Spectral/hp Least-Squares Finite Element Analysis of Isothermal and Non-Isothermal Flows of Generalized Newtonian Fluids

Abstract

Mixed least-squares finite element models with spectral/hp approximations were developed for steady, incompressible, isothermal and non-isothermal flows of generalized-Newtonian fluids obeying the Carreau–Yasuda viscosity model. The finite element model for isothermal flows consists of velocity, pressure, and stress fields as independent variables, and the model for nonisothermal flows consists of temperature and heat flux in addition to the three fields. (hence, called a mixed model). Least-squares models offer an alternative variational setting to the conventional weak-form Galerkin models for the Navier–Stokes equations, and no compatibility conditions on the approximation spaces are necessary when the polynomial order (p) used is sufficiently high (say, p > 3, as determined numerically). Also, the use of the high-order spectral/hp elements in conjunction with a least-squares formulation alleviates various forms of locking which often appear in low-order least-squares finite element models for incompressible viscous fluids, and accurate results can be obtained with exponential convergence. To verify and validate the present model, various benchmark problems of two- and three-dimensional flows are solved. In addition, the effect of different parameters of the Carreau–Yasuda constitutive model on the flow characteristics are studied parametrically.