An Approach for Increasing the Discretization Interval along Radial Coordinate in Terrain Reflections Model

2020 ◽  
Author(s):  
Margarita V. Oreshkina
Keyword(s):  
Radial Coordinate ◽  
Discretization Interval

scholarly journals Abundance of Primordial Black Holes in Peak Theory for an Arbitrary Power Spectrum

2020 ◽  
Author(s):  
Chul-Moon Yoo ◽  
Tomohiro Harada ◽  
Shin’ichi Hirano ◽  
Kazunori Kohri
Keyword(s):  
Mass Spectrum ◽  
Power Spectrum ◽  
Environmental Effect ◽  
Window Function ◽  
Radial Coordinate ◽  
Primordial Black Holes ◽  
Scale Invariant ◽  
Curvature Perturbation ◽  
Arbitrary Power ◽  
Nonlinear Relation

Abstract We modify the procedure to estimate PBH abundance proposed in Ref. [1] so that it can be applied to a broad power spectrum such as the scale-invariant flat power spectrum. In the new procedure, we focus on peaks of the Laplacian of the curvature perturbation △ ζ and use the values of △ ζ and △ △ ζ at each peak to specify the profile of ζ as a function of the radial coordinate while the values of ζ and △ ζ are used in Ref. [1]. The new procedure decouples the larger-scale environmental effect from the estimate of PBH abundance. Because the redundant variance due to the environmental effect is eliminated, we obtain a narrower shape of the mass spectrum compared to the previous procedure in Ref. [1]. Furthermore, the new procedure allows us to estimate PBH abundance for the scale-invariant flat power spectrum by introducing a window function. Although the final result depends on the choice of the window function, we show that the k-space tophat window minimizes the extra reduction of the mass spectrum due to the window function. That is, the k-space tophat window has the minimum required property in the theoretical PBH estimation. Our procedure makes it possible to calculate the PBH mass spectrum for an arbitrary power spectrum by using a plausible PBH formation criterion with the nonlinear relation taken into account.

Droplet spreading and absorption on rough, permeable substrates

2015 ◽  
Vol 784 ◽  
pp. 465-486 ◽  
Author(s):  
Leonardo Espín ◽  
Satish Kumar
Keyword(s):  
Surface Roughness ◽  
Contact Line ◽  
Nonlinear Evolution ◽  
Precursor Film ◽  
Radial Coordinate ◽  
Droplet Spreading ◽  
Liquid Spreading ◽  
Line Region ◽  
Influence Of Substrate ◽  
Contact Line Pinning

Wetting of permeable substrates by liquids is an important phenomenon in many natural and industrial processes. Substrate heterogeneities may significantly alter liquid spreading and interface shapes, which in turn may alter liquid imbibition. A new lubrication-theory-based model for droplet spreading on permeable substrates that incorporates surface roughness is developed in this work. The substrate is assumed to be saturated with liquid, and the contact-line region is described by including a precursor film and disjoining pressure. A novel boundary condition for liquid imbibition is applied that eliminates the need for a droplet-thickness-dependent substrate permeability that has been employed in previous models. A nonlinear evolution equation describing droplet height as a function of time and the radial coordinate is derived and then numerically solved to characterize the influence of substrate permeability and roughness on axisymmetric droplet spreading. Because it incorporates surface roughness, the new model is able to describe the contact-line pinning that has been observed in experiments but not captured by previous models.

scholarly journals Theory of the tertiary instability and the Dimits shift within a scalar model

2020 ◽  
Vol 86 (4) ◽  
Author(s):  
Hongxuan Zhu ◽  
Yao Zhou ◽  
I. Y. Dodin
Keyword(s):  
Turbulent Transport ◽  
Plasma Phys ◽  
Magnetized Plasma ◽  
Radial Coordinate ◽  
Flow Shear ◽  
Zonal Flows ◽  
Drift Wave ◽  
Zonal Velocity ◽  
Instability Growth ◽  
Traditional Paradigm

The Dimits shift is the shift between the threshold of the drift-wave primary instability and the actual onset of turbulent transport in a magnetized plasma. It is generally attributed to the suppression of turbulence by zonal flows, but developing a more detailed understanding calls for consideration of specific reduced models. The modified Terry–Horton system has been proposed by St-Onge (J. Plasma Phys., vol. 83, 2017, 905830504) as a minimal model capturing the Dimits shift. Here, we use this model to develop an analytic theory of the Dimits shift and a related theory of the tertiary instability of zonal flows. We show that tertiary modes are localized near extrema of the zonal velocity $U(x)$ , where $x$ is the radial coordinate. By approximating $U(x)$ with a parabola, we derive the tertiary-instability growth rate using two different methods and show that the tertiary instability is essentially the primary drift-wave instability modified by the local $U'' \doteq {\rm d}^2 U/{\rm d} x^2 $ . Then, depending on $U''$ , the tertiary instability can be suppressed or unleashed. The former corresponds to the case when zonal flows are strong enough to suppress turbulence (Dimits regime), while the latter corresponds to the case when zonal flows are unstable and turbulence develops. This understanding is different from the traditional paradigm that turbulence is controlled by the flow shear $| {\rm d} U / {\rm d} x |$ . Our analytic predictions are in agreement with direct numerical simulations of the modified Terry–Horton system.

A COMPLETE VARIATIONAL WAVE FUNCTION FOR THE GROUND STATE OF 3H AND 3He

1966 ◽  
Vol 44 (9) ◽  
pp. 2095-2110 ◽  
Author(s):  
Marcel Banville ◽  
P. D. Kunz
Keyword(s):  
Wave Function ◽  
Coordinate System ◽  
Body Wave ◽  
Center Of Mass ◽  
Minimum Energy ◽  
Equal Mass ◽  
Radial Coordinate ◽  
Orthogonal Functions ◽  
Gaussian Shape ◽  
Three Body

The three-body wave function for particles of equal mass is expanded in a systematic way by making use of a hyperspherical coordinate system. Apart from the center-of-mass coordinates, three of the variables are the usual Euler angles describing the orientation of the plane defined by the three particles. The other three variables, which describe the shape of the triangle, are represented in terms of a radial coordinate and two angular coordinates. The kinetic energy for these last three coordinates is separable and allows one to expand the three-body wave function in a complete set of orthogonal functions based upon the angular variables. The particular symmetry of the internal part of the wave function under permutations of the three particles is easily represented in terms of the set of functions for one of the angular variables. By choosing a particular set of radial functions one can then obtain the upper limit on the binding energy for the three-body system through the Rayleigh–Ritz variational procedure. The advantage of this particular coordinate system is that all but a few of the variational parameters occur linearly in the wave function, and the minimum energy can be obtained by diagonalizing a small number of the energy matrices. The method is applied to find the lower limit to a standard spin-independent potential of Gaussian shape.

Rotating Flow Over a Disk Sector

1982 ◽  
Vol 49 (1) ◽  
pp. 13-18
Author(s):  
M. Toren ◽  
A. Solan ◽  
M. Ungarish
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Rotating Flow ◽  
Radial Coordinate ◽  
Large Radius ◽  
Full Circle ◽  
Blasius Boundary Layer ◽  
Layer Flow ◽  
Rotating Boundary ◽  
Limiting Values

The rotating boundary layer flow over a plane sector of angle θs and infinite radius is solved. For sufficiently large radius the radial coordinate is eliminated by a Von Karman transformation, leaving a nonaxisymmetric flow in (θ,z), which cyclically changes over the full circle, from a Blasius boundary layer, to a Bodewadt flow, and to a rotating wake. Leading terms of the three-dimensional perturbation of the Blasius flow, and of the rotating wake are given, and the matching over the full circle is outlined for limiting values of θs.

scholarly journals Experiments and Computational Research of Biomass Pyrolysis in a Cylindrical Reactor

2021 ◽  
Vol 64 (1) ◽  
pp. 51-64
Author(s):  
A. V. Mitrofanov ◽  
V. E. Mizonov ◽  
S. V. Vasilevich ◽  
M. V. Malko
Keyword(s):  
Biomass Conversion ◽  
Batch Reactor ◽  
Beech Wood ◽  
Electrical Heating ◽  
Difference Approximation ◽  
Radial Coordinate ◽  
Internal Diameter ◽  
Biomass Pyrolysis ◽  
Ideal Mixing ◽  
Cylindrical Reactor

The article features an experimental study of thermally thin biomass samples (beech wood particles 17×8×6 mm) pyrolysis in a laboratory scale batch reactor. The reactor was a cylindrical steel body with internal diameter of 200 mm and height of 500 mm. The temperature of a lateral surface of the cylinder during the experiment was being kept constant (550 °C) due to electrical heating. The initial loading of the apparatus was about 4 kg with moisture content of about 14 % by weight. During the experiment, the temperature values of the material being pyrolyzed were recorded at two points of the radial coordinate, viz. at the wall of the apparatus and on its axis. A one-dimensional numerical model of the nonstationary process of biomass conversion (heat and mass transfer in combination with the Avrami – Erofeev reaction model) has been proposed and verified. The reactor is represented as a set of a countable number of cylindrical layers, considered as cells (representative meso-volumes) with an ideal mixing of the properties inside. The cylindrical surfaces that form cells are considered to be isothermal. The size of the cells is chosen to be sufficiently large in comparison with the individual particles of the layer, which makes it possible to consider the temperature field inside the cell volume as monotonic. The evolution of the temperature distribution over the radius of a cylindrical reactor is determined on the basis of a difference approximation of the process of non-stationary thermal conductivity. The calculated forecasts and experimental data showed a good agreement, which indicates the adequacy of the developed mathematical model of pyrolysis and makes it possible to recommend it for engineering calculations of biomass pyrolysis. This model can also be useful in improving the understanding of the basic physical and chemical processes occurring in the conditions of biomass pyrolysis.

scholarly journals Узагальнення моделі Голанда і Рейсснера на випадок осьової симетрії

2021 ◽  
pp. 12-19
Author(s):  
Костянтин Петрович Барахов
Keyword(s):  
Stress State ◽  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Bessel Functions ◽  
Classical Model ◽  
Adhesive Bond ◽  
Radial Coordinate ◽  
Round Hole ◽  
Normal Stresses

The purpose of this work is to create a mathematical model of the stress state of overlapped circular axisymmetric adhesive joints and to build an appropriate analytical solution to the problem. To solve the problem, a simplified model of the adhesive bond of two overlapped plates is proposed. The simplification is that the movement of the layers depends only on the radial coordinate and does not depend on the angular one. The model is a generalization of the classical model of the connection of Holland and Reissner in the case of axial symmetry. The stresses are considered to be evenly distributed over the thickness of the layers, and the adhesive layer works only on the shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stress state of the adhesive bond of two plates is solved, one of which is weakened by a round hole, and the other is a round plate concentric with the hole. A load is applied to the plate weakened by a round hole. The discussed area is divided into three parts: the area of bonding, as well as areas inside and outside the bonding. In the field of bonding, the problem is reduced to third- and fourth-order differential equations concerning tangent and normal stresses, respectively, the solutions of which are constructed as linear combinations of Bessel functions of the first and second genera and modified Bessel functions of the first and second genera. Using the found tangential and normal stresses, we obtain linear inhomogeneous Euler differential equations concerning longitudinal and transverse displacements. The solution of the obtained equations is also constructed using Bessel functions. Outside the area of bonding, displacements are described by the equations of bending of round plates in the absence of shear forces. Boundary conditions are met exactly. The satisfaction of marginal conditions, as well as boundary conditions, leads to a system of linear equations concerning the unknown coefficients of the obtained solutions. The model problem is solved and the numerical results are compared with the results of calculations performed by using the finite element method. It is shown that the proposed model has sufficient accuracy for engineering problems and can be used to solve problems of the design of aerospace structures.

AN APPROXIMATE FORMULATION OF THE EFFECTIVE INDENTATION MODULUS OF ELASTICALLY ANISOTROPIC FILM-ON-SUBSTRATE SYSTEMS

2009 ◽  
Vol 01 (03) ◽  
pp. 515-525 ◽  
Author(s):  
T. L. LI ◽  
J. H. LEE ◽  
Y. F. GAO
Keyword(s):  
Applied Load ◽  
Indentation Depth ◽  
Surface Displacement ◽  
Contact Radius ◽  
Radial Coordinate ◽  
Indentation Modulus ◽  
Approximate Formulation ◽  
Punch Contact ◽  
Indenter Shape ◽  
Circular Contact

Frictionless contact between an arbitrarily-shaped rigid indenter and an elastically anisotropic film-on-substrate system can be regarded as being superposed incrementally by a flat-ended punch contact, the shape and size of which are determined by the indenter shape, indentation depth (or applied load) and elastic properties of film and substrate. For typical nanoindentation applications, the indentation modulus can thus be approximated from the response of a circular contact with pressure of the form of [1 - (r/a)2]-1/2, where r is the radial coordinate and a is the contact radius. The surface-displacement Green's function for elastically anisotropic film-on-substrate system is derived in closed-form by using the Stroh formalism and the two-dimensional Fourier transform. The predicted dependence of the effective modulus on the ratio of film thickness to contact radius agrees well with detailed finite element simulations. Implications in evaluating film modulus by nanoindentation technique are also discussed.

scholarly journals Exact Axially Symmetric Solution inf(T)Gravity Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Gamal G. L. Nashed
Keyword(s):  
Vacuum Solution ◽  
Gravity Theory ◽  
Kerr Metric ◽  
Radial Coordinate ◽  
Lorentz Transformations ◽  
Axially Symmetric ◽  
Field Equations ◽  
Tetrad Field ◽  
Kerr Spacetime ◽  
Euler’S Angles

A general tetrad field with sixteen unknown functions is applied to the field equations off(T)gravity theory. An analytic vacuum solution is derived with two constants of integration and an angleΦthat depends on the angle coordinateϕand radial coordinater. The tetrad field of this solution is axially symmetric and the scalar torsion vanishes. We calculate the associated metric of the derived solution and show that it represents Kerr spacetime. Finally, we show that the derived solution can be described by two local Lorentz transformations in addition to a tetrad field that is the square root of the Kerr metric. One of these local Lorentz transformations is a special case of Euler’s angles and the other represents a boost when the rotation parameter vanishes.

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