P-Version least squares finite element formulation for axisymmetric incompressible non-Newtonian fluid flow

Abstract

The paper presents a p-version least squares finite element formulation (LSFEF) for axisymmetric incompressible non-Newtonian fluid flow. The dimensionless form of the equations describing the fluid motion are cast into a set of first order coupled partial differential equations involving non-Newtonian stresses as auxiliary variables. The pressure, velocities (primary variables) and the stresses (auxiliary variables) are interpolated using equal order, C 0 continuity, p-version hierarchical approximation functions. The least squares functional (or error functional) is constructed using the system of coupled first order non-linear partial differential equations (integrated sum of squares of the errors resulting from the individual equations for the entire discretization) without linearization, approximations or assumptions. The minimization of this least squares error functional results in finding a solution vector {δ} for which the partial derivatives of the error functional with respect to the nodal degrees of freedom {δ} becomes zero. This is accomplished by using Newton's method with a line search. The paper presents an implementation of the power law model for non-Newtonian viscosity. Numerical examples are presented to demonstrate the convergence characteristics and the accuracy of the proposed formulation.