Laplace Integral Transformation Research Articles

The propagation of longitudinal stress waves in an infinite rod with a viscoelastic region

The problem of propagation of longitudinal stress waves in an infinite piece-wise homogeneous rod with a viscoelastic region of finite size is solved. A mathematical formulation of the problem and its analytical solution in stresses are obtained. The solution uses the method of the Laplace integral transformation with respect to time. The expressions obtained allow one to determine the stresses in an arbitrary cross-section of the rod at any time. The structure of the solution reflects the process of transformation of the incoming wave with its multiple refraction and reflection at the boundaries of the sections. The solution obtained makes it possible to quantitatively investigate the damping effect of the viscoelastic region and can serve as the basis for selecting the parameters of the insert material when it is used to extinguish the dynamic effects.

Применение «условия Cтефана» для решения тепловых задач в объектах лесного комплекса

A two-phase formulation of the problems of "thermal shock" is considered when two homogeneous half-spaces come into contact at an initial instant of time with different phases and temperatures different from the phase transition temperature. In the absence of convection and thermal sources for constant thermophysical parameters, which can be formulated as the problem of conjugation of two temperature fields on the moving solidification front with additional boundary conditions (Stefan conditions). From a practical point of view, this kind of problem can be the arrival of processes occurring in the objects of the forest complex: in the production of particle board, the processing of ponds and reservoirs, the freezing (thawing) of soils, etc. The solution was carried out using the Laplace integral transformation. The exact analytical dependence obtained in this way explicitly determines the law of interference in each phase. These functions are used for integral transformations. The resulting temperature field corresponds to the known Gaussian distribution, and the velocity of the interphase boundary movement is inversely proportional to the square root of the crystallization time. The data of the approximate numerical calculation carried out for the water-ice system corresponds to a freezing (thawing) rate of approximately 10-3 mm / s. The obtained results can be used for research work in the field of construction thermal physics, geophysics and metallurgy.

Determination of Viscoelastic Stresses in Plates with Inclusions

We describe an approach to the determination of viscoelastic stresses in plates with inclusions. It is based on the method of boundary integral equations and the Laplace integral transformation. The formal solution of the problem of elasticity theory in which the differential operators are replaced with constants is constructed by the method of boundary integral equations and reduced to the solution of a system of algebraic equations. The corresponding system for the viscoelasticity problem is solved with the help of the Laplace integral transformation and a refined numerical-analytic formula for its inversion.

Pressure transient responses study on the hydraulic volume fracturing vertical well in stress-sensitive tight hydrocarbon reservoirs

Based on the characteristics of stimulated zone and un-stimulated zone after hydraulic volume fracturing of vertical well, as well as considering the stress-sensitive characteristic of the tight oil reservoir, the two-zone composite pressure transient responses model is built. The inner zone represents the stimulated reservoir volume region and is modeled as a dual-porosity radial reservoir, while the outer zone represents the un-stimulated reservoir volume region and is modeled as a single porosity reservoir. The model is solved by the method of superposition, Laplace integral transformation, Stehfest numerical inverse algorithm, and the perturbation technique. Thus the analytical solutions to the model are obtained. On the basis of the field example verification of the model, the flow regimes of the hydraulic volume fracturing vertical well are analyzed. The results of the study show that the new model can provide the reliable pressure transient responses model for hydraulic volume fracturing vertical well, and the flow regimes can be divided into five stages, including the wellbore storage stage, transition flow from matrix system to fracture system, radial flow of SRV zone, transition flow from UN-SRV zone to SRV zone, and radial flow of whole reservoir system. The new model and study are conducive to the well test interpretation of hydraulic volume fracturing vertical well in stress-sensitive tight oil reservoir.

Radon transport model into a porous ground layer of finite capacity

The model of radon transfer is considered in a porous ground layer of finite power. With the help of the Laplace integral transformation, a numerical solution of this model is obtained which is based on the construction of a generalized quadrature formula of the highest degree of accuracy for the transition to the original - the function of solving this problem. The calculated curves are constructed and investigated depending on the diffusion and advection coefficients.The work was a mathematical model that describes the effect of the sliding attachment (stick-slip), taking into account hereditarity. This model can be regarded as a mechanical model of earthquake preparation. For such a model was proposed explicit finite- difference scheme, on which were built the waveform and phase trajectories hereditarity effect of stick-slip.

О ЗАДАЧЕ ИДЕНТИФИКАЦИИ НАЧАЛЬНОГО ГИДРОДИНАМИЧЕСКОГО УЧАСТКА ПРИ ЛАМИНАРНОМ ТЕЧЕНИИ НЬЮТОНОВСКОЙ ЖИДКОСТИ В ГОРИЗОНТАЛЬНОМ КОЛЬЦЕВОМ КАНАЛЕ

In the frameworks of physical linearization of the Navier-Stokes equations in a cylindrical coordinates system on the one-way axial force-feed laminar flow of Newtonian fluid, a mathematical model of the flow development in the entrance region of a horizontal annular channel is formulated. The unknown constant gradient of pressure along the channel is connected with the equation of continuity written in an integral form of stability of liquid flow in any cross section of a channel. Use of the one-way integral Laplace transformation along the longitudinal coordinate allowed to obtain an analytical expression of the local hydrodynamic field at the entrance region and determine pressure losses coincided with the classic data. Analysis of the characteristic structure of the hydrodynamic field of dimensionless velocities at the entrance region showed that for small values of the ratio of the radii of the inner and outer coaxial cylindrical tubes constituting the annular channel, asymmetry of the longitudinal velocity profile is observed with a shift of the maximum value towards the surface of a coaxial cylinder of smaller radius, and an increase in the Reynolds number practically linearly increases the length of the hydrodynamic entrance region. Assumption about the absence of drift of the radial coordinate of the maximum velocity in the entrance hydrodynamic region, limited to the so-called "regular" regime, made it possible to identify the length of the entrance hydrodynamic region in the annular channel by the completed expression in an explicit form that correlates with the classical estimates obtained as a result of computational experiments. It is noted that when the ratio of the radii of the inner and outer coaxial cylinders approaches zero or infinity (corresponding to particular cases of a circular tube and a flat channel), the known results for the lengths of the entrance hydrodynamic regions are obtained. Difference between velocity values calculated by the proposed model and experimental values in the region adjacent to the entry section is explained by the fact that kinetic energy of the liquid flow is not accounted for by leveling the pressure inhomogeneity along the channel cross-section. Nonetheless, it is shown that it does not have a significant influence on the length of hydrodynamic entrance region.

Modeling and Determination of the Nonsteady Thermoelastic State of a Two-Layer Thermosensitive Plate

On the basis of a model of thermosensitive bodies, we determine the distribution of the unsteady temperature field and the thermoelastic state caused by this field in a two-layer plate. A solution of the nonlinear nonstationary problem of heat conduction is constructed with the use of the Kirchhoff transformation, the method of linearizing parameter, and the integral Laplace transformation with respect to time. The influence of the temperature dependence of the thermophysical and mechanical characteristics of the materials of layers on the values and nature of the distributions of temperature and thermal stresses caused by this dependence in the plate are numerically analyzed.

Braking Current Spectral Characteristics for Linear Induction Motor with Short Secondary Part

The submitted list of scientific publications and literature indicate, that so far there have never been comprehensively analysed the braking modes of the Linear Induction Motor (LIM) with a short secondary part. The article deals with the theoretical simulation model of such a LIM, containing even, odd or fractional number of the active (excitation) zones. The spectral analysis method is used to calculate the spectral characteristics of the motor braking. After having applied, Fourier and Laplace integral transformations for the non – periodic volumetric density functions of the braking current, there have been derived the analytical expressions of the continuous amplitude spectra. These obtained expressions successfully meet/harmonize the analysis of spectral characteristics of the primary and secondary magnetic field.

An Analytic Model for Dispersion of Rocket Exhaust Clouds: Specifications and Analysis in Different Atmospheric Stability Conditions

This paper brings the first results concerning a new analytic model to evaluate atmospheric dispersion in rocket launch scenarios, namely the GILTTR (Generalized Integral Laplace Transform Technique for Rocket effluent dispersion) model. The model is constituted by three different modules, being them two pre-processors (micrometeorological parameters and deposition parameters) and the dispersion program. The dispersion calculations are made through the GILTT solution of the two-dimensional, time-dependant, advection-diffusion-deposition equation. The results show a set of simulations made using data from the Chuva Project from the Alcantara Rocket Launch Center (Brazil) for both stable and unstable planetary boundary layers (PBL), in order to evaluate model performance and illustration.

Eddy diffusivities for the convective boundary layer derived from LES spectral data

Large Eddy Simulation (LES) spectral data and Taylor statistical diffusion theory are used to obtain Eddy diffusivities in a convective boundary layer. The derivation employs a fitting expression obtained from LES data for the vertical peak frequency. The vertical Eddy diffusivities are well behaved and show similar patterns and magnitudes as those derived from experimental spectral peak frequency data. In addition, this new vertical Eddy diffusivity was introduced into an advection diffusion equation which was solved by Generalized Integral Laplace Transform Technique (GILLT) method and validated with observed contaminant concentration data of the Copenhagen experiment. The results of this new approach are shown to agree with the measurements of Copenhagen.

Pollutant’s Horizontal Dispersion Along and Against Sinusoidally Varying Velocity from a Pulse Type Point Source

An analytical solution of a two-dimensional advection diffusion equation with time dependent coefficients is obtained by using Laplace Integral Transformation Technique. The horizontal medium of solute transport is considered of semi-infinite extent along both the longitudinal and lateral directions. The input concentration is assumed at an intermediate position of the domain. It helps to evaluate concentration level along the flow as well as against the flow through one model only. The source of the input concentration is considered to be of pulse type. In the presence of the source, it is assumed to be decreasing very slowly with time, and just after the elimination of the source it is assumed to be zero. The dispersion coefficient and the advection parameter are considered directly proportional to each other. The analytical solution may be used to predict the solute concentration level with position and time in an open medium as well as in a porous medium. The effect of heterogeneity on the solute transport may also be predicted.

Elastic Half Plane Under the Action of Nonstationary Surface Kinematic Perturbations

A homogeneous isotropic elastic half plane with nonstationary normal displacements on the boundary is considered. With the use of the Laplace and Fourier integral transformations, we propose integral representations for displacements with kernels in the form of surface influence functions. The originals are computed by using an algorithm of joint inversion of these transformations based on the construction of analytic extensions of the transforms. The explicit form of the influence functions is obtained. Some examples of their evaluation are presented.

Unsteady-state diffusion modeling of gas in coal matrix for horizontal well production

In this work, an unsteady-state diffusion model was developed to describe gas transport in coal structures for horizontal well production. The model assumes unsteady-state diffusion of gas through the matrix according to Fick’s law, Darcy flow through the cleat network (natural fractures), and that gas adsorption can be described by the Langmuir equation. Then the model using Laplace integral transformation was solved. Using the new model, dimensionless-type curves for pressure-transient and rate-decline analyses were used to analyze transient transport characteristics. The differences for unsteady-state diffusion between horizontal and vertical well models and the differences for horizontal well production between unsteady-state diffusion and pseudosteady-state diffusion were specifically analyzed. Several scenarios confronting real coal beds were studied and discussed fully through simulating well production under different conditions. The research results showed that the unsteady-state diffusion model would be another choice with which to analyze well tests.

A puff model using a three-dimensional analytical solution for the pollutant diffusion process

A new model for the dispersion of passive puffs is presented (puff model). The model is based on the GILTT (Generalized Integral Laplace Transform Technique) approach, a general technique for solving the advection–diffusion equation. It is a well-known hybrid method that had solved a wide class of direct and inverse problems mainly in the area of Heat Transfer and Fluid Mechanics and the solution is given in series form. It is used a boundary layer parameterization that allows the model to be applied routinely using as input simple ground-level meteorological data. A preliminary performance evaluation is shown in the case of continuous emission from an elevated source in a variable boundary layer (with a time resolution of 10min).

Active control of the strained state of an asymmetric trimorphic beam under nonstationary modes of operation

We consider the problem of active control of the strained state of an asymmetric trimorphic beam with floating end faces. The nonstationary mechanical load applied to the trimorph is an unknown function of time. The electric signal, varying in time according to a law that is determined by the proposed criterion, is supplied to the electrodes of one of the piezolayers working under the mode of inverse piezoelectric effect. Here, the state of the beam is sustained close to unstrained. The formation of the controlling electric signal and the identification of the external mechanical action are carried out on the basis of the known potential difference, which arises between the electrodes of the second piezolayer, working under the mode of direct piezoelectric effect. The problem is solved using the Laplace integral transformation in time. As a result of the analytic transition to the space of originals, we determine the required quantities from the system of Volterra integral equations, which is solved with the help of regularizing algorithms. The results of calculations and their analysis are also presented.

The plane problem of the impact of a rigid body on a half-space modelled by a Cosserat medium

Abstract The formulation and method of solving of a plane time-varying contact problem with moving boundaries is developed for an absolutely rigid impactor and a half-space filled with a Cosserat medium. A new function, the surface influence function for a Cosserat medium, is constructed and investigated. An original algorithm for constructing it, based on the small-parameter method in conjunction with a simultaneous Fourier–Laplace integral transformation, is developed and implemented. The quantitative and qualitative differences between the influence function constructed and the well-known solution of the plane Lamb problem for an elastic half-space are indicated.

The Hamiltonian System Method for the Stress Analysis in Axisymmetric Problems of Viscoelastic Solids

With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.

Dual Porosity and Dual Permeability Modeling of Horizontal Well in Naturally Fractured Reservoir

This article is the first investigation on the dual permeability flow issue for horizontal well-production in a naturally fractured dual-porosity reservoir. Based on the inter-porosity flow from matrix system to fracture system and treating the media directly connected with horizontal wellbore as matrix and fracture systems, we established a model of horizontal well-production and then solved the model using some modern mathematical methods, such as Laplace integral transformation, separation of variables, eigenvalue, and eigenfunction. Later in the article, we obtained the standard log–log type curves using numerical simulation and analyzed the transient flow behavior thoroughly, which showed it is dual porosity and dual permeability flow behavior. The numerical simulation results showed that there are obvious differences between dual permeability and single permeability models. The dual permeability flow behavior accelerates energy supplement during production and reduces the classical matrix-fracture (V-shaped) response. We also showed that type curves characteristics are affected by external boundary conditions, the parameter κ, ω f and λ mf, etc. The research results show that our model would be a good semi-analytical model supplied to users. Because the single permeability modeling ignores the direct fluid supply from matrix to wellbore, we recommend using the dual permeability modeling to make well testing and rate decline interpretation in real case studies.

An Analytical Simple Formula for the Ground Level Concentration from a Point Source

The Advection-Diffusion Equation is solved for a constant pollutant emission from a point-like source placed inside an unstable Atmospheric Boundary Layer. The solution is obtained adopting the novel analytical approach: Generalized Integral Laplace Transform Technique. The concentration solution of the equation is expressed through an infinite series expansion. After setting a realistic scenario through the wind and diffusivity parameterizations, the Ground Level Concentration (GLC) is determined, and an explicit approximate expression is provided for it, allowing an analytically simple expression for the position and value of the maximum. Remarks arise regarding the ability to express value and position of the GLC as explicit functions of the parameters defining the Atmospheric Boundary Layer scenario and the source height.

Puff Models for Simulation of Fugitive Hazardous Emissions in Atmosphere

A puff model for the dispersion of material from fugitive hazardous emissions is presented. For vertical diffusion the model is based on general techniques for solving time dependent advection-diffusion equation: the ADMM (Advection Diffusion Multilayer Method) and GILTT (Generalized Integral Laplace Transform Technique) techniques. Both approaches accept wind and eddy diffusion coefficients with any restriction in their height functions. Comparisons between values predicted by the models against experimental ground-level concentrations (from Copenhagen data set) are shown. The preliminary results confirm the applicability of the approaches proposed and are promising for future work.