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On the thin-film asymptotics of surface tension driven microfluidics

Published online by Cambridge University Press:  19 August 2020

S. N. Calver Open the ORCID record for S. N. Calver [Opens in a new window] ,
E. A. Gaffney ,
E. J. Walsh ,
W. M. Durham  and
J. M. Oliver
S. N. Calver*
Affiliation:
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
E. A. Gaffney
Affiliation:
Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
E. J. Walsh
Affiliation:
Department of Engineering Science, Osney Thermo-Fluids Laboratory, University of Oxford, Osney Mead, OxfordOX2 0ES, UK
W. M. Durham
Affiliation:
Department of Zoology, University of Oxford, South Parks Road, OxfordOX1 3PS, UK Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, SheffieldS3 7RH, UK
J. M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OxfordOX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

Recent technological advances have led to a novel class of microfluidic devices which can be rapidly fabricated by printing a fluid onto a solid substrate with flows generated passively via surface tension. The nonlinear dependence between flow and the heights of the conduits, however, prevent straightforward calculation of the resulting dynamics. In this paper we use matched asymptotic expansions to predict how flow through these devices can be tuned by changing their geometry. We begin with the simple ‘dumbbell’ configuration in which two fluid drops with different sizes are connected by a long, thin and narrow conduit. We calculate the time scale required for one drop to drain into the other and how this depends both on the geometry of the pinned contact line and volume of fluid deposited into the drops. Our model therefore provides the mechanistic basis to design conduits with a particular fluid flux and/or shear stress, which are often key experimental constraints. Our asymptotic predictions are shown to be in excellent agreement with numerical simulations even for moderate aspect ratios (the ratio of conduit width to length). Next, we show how our results for the simple dumbbell configuration can be extended to predict the flow through networks of conduits with multiple drops and nodes, and hence may assist in their design and implementation. This new mathematical framework has the potential to increase the use of surface tension driven microfluidics across a wide range of disciplines as it allows alternate designs to be rapidly assessed.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A6
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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