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We investigate the steady-state fluid--structure interaction between aNewtonian fluid flow and a deformable microtube in two novel geometricconfigurations arising in recent microfluidics experiments. The firstconfiguration is a cylindrical fluidic channel surrounded by an annulus of softmaterial with a rigid outer wall, while the second one is a cylindrical fluidicchannel extruded from a soft rectangular slab of material. In eachconfiguration, we derive a mathematical theory for the nonlinear flowrate--pressure drop relation by coupling lubrication theory for the flow withlinear elasticity for the inner tube wall's deformation. Using the flowconduit's axial slenderness and its axisymmetry, we obtain an analyticalexpression for the radial displacement in each configuration from aplane-strain configuration. The predicted displacement field, and the resultingclosed-form flow rate--pressure drop relation, are each validated againstthree-dimensional direct numerical simulations via SimVascular'stwo-way-coupled fluid--structure interaction solver, svFSI, showing goodagreement. We also show that weak flow inertia can be easily incorporated inthe derivation, further improving the agreement between theory and simulationsfor larger imposed flow rates.

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