Variational multiscale stabilized finite element analysis of non-Newtonian Casson fluid flow model fully coupled with Transport equation with variable diffusion coefficients

• Published in: Computer methods in applied mechanics and engineering Vol. 388; p. 114272
• Main Authors: Rathish Kumar, B.V., Chowdhury, Manisha
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.01.2022; Elsevier BV
• Subjects: Advection–Diffusion–Reaction equation; Aposteriori error estimation; Apriori error estimation; Coefficients; Convergence; Diffusion; Exact solutions; Finite element method; Fluid dynamics; Fluid flow; Mathematical models; Non-Newtonian Casson fluid; Reaction-diffusion equations; Subgrid multiscale stabilized method; Transport equations; Viscosity; Apriori error estimation; Advection–Diffusion–Reaction equation; Subgrid multiscale stabilized method; Non-Newtonian Casson fluid; Aposteriori error estimation
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2021.114272

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. •A multiscale stabilized FE formulation for the Unsteady Non-Newtonian Casson Fluid Flow model coupled with the transient solute transport model is derived.•The viscosity coefficient is considered to be concentration dependent and the diffusion coefficients are spatially variable and time dependent.•The subscales associated with the stabilized multiscale FE scheme are taken to be both space and time dependent and the stabilized FE scheme is made free from subgrid spaces.•Optimal order apriori and aposteriori error estimates are derived for the fully coupled model.•The proposed stabilized method is validated for a benchmark problem.•The theoretical rates of converge of the scheme have been numerically verified.