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SUMMARYA high resolution scheme with improved iterative convergence properties was devised by incorporating total-variation diminishing constraints, appropriate for unsteady problems, into an implicit time-marching method used for steady ow problems. The new scheme, referred to as Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA), has similar accuracy to the well-known SMART scheme, both being formally third-order accurate on uniform meshes for smooth ows. Three demonstration problems are considered: (1) advection of three scalar proÿles, a step, a sine-squared, and a semi-ellipse; (2) Newtonian ow over a backward-facing step; and (3) viscoelastic ow through a planar contraction and around a cylinder. For the case of the viscoelastic ows, in which the high resolution schemes are also used to represent the advective terms in the constitutive equation, it is shown that only the new scheme is able to provide a converged solution to the prescribed tolerance.

Using a numerical technique we demonstrate that the flow of the simplest differential viscoelastic fluid model (i.e., the upper-convected Maxwell model) goes through a bifurcation to a steady asymmetric state when flowing in a perfectly symmetric ''cross-slot'' geometry. We show that this asymmetry is purely elastic in nature and that the effect of inertia is a stabilizing one. Our results are in qualitative agreement with very recent experimental visualizations of a similar flow in the microfluidic apparatus of Arratia et al.

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