  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 stabilized
finite element method. In particular algebraic approach of approximating the
subscales has been considered to arrive at stabilized variational formulation
of the coupled system. This system is strongly coupled since viscosity of the
fluid depends upon the concentration, whose transportation is modelled by
$VADR$ equation. Fully implicit schemes have been considered for time
discretisation. Further more elaborated derivations of both $apriori$ and
$aposteriori$ error estimates for stabilized finite element scheme have been
carried out. Credibility of the stabilized method is also established well
through various numerical experiments, presented before concluding.
