Monolithic weighted least-squares finite element method for non-Newtonian fluids with non-isothermal effects
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Date
2020
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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.
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Keywords
Least-squares, Finite element method, Non-Newtonian, Non-isothermal