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.

中文翻译：

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全耦合纳维-斯托克斯输运模型子网格多尺度稳定有限元方法的先验和后验误差估计

本文通过亚网格多尺度稳定有限元方法研究了瞬态$Navier$-$Stokes$($NS$)流体流动模型和变系数非定常对流-扩散-反应($VADR$)输运模型的全耦合系统。 . 特别是近似子尺度的代数方法已被认为可以达到耦合系统的稳定变分公式。该系统是强耦合的，因为流体的粘度取决于浓度，其传输由 $VADR$ 方程建模。全隐式方案已被考虑用于时间离散化。已经对稳定的有限元方案的 $apriori$ 和 $posteriori$ 误差估计进行了更详细的推导。