COMBINED METHOD OF INTEGRAL TRANSFORMS FOR THE SPHERICALLY SYMMETRIC DROPLET HEATING PROBLEM

2021 ◽  
Vol 10 (9) ◽  
pp. 3141-3164
Author(s):  
Kwassi Anani
Keyword(s):  
Series Solution ◽  
Integral Transform ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Heat Diffusion ◽  
Spherically Symmetric ◽  
Time Step ◽  
Spherical Droplet ◽  
Heat Diffusion Equation ◽  
Gas Environment

In this paper, we analysed the spherically symmetric heat diffusion equation, which governs the temperature distribution inside a heated but non-evaporating droplet. The spherical droplet, with an initial uniform temperature, is assumed at rest in an unsteady gas environment. The classical Fourier sine integral transform (FSIT) and the unilateral Laplace integral transform (LIT) are successively used to solve the resulting initial-boundary value problem, first reduced in a dimensionless form. An explicit solution in the Laplace domain is obtained for the temperature inside the droplet. Then, depending on the time-varying temperature of the gas environment at the immediate vicinity of the droplet, an exact series solution and an approximate analytical solution in short time limits are derived for the droplet internal temperature. In the case of steady gas environment at constant temperature, the standard series solution obtained in the literature for the symmetrical problem of heating or cooling of a solid spherical body, is recovered. The results may be useful for time step analysis in droplets and sprays vaporization models.

scholarly journals A difference scheme with the optimal weight for the diffusion-convection equation

2019 ◽  
pp. 283-292
Author(s):  
А.И. Сухинов ◽  
А.Е. Чистяков ◽  
В.В. Сидорякина ◽  
С.В. Проценко
Keyword(s):  
Difference Scheme ◽  
Approximation Error ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Step Function ◽  
Time Step ◽  
Optimal Weight ◽  
Convection Equation ◽  
Weight Value ◽  
Diffusion Convection

Исследована разностная схема с весами для однородного пространственно-одномерного уравнения диффузии-конвекции. Выполнено исследование погрешности аппроксимации разностной схемы в зависимости от шага по времени на основе разложения функции решения и погрешности аппроксимации по тригонометрическому базису. Разработан алгоритм нахождения оптимального значения веса, обеспечивающий минимум погрешности аппроксимации решения исходной начально-краевой задачи для заданных значений шагов временной сетки. Улучшенная точность построенной схемы с оптимальным весом по сравнению с явной схемой и эффективность алгоритма поиска оптимального значения весового параметра продемонстрированы на примере тестовой задачи. A difference scheme with weights for a homogeneous spatially one-dimensional diffusion-convection equation is studied. An analysis of the approximation error for the difference scheme as a time step function is performed on the basis of the expansion of the solution and approximation error in a trigonometric basis. An algorithm is proposed to find the optimal weight value that ensures the minimum approximation error of the solution to an initial boundary value problem for the given values of the time grid steps. A better accuracy of the constructed scheme with the optimal weight compared to the explicit scheme as well as the efficiency of the algorithm for finding the optimal weight value is shown using a test problem.

scholarly journals AN INTEGRO-DIFFERENTIAL CONSERVATION LAW ARISING IN A MODEL OF GRANULAR FLOW

2012 ◽  
Vol 09 (01) ◽  
pp. 105-131 ◽  
Author(s):  
DEBORA AMADORI ◽  
WEN SHEN
Keyword(s):  
Granular Flow ◽  
Conservation Law ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Standard Theory ◽  
Step Method ◽  
Fractional Step Method ◽  
Time Step ◽  
Global Existence Of Solutions

We study a scalar integro-differential conservation law which was recently derived by the authors as the slow erosion limit of a granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one cannot adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A prioriL∞bound and total variation estimates yield the convergence and global existence of solutions with bounded variation. Furthermore, we present a well-posedness analysis which establishes that these solutions are stable in the L1norm with respect to the initial data.

Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion

2015 ◽  
Vol 12 (03) ◽  
pp. 469-488 ◽  
Author(s):  
Huapeng Li ◽  
Ronghua Pan ◽  
Weizhe Zhang
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Boussinesq Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Slip Boundary Condition ◽  
Heat Diffusion ◽  
2D Boussinesq Equations ◽  
Initial Boundary Value

We consider the initial-boundary value problem (IBVP) of 2D inviscid heat conductive Boussinesq equations with nonlinear heat diffusion over a bounded domain with smooth boundary. Under slip boundary condition of velocity and the homogeneous Dirichlet boundary condition for temperature, we show that there exists a unique global smooth solution to the IBVP for H3initial data. Moreover, we show that the temperature converges exponentially to zero as time goes to infinity, and the velocity and vorticity are uniformly bounded in time.

scholarly journals Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure

2017 ◽  
Vol 473 (2204) ◽  
pp. 20170113 ◽  
Author(s):  
Irene Brito ◽  
Filipe C. Mena
Keyword(s):  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Einstein Equations ◽  
Fluid Distribution ◽  
Spherically Symmetric ◽  
Tangential Pressure ◽  
Elastic Fluid ◽  
Initial Space ◽  
Initial Boundary Value

We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.

3D Numerical Prediction of Thermal Weakening of Granite under Tension

Geosciences ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 10
Author(s):  
Timo Saksala
Keyword(s):  
Tensile Strength ◽  
Rock Fracture ◽  
Critical Time ◽  
Heterogeneous Material ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Numerical Approach ◽  
Numerical Prediction ◽  
Time Step ◽  
Mass Scaling

This paper deals with numerical prediction of temperature (weakening) effects on the tensile strength of granitic rock. A 3D numerical approach based on the embedded discontinuity finite elements is developed for this purpose. The governing thermo-mechanical initial/boundary value problem is solved with an explicit (in time) staggered method while using extreme mass scaling to increase the critical time step. Rock fracture is represented by the embedded discontinuity concept implemented here with the linear (4-node) tetrahedral elements. The rock is modelled as a linear elastic (up to fracture by the Rankine criterion) heterogeneous material consisting of Quartz, Feldspar and Biotite minerals. Due to its strong and anomalous temperature dependence upon approaching the α-β transition at the Curie point (~573 °C), only Quartz in the numerical rock depends on temperature in the present approach. In the numerical testing, the sample is first volumetrically heated to a target temperature. Then, the uniaxial tension test is performed on the cooled down sample. The simulations demonstrate the validity of the proposed approach as the experimental deterioration, by thermally induced cracking, of the rock tensile strength is predicted with a good accuracy.

scholarly journals Parallel implementation of a meshfree method for calculating flows of ideal incompressible fluid

2017 ◽  
pp. 175-186
Author(s):  
В.Н. Говорухин
Keyword(s):  
Parallel Algorithm ◽  
Stream Function ◽  
Parallel Implementation ◽  
Flow Region ◽  
Meshfree Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Ideal Incompressible Fluid ◽  
Time Step ◽  
Initial Boundary Value

Предложен параллельный алгоритм для расчета двумерной динамики невязкой несжимаемой жидкости на вращающейся сфере. Основой алгоритма является бессеточный метод вихрей в ячейках для решения начально-краевой задачи для нестационарных уравнений движения идеальной жидкости в терминах абсолютной завихренности и функции тока. Метод базируется на аппроксимации функции тока отрезком ряда Фурье, приближении поля завихренности ее значениями в частицах и расчете траекторий частиц с использованием псевдосимплектического интегратора. Схема распараллеливания на каждом временном шаге включает в себя расщепление по подмножествам частиц и декомпозицию области течения. Представлено описание алгоритма для вычислительных систем с общей памятью. Эффективность метода и производительность параллельного алгоритма оценены экспериментально при различных параметрах расчета, показана хорошая масштабируемость алгоритма. A parallel algorithm for calculating the two-dimensional dynamics of inviscid incompressible fluids on a rotating sphere is proposed. The algorithm is based on the meshfree vortex-in-cell method for solving an initial boundary value problem for unsteady equations describing the motion of an ideal fluid in terms of the absolute vorticity and stream function. The method is based on the approximation of the stream function using the Fourier series. The vorticity field is defined by its values on a set of particles. The particle trajectories are calculated using a pseudo-symplectic integrator. At each time step, the parallelization involves the splitting into subsets of particles and the decomposition of the flow region. The efficiency of the parallel algorithm and its performance are evaluated experimentally for various parameters of the method. The numerical results show a good scalability of the algorithm.

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