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

Research output: Contribution to journalArticle

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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.
LanguageEnglish
Pages293-322
Number of pages29
JournalIMA Journal of Applied Mathematics
Volume70
Issue number2
Early online date16 Dec 2004
DOIs
Publication statusPublished - 1 Apr 2005

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Wetting
Substrate
Fluid
Fluids
Substrates
Fluxes
Transverse
Horizontal
Lubrication Approximation
Qualitative Behavior
Newtonian Fluid
Circular Cylinder
Inclined
Circular cylinders
Lubrication
Gravity
Gravitation
Power Law
Vertical
Exponent

Keywords

  • lubrication approximation
  • perfectly wetting fluid
  • rivulet

Cite this

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

In: IMA Journal of Applied Mathematics, Vol. 70, No. 2, 01.04.2005, p. 293-322.

Research output: Contribution to journalArticle

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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.

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