  Non-monotonic velocity profiles are an inherent feature of mixing flows
obeying non-slip boundary conditions. There are, however, few known models of
laminar mixing which incorporate this feature and have proven mixing
properties. Here we present such a model, alternating between two non-monotonic
shear flows which act in orthogonal (i.e. perpendicular) directions. Each shear
is defined by an independent variable, giving a two-dimensional parameter space
within which we prove the mixing property over open subsets. Within these
mixing windows, we use results from the billiards literature to establish
exponential mixing rates. Outside of these windows, we find large parameter
regions where elliptic islands persist, leading to poor mixing. Finally, we
comment on the challenges of extending these mixing windows and the potential
for a non-exponential mixing rate at particular parameter values.
