From microchannels to blood vessels: A mathematical modeling perspective on flow in soft conduits
Ivan Christov, Purdue


The interaction between viscous fluid flows and elastic objects is common across many microscale phenomena. I will focus, specifically, on some recent results from and new research directions for my research group, the Transport: Modeling, Numerics & Theory laboratory at Purdue. The interaction between an internal flow and a soft boundary presents an example of a fluid--structure interaction (FSI). This particular type of FSI is relevant to problems from lab-on-a-chip microdevices for rapid diagnostics to blood pressure measurement cuffs. Experimentally, a microchannel or a blood vessel is found to deform into a non-uniform cross-section due to FSIs. Specifically, deformation leads to a non-linear relationship between the volumetric flow rate and the pressure drop (unlike Poiseuille’s law) at steady state. We have developed a perturbative approach to deriving these relations. Specifically, the Stokes equations for vanishing Reynolds number are coupled to the governing equations of an elastic rectangular plate or axisymmetric cylindrical shell. For example, the vessel’s deformation can be captured using Kirchhoff—Love plate theory or Donnell—Sanders shell theory under the assumption of a thin, slender geometry. For the case of shells, an elegant matched asymptotics problem (with a boundary layer and a corner layer) provides a closed-form expression for a deformed microtube's radius. Several mathematical predictions arise from this approach: the flow rate--pressure drop relation, the cross-sectional deformation profile of the soft conduit, and the scaling of the maximum displacement with the flow rate. To verify the mathematical predictions, we perform fully 3D, two-way coupled direct numerical simulations using the commercial software suite ANSYS. The numerical results are first benchmarked against experimental data in the literature. Then, the numerical results are compared against the mathematical predictions, showing excellent agreement. Some extensions to biophysiological situations (e.g., hyperelastic conduits and non-Newtonian fluids) and to unsteady flows (e.g., stop-flow lithography in compliant microchannels) will be discussed.