Asymptotics of a small liquid drop on a cone and plate rheometer

V. Cregan, Stephen B.G. O'Brien, Sean McKee

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

A cone and a plate rheometer is a laboratory apparatus used to measure the viscosity and other related parameters of a non-Newtonian liquid subject to an applied force. A small drop, of order millimetres, of the liquid is located between the horizontal plate and the shallow cone of the rheometer. Rotation of the cone ensues, the liquid begins to flow and the plate starts to rotate. Liquid parameters are inferred based on the difference in the applied rotational force and the resulting rotational force of the plate. To describe the flow of the drop, the initial drop configuration, before rotation commences, must be determined. The equilibrium drop profile is given by the solution to the well-known nonlinear Young-Laplace equation. We formulate asymptotic solutions for the drop profile based on the small Bond number. The modelling of the drop exhibits a rich asymptotic structure consisting of five distinct scalings which are resolved via the method matched asymptotics.
LanguageEnglish
Title of host publicationProgress in Industrial Mathematics at ECMI 2010
EditorsMichael Günther , Andreas Bartel, Markus Brunk, Sebastian Schöps, Michael Striebel
Place of PublicationHeidelberg
Pages449-455
Number of pages7
Volume17
DOIs
Publication statusPublished - 2012

Publication series

NameMathematics in Industry: Progress in Industrial Mathematics at ECMI 2010
PublisherSpringer Verlag
Name
Volume17

Fingerprint

rheometers
cones
liquids
Bond number
Laplace equation
profiles
viscosity
scaling
configurations

Keywords

  • optimisation
  • partial differential equation
  • quantitative finance

Cite this

Cregan, V., O'Brien, S. B. G., & McKee, S. (2012). Asymptotics of a small liquid drop on a cone and plate rheometer. In M. Günther , A. Bartel, M. Brunk, S. Schöps, & M. Striebel (Eds.), Progress in Industrial Mathematics at ECMI 2010 (Vol. 17, pp. 449-455). (Mathematics in Industry: Progress in Industrial Mathematics at ECMI 2010). Heidelberg. https://doi.org/10.1007/978-3-642-25100-9_51
Cregan, V. ; O'Brien, Stephen B.G. ; McKee, Sean. / Asymptotics of a small liquid drop on a cone and plate rheometer. Progress in Industrial Mathematics at ECMI 2010. editor / Michael Günther ; Andreas Bartel ; Markus Brunk ; Sebastian Schöps ; Michael Striebel. Vol. 17 Heidelberg, 2012. pp. 449-455 (Mathematics in Industry: Progress in Industrial Mathematics at ECMI 2010).
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Cregan, V, O'Brien, SBG & McKee, S 2012, Asymptotics of a small liquid drop on a cone and plate rheometer. in M Günther , A Bartel, M Brunk, S Schöps & M Striebel (eds), Progress in Industrial Mathematics at ECMI 2010. vol. 17, Mathematics in Industry: Progress in Industrial Mathematics at ECMI 2010, Heidelberg, pp. 449-455. https://doi.org/10.1007/978-3-642-25100-9_51

Asymptotics of a small liquid drop on a cone and plate rheometer. / Cregan, V.; O'Brien, Stephen B.G.; McKee, Sean.

Progress in Industrial Mathematics at ECMI 2010. ed. / Michael Günther ; Andreas Bartel; Markus Brunk; Sebastian Schöps; Michael Striebel. Vol. 17 Heidelberg, 2012. p. 449-455 (Mathematics in Industry: Progress in Industrial Mathematics at ECMI 2010).

Research output: Chapter in Book/Report/Conference proceedingChapter

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AB - A cone and a plate rheometer is a laboratory apparatus used to measure the viscosity and other related parameters of a non-Newtonian liquid subject to an applied force. A small drop, of order millimetres, of the liquid is located between the horizontal plate and the shallow cone of the rheometer. Rotation of the cone ensues, the liquid begins to flow and the plate starts to rotate. Liquid parameters are inferred based on the difference in the applied rotational force and the resulting rotational force of the plate. To describe the flow of the drop, the initial drop configuration, before rotation commences, must be determined. The equilibrium drop profile is given by the solution to the well-known nonlinear Young-Laplace equation. We formulate asymptotic solutions for the drop profile based on the small Bond number. The modelling of the drop exhibits a rich asymptotic structure consisting of five distinct scalings which are resolved via the method matched asymptotics.

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KW - partial differential equation

KW - quantitative finance

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Cregan V, O'Brien SBG, McKee S. Asymptotics of a small liquid drop on a cone and plate rheometer. In Günther M, Bartel A, Brunk M, Schöps S, Striebel M, editors, Progress in Industrial Mathematics at ECMI 2010. Vol. 17. Heidelberg. 2012. p. 449-455. (Mathematics in Industry: Progress in Industrial Mathematics at ECMI 2010). https://doi.org/10.1007/978-3-642-25100-9_51