Viscoelastic instabilities are of practical importance, and are the subject of growing interest. Reviewed here are the fresh developments as well as earlier work in this area, organized into the following categories: instabilities in Taylor-Couette flows, instabilities in cone-and-plate and plate-and-plate flows, instabilities in parallel shear flows, extrudate distortions and fracture, instabilities in shear flows with interfaces, instabilities in extensional flows, instabilities in multidimensional flows, and thermohydrodynamic instabilities. Emphasized in the review are comparisons between theory and experiment and suggested directions for future work.

Instabilities in viscoelastic flows

nt transient that may be many orders of magnitude longer than time scales associated with instrument inertia. Intrinsic material time scales can be identified with the dynamics of the macro­ molecular chains. Shaping processes for polymeric materials are usually carried out in the liquid state, often over times that are rapid relative to those associated with molecular reorganization; the viscoelasticity is thus relevant to any understanding of flow and structure development. The distinction between the rheology and the fluid mechanics of visco­ elastic materials is vague but is worth stating. The former is concerned with constitutive relations between stress and deformation and may involve physical modeling at a molecular level. Controllability of the flow field is essential when making rheological measurements to evaluate material properties, so the kinematics are generally imposed and the momentum equation is not solved. Viscoelastic fluid mechanics, on the other hand, is the study of motions in which the kinematics cannot be established a priori, and the continuity and momentum equations must be solved together with the constitutive equation for the stress. The equations that must be solved for even the most elementary viscoelastic liquids are considerably more complex than the Navier-Stokes equations, and many unresolved issues

Issues in viscoelastic fluid mechanics

An algorithm to compute the extremal eigenvalues of a non-Hermitian matrix, based on Arnoldi-orthogonalization, is employed in the linear stability analysis of the viscoelastic Poiseuille flow at high Reynolds numbers. It is shown that this algorithm is both computationally efficient and accurate in reproducing the most unstable modes in the pseudo-spectrally discretized eigenspectrum of the original problem. The Upper Convected Maxwell (UCM), Oldroyd-B and Chilcott-Rallison models are considered for the linear stability analysis of the high Reynolds number viscoelastic Poiseuille flow. Results for the UCM model show a large destabilization of the flow compared to the Newtonian limit, even for low values of flow elasticity, ϵ = We / Re , of the order 10 −3 , realized at high Reynolds numbers for O (1) We. These results agree qualitatively with those reported by Porteus and Denn, although some quantitative differences exist, especially at high values of elasticity. Furthermore, it is shown that the number of spectral modes necessary to obtain converged results increases substantially as the flow elasticity is increased. A comparison of the linear stability characteristics of the Oldroyd-B and the UCM models has revealed that the presence of a non-zero solvent viscosity has a pronounced stabilizing effect on the flow. Further stabilization occurs through the introduction of a finite molecular extensibility in the Chilcott-Rallison model.

Linear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm

Analytical solutions are derived for the steady state channel and pipe flows of viscoelastic fluids obeying the complete single-mode Phan-Thien–Tanner (PTT) constitutive equation with a linear stress coefficient in the absence of a solvent viscosity contribution. The results include the profiles of all relevant stresses, the axial velocity and the viscosity across the gap. The three material functions of the single-mode PTT model in steady Couette flow are also given and it is shown that the conditions of the maximum point in the shear stress versus shear rate curve are related to the conditions for existence of steady state solutions in the channel and pipe flows. The range of model parameters for which a classical steady solution exists is established. © 2001 Elsevier Science B.V. All rights reserved.

Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid

Non-Newtonian fluid mechanics is often distinguished from its Newtonian counterpart by the additional requirement that a constitutive equation be specified as part of the problem statement. For some modeling problems important to polymer processing, a wall boundary condition in which provision is made for slip is also a necessary ingredient. This requires that attention be directed to microscopic processes influencing flow behavior on a macroscopic scale. In this paper theoretical and experimental consequences of slip phenomena are reviewed. It is maintained that future progress will depend heavily on productive interplay between fluid mechanics and materials science.

The behavior of complex fluids at solid boundaries

Pseudo-spectral analysis of the stability of pressure-driven flow of a giesekus fluid between parallel planes

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Pseudo-spectral analysis of the stability of pressure-driven flow of a giesekus fluid between parallel planes