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

Abstract

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.

Numerical Investigation of T-Shaped Microfluidic Oscillator with Viscoelastic Fluid

“…For the case of Mesh4, Figure 3 shows that the flow rate at the right outlet was so large that the fluid flow in the left arm reversed to the right side, which was contrary to the results of the other three sets of grids. This random phenomenon in the flow direction was reasonable and consistent with the experimental [ 57 ] and numerical simulation [ 58 ] results of viscoelastic fluid under the microfluidic cross-slot channel, indicating the randomness of viscoelastic fluid in elastic instability.…”

Active open-loop control of elastic turbulence

“…Both control schemes applied to flow patterns and fluid instabilities in Newtonian fluids have extensively been studied [18][19][20][21][22][23][24][25] . Also, passive control of viscoelastic fluid flow has been examined [26][27][28][29][30][31] using either geometric modifications 28,29 including spatially modulated cylinders in a Taylor-Couette geometry 30 and disorder in microfluidic flows to inhibit elastic turbulence 31 , or soft boundaries 27 , as well as thermal control 26 . In contrast, the search for active control strategies appropriate for viscoelastic fluids has so far been limited.…”

Stabilization of purely elastic instabilities in cross-slot geometries

“…This discussion will then be used to qualitatively investigate the effect of these parameters on the important kinematics of the flow field near the corners of the cross-slot geometry, which are known to be regions driving the instability for the viscoelastic case (Davoodi et al. 2019). Similar to the idea used by Davoodi et al.…”

“…One should note that, in this geometry, although a nominally purely elongational flow is observed at the stagnation point, due to the existence of re-entrant corners (Dean & Montagnon 1949; Moffatt 1964; Davies & Devlin 1993; Hinch 1993), a fluid particle may experience a complex mix of shear and extensional deformation as it flows through the domain. Recently, Davoodi, Domingues & Poole (2019) suggested the use of a cylinder at the geometric centre of the cross-slot geometry to investigate the effect on the onset of instability. This geometric modification applies a fundamental difference on the flow field as the finite value of the strain rate at the free-stagnation point in the standard geometry is replaced with zero-strain-rate pinned stagnation points at the surface of the cylinder.…”

“…In Davoodi et al. (2019), using a combination of an analytical approach and supporting numerical simulations, it was shown that, by controlling the size of the cylinder, and consequently the curvature of streamlines, one may be able to control/delay onset of the instability to higher . A well-known dimensionless parameter which rationalizes these types of ‘curved streamline’ instabilities is the parameter introduced by McKinley, Pakdel & Öztekin (1996) (often referred to as the ‘Pakdel–McKinley’ criterion) which is defined as where is a reference velocity, is the relaxation time, is the curvature of the streamline, is the elastic normal stress in the streamwise direction, is the zero-shear-rate viscosity of the fluid and is the magnitude of the shear rate.…”