RHEOLOGICAL AND GEOMETRIC SCALING OF PURELY ELASTIC FLOW INSTABILITIES

We present a new dimensionless criterion that can be used to character ize and unify the critical conditions required for onset of purely ela stic instabilities in a wide range of different flow geometries. This scaling incorporates both the presence of non-zero elastic normal stre sses in the fluid plus the magnitude of the streamline curvature in th e flow, and it can be thought of as the viscoelastic complement of the Gortler number. We present detailed experimental and theoretical evid ence that justifies and generalizes the form of the dimensionless crit erion. We show how this criterion naturally arises from the linearized stability equations governing the viscoelastic flow and apply it to a nalytical and experimental results in a number of standard benchmark p roblems. In geometrically simple flows (e.g. torsional flows such as t hose in a circular Couette cell or a cone-and-plate rheometer) a chara cteristic radius of curvature of the streamlines may be readily identi fied and an analytical solution for the undisturbed base flow can be f ound. However, in the more complex flows characteristic of those found in commercial polymer processing operations, the base flow must typic ally 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 unde rstanding and quantifying the sensitivity of the critical conditions f or onset of elastic instability to dimensionless geometric design para meters such as the aspect ratio of the test cell. The generality of th e scaling is confirmed by applying it to new experimental measurements in a lid-driven cavity and numerical linear stability calculations fo r flow past a cylinder in a channel. We also show how the scaling may be generalized to incorporate, at least qualitatively, the variation i n the critical conditions with other rheological parameters such as ch anges in the solvent viscosity, shear-thinning in the viscometric func tions, a spectrum of relaxation times and a non-zero second normal str ess coefficient. In a number of cases, these modifications and the pre dicted scaling of the critical onset conditions for purely elastic ins tabilities in other complex geometries, such as planar contractions or eccentric rotating cylinders, remain to be confirmed by future experi ments or calculations.