This paper presents a new subgrid multiscale stabilized formulation for non-Newtonian Casson fluid flow model tightly coupled with variable diffusion coefficients Advection–Diffusion–Reaction equation ($VADR$). Both the subscales are chosen to be time dependent and the stabilized formulation has been reached through the complete elimination of the unresolvable scales in terms of the coarse scale solution. Hence the resultant formulation emerges as a set of equations involving only the coarse scale solution instead of multiple scales and makes the formulation simpler to handle. In this study the shear-rate dependent Casson viscosity coefficient is assumed to be a function of the solute mass concentration, resulting in a two way coupling. This paper investigates the stability and the convergence properties of the stabilized finite element solution, without imposing any additional regularity condition other than the admissible space requirement on it. The proposed expressions of the stabilization parameters play a significant role in obtaining optimal order of convergences. During various theoretical derivations the estimations of the non-linear apparent viscosity coefficient have been carefully carried out assuming sufficiently regular exact solution. The performance of the scheme has been numerically validated with lid driven cavity problem. The theoretically established rate of convergence results are appropriately verified through numerical studies. In addition the transient Casson fluid flow behavior in a flow past square cylinder has been analyzed thoroughly.

本文提出了一种新的子网格多尺度稳定公式，用于与可变扩散系数紧密耦合的非牛顿 Casson 流体流动模型 对流-扩散-反应方程（$伏\mathrm{一种}D\mathrm{电阻}$）。两个子尺度都被选择为与时间相关，并且通过完全消除粗尺度解中无法解析的尺度达到了稳定的公式。因此，得到的公式是一组仅涉及粗尺度解而不是多尺度的方程，并使公式更易于处理。在这项研究中，剪切速率相关的卡森粘度系数被假定为溶质质量浓度的函数，导致双向耦合。本文研究了稳定有限元解的稳定性和收敛性，除了允许空间要求外，没有对其施加任何额外的规律性条件。稳定参数的建议表达式在获得最佳收敛顺序方面起着重要作用。在各种理论推导过程中，假设有足够规则的精确解，已经仔细地进行了非线性表观粘度系数的估计。该方案的性能已通过盖子驱动腔问题进行了数值验证。理论上建立的收敛速度结果通过数值研究得到了适当的验证。此外，已经彻底分析了流过方形圆柱体的瞬态 Casson 流体流动行为。该方案的性能已通过盖子驱动腔问题进行了数值验证。理论上建立的收敛速度结果通过数值研究得到了适当的验证。此外，已经彻底分析了流过方形圆柱体的瞬态 Casson 流体流动行为。该方案的性能已通过盖子驱动腔问题进行了数值验证。理论上建立的收敛速度结果通过数值研究得到了适当的验证。此外，已经彻底分析了流过方形圆柱体的瞬态 Casson 流体流动行为。