Alparslan Oztekin

Rheological and geometric scaling of purely elastic flow instabilities

Abstract We present a new dimensionless criterion that can be used to characterize and unify the critical conditions required for onset of purely elastic instabilities in a wide range of different flow geometries. This scaling incorporates both the presence of non-zero elastic normal stresses in the fluid plus the magnitude of the streamline curvature in the flow, and it can be thought of as the viscoelastic complement of the Gortler number. We present detailed experimental and theoretical evidence that justifies and generalizes the form of the dimensionless criterion. We show how this criterion naturally arises from the linearized stability equations governing the viscoelastic flow and apply it to analytical and experimental results in a number of standard benchmark problems. In geometrically simple flows (e.g. torsional flows such as those in a circular Couette cell or a cone-and-plate rheometer) a characteristic radius of curvature of the streamlines may be readily identified and an analytical solution for the undisturbed base flow can be found. However, in the more complex flows characteristic of those found in commercial polymer processing operations, the base flow must typically be determined numerically and the streamline curvature varies in a complex manner throughout the flow. In the former case, we show how our scaling reduces to well-established results in the literature and for the latter case we present a particularly simple approach for understanding and quantifying the sensitivity of the critical conditions for onset of elastic instability to dimensionless geometric design parameters such as the aspect ratio of the test cell. The generality of the scaling is confirmed by applying it to new experimental measurements in a lid-driven cavity and numerical linear stability calculations for flow past a cylinder in a channel. We also show how the scaling may be generalized to incorporate, at least qualitatively, the variation in the critical conditions with other rheological parameters such as changes in the solvent viscosity, shear-thinning in the viscometric functions, a spectrum of relaxation times and a non-zero second normal stress coefficient. In a number of cases, these modifications and the predicted scaling of the critical onset conditions for purely elastic instabilities in other complex geometries, such as planar contractions or eccentric rotating cylinders, remain to be confirmed by future experiments or calculations.

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

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.

Self-similar spiral instabilities in elastic flows between a cone and a plate

Experimental observations and linear stability analysis are used to quantitatively describe a purely elastic flow instability in the inertialess motion of a viscoelastic fluid confined between a rotating cone and a stationary circular disk. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the spatially homogeneous azimuthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetric disturbances in the form of spiral vortices that extend throughout the fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-viscosity fluid show that the resulting secondary flow is composed of logarithmically spaced spiral roll cells that extend across the disk in the self-similar form of a Bernoulli Spiral. Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison.

Instability of a viscoelastic fluid between rotating parallel disks: analysis for the Oldroyd-B fluid

The stability of the viscometric motion of a viscoelastic fluid held between rotating parallel disks with large radii to small-amplitude perturbations is studied for the Oldroyd-B constitutive model. The disturbances are assumed to be radially localized and are expressed in Fourier form so that a separable eigenvalue problem results; these disturbances describe either axisymmetric or spiral vortices, depending on whether the most dangerous disturbance has zero or non-zero azimuthal wavenumber, respectively. The critical value of the dimensionless radius R* for the onset of the instability is computed as a function of the Deborah number De, a dimensionless time constant of the fluid, the azimuthal and radial wavenumbers, and the ratio of the viscosities of the solvent to the polymer solution. Calculations meant to match the experiments of McKinley et al. (1991) for a Boger fluid show that the most dangerous instabilities are spiral vortices with positive and negative angle that start at the same critical radius and travel outward and inward toward the centre of the disk; the axisymmetric mode also becomes unstable at only slightly greater values of R*, or De for fixed R*. The predicted dependence of the value of De for a fixed R* on the gap between the disks agrees quantitatively with the measurements of McKinley et al., when the longest relaxation time for the fluid at the shear rate corresponding to the maximum value of R* is used to define the time constant in the Oldroyd-B model.

Quantitative prediction of the viscoelastic instability in cone-and-plate flow of a Boger fluid using a multi-mode Giesekus model

Abstract Analysis and experiments have shown that the flow of a viscoelastic fluid between a rotating cone and plate is unstable to a three-dimensional time-dependent instability which results in a secondary motion that, close to onset, has the form of a Bernoulli spiral. Here we report results of a linear stability analysis for a multi-mode formulation of the Giesekus constitutive equation with parameters determined by regression to the linear relaxation spectrum and steady-state shear flow properties of the constant viscosity solution of polyisobutylene, polybutene and tetradecane first characterized by Quinzani et al. (J. RheoL, 34 (1990) 705-748). The numerically calculated stability boundaries for the onset of the elastic instability are compared to flow visualization experiments and provide quantitative agreement for both the critical Deborah number and the non-axisymmetric azimuthal structure of the spiral instability. The results are qualitatively similar to the predictions obtained from non-linear models with a single relaxation time, such as a single-mode Giesekus model or the FENE dumbbell model proposed by Chilcott and Rallison; however, the more precise fit of the first normal stress coefficient supplied by the multi-mode model appears necessary for quantitative prediction of the experiments. Quasi-linear models lacking shear-thinning behavior in the first normal stress coefficient Ψ1(γ), such as single- or multi-mode formulations of the Oldroyd-B model, are incapable of even qualitative prediction of the dependence of the stability boundaries on Debroah number observed in experiments with viscoelastic polymer solutions.

Viscoelastic flow around a confined circular cylinder: measurements using high-image-density particle image velocimetry

Abstract Particle image velocimetry (PIV) is used to measure the full-field instantaneous velocity in the inertialess, viscoelastic flow past a confined circular cylinder. Instantaneous fields of velocity vectors are obtained over an entire plane for the creeping motion of a highly viscoelastic fluid. In all cases considered, a fixed aspect ratio (channel half width to cylinder radius) of 16 was used. Our results indicated that at relatively low Deborah number (De), i.e. low flow rates, of 0.6 and 1.2, the flow reached steady-state conditions rapidly (i.e. ≤ 10 s). But as the flow rate is increased (or De is increased), e.g. De = 1.8 and De = 2.4, the viscoelastic flow may require as long as 90 s to reach steady flow. We also conducted experiments for steady, viscoelastic flow past an off-centered cylinder in the channel. Instantaneous full-field velocity and vorticity fields are examined for flow past an eccentrically-placed cylinder between parallel channels for various values of the eccentricity parameter (ϵ). Measurements are made at different values of Deborah number. It is seen from these studies that the spatial structure of the flow field, the vorticity intensity, and the spatial characteristics of vorticity field are all strongly affected by rheological parameter (elasticity) and geometric parameter (eccentricity).

Stability of planar stagnation flow of a highly viscoelastic fluid

Abstract Linear stability analysis is used to predict the onset of instabilities in inertialess viscoelastic planar stagnation flow. Beyond a critical value of the dimensionless flow rate, or Deborah number, the creeping base flow of similarity type, which is valid in the limit of vanishingly small Reynolds numbers, becomes unstable to localized three-dimensional disturbances. Stability calculations of the local similarity type viscoelastic flow in a small region near the stagnation plane are reported for the quasi-linear Oldroyd-B constitutive equation. The stability results for a range of Deborah numbers and viscosity ratio are presented to explore systematically the effects of elasticity and other rheological properties. The onset of instability and the temporal and spatial characteristics of the secondary flow predicted here resemble other purely elastic instabilities measured and predicted for viscoelastic flows in other simple and complex geometries with curved streamlines.

Binary and Ternary Nitrate Solar Heat Transfer Fluids

Current heat transfer fluids for concentrated solar power applications are limited by their high temperature stability. Other fluids that are capable of operating at high temperatures have very high melting points. The present work is aimed at characterizing potential solar heat transfer fluid candidates that are likely to be thermally stable (up to 500 C) with a lower melting point ( 100 C). Binary and ternary mixtures of nitrates have the potential for being such heat transfer fluids. To characterize such eutectic media, both experimental measurements and analytical methods resulting in phase diagrams and other properties of the fluids are essential. Solidus and liquidus data have been determined using a differential scanning calorimeter over the range the compositions for each salt system and mathematical models have been derived using Gibbs Energy minimization. The Gibbs models presented in this paper sufficiently fit the experimental results as well as providing accurate predictions of the eutectic compositions and temperatures for each system. The methods developed here are expected to have broader implications in the identification of optimizing new heat transfer fluids for a wide range of applications, including solar thermal power systems. [DOI: 10.1115/1.4023026]

Instabilities in viscoelastic flow past a square cavity

Abstract Elastic flow transitions in viscoelastic flow past a square cavity adjacent to a channel are reported. The critical conditions for the onset of flow transitions and the qualitative and quantitative characterization of the secondary flows generated by the instability have been examined using streakline photography and instantaneous pressure measurements. Cellular type of instabilities inside the cavity is observed for flow rates beyond a critical value. Small and large scale eddies are observed at high flow rates. The flow inside the cavity and in the channel upstream and downstream of the cavity becomes weakly time-dependent for high flow rates.

Dynamics of viscoelastic jets of polymeric liquid extrudate

Experimental observations are used to quantitatively describe purely elastic instabilities in viscoelastic jets issuing from a capillary and an orifice. Beyond a critical value of the dimensionless flow rate, or Deborah number, the flow becomes unstable and the purely elastic flow transition occurs. The instabilities result in free surface distortion in the form of vertical stripes laid longitudinally along the axis of the jet. The stripes uniformly spaced along the periphery of the jet are observed by photo-imaging measurements for PIB-based Boger fluid. The critical conditions for the onset of surface distortions and spatio-temporal characteristics of the instabilities are reported. Measurements of the extrudate swell are also conducted for different values of capillary diameter to systematically examine the effect of geometry.