Publication: Start-up flow in shallow deformable microchannels
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2020-02-25
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Cambridge University Press
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Abstract
Microfluidic systems are usually fabricated with soft materials that deform due to
the fluid stresses. Recent experimental and theoretical studies on the steady flow
in shallow deformable microchannels have shown that the flow rate is a nonlinear
function of the pressure drop due to the deformation of the upper soft wall. Here,
we extend the steady theory of Christov et al. (J. Fluid Mech., vol. 841, 2018, pp.
267–286) by considering the start-up flow from rest, both in pressure-controlled and
in flow-rate-controlled configurations. The characteristic scales and relevant parameters
governing the transient flow are first identified, followed by the development of an
unsteady lubrication theory assuming that the inertia of the fluid is negligible, and
that the upper wall can be modelled as an elastic plate under pure bending satisfying
the Kirchhoff–Love equation. The model is governed by two non-geometrical
dimensionless numbers: a compliance parameter β, which compares the characteristic
displacement of the upper wall with the undeformed channel height, and a parameter
γ that compares the inertia of the solid with its flexural rigidity. In the limit of
negligible solid inertia, γ → 0, a quasi-steady model is developed, whereby the fluid
pressure satisfies a nonlinear diffusion equation, with β as the only parameter, which
admits a self-similar solution under pressure-controlled conditions. This simplified
lubrication description is validated with coupled three-dimensional numerical
simulations of the Navier equations for the elastic solid and the Navier–Stokes
equations for the fluid. The agreement is very good when the hypotheses behind the
model are satisfied. Unexpectedly, we find fair agreement even in cases where the
solid and liquid inertia cannot be neglected.
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Fluid-structure interaction, Lubrication theory, Microfluidics
Bibliographic citation
Journal of Fluid Mechanics, (2020), v. 885, A25, pp.: 1-26.