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

We use the lubrication approximation to investigate the steadylocally unidirectional gravity-driven draining of a thin rivuletof a perfectly wetting Newtonian fluid with prescribed volumeflux down both a locally planar and a locally non-planar slowlyvarying substrate inclined at an angle to the horizontal. Weinterpret our results as describing a slowly varying rivuletdraining in the azimuthal direction some or all of the way fromthe top ( = 0) to the bottom ( = ) of a large horizontal circularcylinder with a non-uniform transverse profile. In particular,we show that the behaviour of a rivulet of perfectly wettingfluid is qualitatively different from that of a rivulet of anon-perfectly wetting fluid. In the case of a locally planar substrate we find that thereare no rivulets possible in 0 /2 (i.e. there are no sessilerivulets or rivulets on a vertical substrate), but that thereare 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-lawtransverse profile with exponent p > 0 we find, rather unexpectedly,that the behaviour of the possible rivulets is qualitativelydifferent in the cases p < 2, p = 2 and p > 2 as wellas in the cases of locally concave and locally convex substrates.In the case of a locally concave substrate there is always asolution near the top of the cylinder representing a rivuletthat becomes infinitely wide and deep, whereas in the case ofa locally convex substrate there is always a solution near thebottom of the cylinder representing a rivulet that becomes infinitelydeep with finite semi-width. In both cases the extent of therivulet around the cylinder and its qualitative behaviour dependon the value of p. In the special case p = 2 the solution representsa rivulet on a locally parabolic substrate that becomes infinitelywide and vanishingly thin in the limit /2. We also determinethe behaviour of the solutions in the physically important limitsof a weakly non-planar substrate, a strongly concave substrate,a strongly convex substrate, a small volume flux, and a largevolume flux.