{'fraction_non_alphanumeric': 0.020202020202020204, 'fraction_numerical': 0.0, 'mean_word_length': 5.1415929203539825, '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}}
Exact solutions for steady bubbles in a Hele-Shaw cell with rectangular geometry

Exact solutions are presented for an arbitrary number of steadily moving bubbles in a Hele-Shaw channel when surface tension is neglected. According to the symmetry displayed by the bubbles, the solutions are classified into two groups: (i) solutions where the bubbles are symmetrical about the channel centreline and (ii) solutions in which the bubbles have fore-and-aft symmetry. The general solutions are expressed in integral form but in some special cases analytic solutions in terms of elliptic integrals are found. The possible relevance of these exact solutions to experiments is also briefly discussed.