Abstract A stabilized mixed finite element method is presented for a two-way coupled system of partial differential equations governing the transient thermal convection of nanofluids. The method employs a scale separation of the velocity and temperature fields into coarse and fine scales, facilitated by the Variational Multiscale (VMS) framework. This, combined with the consistent linearization of the fine-scale variational equations, yields a stabilized formulation with improved stability and accuracy. The explicit structure of the stabilization terms is derived through the direct application of a bubble function-based approach to the fine-scale variational equations. The formulation ensures full coupling between the mechanical and thermal phases and satisfies the inf-sup condition, enabling the use of equal-order Lagrange basis functions for all unknown fields. The resulting numerical method not only yields quadratic convergence of the nonlinear coupled equations, it also yields optimal convergence rates, both in space and time, for different element types and time marching schemes. The method is applied to transient thermal convection problems that include natural convection in irregular geometries and forced convection under the influence of electromagnetic force field.
Goraya et al. (Thu,) studied this question.
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