Spiral instabilities in the flow of highly elastic fluids between rotating parallel disks

Abstract

Experimental observations and linear stability calculations are presented for the stability of torsional flows of viscoelastic fluids between two parallel coaxial disks, one of which is held stationary while the other is rotated at a constant angular velocity. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the purely circumferential, viscometric base flow becomes unstable with respect to a nonaxisymmetric, time-dependent motion consisting of spiral vortices which travel radially outwards across the disks. Video-imaging measurements in two highly elastic polyisobutylene solutions are used to determine the radial wavelength, wavespeed and azimuthal structure of the spiral disturbance. The spatial characteristics of this purely elastic instability scale with the rotation rate and axial separation between the disks; however, the observed spiral structure of the secondary motion is a sensitive function of the fluid rheology and the aspect ratio of the finite disks.Very near the centre of the disk the flow remains stable at all rotation rates, and the unsteady secondary motion is only observed in an annular region beyond a critical radius, denoted R*1. The spiral vortices initially increase in intensity as they propagate radially outwards across the disk; however, at larger radii they are damped and the spiral structure disappears beyond a second critical radius, R*2. This restabilization of the base viscometric flow is described quantitatively by considering a viscoelastic constitutive equation that captures the nonlinear rheology of the polymeric test fluids in steady shearing flows. A radially localized, linear stability analysis of torsional motions between infinite parallel coaxial disks for this model predicts an instability to non-axisymmetric disturbances for a finite range of radii, which depends on the Deborah number and on the rheological parameters in the model. The most dangerous instability mode varies with the Deborah number; however, at low rotation rates the steady viscometric flow is stable to all localized disturbances, at any radial position.Experimental values for the wavespeed, wavelength and azimuthal structure of this flow instability are described well by the analysis; however, the critical radii calculated for growth of infinitesimal disturbances are smaller than the values obtained from experimental observations of secondary motions. Calculation of the time rate of change in the additional viscous energy created or dissipated by the disturbance shows that the mechanism of instability for both axisymmetric and non-axisymmetric perturbations is the same, and arises from a coupling between the kinematics of the steady curvilinear base flow and the polymeric stresses in the disturbance flow. For finitely extensible dumb-bells, the magnitude of this coupling is reduced and an additional dissipative contribution to the mechanical energy balance arises, so that the disturbance is damped at large radial positions where the mean shear rate is large.Hysteresis experiments demonstrate that the instability is subcritical in the rotation rate, and, at long times, the initially well-defined spiral flow develops into a more complex three-dimensional aperiodic motion. Experimental observations indicate that this nonlinear evolution proceeds via a rapid splitting of the spiral vortices into vortices of approximately half the initial radial wavelength, and ultimately results in a state consisting of both inwardly and outwardly travelling spiral vortices with a range of radial wavenumbers.

Stretching and mixing of non-Newtonian fluids in time-periodic flows

“…Flow instabilities, irregular flow patterns, and nonlinear dynamics have long been observed in purely elastic fluids [8][9][10][11][12][13] and in fluids possessing both elasticity and shear-thinning viscosity. [14][15][16] Most of the nonlinear flow behavior observed in these studies arises from the extra elastic stresses due to the presence of polymer molecules.…”

Elastic turbulence in a polymer solution flow

“…This instability occurs at moderate W i and vanishingly small Re and is driven by the elastic stresses [7,9]. As a result of the instability, secondary, in general oscillatory, vortex flows develop, and flow resistance somewhat increases [6][7][8][9][10]. Flow instabilities in elastic liquids are reviewed in [11,12].…”

“…The curvature ratio was made quite high, d/R = 0.263, in order to provide destabilization of the primary shear flow and development of the secondary vortical fluid motion at lower shear rates [7,10]. (The flow between two plates with small d/R was studied before in context of the purely elastic instability [10].) The whole flow set-up was mounted on top of a commercial viscometer (AR-1000 of TA-instruments), so that we could precisely measure the angular velocity, ω, of the rotating upper plate and the torque applied to it.…”

Oscillatory torsional flow of a viscoelastic solid

“…The general torsional deformation for an isotropic material has been considered by Rivlin (1948), Klingbeil and Shield (1966), and its stability has been discussed by PhanThien (1988). The torsional flow for the Maxwell/Oldroyd class of fluids has already been considered by many researchers, primarily in connection with the flow stability (linearised, Phan-Thien 1983; Phan-Thien and Huilgol 1988; spiral disturbances, McKinley et al 1991;Ö ztekin and Brown 1993;Byars et al 1994; bounded flow domain, Avagliano and Phan-Thien 1996, 1999. We now show that the following deformation in cylindrical coordinates ðr; h; zÞ,…”

“…The torsional flow for these fluids also loses its stability when the Deborah number (product of the relaxation time and the angular velocity of the top plate) is greater than a critical value. This instability has been investigated in detail, both experimentally (Magda and Larson 1988;McKinley et al 1991), and theoretically with different disturbance modes, including a bounded domain (Ö ztekin and Brown 1993;Byars et al 1994;Avagliano and Phan-Thien 1996, 1999. In this work, we show that the torsional flow of a viscoelastic solid model has a similarity solution, including the fluid inertia.…”