Solution of Moore–Gibson–Thompson Equation of an Unbounded Medium with a Cylindrical Hole

In the current article, in the presence of thermal and diffusion processes, the equations governing elastic materials through thermodiffusion are obtained. The Moore–Gibson–Thompson (MGT) equation modifies and defines the equations for thermal conduction and mass diffusion that occur in solids. This modification is based on adding heat and diffusion relaxation times in the Green–Naghdi Type III (GN-III) models. In an unbounded medium with a cylindrical hole, the built model has been applied to examine the influence of the coupling between temperature and mass diffusion and responses. At constant concentration as well as intermittent and decaying varying heat, the surrounding cavity surface is traction-free and is filled slowly. Laplace transform and Laplace inversion techniques are applied to obtain the solutions of the studied field variables. In order to explore thermal diffusion analysis and find closed solutions, a suitable numerical approximation technique has been used. Comparisons are made between the results obtained with the results of the corresponding previous models. Additionally, to explain and realize the presented model, tables and figures for various physical fields are presented.

THE ANALYSIS OF THERMAL FIELDS AT THE PRIMARY STAGE OF THE STEAM-ASSISTED GRAVITY DRAINAGE PROCESS

This article analyzes the temperature distribution in a producer well at the primary stage of the steam-assisted gravity drainage process. The increase in share of hard-to-recover reserves requires using steam-assisted gravity drainage (SAGD). Its successful application, in turn, depends on warming up the inter-well zone, which demands steam circulation in both wells at the primary stage of the process. The duration of this stage affects the transition to oil production and the profitability of the process, which emphasizes the importance of analyzing thermal fields at this stage to assess its duration. The existing research does not allow estimating the temperature in the producer, using the correct formulation of the problem. This paper presents the temperature distribution in a producer for SAGD for classical and chess well patterns for the first time. The aim of the work is to choose a development system for the minimum duration of primary stage of SAGD. For this purpose, the fundamental solution of the non-stationary heat equation for a continuous stationary point source in an unbounded medium is used. The estimation of temperature, at which oil becomes mobile, allows determining the primary stage duration. The authors compare the classical and chess well patterns. In addition, they have obtained the temperature distribution in producer. The results show that classical well pattern provides faster heating of inter-well zone. It is determined that the closest injection well has the greatest influence on the temperature in the producing well.

Wave Propagation in an Unbounded Magneto-Thermoelastic Rotating Medium Permeated by a Heat Source

Matrix method of solution is applied to determine generalized thermoelastic wave propagation in an unbounded medium due to periodically varying heat source under the influence of magnetic field. Green–Lindsay (GL) model of generalized thermoelasticity for finite wave propagation is considered along with a magnetic field for a rotating medium with uniform velocity. Basic equations are solved by eigenvalue approach method after compiling in a form of vector–matrix linear differential equation in Laplace transform domain. Finally inverting the perturbed magnetic field and other field variables by a suitable numerical method, the results are analyzed by depicting several graphs in space–time domain.

A screw dislocation near a damaged arbitrary inhomogeneity–matrix interface

In the literature, the analytical solutions concerned with the interaction between screw dislocation and surfaces/interfaces have been mainly limited to simple geometries and perfect interfaces. The focus of the current work is to provide an approach based on a rigorous semi-analytical theory suitable for treatment of such surfaces/interfaces that concurrently have complex geometry and imperfect bonding. The proposed approach captures the singularity of the elastic fields exactly. A vast variety of the pertinent interaction problems such as dislocation near a multi-inhomogeneity with arbitrary geometry bonded imperfectly to a matrix, dislocation near the free boundaries of a finite elastic medium of arbitrary geometry, and so on is considered. In the present approach the out-of-plane component of the displacement in each domain is decomposed as the displacement corresponding to a screw dislocation in a homogeneous elastic body of infinite extent and the disturbance displacement due to the interaction. Subsequently, the disturbance displacement in each medium is expressed in terms of eigenfunction expansion. Damaged interfaces are modeled by a spring layer of vanishing thickness, and the amount of damage is controlled via the stiffness of the spring. For the illustration of the robustness of the proposed methodology a variety of examples including the interaction of a screw dislocation with a circular as well as a star-shaped inhomogeneities, two interacting inhomogeneities, imperfectly bonded to an unbounded medium are given. Also, examples for highlighting the effect of free surfaces in the case of finite domains are provided. It is revealed that in the cases where matrix is stiffer than the inhomogeneity and the dislocation is inside the inhomogeneity, or the other way around, then the amount of interface damage can change the sign of the image force.

Механизмы отражения и преломления пучка заряженных частиц в рассеивающей среде

The mathematical model describing the reflection and refracting phenomena at incline border of scattering medium and vacuum for a moving charged particle traversing this border is suggested. To build such a model, the distribution function for multiple Coulomb scattering processes in unbounded medium is used. Formulae for refraction angle and reflection coefficient are derived followed by numerical calculations and appropriate plots.

Spherical cavity in the layer subjected to dynamic load

The problem of wave motion with spherical symmetry is analysed. For this purpose, a spherical cavity surrounded by layer is considered as a mathematical model which can be used for modelling various phenomena in solid mechanics. The additional layer is also spherical and the outer space is described as unbounded medium. All layers are isotropic, homogeneous and linearly elastic, although the presented formulation allows inclusion of weak nonlinearities. Analytical solutions for displacement and stresses (radial and circumferential) are presented along with some discussion of possible model extensions.

On Mindlin’s isotropic strain gradient elasticity: Green tensors, regularization, and operator-split

The theory of Mindlin’s isotropic strain gradient elasticity of form II is reviewed. Three-dimensional and two-dimensional Green tensors and their first and second derivatives are derived for an unbounded medium. Using an operator-split in Mindlin’s strain gradient elasticity, three-dimensional and two-dimensional regularization function tensors are computed, which are the three-dimensional and two-dimensional Green tensors of a tensorial Helmholtz equation. In addition, a length scale tensor is introduced, which is responsible for the characteristic material lengths of strain gradient elasticity. Moreover, based on the Green tensors of Mindlin’s strain gradient elasticity, point, line and double forces are studied.