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In this paper a fully coupled system of transient $Navier$-$Stokes$ ($NS$)fluid flow model and variable coefficient unsteady Advection-Diffusion-Reaction($VADR$) transport model has been studied through subgrid multiscale stabilizedfinite element method. In particular algebraic approach of approximating thesubscales has been considered to arrive at stabilized variational formulationof the coupled system. This system is strongly coupled since viscosity of thefluid depends upon the concentration, whose transportation is modelled by$VADR$ equation. Fully implicit schemes have been considered for timediscretisation. Further more elaborated derivations of both $apriori$ and$aposteriori$ error estimates for stabilized finite element scheme have beencarried out. Credibility of the stabilized method is also established wellthrough various numerical experiments, presented before concluding.

Apriori and aposteriori error estimation of Subgrid multiscale stabilized finite element method for fully coupled Navier-Stokes Transport model