Search

Narrow escape theory deals with the first passage of a particle diffusing in a cavity with small circular windows on the cavity wall to one of the windows. Assuming that (i) the cavity has no size anisotropy and (ii) all windows are sufficiently far away from each other, the theory provides an analytical expression for the particle mean first-passage time (MFPT) to one of the windows. This expression shows that the MFPT depends on the only global parameter of the cavity, its volume, independent of the cavity shape, and is inversely proportional to the product of the particle diffusivity and the sum of the window radii. Amazing simplicity and universality of this result raises the question of the range of its applicability. To shed some light on this issue, we study the narrow escape problem in a cylindrical cavity of arbitrary size anisotropy with two small windows arbitrarily located on the cavity side wall. We derive an approximate analytical solution for the MFPT, which smoothly goes from the conventional narrow escape solution in an isotropic cavity when the windows are sufficiently far away from each other to a qualitatively different solution in a long cylindrical cavity (the cavity length significantly exceeds its radius). Our solution demonstrates the mutual influence of the windows on the MFPT and shows how it depends on the inter-window distance. A key step in finding the solution is an approximate replacement of the initial three-dimensional problem by an equivalent one-dimensional one, where the particle diffuses along the cavity axis and the small absorbing windows are modeled by delta-function sinks. Brownian dynamics simulations are used to establish the range of applicability of our approximate approach and to learn what it means that the two windows are far away from each other. [ABSTRACT FROM AUTHOR]

We present a novel approach to control the blending of line and polygon skeleton-based convolution surfaces using locally varying Ratio of Support radius and Thickness (RST). With our method, solutions for local convolution surface approximation with prescribed surface thickness and support radii can be derived analytically. In addition, iso-surface shrinkage can be avoided by offsetting the endpoints of line skeletons and the edges of polygon skeletons. Our RST-based blending for convolution surfaces is local and can generate desired blending effects while approximating shapes with a specified thickness. Moreover, our method is intuitive and users can control the blending by adjusting the skeletal radius or the support radius of the finite support kernel independently. As our blending utilizes convolution integration only without requiring any extra composition operators, it allows for successive convolution blending operations to create complex shapes. [Display omitted] • Blending control for convolution surfaces using locally varying RSTs. • Analytical solutions for surface thickness approximation with prescribed RSTs. • Numerical solutions for approximating the skeleton boundaries with given RSTs. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Aeronautical Materials is the property of Editorial Board of Journal of Aeronautical Materials and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The deep borehole heat exchanger systems (DBHE) are offering potential low carbon heating for buildings. However, the study of DBHE is hindered by both modeling and system complexity. Although some analytical models of DBHE are presented in literature, the involved expression of complex integral calculation on the contrary slows down the computation speed and many calculations are redundant in algorithm. In this study, a Response Factor Matrix method (RFM) is proposed based on simple analytical expression and well-designed data structure for DBHE. The RFM can be prefabricated and then used in each time step updates of temperature field. A superposition algorithm is designed for fast computation of DBHE array. The proposed model and algorithms are verified by measurements from two independent engineering projects. Based on the new tool, both thermal radius and heating capacity of DBHE array are investigated in details. The step-ascending pattern of thermal radius is recognized. The correlation of borehole spacing and system heating capacity is identified. In the end, the computational efficiency of the new tool is justified and some outlooks are proposed for a further development and applications. • A fast computational model is proposed for single and multiple deep borehole system. • The proposed new model is verified through comparison with experimental data. • The thermal impact radius of deep borehole system is investigated. • The spacing of deep borehole array is critical for system heat capacity. [ABSTRACT FROM AUTHOR]

The radius evolution for the synchronous collapse motion of a circular vapor bubble cluster in an ideal and infinite fluid is studied. With the assumption that all bubbles are evenly distributed around the circumference and have the same initial conditions, we establish the associated governing equation for this case from a modified Rayleigh–Plesset equation. The inverted analytical solution in the form of t = t (r) and analytical approximations in the form of r = r (t) for the radius evolution to the initial value problem of the dimensionless governing equation are derived. These novel analytical approximations extend the previous ones for a single bubble by adding the effects of surface tension and multiple bubbles. In addition, we point out that the analytical results and dynamic behaviors for multiple bubbles will degenerate into the corresponding ones for single bubbles when the radius of the bubble cluster approaches to infinity. • The governing equation for the synchronous collapse motion of a circular vapor bubble cluster is established. • The inverted analytical solution and analytical approximations are derived. • The effects of surface tension and multiple bubbles are included in the obtained solutions. • The limiting behaviors of the circular vapor bubble cluster are discussed. [ABSTRACT FROM AUTHOR]

In this work, the effects of externally applied axial pressure gradients and transverse magnetic fields on the electrokinetic energy conversion (EKEC) efficiency and the streaming potential of nanofluids through a microannulus are studied. The analytical solution for electro-magneto-hydro-dynamic (EMHD) flow is obtained under the condition of the Debye-Hückel linearization. Especially, Green's function method is used to obtain the analytical solutions of the velocity field. The result shows that the velocity distribution is characterized by the dimensionless frequency Ω, the Hartmann number Ha, the volume fraction of the nanoparticles φ, the geometric radius ratio a, and the wall ζ potential ratio b. Moreover, the effects of three kinds of periodic excitations are compared and discussed. The results also show that the periodic excitation of the square waveform is more effective in increasing the streaming potential and the EKEC efficiency. It is worth noting that adjusting the wall ζ potential ratio and the geometric radius ratio can affect the streaming potential and the EKEC efficiency. [ABSTRACT FROM AUTHOR]

• Solution for in-service shallow-buried twin tunnels subjected to surcharge loads is presented. • The Flamant solution and the complex variable method are iteratively combined. • Twin tunnels of arbitrary geometry and surcharge loads of different form are considered. This paper presents an analytical solution for the stress and displacement for shallow twin tunnels subjected to surcharge loads. Multi-interactions, including the interaction between liners and geomaterial, the interaction between twin tunnels, and the interaction between surcharge loads and twin tunnels, are considered in the proposed solution. The solution is elaborated using an iterative three-step procedure based on the Schwarz alternating method. The complex variable method and the Flamant solution and its modification are employed in the derivation. Verification of the analytical solution is obtained through comparisons to a corresponding numerical solution and an existing case study. The close correspondence indicates the accuracy and generality of the proposed solution. A parametric investigation is performed to investigate the influence of surcharge load (including load position, range and distribution), elastic parameters of geomaterial, and liner geometry (including tunnel radius and liner thickness). Based on the parametric investigation, several advices for surcharge loads and liners are provided to help tunnel designers to avoid possible damages for shallow twin tunnels. [ABSTRACT FROM AUTHOR]

• Combined impact of conduction, dispersion and buoyancy-driven flow on thermal losses of ATES is assessed. • Conduction always occurs and dominates dispersion effects, buoyancy-driven flow effects depend on conditions. • Storage geometry (A/V and L/R th) influences the recovery efficiency for all storage temperatures. • For conduction dominated conditions, the geometry is improved by decreasing A/V (smallest at L/R th =2). • For conditions with high potential for buoyancy-driven flow, increasingly smaller L/R th (<1) decreases thermal losses. The technical and economic success of an Aquifer Thermal Energy Storage (ATES) system depends strongly on its thermal recovery efficiency, i.e. the ratio of the amount of energy that is recovered to the energy that was injected. Typically, conduction most strongly determines the thermal recovery efficiency of ATES systems at low storage temperatures (<25 °C), while the impact of buoyancy-driven flow can lead to high additional heat losses at high storage temperatures (>50 °C). To date, however, it is unclear how the relative contribution of these processes and mechanical dispersion to heat losses across a broad temperature range is affected by their interaction for the wide range of storage conditions that can be encountered in practice. Since such process-based insights are important to predict ATES performance and support the design phase, numerical thermo-hydraulic ATES simulations were conducted for a wide range of realistic operational storage conditions ([15–90 °C], [50,000–1,000,000 m3/year]) and hydrogeological conditions (aquifer thickness, horizontal hydraulic conductivity, anisotropy). The simulated heat loss fractions of all scenarios were evaluated with respect to analytical solutions to assess the contribution of the individual heat loss processes. Results show that the wide range of heat losses (10–80 % in the 5th year) is the result of varying contributions of conduction, dispersion and buoyancy-driven flow, which are largely determined by the geometry of the storage volume (ratio of screen length / thermal radius, L/R th) and the potential for buoyancy-driven flow (q 0) as affected by the storage temperature and hydraulic conductivity of the aquifer. For ATES systems where conduction dominates the heat losses, a L/R th ratio of 2 minimizes the thermal area over volume ratio (A/V) and resulting heat losses for a given storage volume. In contrast however, the impact of dispersion decreases with L/R th and particularly for ATES systems with a high potential for buoyancy-driven flow (q 0 > 0.05 m/d), increasingly smaller L/R th ratios (<1) strongly reduce the heat losses due to tilting. Overall, the results of this study support the assessment of thermal recovery efficiencies for particular aquifer and storage conditions, thereby aiding the optimization of initial ATES designs. [Display omitted] [ABSTRACT FROM AUTHOR]

This paper incorporated two important characteristics of clay, namely its rate-dependent and strain-softening behavior, into a theoretical framework based on cavity expansion. The modified Tresca model for clay proposed by Einav and Randolph was selected to describe the effects of the rate dependency and gradual softening of clay as it is sheared and remolded. First, the modified Tresca model was reduced to a strain-softening model without considering the rate effect. The exponential degradation (ED) function of the strength reduction proposed by Einav and Randolph was simplified further to a bilinear degradation (SBD) function. This step allowed an approximate closed-form solution to be obtained for the cavity expansion response, including the plastic zone radius, stress distribution, and cavity pressure-expansion relation, of the SBD model. The approximate closed-form solution for the SBD model was compared with the numerical integration solution for the ED model, and the comparison showed that the approximate closed-form solution is relatively accurate. Subsequently, the rate effects were incorporated into the SBD and ED strain-softening models, which are referred to as the rate-dependent (RD)-SBD and RD-ED models. Numerical integration solutions for cavity expansion in the RD-SBD and RD-ED models were obtained. A series of parametric analyses was conducted to investigate the effects of the cavity expansion velocity, strength reduction, and rate-dependent parameters on the cavity expansion responses. A closed-form solution for the limit expansion pressure (LEP) of the cavity expansion was obtained through a simple regression analysis based on the parametric results. The LEP solution was used to interpret the results of the cone penetration test (CPT) test, and a modified expression for the cone penetration resistance factor that incorporates the rate effect and strain softening was given. This simple modification provides a convenient way to evaluate the influence of the two factors on the CPT test results and allows the clay properties to be captured appropriately. [ABSTRACT FROM AUTHOR]

Finite element (FEM) based software is frequently used in practice for tunnel design, alongside the traditional analytical and empirical solutions. Design is not the only challenge in this kind of projects, there are other important factors such as considering what is necessary to develop an efficient construction plan under a schedule and foreseeing possible changes that modify the original design. The use of two-dimensional (2D) FEM is one of the main tools used in the industry. However, 2D results yield not an entirely accurate analysis, since the behaviour of the infrastructure is considered as threedimensional (3D). This paper presents nomographs and a 3D and 2D relationship, to rapidly estimate values of elastic 3D and 2D displacements produced in the periphery of a deep circular tunnel, inside soils of different rigidities. Graphics given correspond to five different radii and for a 100 m excavation length. Nomographs were obtained from RS3© and RS2© FEM simulations and according to the elastic theory. Geotechnical parameters correspond to a constant friction angle, cohesion and soil specific weight. FEM analysis was made using the Mohr-Coulomb model, considering isotropic conditions. [ABSTRACT FROM AUTHOR]

For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant. [ABSTRACT FROM AUTHOR]

Based on the elementary solutions and new integral equations, a new analytical-numerical method is proposed to calculate the interacting stresses of multiple circular holes in an infinite elastic plate under both remote stresses and arbitrarily distributed stresses applied to the circular boundaries. The validity of this new analytical-numerical method is verified by the analytical solution of the bi-harmonic stress function method, the numerical solution of the finite element method, and the analytical-numerical solutions of the series expansion and Laurent series methods. Some numerical examples are presented to investigate the effects of the hole geometry parameters (radii and relative positions) and loading conditions (remote stresses and surface stresses) on the interacting tangential stresses and interacting stress concentration factors. The results show that whether the interference effect is shielding (k < 1) or amplifying (k > 1) depends on the relative orientation of holes (α) and remote stresses (

This paper concerned an analytical method to obtain exact solutions for displacements of submerged hemi-ellipsoidal shells. Based on the linear membrane theory, the membrane forces and strain components were defined in terms of the radius of a parallel circle. General solutions for normal and tangential displacements were established, which could be written in the simplest form when considered at the apex and equatorial points. To investigate the accuracy of the results, a finite-element modeling technique was used to verify the analytical solutions. The fundamentals of differential geometry and the principle of virtual work were used to develop the stiffness matrix for nonlinear finite-element procedures. This led to the numerical results for validation of the analytical solutions. The effect of shape ratios, thickness, and seawater depth on the forces and displacements of submerged hemi-ellipsoidal shells were also presented in this study. [ABSTRACT FROM AUTHOR]

The mode-III fracture properties of the radial multi cracks on the edge of a nanoscale circular hole were theoretically investigated. Based on the Gurtin-Murdoch surface elasticity theory and the conformal mapping technique, the analytical solutions of the stress fields around the hole and cracks were given, and the closed form solution of the stress intensity factor at the crack tip was obtained. The size effect of the stress intensity factor was analyzed based on the solution. The effects of the crack number, the ratio of crack/hole radius and the surface defects on the stress intensity factor were discussed. The results indicate that, the dimensionless stress intensity factor has remarkable size-dependent effects when the sizes of the cracked hole are at the nanoscale. The variation of the stress intensity factor with the number of cracks is influenced by the crack/hole size ratio. The effects of the crack/hole size ratio on the stress intensity factors are restricted by the surface defects, and the effects of the surface properties on the stress intensity factor are also limited by the crack/hole size ratio. The surface effects on the stress intensity factors are independent of the number of cracks. [ABSTRACT FROM AUTHOR]

We consider Y-system functional equations of the form Y n (x + i) Y n (x - i) = ∏ m = 1 N 1 + Y m (x) G nm and the corresponding nonlinear integral equations of the thermodynamic Bethe ansatz. We prove an existence and uniqueness result for solutions of these equations, subject to appropriate conditions on the analytical properties of the Y n , in particular the absence of zeros in a strip around the real axis. The matrix G nm must have non-negative real entries and be irreducible and diagonalisable over R with spectral radius less than 2. This includes the adjacency matrices of finite Dynkin diagrams, but covers much more as we do not require G nm to be integers. Our results specialise to the constant Y-system, proving existence and uniqueness of a strictly positive solution in that case. [ABSTRACT FROM AUTHOR]

There is little analytical theory for the behavior of solutions of the Kuramoto–Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the linearized behavior depends on the size of the torus—in particular, for different sizes of the domain, there are different numbers of linearly growing modes. We prove that small solutions exist for all time if there are no linearly growing modes, proving also in this case that the radius of analyticity of solutions grows linearly in time. In the general case (i.e., in the presence of a finite number of growing modes), we make estimates for how the radius of analyticity of solutions changes in time. [ABSTRACT FROM AUTHOR]

The yybar diagram for Gaussian beam optics is employed to model the behavior of an interferometer system testing very small radius parts. The model was developed to overcome the limitations and known inconsistencies of a paraxial optics representation used to evaluate a calibration method for testing cylindrical wave optics using a fiber reference test. Gaussian beam analysis inherently contains physical optics conditions, and the yybar diagram method provides both an intuitive and powerful framework to generate analytical solutions. Particularly, we show how to model an interferometric test from a Fizeau transmission sphere (TS), to a small test ball and back to the TS, and yield test ball radius limits as a function of the test wavelength and TS F∕#. A computation of error estimates for measuring the radius of curvature of the test balls is also presented. [ABSTRACT FROM AUTHOR]

The purpose of this research is to present the steps of one-dimensional differential transformation method (DTM) to find the series solutions for the vibrations of a circular membrane under the specified initial and boundary conditions. The problems will be studied in the both cases of vibrations depending only on radius and of the vibrations depending on both radius and angle. We illustrate four examples of problems which the exact solutions can be solve analytically and compare them to the DTM results, to show that the DTM is reliable and of high accuracy. This work shows that the DTM is easier to use than the analytical method from the point of view of programming. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the propagation of abruptly autofocusing circular Airy beams (CAB) in linear index potential both analytically and numerically. Based on the initial field located far away from the center, we get the approximate analytical solutions to describe the trajectory and focal length of CAB in linear index potential. The modified focal length formula agrees well with numerical results for a wide range of the initial radius of CAB. Furthermore, the approximate method is also applicable to some dynamic potential varying along the propagation coordinate. [ABSTRACT FROM AUTHOR]

Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Grujić and Kalisch. [ABSTRACT FROM AUTHOR]

Utilizing the analytical model of Titov & Demomulin (1999) to set a toroidal current channel in force-free equilibrium, partially submerged under the solar photosphere to model a solar prominence, we studied the threshold of torus instability for a range of different geometries and external toroidal field strengths. Four parameters of the equilibrium have been varied: minor radius and footpoint distance of the current channel, strength of the external toroidal field, and sunspot distance. The sunspot distance determines the height profile of the external poloidal field's decay index, thus, determines the torus-unstable height range. We found that the critical decay index at the torus instability threshold increases (corresponding to a more stable situation) when the strength of the external toroidal field or the radius of the current channel increase. For given apex height, the threshold does not depend significantly on the footpoint distance of the current channel. [ABSTRACT FROM AUTHOR]

We propose a new approach to unambiguous determination of parameters of positive andnegative electric streamer discharges (streamers). We derive several relations betweenstreamer parameters which allow us to express them in terms of the streamer length L andthe external electric field Ee, as functions of streamer radius a. In particular, wefind that the streamer velocity V (a) has a maximum at a certain value of radiusas. We interpret the streamer as a nonlinear instability, whose behavior is determined bymaximizing its growth rate, proportional to V. The radius of the streamer is therefore equal toas, which also fixes all other streamer parameters. Thus, we prove that the streamerparameters are completely determined by Ee and L and suggest a relatively simple way tocalculate them that does not require hydrodynamic simulations. The parameters of streamersin air at sea level conditions, calculated in the proposed way, are compared and found to becompatible with experimental data [e.g., velocities obtained by Allen and Mikropoulos, 1999,doi:10.1088/0022-3727/32/8/012]. We also reproduce the correct values of streamer thresholdfields of ∼0.45 MV/m for positive streamers and ∼0.75–1.25 MV/m for negativestreamers. An analytical solution of the above system gives a quick but less accurate answer.However, the qualitative behavior of parameters may be demonstrated in this way. [ABSTRACT FROM AUTHOR]

The nonlinear saturation amplitude (NSA) of the fundamental mode in the classical Rayleigh—Taylor instability with a cylindrical geometry for an arbitrary Atwood number is analytically investigated by considering the nonlinear corrections up to the third order. The analytic results indicate that the effects of the initial radius of the interface (r0) and the Atwood number (A) play an important role in the NSA of the fundamental mode. The NSA of the fundamental mode first increases gently and then decreases quickly with increasing A. For a given A, the smaller the r0/λ (λ is the perturbation wavelength), the larger the NSA of the fundamental mode. When r0/λ is large enough (r0 ≫ λ), the NSA of the fundamental mode is reduced to the prediction in the previous literatures within the framework of the third-order perturbation theory. [ABSTRACT FROM AUTHOR]