Numerical simulations of heat explosion with convection in porous media

International audienceIn this article, we study the interaction between natural convection and heat explosion in porous media. The model consists of the heat equation with a nonlinear source term describing heat production due to an exothermic chemical reaction coupled with the Darcy law. Stationary and oscillating convection regimes and oscillating heat explosion are observed. The models with quasi-stationary and unstationary Darcy equation are compared

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d-dimensional problem in space into d+1-dimensional one in the space-time domain by combining the d-dimensional vector space variable and 1-dimensional time variable in one d+1-dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ-method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d-dimensional problem as a d+1-dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated

Mathematical modeling of thermal explosion with natural convection: a brief survey

Thermal (or heat) explosion occurs in a reacting medium if the heat production due to an exothermic chemical reaction exceeds the heat loss through the boundary. In the mathematical approximation heat explosion is characterized by an unbounded temperature growth. If the reaction occurs in a liquid or gaseous medium, then a nonuniform temperature distribution can lead to natural convection. The interaction of heat explosion with natural convection can result in various regimes with a bounded temperature distribution (stationary, periodic, chaotic) and to a transition to heat explosion. The latter can be accompanied by a monotonic temperature growth or by temperature oscillations (oscillating heat explosion). This paper presents a review on mathematical modelling of heat explosion with natural convection in a homogeneous fluid and in a porous medium

Dynamics of Convective Thermal Explosion in Porous Media

International audienceIn this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion

Heat Explosion In Porous Media Using Radial Basis Functions

The paper is devoted to the numerical investigation of the interaction between natural convection and heat explosion in a fluid-saturated porous media in a rectangular domain. The model consists of Darcy equations for an incompressible fluid in a porous medium coupled with the nonlinear heat equation. Numerical simulations are performed using the radial basis functions method (RBFs). We study the bifurcation of the periodic oscillation of the response born by Hopf bifurcation. First, a symmetry-breaking bifurcations observed; then is followed by successive period-doubling bifurcations leading to chaos

A Meshfree Method for Heat Explosion Problems with Natural Convection in Inclined Porous Media

This paper investigates the interaction between natural convection and heat explosion in porous media. A meshless collocation method based on multiquadric radial basis functions has been applied to study the problem in an inclined two-dimensional porous media. The governing equations consist of coupling the Darcy equations in the Boussinesq approximation of low density variations to the heat equation with a nonlinear chemical source term. The numerical results obtained are in good agreement with some previous studies that consider the vertical direction. A complex behaviour of solutions is observed, including periodic and aperiodic oscillations. We have shown that a small inclination of the container stabilizes the reactive fluid and can prevent thermal explosion

A Meshfree Method for Heat Explosion Problems with Natural Convection in Inclined Porous Media

This paper investigates the interaction between natural convection and heat explosion in porous media. A meshless collocation method based on multiquadric radial basis functions has been applied to study the problem in an inclined two-dimensional porous media. The governing equations consist of coupling the Darcy equations in the Boussinesq approximation of low density variations to the heat equation with a nonlinear chemical source term. The numerical results obtained are in good agreement with some previous studies that consider the vertical direction. A complex behaviour of solutions is observed, including periodic and aperiodic oscillations. We have shown that a small inclination of the container stabilizes the reactive fluid and can prevent thermal explosion

Heat Explosion In Porous Media Using Radial Basis Functions

The paper is devoted to the numerical investigation of the interaction between natural convection and heat explosion in a fluid-saturated porous media in a rectangular domain. The model consists of Darcy equations for an incompressible fluid in a porous medium coupled with the nonlinear heat equation. Numerical simulations are performed using the radial basis functions method (RBFs). We study the bifurcation of the periodic oscillation of the response born by Hopf bifurcation. First, a symmetry-breaking bifurcations observed; then is followed by successive period-doubling bifurcations leading to chaos

A stabilized meshless method for time-dependent convection-dominated flow problems

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