{'fraction_non_alphanumeric': 0.021176470588235293, 'fraction_numerical': 0.0, 'mean_word_length': 5.096774193548387, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Numerical study on the bulk instability of constant-current conduction in cation-exchange membranes.

Numerical simulation of two-dimensional bulk instability of the one-dimensional conduction state in an electrolyte confined between a pair of cation-exchange membranes subjected to an external voltage is conducted under an assumption of constant current. By employing variable grids, we resolve the problem of sensitive dependence of the numerical solutions to the grid refinement in particular at a current close to the limiting value. In fact, the full range of parameters, i.e., Schmidt number, Peclet number, diffusivity ratio, and current, is considered in this study in obtaining the stability chart. It turns out that the Schmidt number exerts almost no influence on the results. From the neutral curves in the chart of the parameter space (current versus diffusivity ratio, i.e., cation diffusivity divided by anion diffusivity) we confirm that the system tends to be unstable at high Peclet numbers, high currents, and low diffusivity ratios. As the diffusivity ratio is increased the instability mode switches from the monotonic to the oscillatory type, and the critical diffusivity ratio for the switching is found to be decreased as the Peclet number decreases. At the switching, no jump in the neutral curves is found, contrary to the earlier result, because the wave number is set free to change in this study. The stability chart obtained in this study represents the true boundaries in discriminating the stable from the unstable parameter sets because the critical eigenvalues constituting the chart are sought for the entire possible range of the wave numbers and the frequencies.