A least‐squares finite element formulation to solve incompressible non‐Newtonian fluid flow

Abstract

AbstractThe Modeling of non‐Newtonian fluids is an essential issue for many industrial applications, for example in the field of chemistry, bioengineering and medical science. In this contribution we present a least‐squares (LS) finite element approach to model steady flow of incompressible non‐Newtonian fluids based on the Navier‐Stokes equations. The application of the least‐squares finite element method (LSFEM) especially in the case of fluid mechanics is motivated by some theoretical advantages compared to the well‐known (mixed) Galerkin method. The LSFEM is not restricted to the LBB‐condition and results in a minimization problem with symmetric positive definite equation systems also for differential equations with non self‐adjoint operators. Furthermore, the LSFEM provides an inherent a posteriori error estimator without additional costs, which enables an efficiency enhancement through adaptive mesh refinements.In this contribution the first‐order LS formulation in terms of stresses and velocities, as introduced in [4], is extended to consider the nonlinear dependence of the viscosity on the shear rate of the fluid. Therefore, the well‐known Carreau model for generalized Newtonian fluids, see for instance [1], is investigated. For the approximation of the independent variables, velocities and stresses, we apply conforming discretizations in H1 and H(div) using standard Lagrange interpolation polynomials and vector‐valued Raviart‐Thomas interpolations functions. A flow through a square domain with exact solution is considered to validate the proposed schemes with respect to accuracy and efficiency.