Nonlinear flow rate–pressure drop relations in slender compliant microtubes

We investigate the steady-state fluid–structure interaction between a Newtonian fluid flow and a deformable microtube, in two new configurations inspired by microfluidics experiments. Specifically, 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 of the tube wall from a plane-strain configuration. We apply the theory to two novel geometries: (i) a cylindrical fluidic channel enclosed by an annulus of soft material with a rigid outer wall and (ii) a cylindrical fluidic channel extruded from a large rectangular slab of soft material. The predicted displacement fields, and the resulting closed-form flow rate--pressure drop relations, are each validated against 3D direct numerical simulations via SimVascular's two-way-coupled fluid–structure interaction solver, svFSI, showing good agreement. We also incorporate weak flow inertia into theory, further improving the agreement between theory and simulations for larger imposed flow rates.