Towards a theory of soft hydraulics of complex fluids flows through compliant conduits

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 the flow forces and the inherently compliant conduits of such devices requires careful consideration. Mechanical compliance of these devices has now become a paradigm, enabling new approaches to microrheological measurements, new modalities of micromixing, and improved sieving of micro- and nano-particles, to name a few applications. While the theory of soft hydraulics of Newtonian fluid flows through compliant conduits has matured, the case of complex fluids remains largely open due to a lack of both theoretical and experimental results. In this talk, I will describe how to construct tractable, reduced models of soft hydraulics of complex fluids, taking into account their shear-dependent viscosity, the hydrodynamic pressure gradients during flow, and the elastic response (bulging and deformation) of the soft conduits due to flow within, including the effect of cross-sectional conduit geometry on the resulting fluid--structure interaction. First, the relationship between volumetric flow rate and axial pressure gradient is needed. This relationship is challenging (or impossible) to obtain in closed analytical form for most non-Newtonian fluids with shear-dependent viscosity. Notable exceptions are the power-law and Ellis models for the effective viscosity. Then, the channel height or tube radius in the pressure gradient--flow rate relation is to be replaced with an expression, suitably derived from the theory of elasticity, that accounts for the pressure-induced deformation of the height/radius of the conduit. Then, the entire 3D coupled, multiphysics problem is reduced to a single ODE, which can often be turned into a quadrature.