A rivulet of a power-law fluid with constant width draining down a slowly varying substrate

F.H.H. Al Mukahal, S.K. Wilson, B.R. Duffy

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

The flow of a slowly varying rivulet of a power-law fluid with prescribed constant semi-width (ie with pinned contact lines) but slowly varying contact angle down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder, is described. The solution for a perfectly wetting fluid (which can never have pinned contact lines) is obtained, and it is shown that, despite having the same local behaviour, the global behaviour of a rivulet of a non-perfectly wetting fluid is qualitatively different from that of a slowly varying rivulet with prescribed constant contact angle but slowly varying semi-width. Specifically, it is described how the contact lines of a sufficiently narrow rivulet can remain pinned as it drains all the way from the top to the bottom of the cylinder, but how the contact lines of a wider rivulet de-pin at a critical position on the lower half of the cylinder, and how thereafter it drains to the bottom of the cylinder with zero contact angle and slowly varying semi-width. How the shape of the rivulet and the velocity within it depend on the power-law index N is described in detail. In particular, it is shown that whereas neither the shape of the rivulet nor the velocity within it vary monotonically with N, its mass always decreases monotonically with N. Despite the limitations of the power-law model, the present results provide rare analytical insight into non-Newtonian rivulet flow, and, in particular, are a useful benchmark for the study of rivulet flow of more realistic non-Newtonian fluids.
LanguageEnglish
Pages30-39
Number of pages10
JournalJournal of Non-Newtonian Fluid Mechanics
Volume224
Early online date13 Aug 2015
DOIs
Publication statusPublished - Oct 2015

Fingerprint

Power-law Fluid
Contact Line
drainage
Contact Angle
Substrate
Contact angle
Fluids
fluids
Wetting
Substrates
Contacts (fluid mechanics)
wetting
Power Law
Zero angle
Non Newtonian flow
Non-Newtonian Flow
Fluid
Non-Newtonian Fluid
circular cylinders
Circular Cylinder

Keywords

  • rivulet
  • power-law fluid
  • pinned and de-pinned contact lines

Cite this

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title = "A rivulet of a power-law fluid with constant width draining down a slowly varying substrate",
abstract = "The flow of a slowly varying rivulet of a power-law fluid with prescribed constant semi-width (ie with pinned contact lines) but slowly varying contact angle down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder, is described. The solution for a perfectly wetting fluid (which can never have pinned contact lines) is obtained, and it is shown that, despite having the same local behaviour, the global behaviour of a rivulet of a non-perfectly wetting fluid is qualitatively different from that of a slowly varying rivulet with prescribed constant contact angle but slowly varying semi-width. Specifically, it is described how the contact lines of a sufficiently narrow rivulet can remain pinned as it drains all the way from the top to the bottom of the cylinder, but how the contact lines of a wider rivulet de-pin at a critical position on the lower half of the cylinder, and how thereafter it drains to the bottom of the cylinder with zero contact angle and slowly varying semi-width. How the shape of the rivulet and the velocity within it depend on the power-law index N is described in detail. In particular, it is shown that whereas neither the shape of the rivulet nor the velocity within it vary monotonically with N, its mass always decreases monotonically with N. Despite the limitations of the power-law model, the present results provide rare analytical insight into non-Newtonian rivulet flow, and, in particular, are a useful benchmark for the study of rivulet flow of more realistic non-Newtonian fluids.",
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A rivulet of a power-law fluid with constant width draining down a slowly varying substrate. / Al Mukahal, F.H.H.; Wilson, S.K.; Duffy, B.R.

In: Journal of Non-Newtonian Fluid Mechanics, Vol. 224, 10.2015, p. 30-39.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A rivulet of a power-law fluid with constant width draining down a slowly varying substrate

AU - Al Mukahal, F.H.H.

AU - Wilson, S.K.

AU - Duffy, B.R.

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N2 - The flow of a slowly varying rivulet of a power-law fluid with prescribed constant semi-width (ie with pinned contact lines) but slowly varying contact angle down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder, is described. The solution for a perfectly wetting fluid (which can never have pinned contact lines) is obtained, and it is shown that, despite having the same local behaviour, the global behaviour of a rivulet of a non-perfectly wetting fluid is qualitatively different from that of a slowly varying rivulet with prescribed constant contact angle but slowly varying semi-width. Specifically, it is described how the contact lines of a sufficiently narrow rivulet can remain pinned as it drains all the way from the top to the bottom of the cylinder, but how the contact lines of a wider rivulet de-pin at a critical position on the lower half of the cylinder, and how thereafter it drains to the bottom of the cylinder with zero contact angle and slowly varying semi-width. How the shape of the rivulet and the velocity within it depend on the power-law index N is described in detail. In particular, it is shown that whereas neither the shape of the rivulet nor the velocity within it vary monotonically with N, its mass always decreases monotonically with N. Despite the limitations of the power-law model, the present results provide rare analytical insight into non-Newtonian rivulet flow, and, in particular, are a useful benchmark for the study of rivulet flow of more realistic non-Newtonian fluids.

AB - The flow of a slowly varying rivulet of a power-law fluid with prescribed constant semi-width (ie with pinned contact lines) but slowly varying contact angle down a slowly varying substrate, specifically the flow in the azimuthal direction around the outside of a large horizontal circular cylinder, is described. The solution for a perfectly wetting fluid (which can never have pinned contact lines) is obtained, and it is shown that, despite having the same local behaviour, the global behaviour of a rivulet of a non-perfectly wetting fluid is qualitatively different from that of a slowly varying rivulet with prescribed constant contact angle but slowly varying semi-width. Specifically, it is described how the contact lines of a sufficiently narrow rivulet can remain pinned as it drains all the way from the top to the bottom of the cylinder, but how the contact lines of a wider rivulet de-pin at a critical position on the lower half of the cylinder, and how thereafter it drains to the bottom of the cylinder with zero contact angle and slowly varying semi-width. How the shape of the rivulet and the velocity within it depend on the power-law index N is described in detail. In particular, it is shown that whereas neither the shape of the rivulet nor the velocity within it vary monotonically with N, its mass always decreases monotonically with N. Despite the limitations of the power-law model, the present results provide rare analytical insight into non-Newtonian rivulet flow, and, in particular, are a useful benchmark for the study of rivulet flow of more realistic non-Newtonian fluids.

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T2 - Journal of Non-Newtonian Fluid Mechanics

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SN - 0377-0257

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