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On the critical solutions in coating and rimming flow on a uniformly rotating horizontal cylinder

Wilson, S.K. and Hunt, R. and Duffy, B.R. (2002) On the critical solutions in coating and rimming flow on a uniformly rotating horizontal cylinder. Quarterly Journal of Mechanics and Applied Mathematics, 55 (3). pp. 357-383. ISSN 0033-5614

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Abstract

We use a combination of analytical and numerical techniques to re-examine the question posed by Moffatt [Journal de Mæ#169;canique 16 (1977) 651-673] of determining the critical weights of fluid that can be maintained per unit length in a steady, smoothly varying, two-dimensional film on either the outside ('coating flow') or the inside ('rimming flow') of a rotating horizontal cylinder. We use a pseudospectral method to obtain highly accurate numerical solutions for steady Stokes flow on a cylinder and hence to calculate the critical weights. In particular, these calculations reveal that the behaviour of the critical solutions in the thin-film limit 0 (where is the aspect ratio of the film) in an inner region near the horizontal on the ascending side of the cylinder (where Moffatt's leading-order outer solution has a corner) are not captured by naive outer asymptotic solutions in integer powers of . Motivated by these numerical results we obtain the uniformly valid critical asymptotic solutions in the thin-film limit to sufficient accuracy to enable us to calculate the critical fluxes and weights to accuracies o(4/3 (log )-3) and o(4/3 (log )-2) relative to Moffatt's leading-order values, respectively. We find that our asymptotic solutions for the critical weights are in good agreement with the numerically calculated results over a wide range of values of . In particular, our numerical and asymptotic calculations show that, even in the absence of surface-tension effects, the corner predicted by Moffatt's leading-order outer solution never actually occurs. In practice the higher-order terms obtained in the present work dominate the formally lower-order term that can be obtained straightforwardly without a detailed knowledge of the solution in the inner region, and so these higher-order terms must be included in order to obtain accurate corrections to Moffatt's leading-order value of the critical weight. In particular, in practice the critical weights in both coating and rimming flow always exceed Moffatt's value.