Soft hydraulics: Theory of flow in deformable microchannels

The hydraulic resistance of conduits of any cross-section can be calculated from exact unidirectional flow solutions of the steady Stokes equations. Recently, however, experiments on internal flows in channels with soft boundaries have showm that wall deformation leads to a nonlinear relationship between the volumetric flow rate and the pressure drop. Thus, the soft hydraulic resistance is not simply a constant dependent on the cross-sectional shape. We propose a perturbative approach to solving soft hydraulics problems. The Stokes equations are coupled to the equations of linear elasticity. For a long and slender geometry, the flow problem is reduced to lubrication theory. The deformation of the elastic wall is reduced to a two-dimensional problem in each flow-wise cross-section. Closed-form solutions for the deformation (either from the full elasticity problem or through simplifications via plate theory) allow us to predict the resistance of soft hydraulic elements. Our theory compares favorably to microscale flow experiments, as well as to three-dimensional two-way coupled direct numerical simulations. The effect of non-Newtonian fluid rheology will also be addressd.