Sub - cell approximations for viscoelastic flows - filament stretching

The accuracy, stability and consistency of new stress interpolation schemes is investigated, based upon sub-cell approximations. This includes the contrast of two alternative hybrid spatial discretisations: a cell-vertex finite element/volume (fe/fv) scheme and a finite element equivalent (fe). Here, the interest is to explore the consequences of utilizing conventional methodology and to demonstrate resulting drawbacks in the presence of complex stress equation source terms. Alternative strategies worthy of consideration are presented for a constant shear viscosity model, that of Oldroyd-B, with strain-hardening and unbounded extensional properties. We demonstrate how high-order accuracy may be achieved by respecting consistency in our algorithmic constructions. Both feand fv-spatial discretisations are embedded within this methodology. Linear interpolation for stress, of either feor fv-form on triangular sub-cells, is referenced within parent triangular finite elements in two dimensions. Finite element discretization is employed for the momentum and continuity system, via a second-order pressurecorrection scheme. In this regard, a complex filament-stretching flow with a free-surface is selected to compute the transient evolution of kinematic and stress fields. Shortcomings of various up-winding schemes are discussed, whilst dealing with such free-surface type problems and appropriate strategies for dealing with these types of problems are outlined.