Flow of shear-thinning liquids in channels with superhydrophobic surfaces

We investigate the influence of shear-thinning on fully-developed flow in channels with a spanwise-periodic array of longitudinal superhydrophobic surfaces (SHSs) on each wall. The Carreau constitutive equation is used to model shear thinning, and the influence of shear thinning on the volume flow rate and slip length are examined. For weakly shear-thinning fluids where the maximum possible reduction in the viscosity is small, asymptotic analysis is used to derive integral expressions for the flow rate and slip length which can be evaluated using the Newtonian solution. As observed in previous related studies, there is a particular Carreau number where the slip length is maximum and the SHSs are most effective. Numerical simulations of the full momentum equation are used for strongly shear-thinning liquids, and Carreau model parameters are chosen to match measurements for whole blood and a xanthan gum solution. The slip length again reaches a maximum for a particular Carreau number, and the explanation for these peaks is similar for both weakly-and strongly shear thinning liquids. At small Carreau numbers, high values of shear are needed to reduce the viscosity and increase the flow rate. Such shear is generated at the SHS groove edges where there is an abrupt change in the boundary conditions. In the absence of SHSs, there is little shear thinning at low Carreau numbers since the shear rate is relatively small. When the Carreau number is large, there is less advantage to using SHSs as the shear generated at the solid wall of a conventional channel is sufficiently large for tangible viscosity reduction and flow enhancement. This difference for low and large Carreau numbers results in a peak slip length at an "intermediate"Carreau number (typically between 1 and 3). Reducing the solid fraction leads to large increases in the shear generated near the solid wall which, in turn, leads to substantial increases in the slip length. Modifying the deformation of the gas/liquid interface produces trends similar to those observed with Newtonian liquids. For the aspect ratios considered here, when the interface deforms into the SHS groove and the channel cross-sectional area increases, the flow rate and slip length both increase. The opposite occurs when the interface deforms away from the groove towards the liquid. The change in the slip length relative to the Newtonian slip length shows little sensitivity to the magnitude of the interface deformation indicating that the influence of shear thinning is not strongly affected by the interface deformation for the cases considered here.