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A NEW APPROACH FOR THE DIFFUSION OF A SINGLE CLASS OF PARTICLES UNDER TWO DISTINCT ENERGY STATES

DOI: 10.1615/ICHMT.2017.CHT-7.1640
pages 1563-1580

Luiz Bevilacqua
A. L. Coimbra Institute, Graduate Course in Engineering, Federal University of Rio de Janeiro, Brazil

Maosheng Jiang
A. L. Coimbra Institute, Graduate Course in Engineering, Federal University of Rio de Janeiro, Brazil

Antonio J. Silva Neto
Department of Mechanical and Aerospace Engineering North Carolina State University Raleigh, NC 27695-7910, USA; Polytechnique Institute, State University of Rio de Janeiro, Brazil

Diego C. Knupp
Polytechnique Institute, State University of Rio de Janeiro, Brazil

Augusto C. N. Rodrigues Galeão
National Lboratory for Scientific Computation, Brazil

Resumo

The main aim of this paper is to introduce the behavior of a new class of diffusion phenomena in two dimensional domains. This new class considers two interdependent energy states. For the principal state the classical flux potential applies. For the secondary energy state a new flux potential was introduced which depends on the third derivative of the concentration with respect to the space variables. Consequently the particles are divided into two simultaneous fluxes. The secondary flux, derived from the new potential, is subsidiary to the principal flux in the sense that it exists if and only if the principal flux is activated. The new diffusion equation requires the introduction of two new parameters namely the flux distribution parameter, or flux partition β and a new physical coefficient R that we call reactivity coefficient. It was shown in previous papers that the bi-flux approach for one dimensional media introduces delays or acceleration in the scattering process. Under certain conditions it is possible that the primary and the secondary fluxes run in opposite directions. In these cases it is admissible to have in a given spatial domain, increasing density, rarefaction or stagnation depending on the inflow/outflow ratio. This flexibility to model the dynamics of motion allows for a better representation of the effect of external fields on the moving particles. It is considered here the existence of two main energy states induced by external fields splitting the flux into two main streams. This is particularly important for temperature sensitive particles scattering on substrata subjected to non-uniform temperature fields. It was clearly shown through the solution of the inverse problem that it is expected a relation of the form β = F(R). As a consequence of the introduction of the second potential the solutions of a large class of problems suggest that the concentration tends to grow in regions where R is large. Therefore secondary flux plays an important role on the concentration distribution. Some examples are presented to illustrate the peculiar evolution processes obtained with this theory. Population dynamics and capital flow may also be modeled with this theory with improved accuracy.

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