M. Thompson

Existence Theory for Radiative Flows

We consider the coupling of radiative heat transfer equations and the energy equation for the temperature Tof a compressible fluid occupying a bounded convex region D with smooth boundary. Using the technique of upper and lower sequences associated with integro-parabolic equations, we establish the existence and uniqueness of a solution T, 0 ≤ Λ− ≤ T(x,t) ≤ Λ+ < ∞ with corresponding radiative intensity I(x,Ω,ν,t) where the total incident radiation satifies ∫S2 I(x,Ω,ν,t)dΩ = Sg B(ν,t)+Sb B(ν,Tb), and where Sb and Sg are positivity preserving linear operator, Tb is the external temperature of the boundary, and B is Plancks function. We also establish certain energy estimates for T.

Qualitative analysis of the SN approximations of the transport equation and combined conduction-radiation heat transfer problem in a slab

Abstract A review is presented on the recent work carried out by our research group on mathematical aspects of the existence and uniqueness of solutions of the one-dimensional steady-state transport equation as well as an analytical study of the nonlinear radiative transfer equation in a slab. Dependence on control parameters about the solutions of the one-dimensional S N , approximations of the transport equation is also addressed.

Solution of the one-dimensional transport equation by the vector Green function method: Error bounds and simulation

In this work we solve the general anisotropic transport equation for an arbitrary source with semi-reflexive boundary conditions. First we present a complete existence theory for this problem in the space of continuous functions and in the space of @a-Holder continuous functions. As a result of our analysis we construct integral operators which we discretize in a finite dimensional functional space, yielding a new robust numerical method for the transport equation, which we call Greens function decomposition method (GFD). As well, we demonstrate a convergence theorem providing error bounds for the reported method. Finally we provide numerical results and applications.

Weak solutions for the electrophoretic motion of charged particles

We introduce a weak formulation for a system of electrostatic and hydrodynamic equations modelling the electrophoretic motion of charged particles in ionized fluids. We obtain a local in time existence theorem, using the results established in (11) and properties of the solutions of the Poisson-Boltzmann equation. These properties follows from singular integral operators techniques.

Eigenvalues of the anisotropic transport equation in a slab

The critical eigenvalues of the transport equation play an important role in the description of the dynamics of transport problems both in nuclear physics as well as in radiative transport theory. This article treats the problem of calculating numerically the critical spectrum of the transport equation with semireflecting boundary conditions. The eigenvalue problem is solved using spectral methods and numerical results are presented. The scattering kernel is considered to be one of three types, namely, isotropic, linearly anisotropic, or Rayleigh scattering, even although more general kernels could be considered.

Existence Theory for the Solution of a Stationary Nonlinear Conductive‐Radiative Heat‐Transfer Problem in Three Space Dimensions

Abstract In this work we show that a stationary nonlinear coupled radiative‐conductive heat‐transfer problem in a convex bounded region with piecewise differentiable boundary in three dimensions under fairly general boundary conditions has a unique solution in a segment of a positive cone in a certain function space.

Green’s Function Decomposition Method for Transport Equation

The Green’s Function Decomposition Method is a methodology to solve the trans-port equation in a slab with specular reflexion at the boundaries. Nomerical solutions face in general at least three difficulties: (1) the domain is not finite; (2) the scattering kernel is not a nonnegative function and may assume large values; (3) the reflection coefficient may not vary smoothly with the angular variable. The first difficult is overcome by truncating the domain into a finite interval taking into account some analitical estimates. The second difficulty means that well-known iterative methods will not converge easily outside the spectral radius. The third difficulty implies a large number of ordinates in case of angular discretization. The present method makes use of the Green’s Function Decomposition Method (GFD) with the following features: (1) It is not iterative. (2) It does not involve any discretization of the angular variable. In this work we present the GFD method to solve numerically the transport equation in a slab with anisotropic scattering kernel and specular reflection at the boundary. We present the original problem and solve it by reformulation as an integral operator equation. Finally, the integral operators involved are discretized yielding a finite approximation of the problem which can be solved numerically. We present numerical results for a broad range of applications.

MOTION OF A CHARGED PARTICLE IN IONIZED FLUIDS

We discuss the motion of a charged macromolecule in an ionized fluid in a finite region governed by the interaction of the electrical field described by the Poisson–Boltzmann equation and Stokes equation. Both the surface of the macromolecule and the external surface of the container are supposed to be sufficiently regular. The case of a fluid occupying an infinite region is also discussed assuming an approximation hypothesis on the electrical field.

Error estimates and existence of solutions for an atmospheric model of Lorenz on periodic domains

Abstract This paper deals with equations governing an atmospheric model of Lorenz derived from the incompressible f -plane shallow-water equations under a time independent mass forcing. The horizontal and vertical motions of the fluid are damped diffusively by coefficients ν and κ , where κ is taken to be a turbulent viscosity. For small initial data, we prove the existence of weak and strong solutions of such a problem. Uniqueness is proved only for strong solutions. Furthermore, we prove that the solutions to the system of equations of the approximating Galerkin problem converge to that of the original systems and error estimates are established.

Network formation as a cluster-cluster diffusion-limited aggregation process, 2 Modelling the polyethylene crosslinking process using Monte Carlo and graph techniques

A Monte Carlo simulation of the network formation of polyethylene radicals has been carried out using the reaction modelling scheme and a graph exploration algorithm based on the breadth-first search technique. The results are obtained in a three dimensional cubic continuum space simulation with periodic boundary conditions. Results for three different polyethylene concentrations are reported. The structural evolution of the studied system was followed in terms of: number of reactions, molecular mass, aggregate dimension and fractal dimension analysis.