Flow rate--pressure drop relations for new configurations of slender compliant tubes arising in microfluidics experiments

TL;DR

This paper develops analytical models for flow rate-pressure drop relations in novel microfluidic compliant tube configurations, validated by numerical simulations, enhancing understanding of fluid-structure interactions in microfluidics.

Contribution

It introduces new theoretical models for two innovative microfluidic geometries involving compliant tubes, validated through numerical simulations, advancing microfluidic flow analysis.

Findings

Analytical expressions accurately predict flow-pressure relations.

Good agreement between theory and 3D numerical simulations.

Weak flow inertia effects can be incorporated for larger flow rates.

Abstract

We investigate the steady-state fluid--structure interaction between a Newtonian fluid flow and a deformable microtube in two novel geometric configurations arising in recent microfluidics experiments. The first configuration is a cylindrical fluidic channel surrounded by an annulus of soft material with a rigid outer wall, while the second one is a cylindrical fluidic channel extruded from a soft rectangular slab of material. In each configuration, we derive a mathematical theory for the nonlinear flow rate--pressure drop relation by coupling lubrication theory for the flow with linear elasticity for the inner tube wall's deformation. Using the flow conduit's axial slenderness and its axisymmetry, we obtain an analytical expression for the radial displacement in each configuration from a plane-strain configuration. The predicted displacement field, and the resulting closed-form flow rate--pressure drop relation, are each validated against three-dimensional direct numerical simulations via SimVascular's two-way-coupled fluid--structure interaction solver, svFSI, showing good agreement. We also show that weak flow inertia can be easily incorporated in the derivation, further improving the agreement between theory and simulations for larger imposed flow rates.