sigma$, the $\theta$ boundary, model which we will refer to as $\theta$ electrodynamics ($\theta$ ED). This model provides a correct low energy effective action for describing topological insulators (TI). In this work we construct the static GF in $\theta$ ED for different geometrical configurations of the $\theta$ boundary, namely: planar, spherical and cylindrical $\theta$ interfaces. Also we adapt the standard Green theorem to include the effects of the $\theta$ boundary. These are the most important results of our work, since they allow to obtain the corresponding static electric and magnetic fields for arbitrary sources and arbitrary boundary conditions in the given geometries. Also, the method provides a well defined starting point for either analytical or numerical approximations in the cases where the exact analytical calculations are not possible. Explicit solutions for simple cases in each of the aforementioned geometries for $\theta$ boundaries are provided. The adapted Green theorem is illustrated by studying the problem of a point like electric charge interacting with a planar TI with prescribed boundary conditions. Our generalization, when particularized to specific cases, is successfully compared with previously reported results, most of which have been obtained by using the methods of images., Comment: 24 pages, 4 figures, accepted for publication in PRD. arXiv admin note: text overlap with arXiv:1511.01170

This paper studies the large-t behavior of the boundary generated by the method of images for the first-passage-time problem. We show that this behavior is characterized by certain properties of the Laplace transform of the input measure. Such properties also determine the asymptotic behavior of the first-passage-time density. Most of the paper assumes a positive input measure, which generates a concave boundary. The last section, however, discusses a non-positive measure. We obtain a sufficient condition for the boundary to be convex.

In the present study, mathematical model of the sound radiated from an impact oscillator that incorporates the Hertzian contact is analyzed. Transient sound generated from the contact of a sphere with a reflecting plane is obtained theoretically and it is compared with the previous experimental results. The acoustic noise radiated from a spherical mass colliding with a wall–floor intersection is obtained using the method of images. This method is used to predict the acoustic noise generated by a single unit impact damper. A nonlinear model is developed to investigate the effect of mass ratio, coefficient of restitution and gap size on behavior of the single unit impact damper. Finally, damping and acoustic behavior of the impact damper are illustrated in a user-oriented diagram to find a low noise system, which can strongly suppress the undesired vibrations.

The vibration motors for smart phones with haptic interfaces require higher reaction rate and longer service life than what a typical rotational motor with eccentric mass provides. In this paper, a new type of a linear vibration motor is proposed. A voice-coil actuator is employed for high reaction rate. Permanent magnets replace mechanical springs, which would help increase the service life. For analyzing and modeling the proposed vibration motor, we use the method of equivalent current sheet and the method of images. These methods are validated against finite element analyses and experiments. A prototype motor is designed and fabricated. Tests with the prototype show the feasibility of the proposed linear vibration motor.

As left-handed materials and metamaterials are becoming more prevalent, we examine the effect of negative permeability upon levitation force. We first consider two half spaces of differing permeability and a point magnetic source, so that the method of images may be employed. We determine that the resulting force may be larger than for conventional magnetic materials. We then illustrate the inclusion of a finite sample thickness.

Vortex methods using vorticity–velocity formulations have become an increasingly powerful and popular means of studying complex fluid flow systems. The problem of combining an integral equation method and grid-free discrete vortex method DVM when studying three-dimensional wall-bounded flows is considered. While the normal boundary condition is satisfied by means of a boundary integral equation BIE, we also consider the problem of recovering pressure from given vorticity and velocity fields when using Lagrangian DVMs in terms of a BIE. For validation purposes, vortical flow past a sphere and past a flat plate are considered, for which the commonly used method of images is available. Results of near-wall boundary-layer flow simulations are then presented as an illustration of the numerical scheme. The importance of hairpin vortices is highlighted. Finally, results on wall compliance fluid flow are displayed emphasizing the versatility of the numerical method.

We develop a multiresolution representation of a class of integral operators satisfying boundary conditions on simple domains in order to construct fast algorithms for their application. We also elucidate some delicate theoretical issues related to the construction of periodic Greenʼs functions for Poissonʼs equation. By applying the method of images to the non-standard form of the free space operator, we obtain lattice sums that converge absolutely on all scales, except possibly on the coarsest scale. On the coarsest scale the lattice sums may be only conditionally convergent and, thus, allow for some freedom in their definition. We use the limit of square partial sums as a definition of the limit and obtain a systematic, simple approach to the construction (in any dimension) of periodized operators with sparse non-standard forms. We illustrate the results on several examples in dimensions one and three: the Hilbert transform, the projector on divergence free functions, the non-oscillatory Helmholtz Greenʼs function and the Poisson operator. Remarkably, the limit of square partial sums yields a periodic Poisson Greenʼs function which is not a convolution. Using a short sum of decaying Gaussians to approximate periodic Greenʼs functions, we arrive at fast algorithms for their application. We further show that the results obtained for operators with periodic boundary conditions extend to operators with Dirichlet, Neumann, or mixed boundary conditions.

The Method of Images poses an important difficulty when used to solve the problem of a charge in the presence of a grounded conducting sphere. This arises from the fact that the sum of the inducing charge and the image charge is different from zero. As a consequence, there is a monopole field far from the system, and any ground wire physically connected to the sphere will carry an electric current, changing the initial balance of charges until a new equilibrium is reached. The approach taken in this paper assumed an infinite straight wire connecting the sphere to ground. The charge distribution over the surface of the conductors was calculated, and the results analyzed. It was shown that the thinner the wire, the lower will be its total charge, and the closer will be the calculated charge density at the surface of the sphere to the conventional solution by the Method of Images.

Electrostatic interactions between multiple colloids in ionic fluids are attracting much attention in studies of biological and soft matter systems. The evaluation of the polarization surface charges due to the spherical dielectric discontinuities poses a challenging problem to highly efficient computer simulations. In this paper, we propose a new method for fast calculating the electric field of spaced spheres using the multiple reflection expansion. The method uses a technique of recursive reflections among the spherical interfaces based on a formula of the multiple image representation, resulting in a simple, accurate and close-form expression of the surface polarization charges. Numerical calculations of the electric potential energies of charged spheres demonstrate the method is highly accurate with small number of reflections, and thus attractive for the use in practical simulations of related problems such as colloid suspension and macromolecular interactions.

In the last years the authors have developed a numerical formulation based on the Boundary Element Method for the analysis of grounding systems embedded in uniform soils. This approach has been implemented in a CAD system that currently allows to analyze real grounding grids in real-time in personal computers. The extension of this approach for the grounding analysis in layered soils is straightforward by application of the method of images. However in some practical cases the resulting series have a poor rate of convergence; consequently, the analysis of real earthing grids in multilayer soils requires an out of range computational cost. In this paper we present a CAD system based on this BEM numerical formulation for grounding analysis in multilayer soils that include an efficient technique based on the Aitken acceleration in order to improve the rate of convergence of the involved series expansions. Finally, we show some examples by using the geometry of real grounding systems.

Image charge approximations are developed for electric potentials in the Poisson-Boltzmann theory in inhomogeneous media consisting of dielectrics or electrolyte solutions such as the layered structure in a membrane or cylindrical ion-channels. The image charges are obtained either by a least square fitting between the potential of unknown images and the exact reaction potential (for the layered media or cylindrical region) or by a Prony fitting to the Fourier transform of the exact potential (layered media only) and a Sommerfeld-type identity, which yields the locations and strengths of the image charges. Next, combining the results for the two geometries, the image charge approximation for the reaction potential, due to a charge inside an ion-channel model, is obtained, which accounts for the polarization of the region outside the ion-channel (consisting of a membrane and electrolyte solutions below and above). Such an approximation to the reaction field in the ion-channel model can be used for an explicit/implicit hybrid treatment of electrostatics interaction in modeling ion-channels. Numerical tests show that the proposed method has an attractive performance in computing electrostatic interactions of source charges inside the ion-channel model via a simple summation of pairwise interactions among source and image charges.

Recently, a new technique has been proposed by one of the authors for solving diffraction problems belonging to certain class. The class includes 2D problems that can be reduced to propagation problems on branched surfaces by applying the method of images. One of the simplest problems belonging to this class is diffraction by an infinitely thin ideal strip. Within the framework of the new method, a "spectral equation" is derived, which is an ordinary differential equation for the components of the directivity of the scattered field. The coefficients of the spectral equation are known up to several numerical parameters, and these are found by a complicated numerical procedure from the a-priori known monodromy data for the equation. In the current paper the numerical procedure is described and its accuracy and efficiency are analyzed.

We give a brief description of how Maxwell's equations are expressed in the language of differential forms and use this to provide an elegant demonstration of how the method of images (well known in electrostatics) also works for electrodynamics in the presence of an infinite plane conducting boundary.

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model for water availability from surface runoff in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated.

Under some conditions, the method of images (well known in electrostatics) may be implemented in magnetostatic problems too, giving an excellent example of the usefulness of formal analogies in the description of physical systems. In this paper, we develop a quantitative model for the magnetic interactions underlying the so-called Geomag™ paradox and describe a quantitative experimental investigation to validate the model. The validity ranges of some approximations involved in this problem are quantitatively discussed and the advantages of a dimensionless formulation of the interaction are pointed out. This work offers many educational suggestions suitable for university students.

This paper is devoted to discussing the behavior of an unbiased 1D random walk in a semi-infinite lattice confined by a certain type of boundary, termed totally reflecting, and in particular to proving the validity of the so-called “reflection principle” for computing the probability mass function in such systems. This is motivated by the publication of an earlier paper by [Orlowski, Phys. Status Solidi B 239, 158–161 (2003)], which denied the validity of the reflection principle, and proposed a different, inconsistent method of computing the walker's probability mass, and also by the fact that earlier authors published arguments for the reflection principle that themselves contained errors or were not entirely rigorous. This paper provides a new, rigorous argument for the validity of the reflection principle, and also shows where the error in Orlowski's analysis lies. It further contextualizes these arguments with respect to some existing literature on similar systems where the probability of reflection may be less than unity, and discusses the proper relationship between the discrete random walk, the diffusion equation, and approches which employ a different, master equation-based discretization technique. The spatial relationship between the discrete and continuous formulations is discussed, and an existing derivation [van Kampen and Oppenheim, J. Math. Phys. 13, 842–849 (1972)] is extended. Finally, it is shown that the reflection principle, as outlined in this paper, preserves initially uniform concentrations for all time (in contrast to Orlowski's proposed method).

The hydrodynamic properties of free surface vortices at hydraulic intakes were investigated. Based on the axisymmetric Navier–Stokes equations and empirical assumptions, two sets of formulations for the velocity distributions and the free surface profiles are proposed and validated against measurements available in the literature. Compared with previous formulae, the modifications based on Mih's formula are found to greatly improve the agreement with the experimental data. Physical model tests were also conducted to study the intake vortex of the Xiluodu hydroelectric project in China. The proposed velocity distribution formula was applied to the solid boundary as considered by the method of images. A good agreement was again observed between the prediction and the measurements.

The problem of computing the capacitance coupling in Very Large Scale Integrated (VLSI) circuits is studied in this work. The proposed method is an approximate extended version of the method of images. The initial problem is formulated here as an optimization problem for the solution of which a genetic algorithm (GA) is employed. The proposed method is fast, general, does not rely on fitting techniques and is applicable to an arbitrary 2D or 3D geometry configuration of conductors. Extensive simulation results are presented for several practical case studies. Comparative results are given with other methods from literature and a commercial tool employing the Finite Element Method (FEM). The results show that the capacitance value computed by our method is in close agreement to the value obtained by the other methods from literature and also by the commercial tool with the average difference ranging between 2% and 5% while demonstrating better scalability as the problem complexity rises.

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let $\Cal H$ be the half plane to the left of $\ell$. We consider lozenge tilings of $\Cal H$ that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images., Comment: 34 pages, AmS-TeX

The distributed dislocation technique is used to analyze the defects in the forms of cracks in a half-plane under timeharmonic anti-plane excitation. The stress fields are obtained for a half- plane containing a Volterra screw dislocation. The method of images is utilized to derive the solution of a screw dislocation under time-harmonic conditions for an elastic half-plane from the solution of infinite planes. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surfaces of smooth cracks. The integral equations are of Cauchy singular type which are solved numerically for several different cases of crack configurations and arrangements. The displacement and stress components are obtained for a half-plane under concentrated anti-plane, time-harmonic traction. The dislocation solution is employed to analyze a half-plane weakened by cracks under anti-plane point load. The solution may be viewed as the Green's function for a half-plane under arbitrary self-equilibrating traction. The effects of load frequency and crack orientation on the stress intensity factors are studied.

Volume fracturing is the key technology that has made the development of tight oil formations economical. Well testing is often used to obtain formation parameters in oil field after volume fracturing in tight oil formations, which is time-consuming and accuracy is low. Based on a new analytical solution methodology, this paper presents a simple and efficient approach by matching the production data to obtain formation parameters, just like formation permeability, fracture conductivity, dimensionless conductivity, propped length and so on. The numerical solution is applicable for finite-conductivity vertical fractures in rectangular shaped reservoirs. The mathematical formulation is based on the method of images with no flow boundaries for symmetrical patterns.

Using conformal mapping, fluid motion inside the cochlear duct is derived from fluid motion in an infinite half plane. The cochlear duct is represented by a two-dimensional half-open box. Motion of the cochlear fluid creates a force acting on the cochlear partition, modeled by damped oscillators. The resulting equation is one-dimensional, more realistic, and can be handled more easily than existing ones derived by the method of images, making it useful for fast computations of physically plausible cochlear responses. Solving the equation of motion numerically, its ability to reproduce the essential features of cochlear partition motion is demonstrated. Because fluid coupling can be changed independently of any other physical parameter in this model, it allows the significance of hydrodynamic coupling of the cochlear partition to itself to be quantitatively studied. For the model parameters chosen, as hydrodynamic coupling is increased, the simple resonant frequency response becomes increasingly asymmetric. The stronger the hydrodynamic coupling is, the slower the velocity of the resulting traveling wave at the low frequency side is. The model's simplicity and straightforward mathematics make it useful for evaluating more complicated models and for education in hydrodynamics and biophysics of hearing.

Though there is no exact analytic solution for the capacitance between two cylinders near a dielectric interface, the method of images is used to derive analytic expressions that are accurate for widely separated cylinders, cylinders in close proximity, and widely separated cylinders near a dielectric interface. These solutions are compared with numerical simulations to verify the utility of the derived expressions. The necessity for numerical simulation in the general case is demonstrated. These analytic solutions can be incorporated into self-capacitance terms for solenoids and transmission lines.

It is very difficult to evaluate the productivity of horizontal wells in fault block reservoirs due to the influence of fault sealing. On the basis of the method of images and source-sink theory, a semianalytical model coupling reservoir and finite conductivity horizontal wellbore flow dynamics was built, in which the influence of fault sealing was taken into account. The distribution of wellbore flow and radial inflow profiles along the horizontal interval were also obtained. The impact of the distance between the horizontal well and the fault on the well productivity was quantitatively analyzed. Based on this analysis, the optimal distance between the horizontal well and the fault in banded fault block reservoirs could be determined. According to the field application, the relative error calculated by the model proposed in this paper is within ±15%. It is an effective evaluation method for the productivity of horizontal wells in fault block reservoirs. The productivity of the horizontal well increases logarithmically as the distance between the horizontal well and the fault increases. The optimal distance between the horizontal well and the fault is 0.25–0.3 times the horizontal well length.

A panel method is applied to calculate the sinkage and trim of a ship sailing in a shallow channel. The hull surface, free surface and channel wall surfaces are discretized into panels on which Rankine sources of constant strengths are distributed. An iterative scheme is used to deal with the nonlinear boundary conditions on free surface. The boundary condition on water bottom is satisfied by the method of images, whereas the radiation condition is satisfied by raised panels above free surface. The vertical force and pitching moment are obtained by integrating the hydrodynamic pressure over the hull surface, and the sinkage and trim are calculated according to dynamic equilibrium. Numerical calculations are performed for Series 60 (CB=0.6) ship. The numerical results are in good agreement with experimental data.

This paper presents an analytical method for approximating the eddy current and hysteresis losses in a system of multiple sheathed cables placed in any configuration inside a steel pipe. The method is based on the theory of images, which this paper expands to apply to pipes having high magnetic permeability and high electric conductivity at the same time. The method of images, in combination with approximating the cable conductors and sheaths as multiple physical filaments, is used to compute the final current distributions in the cables and pipe and, thus, the associated losses. The accuracy of the proposed method is assessed against numerical solutions obtained using the Maxwell finite-element program by Ansoft.

Darcian steady 2-D flow to a point sink (vertical well) placed eccentrically with respect to two circles demarcating zones of contrasting permeability is studied by the methods of complex analysis and numerically by MODFLOW package. In the analytical approach, two conjugated Laplace equations for a characteristic flow function are solved by the method of images, i.e. the original sink is mirrored about two circles that generates an infinite system of fictitious sinks and source. The internal circle of the annulus models formation damage (gravel pack) near the well and the ring-shaped zone represents a pristine porous medium. On the external circle the head (pressure) is fixed and on the internal circle streamlines are refracted. The latter is equivalent to continuity of pressure and normal component of specific discharge that is satisfied by the choice of the intensity and loci of fictitious sinks. Flow net and dependence of the well discharge on eccentricity are obtained for different annulus radii and permeability ratios. A non-trivial minimum of the discharge is discovered for the case of the ring domain permeability higher than that of the internal circle. In the numerical solution, a finite difference code is implemented and compared with the analytical results for the two-conductivity zone. Numerical solution is also obtained for an aquifer with a three-conductivity zonation. The case of permeability exponentially varying with one Cartesian coordinate within a circular feeding contour is studied analytically by series expansions of a characteristic function obeying a modified Helmholtz equation with a point singularity located eccentrically inside the feeding contour. The coefficients of the modified Bessel function series are obtained by the Sommerfeld addition theorem. A trivial minimum of the flow rate into a small-radius well signifies the trade-off between permeability variation and short-cutting between the well and feeding contour.

Hydrodynamic properties of the surface vortex have been investigated. Based on the Navier-Stokes equations, three sets of the new formulations for the tangential velocity distributions are derived, and verified against the experimental measurements in the literature. It is shown that one modification greatly improves the agreement with the experimental data. Physical model experiments were carried out to study the intake vortex related to the Xiluodu hydropower project. The velocity fields were measured using the Particle Tracking Velocimetry (PTV) technique. The proposed equation for tangential velocity distribution is applied to the Xiluodu project with the solid boundary being considered by the method of images. Good agreement has been observed between the formula prediction and the experimental observation.

In this contribution, the phenomenon of wall-induced acoustic radiation force and torque on an active cylindrically radiating acoustic source, undergoing axisymmetric or asymmetric harmonic vibrations (i.e., using a superposition of different vibrational source modes) near a flat rigid boundary in a non-viscous fluid is demonstrated using the modal expansion method, the translational addition theorem, and the method of images. These physical phenomena arise due to the multiple interactions with the boundary. Assuming axisymmetric or asymmetric modal oscillations of the source, conditions are found where the longitudinal and transversal radiation force functions as well as the axial torque component vanish, achieving complete translational or rotational motion suppression of the pulsating source. These components also take positive or negative values, depending on the source size and distance from the flat rigid wall, suggesting pushing or pulling effects towards the boundary, and possible source rotation clockwise or counter-clockwise. The analytical formalism developed here can be helpful in predicting emergent phenomena related to applications in underwater acoustics, particle manipulation of active carriers or ultrasound contrast agents located near a boundary, and possibly other applications in fluid dynamics and related topics.In this contribution, the phenomenon of wall-induced acoustic radiation force and torque on an active cylindrically radiating acoustic source, undergoing axisymmetric or asymmetric harmonic vibrations (i.e., using a superposition of different vibrational source modes) near a flat rigid boundary in a non-viscous fluid is demonstrated using the modal expansion method, the translational addition theorem, and the method of images. These physical phenomena arise due to the multiple interactions with the boundary. Assuming axisymmetric or asymmetric modal oscillations of the source, conditions are found where the longitudinal and transversal radiation force functions as well as the axial torque component vanish, achieving complete translational or rotational motion suppression of the pulsating source. These components also take positive or negative values, depending on the source size and distance from the flat rigid wall, suggesting pushing or pulling effects towards the boundary, and possible source rotation ...

In celebration of the Sixth International Conference on Lattice Path Counting and Applications, it is befitting to review the history of lattice path enumeration and to survey how the topic has progressed thus far. We start the history with early games of chance specifically the ruin problem which later appears as the ballot problem. We discuss Andre's Reflection Principle and its misnomer, its relation with the method of images and possible origins from physics and Brownian motion, and the earliest evidence of lattice path techniques and solutions. In the survey, we give representative articles on lattice path enumeration found in the literature in the last 35 years by the lattice, step set, boundary, characteristics counted, and solution method. Some of this work appears in the author's 2005 dissertation.

Left–right symmetry breaking is critical to vertebrate embryonic development; in many species this process begins with cilia-driven flow in a structure termed the ‘node’. Primary ‘whirling’ cilia, tilted towards the posterior, transport morphogen-containing vesicles towards the left, initiating left–right asymmetric development. Recent theoretical models based on the point-force stokeslet and point-torque rotlet singularities, explaining how rotation and surface-tilt produce directional flow are reviewed. Analysis of image-singularity systems enforcing the no-slip condition shows how tilted rotation produces a far-field ‘stresslet’ directional flow, and how time-dependent point-force and time-independent point-torque models are in this respect equivalent. Associated slender-body-theory analysis is reviewed; this approach enables efficient and accurate simulation of three-dimensional time-dependent flow, time-dependence being essential in predicting features of the flow such as chaotic advection, which have subsequently been determined experimentally. A new model for the nodal flow utilising the regularized stokeslet method is developed, to model the effect of the overlying Reichert’s membrane. Velocity fields and particle paths within the enclosed domain are computed and compared with the flow profiles predicted by previous ‘membrane-less’ models. Computations confirm that the presence of the membrane produces flow-reversal in the upper region, but no continuous region of reverse flow close to the epithelium. The stresslet far-field is no longer evident in the membrane model, due to the depth of the cavity being of similar magnitude to the cilium length. Simulations predict that vesicles released within one cilium length of the epithelium are generally transported to the left via a ‘loopy drift’ motion, sometimes involving highly unpredictable detours around leftward cilia. Particles released just above the cilia tips were not predicted to reach to the extreme edges of the node, but rather are returned to the right by the counterflow. Flow to the right and left of the cilia array is of very small magnitude, suggesting that effective transport of particles to the extremities of the node requires cilia to be distributed all the way to the edges. There is no continuous layer of rightward flow close to the epithelium, except for a region close to the posterior edge of the node. Future work will involve investigating issues such as the precise shape of the node and cilia distribution and the effect of advection and diffusion on morphogens, hence explaining more fully the role of fluid mechanics in this vital developmental process.

This paper presents a method to analytically model the self and mutual phase inductances of a brushless dc motor with a Faulhaber winding. The method of images is used to “remove” the motor iron parts in the model in order to analyze the winding completely in the air. The analytical model is verified using 3-D finite-element method simulations and measurements.

Pulsed wire technique is a fast and accurate method for the measurement of first and second field integrals of undulators used in free-electron lasers and synchrotron light sources. In this paper, we present a theoretical analysis of this technique by finding out the analytic solution of the differential equation for the forced vibration of the wire taking dispersion due to stiffness into account. Method of images is used to extend these solutions to include reflections at the ends. For long undulators, the effect of dispersion of the acoustic wave in the wire could be significant and our analysis provides a method for the evaluation of the magnetic field profile even in such cases taking the effect due to dispersion into account in an exact way.