Reduced models of unidirectional flows in compliant rectangular ducts at finite Reynolds number

Abstract

Soft hydraulics, which addresses the interaction between an internal flow and a compliant conduit, is a central problem in microfluidics. We analyze Newtonian fluid flow in a rectangular duct with a soft top wall at steady state. The resulting fluid--structure interaction (FSI) is formulated for both vanishing and finite flow inertia. At the leading-order in the small aspect ratio, the lubrication approximation implies that the pressure only varies in the streamwise direction. Meanwhile, the compliant wall's slenderness makes the fluid--solid interface behave like a Winkler foundation, with the displacement fully determined by the local pressure. Coupling flow and deformation and averaging across the cross-section leads to a one-dimensional reduced model. In the case of vanishing flow inertia, an effective deformed channel height is defined rigorously to eliminate the spanwise dependence of the deformation. It is shown that a previously-used averaged height concept is an acceptable approximation. From the one-dimensional model, a friction factor and the corresponding Poiseuille number are derived. Unlike the rigid duct case, the Poiseuille number for a compliant duct is not constant but varies in the streamwise direction. Compliance can increase the Poiseuille number by a factor of up to four. The model for finite flow inertia is obtained by assuming a parabolic vertical variation of the streamwise velocity. To satisfy the displacement constraints along the edges of the channel, weak tension is introduced in the streamwise direction to regularize the Winkler-foundation-like model. Matched asymptotic solutions of the regularized model are derived.

arXiv comment17 pages, 6 figures; accepted for publication in Physics of Fluids as a contribution to the special issue "Tribute to Frank M. White on his 88th Anniversary"l v2: corrects a minor typos

References in corpus (17)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Soft lubrication: the elastohydrodynamics of non-conforming and conforming contacts
• Soft lubrication
• Flow rate--pressure drop relation for deformable shallow microfluidic channels
• Fluid-Structure Interaction in Deformable Microchannels
• Three-dimensional flows in slowly-varying planar geometries
• Regimes of Soft Lubrication
• Validity of Winkler's mattress model for thin elastomeric layers: Beyond Poisson's ratio
• Theory of the flow-induced deformation of shallow compliant microchannels with thick walls
• Static response of deformable microchannels: A comparative modelling study
• Unsteady fluid--structure interactions in a soft-walled microchannel: A one-dimensional lubrication model for finite Reynolds number
• Elastohydrodynamics of a pre-stretched finite elastic sheet lubricated by a thin viscous film with application to microfluidic soft actuators
• Elastic deformations driven by non-uniform lubrication flows
• Influence of non-hydrodynamic forces on the elastic response of an ultra-thin soft coating under fluid-mediated dynamic loading
• Hydrodynamic Bulge Testing: Materials Characterization without Measuring Deformation
• A 1D model of liquid laminar flows with large Reynolds numbers in tapered microchannels
• Soft Hydraulics in Channels with Thick Walls: The Finite-Reynolds-Number Base State and Its Stability

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• Experimental investigation of the flow rate--pressure drop relation of a viscoelastic Boger fluid in a deformable channel
• Oscillatory flows in three-dimensional deformable microchannels