3-D least-squares finite element analysis of flows of generalized Newtonian fluids

A mixed least-squares finite element model with spectral/hp approximations was developed to analyze three-dimensional, steady, and incompressible flows of generalized Newtonian fluids. The Carreau-Yasuda constitutive model was used for viscosity model. Velocity, pressure, and stress are the field variables of the finite element model (hence, called a mixed model). The least-squares formulation offers a variational setting for the Navier-Stokes equations; hence no compatibility of the approximation spaces used for the velocity, pressure, and stress fields is imposed if the polynomial order is sufficiently high. Also, using high-order spectral/hp elements in the least-squares formulation for the Navier-Stokes equations alleviates various forms of locking and accurate results can be obtained with exponential convergence. To verify the present formulation and computational module, the method of manufactured solutions and two different benchmark problems (namely, lid-driven cavity flow and backward-facing step flow) are used. Then the effect of different parameters of the Carreau-Yasuda model on the flow behavior is studied.