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REACTION−DIFFUSION PHENOMENA WITH RELAXATION

DOI: 10.1615/ICHMT.2012.CHT-12.200
pages 335-355

Juan I. Ramos
Escuela de Ingenierias Industriales, Universidad de Malaga, Dr. Ortiz Ramos, s/n 29071 Malaga, Spain

Abstract

Relaxation phenomena in two-dimensional nonlinear activator-inhibitor processes governed by reaction-diffusion equations are studied by means of a time-linearized implicit method based on the time-linearization and discretization of the time derivatives that result in linear elliptic equations at each time level. These two-dimensional elliptic equations are then factorized in terms of one-dimensional operators which are discretized by means of second-order accurate finite differences, and the second-order approximate factorization errors are neglected. The numerical method uses the dependent variables and its first-order derivative as unknowns, is self-starting and, for relaxation times equal to zero, reduces to the well-known second-order accurate, time-linearized Crank-Nicholson procedure; therefore, the method may be used to study conventional multi-dimensional reaction-diffusion equations as well as relaxation phenomena in reaction-diffusion characterized by hyperbolic equations. Applications of the method to activator-inhibitor problems governed by reaction-diffusion equations with relaxation and in the presence of a solenoidal velocity field that corresponds to a modified Lamb-Oseen vortex indicate that, for small vortex strengths, the spiral wave is almost unaffected by the flow except near its tip, whereas for strong vortices and core radii on the order of the drift of the spiral wave, substantial changes in the spiral wave morphology are observed depending on whether the vortex rotates in a clockwise or counter-clockwise direction. For counter-clockwise rotations, it is shown that spiral wave arm may exhibit a bulge in the activator concentration on the wave's inner side, while the inhibitor concentration may exhibit a bulge in the inner side and a valley in the outer side. For clockwise vortex motion, it is shown that the spiral wave may thicken on account of the opposing flow field and its tip may acquire a pointed feature. It is also shown that, for the conditions considered in this work, the spiral wave does not break up; neither does it form tiles and stripes as it has been found with other velocity fields.

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