  This article presents stability and convergence analyses of subgrid
multiscale stabilized finite element formulation of non-Newtonian power-law
fluid flow model strongly coupled with variable coefficients
Advection-Diffusion-Reaction ($VADR$) equation. Considering the highly
non-linear viscosity coefficient as solute concentration dependent makes the
coupling two way. The stabilized formulation of the transient coupled system is
developed based upon time dependent subscales, which ensures inherent
consistency of the method. The proposed algebraic expressions of the
stabilization parameters appropriately shape up the apriori and aposteriori
error estimates. Both the shear thinning and shear thickening properties,
indicated by different power-law indices are properly highlighted in
theoretical derivations as well as in numerical validations. The numerical
experiments carried out for different combinations of small and large Reynolds
numbers and power-law indices establish far better performance of time
dependent $ASGS$ method in approximating the solution of this coupled system
for all the cases over the other well known stabilized finite element methods.
