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 Görtler 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.
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