Monolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects

Abstract

We study the monolithic finite element method, based on the least-squares minimization principles for the solution of non-Newtonian fluids with non-isothermal effects. The least-squares functionals are balanced by the linear and nonlinear weighted functions and the residuals comprised of L2-norm only. The weighted functions are the function of viscosities and proved significant for optimal results. The lack of mass conservation is an important issue in LSFEM and is studied extensively for the diverse range of weighted functions. Therefore, we consider only inflow/outflow problems. We use the Krylov subspace linear solver, i.e. conjugate gradient method, with a multigrid method as a preconditioner. The SSOR-PCG is used as smoother for the multigrid method. The Gauss-Newton and fixed point methods are employed as nonlinear solvers. The LSFEM is investigated for two main types of fluids, i.e. Newtonian and non-Newtonian fluids. The stress-based first-order systems, named SVP formulations, are employed to investigate the Newtonian fluids. The different types of quadratic finite elements are used for the analysis purposes. The nonlinear Navier-Stokes problem is investigated for two mesh configurations for flow around cylinder problem. The coefficients of lift/drag, pressure difference, global mass conservation are analyzed. The comparison of linear and nonlinear solvers, based on iterations, is presented as well. The analysis of non-Newtonian fluids is divided into two parts, i.e. isothermal and non-isothermal. For the non-Newtonian fluids, we consider only Q2 finite elements for the discretization of unknown variables. The isothermal non-Newtonian fluids are investigated with SVP-based formulations. The power law and Cross law viscosity models are considered for investigations with different nonlinear weighted functions. We study the flow parameters for flow around cylinder problem along with the mass conservation for shear thinning and shear thickening fluids. To study the non-isothermal non-Newtonian fluids, we introduced a new first-order formulation which includes temperature and named it as SVPT formulation. The non-isothermal effects are obtained due to the additional viscous dissipation in the fluid flow and from the preheated source as well. The flow around cylinder problem is analyzed for a variety of flow parameters for Cross law fluids. It is shown that the MPCG solver generates very accurate results for the coupled and highly complex problems.