In this paper, a multiple-relaxation-time color-gradient lattice Boltzmann model is used to simulate the deformation and breakup of a confined droplet in a simple shear flow with power-law rheology. This model is demonstrated to be accurate in simulating power-law fluids with a broad range of power-law indices. Through a preliminary study, we find that the non-Newtonian rheology of the matrix fluid has a more significant effect on the droplet deformation than that of the droplet. Compared to the Newtonian case, the non-Newtonian rheology is found to strengthen the droplet deformation in the shear-thickening matrix fluid but weaken the deformation in the shear-thinning matrix fluid. The extent of droplet deformation increases with increasing power-law index, and the rate of increase is more rapid near the smallest and largest indices considered than for intermediate values. We then systematically investigate the influence of the capillary number, geometrical confinement, and viscosity ratio on the deformation of a Newtonian droplet in power-law matrix fluids. The non-Newtonian effect on droplet deformation increases with increasing the capillary number or the wall confinement. In all the matrix fluids considered, the viscosity ratio noticeably affects the droplet deformation only when the capillary number is not less than 0.15, and the maximum deformation occurs at the viscosity ratio of unity for a constant capillary number. Finally, the critical capillary number, above which the droplet breakup occurs, is investigated for various confinement ratios in three different power-law matrix fluids. As the confinement ratio increases, the critical capillary number exhibits an overall increasing trend in the shear-thinning matrix fluid and an overall decreasing trend in the shear-thickening matrix fluid, both distinct from that in Newtonian case where the critical capillary number first decreases and then increases. The mode of droplet breakup is found to depend on the confinement ratio and the power-law index of the matrix fluid, and a high confinement ratio or a low power-law index favors the ternary breakup. The present study can provide useful suggestions and guidance for precise control of droplet behavior in microfluidic applications where non-Newtonian rheology is often encountered.

Modern wind turbine airfoil designs are increasingly emphasizing low sensitivity to the leading edge roughness in addition to good aerodynamic performances under varying wind conditions. In this study, a multi-point robust design optimization method has been systematically established for the wind turbine airfoil. The objective is to not only maximize the lift-to-drag ratio and lift coefficient at and near the design point, but also minimize their sensitivity to the leading edge roughness associated with the geometry profile uncertainty. The geometry parameterization of the airfoil is conducted by the Bezier curves. In the robust optimization, the multi-objective genetic algorithm, Monte Carlo simulation technique, and artificial neural network model are used. The results show that both the robust Pareto optimal (RPO) and deterministic Pareto optimal (DPO) airfoils have higher lift-to-drag ratio and lift coefficient than the original design FX 63-137 at and near the design point. A smaller maximum camber and larger radii near the leading edge help the RPO airfoil outperform both the DPO and original ones in terms of low sensitivity to the leading edge roughness. This study may be useful to the future development of the wind turbine airfoils with higher efficiency and reliability.