##

The inertio-elastic planar entry flow of low-viscosity elastic fluids in
micro-fabricated geometries

**
**
**Lucy E. Rodd**^{a}^{, }^{c}^{, }^{}^{, }^{}, Timothy P. Scott^{c}, David V.
Boger^{a}, Justin J. Cooper-White^{a}^{, }^{b} and
Gareth H. McKinley^{c}

^{a}Department of Chemical and Biomolecular Engineering,
The University of Melbourne, Vic. 3010, Australia

^{b}Division of Chemical Engineering, The University of
Queensland, Brisbane, Qld 4072, Australia

^{c}Hatsopoulos Microfluids Laboratory, Department of
Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA
02139, USA

Received 4 January 2005; revised 13 April 2005; accepted 13
April 2005. Available online 22 September 2005.

## Abstract

The non-Newtonian flow of dilute aqueous polyethylene oxide (PEO) solutions
through micro-fabricated planar abrupt contraction–expansions is investigated.
The small lengthscales and high deformation rates in the contraction throat lead
to significant extensional flow effects even with dilute polymer solutions
having time constants on the order of milliseconds. By considering the
definition of the elasticity number, *El* = *Wi*/*Re*, we show
that the lengthscale of the geometry is key to the generation of strong
viscoelastic effects, such that the same flow behaviour cannot be reproduced
using the equivalent macro-scale geometry using the same fluid. We observe
significant vortex growth upstream of the contraction plane, which is
accompanied by an increase of more than 200% in the dimensionless extra pressure
drop across the contraction. Streak photography and video-microscopy using
epifluorescent particles shows that the flow ultimately becomes unstable and
three-dimensional. The moderate Reynolds numbers (0.44 ≤ *Re* ≤ 64)
associated with these high Weissenberg number (0 ≤ *Wi* ≤ 548)
micro-fluidic flows results in the exploration of new regions of the
*Re*–*Wi* parameter space in which the effects of both elasticity and
inertia can be observed. Understanding such interactions will be increasingly
important in micro-fluidic applications involving complex fluids and can best be
interpreted in terms of the elasticity number, *El* = *Wi*/*Re*,
which is independent of the flow kinematics and depends only on the fluid
rheology and the characteristic size of the device.

**Keywords: **Contraction–expansion; Polyethylene oxide (PEO);
Elasticity number; Flow instability