Soft hydraulics of non-Newtonian fluids flows through compliant conduits

The interplay between hydrodynamic forces and the confining boundaries of compliant conduits brings about rich physics involving fluid--elastic structure interactions. For example, microfluidic devices for lab-on-a-chip technologies are molded from soft polymeric materials. The mechanical compliance of channels in such devices is now recognized as an advantage to be exploited to enable, for example, new approaches to microrheological measurements and new modalities of micromixing. In this context, complex fluids exhibiting non-Newtonian rheological behavior arise. While hydraulics of Newtonian fluids in rigid conduits is now a textbook problem, a theory of soft hydraulics of non-Newtonian fluids remains elusive. In this talk, I will describe how to construct such a theory yielding reduced models that take into account shear-dependent viscosity, hydrodynamic pressure gradients during flow, and the elastic response (bulging and deformation) of the soft conduits due to flow within. 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, notable exceptions are the power-law and Ellis models for the shear-dependent viscosity. Then, I will show how a perturbative approach justifies replacing the channel height or tube radius in the pressure gradient--flow rate relation with an expression that accounts for the pressure-induced deformation of the conduit. The latter can be derived from the analysis of a suitable elasticity problem, for a given cross-sectional conduit geometry. I will conclude with examples of how the proposed theoretical approach reduces 3D coupled, multiphysics problems to a single ODE, which can often be integrated.