On the thin-film asymptotics of surface-tension-driven microfluidics

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 non-linear 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.