Abstract:
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
[http://tmnt-lab.org] 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 derive this relation via perturbation methods. The Stokes equations for vanishing Reynolds number are coupled to the governing equations of an elastic rectangular plate or cylindrical shell. Specifically, the vessel’s deformation is captured using Donnell—Sanders shell theory or Kirchhoff—Love plate theory under the assumption of a thin, slender geometry. 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.