RT Journal Article
T1 Start-up flow in shallow deformable microchannels
A1 Martínez Calvo, Alejandro
A1 Sevilla Santiago, Alejandro
A1 Peng, Gunnar G.
A1 Stone, Howard A.
AB Microfluidic systems are usually fabricated with soft materials that deform due tothe fluid stresses. Recent experimental and theoretical studies on the steady flowin shallow deformable microchannels have shown that the flow rate is a nonlinearfunction 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 andin flow-rate-controlled configurations. The characteristic scales and relevant parametersgoverning the transient flow are first identified, followed by the development of anunsteady lubrication theory assuming that the inertia of the fluid is negligible, andthat the upper wall can be modelled as an elastic plate under pure bending satisfyingthe Kirchhoff–Love equation. The model is governed by two non-geometricaldimensionless numbers: a compliance parameter β, which compares the characteristicdisplacement of the upper wall with the undeformed channel height, and a parameterγ that compares the inertia of the solid with its flexural rigidity. In the limit ofnegligible solid inertia, γ → 0, a quasi-steady model is developed, whereby the fluidpressure satisfies a nonlinear diffusion equation, with β as the only parameter, whichadmits a self-similar solution under pressure-controlled conditions. This simplifiedlubrication description is validated with coupled three-dimensional numericalsimulations of the Navier equations for the elastic solid and the Navier–Stokesequations for the fluid. The agreement is very good when the hypotheses behind themodel are satisfied. Unexpectedly, we find fair agreement even in cases where thesolid and liquid inertia cannot be neglected.
PB Cambridge University Press
SN 0022-1120
YR 2020
FD 2020-02-25
LK https://hdl.handle.net/10016/35721
UL https://hdl.handle.net/10016/35721
LA eng
NO The authors are grateful to J. Rivero-Rodríguez and B. Scheid for key numericaladvice, to I. C. Christov for pointing out a mistake in figure 2 of an earlier versionof the manuscript, and to R. Zaera for helpful discussions. A.M.-C. and A.S. thank theSpanish MINECO, Subdirección General de Gestión de Ayudas a la Investigación, forits support through projects DPI2014-59292-C3-1-P and DPI2015-71901-REDT, andthe Spanish MCIU-Agencia Estatal de Investigación through project DPI2017-88201-C3-3-R. These research projects have been partly financed through FEDER Europeanfunds. A.M.-C. also acknowledges support from the Spanish MECD through the grantFPU16/02562 and to its associated programme Ayudas a la Movilidad 2018 during hisstay at the Complex Fluids Group in Princeton. H.A.S. thanks the NSF for supportvia CMMI-1661672 and through Princeton University’s Material Research Science andEngineering Center DMR-1420541.
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RD 27 jul. 2024