Motivated by the oil and gas industry, results of a numerical study into the displacement of one generalized Newtonian fluid by another along a plane channel are presented. Here, the fully 2-dimensional displacements that occur along an azimuthal slice of the annulus are considered. These can either be interpreted as displacements of immiscible or miscible fluids. In this talk we concentrate on immiscible fluids using the diffuse interface method. The focus of our study is on the potentially beneficial effects of a non-monotone variation in fluid rheologies as might be caused, for example, by chemical reaction or fluid incompatibility.