A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate
Wilson, S.K. and Duffy, B.R. (2005) A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate. IMA Journal of Applied Mathematics, 70 (2). pp. 293322. ISSN 14643634 (https://doi.org/10.1093/imamat/hxh035)
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Abstract
We use the lubrication approximation to investigate the steady locally unidirectional gravitydriven draining of a thin rivulet of a perfectly wetting Newtonian fluid with prescribed volume flux down both a locally planar and a locally nonplanar slowly varying substrate inclined at an angle to the horizontal. We interpret our results as describing a slowly varying rivulet draining in the azimuthal direction some or all of the way from the top ( = 0) to the bottom ( = ) of a large horizontal circular cylinder with a nonuniform transverse profile. In particular, we show that the behaviour of a rivulet of perfectly wetting fluid is qualitatively different from that of a rivulet of a nonperfectly wetting fluid. In the case of a locally planar substrate we find that there are no rivulets possible in 0 /2 (i.e. there are no sessile rivulets or rivulets on a vertical substrate), but that there are infinitely many pendent rivulets running continuously from = /2 (where they become infinitely wide and vanishingly thin) to = (where they become infinitely deep with finite semiwidth). In the case of a locally nonplanar substrate with a powerlaw transverse profile with exponent p > 0 we find, rather unexpectedly, that the behaviour of the possible rivulets is qualitatively different in the cases p < 2, p = 2 and p > 2 as well as in the cases of locally concave and locally convex substrates. In the case of a locally concave substrate there is always a solution near the top of the cylinder representing a rivulet that becomes infinitely wide and deep, whereas in the case of a locally convex substrate there is always a solution near the bottom of the cylinder representing a rivulet that becomes infinitely deep with finite semiwidth. In both cases the extent of the rivulet around the cylinder and its qualitative behaviour depend on the value of p. In the special case p = 2 the solution represents a rivulet on a locally parabolic substrate that becomes infinitely wide and vanishingly thin in the limit /2. We also determine the behaviour of the solutions in the physically important limits of a weakly nonplanar substrate, a strongly concave substrate, a strongly convex substrate, a small volume flux, and a large volume flux.
ORCID iDs
Wilson, S.K. ORCID: https://orcid.org/0000000178419643 and Duffy, B.R. ORCID: https://orcid.org/0000000326877938;

Item type: Article ID code: 2211 Dates: DateEvent1 April 2005Published16 December 2004Published OnlineKeywords: lubrication approximation, perfectly wetting fluid, rivulet, Physics, Applied Mathematics Subjects: Science > Physics Department: Faculty of Science > Mathematics and Statistics Depositing user: Strathprints Administrator Date deposited: 07 Dec 2006 Last modified: 23 Nov 2023 14:23 URI: https://strathprints.strath.ac.uk/id/eprint/2211