{'fraction_non_alphanumeric': 0.029175050301810865, 'fraction_numerical': 0.01609657947686117, 'mean_word_length': 4.887573964497041, '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}}
The Thickness of a Marangoni-Driven Thin Liquid Film Emerging from a Meniscus

In this paper, we revisit experiments in which a thin liquid film forms at the tip of a capillary meniscus due to the presence of a thermally induced Marangoni shear stress. Starting from basic fluid mechanical equations, we devise a model for which numerical solutions show that it is able to accurately predict the thickness of the film, in contrast to earlier theories [Fanton, Cazabat, and Quere, it Langmuir, 12 (1996), pp. 5875--5880], where discrepancies with the experimental data of up to 20% were observed. We then find that only a few terms of our model yield a significant contribution to the film thickness. For the resulting reduced model, we derive an approximate, matched asymptotic solution, valid for small capillary numbers, which, to leading order, recovers the result of the older theory. Extending the calculation to the next order yields a correction that substantially improves the accuracy.