Secondary flows due to finite aspect ratio in inertialess viscoelastic Taylor–Couette flow

Abstract

Both in rheometry and in fundamental fluid mechanics studies, the Taylor–Couette geometry is used frequently to investigate viscoelastic fluids. In order to ensure a constant shear rate in the gap between the inner and outer cylinders, such studies are usually restricted to the small-gap limit where the assumption of a linear velocity distribution is well justified. In conjunction with a sufficiently large aspect ratio$\unicode[STIX]{x1D6EC}$(i.e. ratio of cylinder height to gap), the flow is then assumed to be viscometric. Here we demonstrate, using a perturbation technique with the curvature ratio (i.e. ratio of the half-gap to the mid-radius of the cylinders) as the perturbation parameter, full nonlinear simulations using a finite-volume technique, and supporting experiments, that, even in the creeping-flow (inertialess) narrow-gap limit, for viscoelastic fluids end effects due to finite aspect ratio always give rise to a secondary motion. Using the constant-viscosity Oldroyd-B model we are able to show that this secondary motion, as has been observed in related pressure-driven flows with curvature, such as the viscoelastic Dean flow, is solely a consequence of the combination of gradients of the first normal-stress difference and curvature. Our results show that end effects can significantly change the flow characteristics, especially for small aspect ratios, and this may have important consequences in some situations such as the onset criteria for purely elastic instabilities.

Active open-loop control of elastic turbulence

“…Additionally, in experiments the aspect ratio of the Taylor–Couette cell influences the onset of the elastic instability. Due to high shear gradients created in the corner between the driving wall and the static wall, a secondary flow with curved streamlines arises in viscoelastic fluids 28 , which is influenced by the aspect ratio. As a consistency check, we have compared results from our two-dimensional simulations to results of a simulation in a three-dimensional Taylor–Couette setup (see supplemental material ).…”

“…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.…”

Efficient mixing enhancement by orthogonal injection of shear-thinning fluids in a micro serpentine channel at low Reynolds numbers

“…This elasticity significantly enhance the shear rate at the wall and homogenizes the viscosity profiles which leading to a perfect mixing as reported by Bodiguel et al [52]. In regard to the secondary flow, the coupling of analogous Dean vortices and the shear-thinning effects in the viscoelastic flows cause the augment of mixing performance since that in the viscoelastic flows, the analogous viscoelastic secondary flow can be generated even at lower Dean numbers, as presented by Poole and his co-workers [53][54][55] in a serpentine channel.…”

Stabilization of purely elastic instabilities in cross-slot geometries

“…Similar to the idea used by Davoodi et al. (2018) for normalization of the aspect ratio, here, a modified form of the viscosity ratio parameter is defined as Using this definition, when the viscosity ratio changes from zero to infinity, the modified form of the viscosity ratio varies from zero to one, respectively. For the 1-D problem, using (6.15), one can say that, by changing the viscosity ratio parameter, the term should show two roots at and .…”