The hydraulic resistance of conduits of arbitrary (but fixed) cross-section can be calculated from exact unidirectional flow solutions of the steady Stokes equations (e.g., as in Happel and Brenner's ``Low Reynolds number hydrodynamics''). Recently, however, experiments on internal flows in channels with soft boundaries have shown that wall deformation leads to a nonlinear relationship between the volumetric flow rate and the pressure drop. Thus, the ``soft'' hydraulic resistance of a deformable conduit is no longer constant, rather it depends in a nontrivial way on the pressure drop through the induced (slowly-varying) cross-sectional deformation. I will discuss a perturbative approach to solving such soft hydraulics problems. The Stokes equations are coupled to the equations of linear elasticity. In the distinguished limit of a long and slender geometry, the flow problem is reduced to lubrication theory and easily solved. The deformation of the elastic body can likewise be reduced to an independent two-dimensional problem in each flow-wise cross-section. The hydrodynamic pressure provides the load, tightly coupling the fluid and elastic problems. Closed-form solutions for the deformation (either from the ``full'' elasticity problem or through simplifications via plate theory) allow us to determine analytical expressions for the soft hydraulic resistance. I will highlight how special attention must be paid in determining leading-order balances for the elasticity problem, depending on the structure's thickness. Finally, I will show that the proposed theory compares favorably to microscale flow experiments, as well as to three-dimensional two-way coupled simulations of fluid--structure interactions. Finally, I will discuss applications of the proposed theory to soft materials characterization (measuring the Young's modulus) at the microscale, without the need for deformation measurements.
Ivan Christov received his Ph.D. in Engineering Sciences and Applied Mathematics from Northwestern University. Subsequently, he was awarded an NSF Mathematical Sciences Postdoctoral Research Fellowship and spent two years with the Complex Fluids Group at Princeton University, working on interfacial instabilities and fluid--structure interactions at low Reynolds number. Following that, he spent two and a half years as the Richard P. Feynman Distinguished Postdoctoral Fellow in Theory and Computing at the Center for Nonlinear Studies at Los Alamos National Laboratory, working on problems of granular materials and porous media flow related to geophysics and unconventional energy utilization. Previously, he has interned at the U.S. Naval Research Laboratory and the ExxonMobil Upstream Research Company. His research interests are primarily in the area of modeling and numerical simulation of transport phenomena with an emphasis on complex and nonlinear systems. He is now an Assistant Professor of Mechanical Engineering at Purdue University directing the Transport: Modeling, Numerics & Theory laboratory, where advanced mathematical concepts and experimental results are combined with mechanistic intuition towards the modeling of multiphysics flows in order to make progress on fundamental problems; specifically, regarding transport as a means of effecting mixing or for mitigating separation. His research is supported by the National Science Foundation and the American Chemical Society's Petroleum Research Fund.
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