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

Hydrodynamic and hydromagnetic stability

Publisher Summary Heat transfer in flowing molten polymers is largely influenced by rheology–– the rheological properties of the polymer and by the flow geometry. The rheology of steady shear flow can treat most of the heat transfer problems completely. This chapter discusses the heat transfer problem, and classifies the heat transfer and viscous dissipation in molten polymers. The heat transfer in shear flow is analyzed and a large emphasis is laid on replacing the commonly used idealized boundary conditions–– constant wall temperature or constant wall heat flux by more general conditions. The heat transfer at the wall is described by an outer temperature difference and the Biot number that is used successfully for describing the boundary conditions for temperature calculations in solids. The Biot number is appropriate for describing the boundary conditions between isothermal and adiabatical, as they occur in real processes. A unifying concept is developed that makes it possible to comprise the most important shear flow cases into a single one that can be solved with one numerical program. The nonviscometric flow in channels and flow with free boundaries is also discussed. An example of heat transfer in unsteady unidirectional shear flow is also provided.

Viscous Dissipation in Shear Flows of Molten Polymers

Dynamic shear bands: an investigation using high speed optical and infrared diagnostics

. Viscous heating plays an important role in the dynamics of fluids with strongly temperature-dependent viscosity because of the coupling between the energy and momentum equations. The heat generated by viscous friction produces a local temperature increase near the tube walls with a consequent decrease of the viscosity which may dramatically change the temperature and velocity profiles. These processes are mainly controlled by the Peclet number, the Nahme number, the flow rate and the thermal boundary conditions. The problem of viscous heating in fluids was investigated in the past for its practical interest in the polymer industry, and was invoked to explain some rheological behaviours of silicate melts, but was not completely applied to study magma flows. In this paper we focus on the thermal and mechanical effects caused by viscous heating in tubes of finite lengths. We find that in magma flows at high Nahme number and typical flow rates, viscous heating is responsible for the evolution from Poiseuille flow, with a uniform temperature distribution at the inlet, to a plug flow with a hotter layer near the walls. When the temperature gradients  induced by viscous heating are very pronounced, local instabilities may occur and the triggering of secondary flows is possible. For completeness, this paper also describes magma flow in infinitely long tubes both at steady state and in transient phase.

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Viscous heating in fluids with temperature-dependent viscosity: implications for magma flows

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.

The stability of plane Couette flow with viscous heating

It has long been recognized that interesting features of viscoelastic materials can be revealed through study of the response of such materials to unsteady inputs. Indeed, most of the early work dealing with measurement of viscoelastic properties consisted of a study of the response of specimens to imposition of small-amplitude oscillatory stress or strain (1). Though much of the early work dealt with viscoelastic polymer solids, oscillatory testing has been extended to liquids, and there is now an extensive literature on the subject [see, for example, (2–5)]. However, by far the greatest interest to date has been in analyses and experiments which are restricted to a linear relation between input (for example, strain or strain rate) and output (for example, stress). This is the province of linear viscoelasticity, a subject which is now well developed and which has been quite successful for study of solid-like materials. Linear theory is insufficient, however, for characterization of many situations involving flow of liquid-like polymer melts, polymer solutions, and suspensions.

Charles A. Goldstein

Papers

The stability of plane Couette flow with viscous heating

Nonlinear effects in the unsteady flow of viscoelastic fluids