A penalty finite element model for axisymmetric flows of power-law and white-metzner fluids

A finite element model based on the penalty function formulation of the equations governing unsteady axisymmetric flows of viscous incompressible fluids obeying power-law and White-Metzner constitutive relations is developed. The formulation accounts for inertial (or convective) terms. For power-law fluids, two different finite element models are developed: one based on the velocity formulation (i.e., only velocities as nodal variables) and the other based on a mixed formulation involving velocities and stress components. For the White-Metzner model, only the mixed model involving the velocities and extra stress components can be developed. The pressure variable does not enter the finite element model because of the application of the penalty method to introduce the incompressibility constraint. However, the pressure can be post-computed once the velocities are obtained.

The finite element models are used to analyze several plane and axisymmetric flows of power-law and White-Metzner fluids. The effect of boundary conditions, power-law model, inertia terms on the velocity profiles is investigated. The numerical solutions agree, qualitatively, with the known experimental and numerical results. The finite element models developed here can be easily modified to include thermal effects and other constitutive models.