Control of a purely elastic symmetry-breaking flow instability in cross-slot geometries

• Published in: Journal of fluid mechanics Vol. 881; pp. 1123 - 1157
• Main Authors: Davoodi, Mahdi, Domingues, Allysson F., Poole, Robert J.
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 25.12.2019
• Subjects: Aquatic reptiles; Asymmetry; Broken symmetry; Computational fluid dynamics; Computer simulation; Control stability; Cylinders; Deformation; Diameters; Elastic instability; Elasticity; Extensibility; Flow distribution; Flow rates; Flow stability; Flow velocity; Fluid flow; Fluid mechanics; Fluids; Geometry; Inertia; Instability; Mathematical models; Newtonian fluids; Non Newtonian fluids; Passive control; Rheometry; Simulation; Stability analysis; Stabilizing; Stagnation point; Streamlines; Symmetry; Theoretical analysis; Velocity; Viscoelasticity; Viscosity; Viscosity ratio
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2019.781

The cross-slot stagnation point flow is one of the benchmark problems in non-Newtonian fluid mechanics as it allows large strains to develop and can therefore be used for extensional rheometry measurements or, once instability arises, as a mixing device. In such a flow, beyond a critical value for which the ratio of elastic force to viscous force is high enough, elasticity can break symmetry even in the absence of significant inertial forces (i.e. creeping flow), which is an unwanted phenomenon if the device is to be used as a rheometer but beneficial from a mixing perspective. In this work, a passive control mechanism is introduced to the cross-slot by adding a cylinder at the geometric centre to replace the ‘free’ stagnation point with ‘pinned’ stagnation points at the surface of the cylinder. In the current modified geometry, effects of the blockage ratio (the ratio of the diameter of the cylinder to the width of the channel), the Weissenberg number (the ratio of elastic forces to viscous forces) and extensibility parameters ( $\unicode[STIX]{x1D6FC}$ and $L^{2}$ ) are investigated in two-dimensional numerical simulations using both the simplified Phan-Thien and Tanner and finitely extensible nonlinear elastic models. It is shown that the blockage ratio for fixed solvent-to-total-viscosity ratio has a stabilizing effect on the associated symmetry-breaking instability. The resulting data show that the suggested modification, although significantly changing the flow distribution in the region near the stagnation point, does not change the nature of the symmetry-breaking instability or, for low blockage ratio, the critical condition for onset. Using both numerical and physical experiments coupled with a supporting theoretical analysis, we conclude that this instability cannot therefore be solely related to the extensional flow near the stagnation point but it is more likely related to streamline curvature and the high deformation rates towards the corners, i.e. a classic ‘curved streamlines’ purely elastic instability. Our work also suggests that the proposed geometric modification can be an effective approach for enabling higher flow rates to be achieved whilst retaining steady symmetric flow.