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A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate

Wilson, S.K. and Duffy, B.R. (2004) A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate. IMA Journal of Applied Mathematics, 70 (2). pp. 293-322. ISSN 0272-4960

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Abstract

We use the lubrication approximation to investigate the steady locally unidirectional gravity-driven draining of a thin rivulet of a perfectly wetting Newtonian fluid with prescribed volume flux down both a locally planar and a locally non-planar 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 non-uniform 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 non-perfectly 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 semi-width). In the case of a locally non-planar substrate with a power-law 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 semi-width. 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 non-planar substrate, a strongly concave substrate, a strongly convex substrate, a small volume flux, and a large volume flux.