  We study the effect of the grafted polyelectrolyte layer on the flow
dynamics, and its consequences on underlying mixing in the rotating
microfluidic channel. For this analysis, the method used by Sadeghi et al. (J.
Fluid Mech., vol. 887, 2020, pp. A13; Phys. Rev. Fluids., vol. 4 (6), 2019,
063701-23), is modified by incorporating the non-linear effect stemming from
the polyelectrolyte layer induced electrostatics to solve the coupled system of
equations, integrated with the non-homogeneous boundary conditions. This method
is used to obtain the velocity distribution in the asymptotic limit of
geostrophic plug flow under the framework of variational calculus approach. We
analyze the mixing dynamics from the perspective of both qualitative assessment
and quantitative evaluation. For the qualitative estimation, we focus on the
Poincar\'e map analysis, while the entropy of mixing approach is used for the
mixing quantification. Results show that the grafted polyelectrolyte layer in
contact with the ionic solution leads to the development of an electrical
double layer, which upon interacting with the external electric field,
strengthens the electroosmotic pumping in the fluidic channel. Such
polyelectrolyte layer modulated strong electroosmotic pumping together with its
intrinsic feature of offering a frictional drag to the underlying transport
helps to modulate the primary as well as the secondary flows in the channel
under the influence of rotational forces. With an alteration in the
electroosmotic pumping and frictional drag force, tuneable through the
thickness of the grafted polyelectrolyte layer, we obtain different types of
secondary flow vortex configurations viz., a standard double-vortex,
dumbbell-shaped vortex and the transition state between the formers. A
significant change in the structure and strengths of these vortices modulates
the chaotic mixing in the present configuration
