### Abstract

Language | English |
---|---|

Pages | 293-322 |

Number of pages | 29 |

Journal | IMA Journal of Applied Mathematics |

Volume | 70 |

Issue number | 2 |

Early online date | 16 Dec 2004 |

DOIs | |

Publication status | Published - 1 Apr 2005 |

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### Keywords

- lubrication approximation
- perfectly wetting fluid
- rivulet

### Cite this

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**A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate.** / Wilson, S.K.; Duffy, B.R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate

AU - Wilson, S.K.

AU - Duffy, B.R.

PY - 2005/4/1

Y1 - 2005/4/1

N2 - 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.

AB - 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.

KW - lubrication approximation

KW - perfectly wetting fluid

KW - rivulet

U2 - 10.1093/imamat/hxh035

DO - 10.1093/imamat/hxh035

M3 - Article

VL - 70

SP - 293

EP - 322

JO - IMA Journal of Applied Mathematics

T2 - IMA Journal of Applied Mathematics

JF - IMA Journal of Applied Mathematics

SN - 0272-4960

IS - 2

ER -