An exact solution of linearized flow of an emitting, absorbing and scattering grey gas

The governing equations for the problem of linearized flow through a normal shock wave in an emitting, absorbing, and scattering grey gas are reduced to two linear coupled integro-differential equations. By separation of variables, these equations are further reduced to an integral equation similar to that which arises in neutron-transport theory. It is shown that this integral equation admits both regular (associated with discrete eigenfunctions) and singular (associated with continuum eigenfunctions) solutions to form a complete set. The exact closed-form solution is obtained by superposition of these eigen-functions. If the gas downstream of a strong shock is absorption–emission dominated, the discrete mode of the solution disappears downstream. The effects of isotropic scattering are discussed. Quantitative comparison between the numerical results based on the exact solution and on the differential approximation are presented.

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A Quantitative Evaluation of Two Classical Approximations Used to Predict the Extent of Vertical Hydraulic Fractures

Journal of Energy Resources Technology ◽

10.1115/1.3230965 ◽

1983 ◽

Vol 105 (4) ◽

pp. 512-527 ◽

Cited By ~ 2

Author(s):

M. B. Rubin

Keyword(s):

Integral Equation ◽

Exact Solution ◽

Hydraulic Fracture ◽

Closed Form Solution ◽

Solution Procedure ◽

Form Solution ◽

Critical Parameters ◽

Hydraulic Fractures ◽

Fracture Width ◽

Special Case

An integral equation was developed to predict the critical parameters (fracture width and length) associated with the propagation of a vertical hydraulic fracture and a numerical solution procedure was developed. The effects of the classical approximations of pressure and fracture width were investigated both separately and together. It was found that the effects associated with the pressure approximation were relatively insignificant, whereas those associated with the fracture width approximation were significant, particularly when the formation was only moderately permeable. Finally, an exact closed-form solution of the integral equation was developed for a special case. It was shown that when the formation is only moderately permeable, this solution provides a better approximation of the exact solution than the classical solution of Carter [2].

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Analytic Solution for Systems of Two-Dimensional Time Conformable Fractional PDEs by Using CFRDTM

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2019/7869516 ◽

2019 ◽

Vol 2019 ◽

pp. 1-7

Author(s):

Abir Chaouk ◽

Maher Jneid

Keyword(s):

Exact Solution ◽

Approximate Solution ◽

Analytic Solution ◽

Closed Form Solution ◽

Form Solution ◽

Two Dimensional ◽

Fractional Pdes ◽

Differential Transform ◽

Computational Work ◽

Two-Group Neutron Transport Theory with Anisotropic Scattering

Zeitschrift für Naturforschung A ◽

10.1515/zna-1970-0502 ◽

1970 ◽

Vol 25 (5) ◽

pp. 587-594

Author(s):

K. O. Thielheim ◽

K. Claussen

Keyword(s):

Integral Equation ◽

Transport Theory ◽

Neutron Transport ◽

Full Range ◽

Completeness Theorem ◽

Anisotropic Scattering ◽

Neutron Transport Theory ◽

Coeffi Cients ◽

Group Transport ◽

Homogeneous Media

Abstract Two-group transport theory with anisotropic scattering in infinite homogeneous media is pre-sented in this paper. The kernel of the integral equation is expanded into a finite series of Legendre polynomials. Eigenfunctions and eigenvalues of the transformed integral equation are found and the number of discrete eigenvalues is calculated. The full-range completeness theorem as well as the orthogonality and normalization relations are presented. As an example the expansion coeffi-cients of the infinite-medium Green's function are explicitly calculated.

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The Axially Loaded Circular Cylinder

Journal of Applied Mechanics ◽

10.1115/1.3640548 ◽

1962 ◽

Vol 29 (2) ◽

pp. 318-320

Author(s):

H. D. Conway

Keyword(s):

Exact Solution ◽

Fourier Transform ◽

Circular Cylinder ◽

Closed Form ◽

Cross Sections ◽

Closed Form Solution ◽

Form Solution ◽

Concentrated Force ◽

Transform Method ◽

Fourier Transform Method

Commencing with Kelvin’s closed-form solution to the problem of a concentrated force acting at a given point in an indefinitely extended solid, a Fourier transform method is used to obtain an exact solution for the case when the force acts along the axis of a circular cylinder. Numerical values are obtained for the maximum direct stress on cross sections at various distances from the force. These are then compared with the corresponding stresses from the solution for an infinitely long strip, and in both cases it is observed that the stresses are practically uniform on cross sections greater than a diameter or width from the point of application of the load.

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Applications of the symplectic method in quasi-static analysis for viscoelastic functionally graded materials

Engineering Computations ◽

10.1108/ec-02-2016-0063 ◽

2017 ◽

Vol 34 (4) ◽

pp. 1314-1331 ◽

Cited By ~ 1

Author(s):

W.X. Zhang ◽

R.G. Liu ◽

Y. Bai

Keyword(s):

Functionally Graded Materials ◽

Correspondence Principle ◽

Separation Of Variables ◽

Closed Form Solution ◽

Form Solution ◽

Functionally Graded ◽

Symplectic Method ◽

Graded Materials ◽

Content Type ◽

Displacement Constraints

Purpose For general quasi-static problems of viscoelastic functionally graded materials (VFGMs), the correspondence principle can be applied only for simple structures with a closed form solution of the corresponding elastic problem exists. In this paper, a new symplectic approach, according to the correspondence principle between linearly elastic and viscoelastic solids, is proposed for quasi-static VFGMs. Design/methodology/approach Firstly, by employing the method of separation of variables, all the fundamental eigenvectors of the governing equations are obtained analytically. Then, the satisfactions of boundary conditions prescribed on the ends and laterals are discussed based on the variable substitution and the eigenvector expansion methods. Findings In the numerical examples, some boundary condition problems are given. The results show the local effects due to the displacement constraints. Originality/value The paper provides an innovative technique for quasi-static problems of VFG Ms. Its correctness and the efficiency are well suported by numerical results.

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Two-Group Neutron Transport Theory for Plane and Spherical Geometry

Zeitschrift für Naturforschung A ◽

10.1515/zna-1970-1002 ◽

1970 ◽

Vol 25 (10) ◽

pp. 1370-1374 ◽

Cited By ~ 1

Author(s):

K. O. Thielheim ◽

W. Blöcker

Keyword(s):

Integral Equation ◽

Singular Integral ◽

Fredholm Integral Equation ◽

Transport Theory ◽

Neutron Transport ◽

Spherical Geometry ◽

Singular Integral Equations ◽

Neutron Transport Theory ◽

Critical Problems ◽

Expansion Coefficients

Abstract Two-Group neutron transport theory is applied to critical problems in plane and spherical geometry. The neutron flux and the density transform for plane and spherical geometry respectively are expanded into singular eigenfunctions of the transport equation. With aid of the theory of singular integral equations the problem is reduced to one Fredholm integral equation for the expansion coefficients. The critical equations are presented as additional conditions.

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Exact Analytical Hyperbolic Temperature Profile in a Three-Dimensional Media Under Pulse Surface Heat Flux

Journal of Mechanics ◽

10.1017/jmech.2015.97 ◽

2015 ◽

Vol 32 (3) ◽

pp. 339-347 ◽

Cited By ~ 2

Author(s):

M. R. Talaee ◽

V. Sarafrazi ◽

S. Bakhshandeh

Keyword(s):

Heat Flux ◽

Numerical Solutions ◽

Rectangular Cross Section ◽

Separation Of Variables ◽

Three Dimensional ◽

Closed Form Solution ◽

Biological Tissues ◽

Form Solution ◽

Hyperbolic Heat Conduction ◽

Duhamel Integral

AbstractIn this paper three-dimensional hyperbolic heat conduction equation in a cubic media with rectangular cross-section under a pulsed heat flux on the upper side has been solved analytically using the method of separation of variables and the Duhamel integral. The closed form solution of both Fourier and non-Fourier profiles are introduced with both modes of steady and pulsed fluxes. The results show the considerable difference between the Fourier and Non-Fourier temperature profiles. Then the answer procedure is used for modeling of interaction of a cubical tissue under a short laser pulse heating. The effects of pulse duration and laser intensity are studied analytically. Furthermore the results can be applied as a verification branch for other numerical solutions or laser treatments of biological tissues.

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Calculation of Discharge Current Waveforms in High Voltage Spark Sources. II. Extensions and Limitations of Closed-Form Solution

Applied Spectroscopy ◽

10.1366/0003702824639051 ◽

1982 ◽

Vol 36 (1) ◽

pp. 25-29 ◽

Cited By ~ 2

Author(s):

Alexander Scheeline ◽

T. V. Tran

Keyword(s):

Exact Solution ◽

Numerical Methods ◽

Closed Form ◽

High Voltage ◽

Discharge Current ◽

Closed Form Solution ◽

Approximate Model ◽

Form Solution ◽

Dynamic Impedance ◽

Numerical Integration Methods

Simulation of gap breakdown and dynamic impedance effects in high voltage spark sources is performed using an algebraically exact solution to an approximate model of source behavior. The importance of diode shunt capacitance in determining gap breakdown behavior is shown. Limitations in generality and implicit use of numerical methods in dynamic situations lead naturally to consideration of numerical integration methods. Comparisons to hardware sources are made.

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Monte Carlo Simulation Compared With Classic Deterministic Solutions for Neutron Transport and Diffusion

18th International Conference on Nuclear Engineering: Volume 2 ◽

10.1115/icone18-29971 ◽

2010 ◽

Author(s):

Zafar Ullah Koreshi ◽

Sadaf Siddiq

Keyword(s):

Monte Carlo ◽

Green’S Function ◽

Green's Function ◽

Transport Theory ◽

Fourier Transforms ◽

Neutron Transport ◽

Isotropic Scattering ◽

Finite Slab ◽

Deterministic Methods ◽

Mc Simulation

The Monte Carlo (MC) simulation method, known to handle complex problems which may be formidable for deterministic methods, will always require validation with classic problems that have evolved historically from deterministic methods [1–5] based on Chandrasekhar’s method in radiative transfer, Fourier transforms, Green’s functions, Weiner-Hopf method etc which are restricted to simple geometries, such as infinite or semiinfinite media, and simple scattering laws too for practical application. This work compares deterministic results with MC simulation results for neutron flux in a slab. We consider mono-energetic transport problem in an infinite medium and in a 1-D finite slab with isotropic scattering. The transport theory solutions used in infinite geometry are the Green’s function solution and the spherical harmonics (P1, P3) solutions, while for the 1-D finite slab, we refer to a transport benchmark for which an exact solution is available. For diffusion theory, we consider the Green’s function infinite geometry solution, and the exact and eigen-function numerical solution for finite geometry (1-D slab). The objective of this work is to illustrate the results from all the methods considered especially near the source and boundaries, and as a function of the scattering probability. The results are plotted for six elements that include a strong absorber, such as gadolinium, and a strong “scaterrer” such as aluminium. The present work is didactic and focuses on problems which are simple enough, yet important, to illustrate the conceptual difference and computational complexity of the deterministic and stochastic approaches.

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An analytical thermal model of a railway vehicle brake shoe

Proceedings of the Institution of Mechanical Engineers Part F Journal of Rail and Rapid Transit ◽

10.1177/09544097211022107 ◽

2021 ◽

pp. 095440972110221

Author(s):

SME Ghafelehbashi ◽

MR Talaee

Keyword(s):

Cast Iron ◽

Thermal Model ◽

Separation Of Variables ◽

Closed Form Solution ◽

Form Solution ◽

Frictional Heat ◽

Railway Vehicle ◽

Brake Shoe ◽

Good Ability ◽

Analytical Thermal Model

Estimation of temperature distribution in brake shoes is very important in order to prevent thermal damages such as cracks, fading etc. In this research, the thermal model of brake shoe in a train is introduced considering time-dependent frictional heat load in two modes of emergency and continuous braking. The governing heat conduction equation is solved in polar coordinate by using the method of separation of variables combined with Duhamel integral and a closed-form solution is introduced. Results show the good ability of analytical solution to estimate exact temperature profile in composite and cast iron brake shoes. The ability of the solution for estimation of braking situation to prevent from melting criteria is demonstrated. The composite brake shoe will reach to the melting point of 590 °C in continuous braking at the speed of about 120 km/h and the fading limit in cast iron shoe is occurred after 8 sec in emergency braking mode. The introduced thermal model can be applied as a verification branch of other works and can reduced the huge costs of experimental tests of brake shoes.

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