Fronts pinned in the local widening of the 3D cylindrical domain.

Abstract

<p>The width of the simulation domain is 2. (A) An example stable stationary solution for <i>D</i> = 0.02 and <i>f</i>(<i>u</i>) = (1 − <i>u</i>)(<i>u</i> + 1)(<i>u</i> + <i>ϵ</i>), <i>ϵ</i> = <i>ϵ</i>_<i>max</i> = 0.264. Cross-section plane containing axis of symmetry is shown. The surface <i>f</i>(<i>u</i>) = 0 defines the position of the stationary front. (B-C) the curvature <i>κ</i> = 1/<i>R</i> of the stationary fronts calculated from numerical simulations for <i>f</i>(<i>u</i>) = (1 − <i>u</i>)(<i>u</i> + 1)(<i>u</i> + <i>ϵ</i>), in the domain with the diameter ratio 0.2 (B) and 0.4 (C) for five values of <i>D</i>: 0.00125, 0.005, 0.02, 0.08, 0.32 (colors from dark blue to red) versus the analytical result given by <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190372#pone.0190372.e018" target="_blank">Eq (8)</a> (black line overlapping with dark blue line) in the <i>D</i> → 0 limit. The front surface position is determined by its radius of curvature via <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0190372#pone.0190372.e012" target="_blank">Eq (3)</a>. Inserts in (B) and (C) show cross-section of the stationary front surfaces with the cylinder symmetry plane for <i>D</i> = 0.005 and six values of <i>ϵ</i>/<i>ϵ</i>_max: 0.1, 0.2, 0.4, 0.6, 0.8, 1; <i>ϵ</i>_max = 0.134 for diameter ratio 0.2 (B) and <i>ϵ</i>_max = 0.0832 for diameter ratio 0.4 (C).</p