A variational multiscale method for natural convection of nanofluids

•A stabilized method is presented for nonlinear coupled system of mixed-field and convection-dominated partial differential equations.•A systematic procedure is described to derive sub-grid scale (SGS) physics models where coupling at the SGS level plays a critical role in the modeling of anisotropy across scales in the thermal fluids.•Optimal spatial convergence rates on structured meshes with linear quadrilateral and triangular elements are achieved.•The formulation is variationally consistent and yields quadratic rate of convergence in nonlinear iterations of the Newton-Raphson method. The notion of enhanced thermal convection via particle laden fluids has been around for a long time. Technological challenges associated with the development of micro to nano particles with desired properties and their uniform dispersion in the base fluid have been a bottleneck. Relatively recently, the advent of modern manufacturing techniques from micro to nanoscales have rekindled interest in this class of fluids for innovative applications in advanced engineering systems. Buoyancy-induced convection and heat transfer involves conservation laws of mass, momentum, and energy. The mathematical model is comprised of two-way coupled system of mixed-field and convection-dominated partial differential equations. A stabilized method for nonlinearly coupled system is presented, and a systematic approach to develop the sub-grid scale (SGS) physics-based models is described. Explicit structure of the stabilization tensor is derived and it is shown to preserve nonlinear coupling in the SGS models that plays a critical role when nonlinear coupling of mechanical and thermal fields leads to anisotropy across the scales. The formulation is variationally consistent and results in optimal spatial convergence rates on structured meshes for linear triangles and bilinear quadrilaterals. Consistent linearization of the nonlinear system of equations yields quadratic rate of convergence of nonlinear iterations in the Newton-Raphson method. The method is tested on problems with increasing level of complexity to highlight the mathematical attributes of the method and its range of applicability. [Display omitted]