Microfluidic devices manufactured from soft polymeric materials have emerged as a paradigm for cheap, disposable and easy-to-prototype fluidic platforms for integrating chemical and biological assays and analyses. The interplay between fluid flow forces and the tiny (the size of a human hair) compliant (elastic) pipes and channels inside a microfluidic devices requires careful consideration. Although microfluidics is leading to disruptive change across a number of industries, fundamental questions remain unanswered. As renowned Harvard chemist George Whitesides noted over a decade ago, microfluidics is "in its infancy ... and extending those manipulations to small volumes, with precise dynamic control over concentrations, while discovering and exploiting new phenomena occurring in fluids at the microscale level, must, ultimately, be very important."

This NSF award enabled the discovery of so-called "flow rate--pressure drop relations", without fitting parameters and capable of quantitatively rationalizing experiments, for a variety of deformable pipe and channel types that arise in micro and biofluid mechanics applications. The relationship between the flow rate (volume of fluid passing through a pipe's cross-section per unit time) and the pressure drop (the force per unit cross-sectional area required to drive and maintain this flow) is the key fundamental relationship required to design microscopic piping systems inside microfluidic devices. This new research area catalyzed by this NSF award, at the intersection of low-Reynolds-number hydrodynamics and soft matter physics, has been termed soft hydraulics by PI Christov in his invited Topical Review in Journal of Physics: Condensed Matter 34 (2022) 063001. Much like Poiseuille's eponymous law is the building block of the classical field of hydraulics, and taught to undergraduate science and engineering for centuries, the generalizations of Poiseuille's law discovered in the course of the research supported by this NSF award will enable rapid and accurate design of soft hydraulic systems for novel lab-on-a-chip technologies (e.g., low-cost and disposable medical diagnostics) suitable for global public health challenges.

PI Christov and his team have been highly productive in the course of the research supported by this NSF award. Specifically, 18 peer-reviewed journal publications resulted from this work, disseminating the results from this NSF award in the top scientific journal in fluid mechanics, including the Journal of Fluid Mechanics, the Journal of Non-Newtonian Fluid Mechanics, Physical Review Fluids, and the Proceedings of the Royal Society A, amongst others. The PI and the graduate students participating in this research have also presented their research to international scientific audiences at the 70th through 75th Annual Meetings of the American Physical Society's Division of Fluid Dynamics, amongst other conferences.

From the journal publications resulting from the research supported by this NSF award, one is worth highlighting for its impact in the field and popular interest. In an "Editor's Choice" article published in the Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM)^* 101 (2021) e201900309 and highlighted by Purdue News, PI Christov and PhD Student Vishal Anand rewrote the book on the fluid mechanics of blood vessels. Specifically, using asymptotic methods, they reduced the coupled fluid and solid mechanics problems to a solvable equation, which establishes a new (soft hydraulic) law for long and slender elastic microtubes. This new law, based on rigorous theory, updated ad hoc expressions found in biomechanics textbooks.

* Founded by the renowned Harvard engineer Richard von Mises, ZAMM was the journal that published Ludwig Prandtl's revolutionary boundary layer theory in the early 1900s.

PI Christov's activities under this NSF award have led to an impact on the development of human resources. Specifically, two PhD students (one female) were supported by the NSF award and completed their PhD dissertations on research related to microscale fluid--structure interactions. Three postdocs (one female) had the opportunity to collaborate with the PI and graduate students on this research. Two of these postdocs have moved on to faculty positions (one tenure-track). Three Master's students were able to interact with the PhD student and postdocs working on this research, completing theses related to microscale fluid--structure interactions. Amongst several undergraduate researchers who were able to interact with PI Christov and his team on this research, two of them became published authors and both continued onto PhD programs in engineering.

PI Christov's activities under this NSF award have also led to an impact teaching and educational experiences. For example, PI Christov incorporated simplified exercises and challenging homework problems on flows through deformable conduits in the lubrication theory portion of his graduate course on fluid mechanics at Purdue. Further, in 2019, PI Christov was invited to the Indian Institute of Technology Kharagpur to deliver a series of lectures on flows through deformable confinements, which led to the creation of a set of lecture notes on this novel topic. Finally, the PI and his students developed several Python-based Jupyter notebooks suitable for interactive demonstrations and for teaching of concepts related to microscale fluid--structure interactions.