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

This review focuses on the theory of oxygen transport to tissue and presents the state of the art in mathematical modeling of transport phenomena. Results obtained with the classic Krogh tissue-cylinder model and recent advances in mathematical modeling of hemoglobin-oxygen kinetics, the role of hemoglobin and myoglobin in facilitating oxygen diffusion, and the role of morphologic and hemodynamic heterogeneities in oxygen transport in the microcirculation are critically discussed. Mathematical models simulate different parts of the pathway of oxygen molecules from the red blood cell, through the plasma, the endothelial cell, other elements of the vascular wall, and the extra- and intracellular space. Special attention in the review is devoted to intracapillary transport, which has been the subject of intensive theoretical research in the last decade. Models of pre- and postcapillary oxygen transport are also discussed. Applications to specific organs and tissues are reviewed, including skeletal muscle, myocardium, brain, lungs, arterial wall, and skin. Unresolved problems and major gaps in our knowledge of the mechanisms of oxygen transport are identified.

Theory of oxygen transport to tissue.

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

The origins, uses and evaluation of constitutive equations for the stress tensor of polymeric liquids are discussed.

Constitutive equations for polymeric liquids

Certain polymers exhibit two distinct branches in their capillary flow curves (wall shear stress versus apparent wall shear rate). This gives rise to oscillatory flow in constant‐piston‐speed rheometers and to flow curve hysteresis in controlled‐pressure rheometers. These curious phenomena have attracted considerable interest over a period of many years, but their basic mechanisms are still the subject of debate. Building on previous work we have developed a model that predicts all the essential features of the curves of pressure and flow rate versus time in the oscillatory flow regime. Fluid compressibility and the second branch of the flow curve are necessary features of the model, but fluid elasticity is found not to be an essential element. While our macroscopic measurements do not prove it conclusively, our data lead us to believe that on the high‐flow‐rate branch of the flow curve there is slip along a cylindrical fracture surface near the wall. The jump to the high‐flow branch occurs when this fracture occurs, at an upper critical value of the shear stress, while the jump back to the low‐flow branch occurs when adhesion is established at the fracture surface at a lower critical shear stress.

https://sor.scitation.org/doi/pdf/10.1122/1.550320

Role of slip and fracture in the oscillating flow of HDPE in a capillary

We examine the behavior of viscoelastic fluid models which exhibit local extrema of the steady shear stress. For the Johnson-Segalman and Giesekus models, a variety of steady singular solutions with jumps in shear rate are constructed and their stability to one dimensional disturbances analyzed. It is found that flow-rate versus imposed stress curves in slit-die flow fit experimental observation of the “spurt” phenomenon with some precision. The flow curves involve linearly stable singular solutions, but some assumptions on the dynamics of the spurt process are required. These assumptions are tested by a semi-implicit finite element solution technique which allows solutions to be efficiently integrated over the very long time-scale involved. The Johnson-Segalman model with added Newtonian viscosity is used in the calculations. It is found that the assumptions required to model spurf are satisfied by the dynamic model. The dynamic model also displays a characteristic “latency time” before the spurt ensues and a characteristic “shape memory” hysteresis in load/unload cycles. These as well as other features of the computed solutions should be observable experimentally. We conclude that constitutive equations with shear stress extrema are not necessarily flawed, that their predicted behavior may appear to be arrested “wall slip”, and that such behavior may actually have been observed already.

Spurt phenomena of the Johnson-Segalman fluid and related models

We investigate the linear stability problem of convection by the general Oldroyd B fluid and its Maxwell limit in the presence of rigid or free boundaries and fixed temperature or fixed flux. Comparison with recent results by Rosenblat [9] for the analytically accessible case of free boundary conditions shows a qualitative similarity in the shape of the neutral stability curves. But while Newtonian and Jeffreys (general Oldroyd B) fluids are sharply stabilized by the presence of rigid boundaries, the Maxwell fluid is largely unaffected at even moderately large Prandtl number. The reasons for this are discussed. Also, a discrepancy between the earlier works by Vest and Arpaci [3], and Sokolov and Tanner [4], which treat the case of a Maxwell fluid, is found to be due to algebraic error, and not multivaluedness of the stress-strain rate relation as earlier suggested by Eltayeb [6].

On the convected linear stability of a viscoelastic oldroyd B fluid heated from below

The spurt phenomenon is a flow instability which occurs in pressure-driven parallel shear flows of viscoelastic liquids. This phenomenon is characterized by an abrupt increase in the volumetric throughput at a critical value of the driving pressure gradient. Recently, non-monotone (steady shear response) constitutive equations have been proposed to model this phenomenon. We analyze the startup problem for the Giesekus model, both asymptotically and numerically, and compare the results to those obtained for other models.

Phase space analysis of the spurt phenomenon for the giesekus viscoelastic fluid model

Evaporation of a Dichloromethane liquid film is explored with an evolution equation describing film dynamics. The film is subject to different initial conditions, smooth and uniform random perturbation. Two different gravity environments (Earth and zero gravity) and two different domain shapes (square and rectangular) have been used. The occurrence of long wave instabilities affecting film dynamics is noted in each of these cases. The evaporating Dicholormethane liquid film is destabilized via long wave instabilities in zero gravity. The thermocapillary patterns formed due to long wave destabilization show a coupling to the initial conditions and domain shape. A criterion for the occurrence of long wave instabilities based on the growth rate of perturbations is described. This equation considers a non-stationary film thickness. It predicts that long wave instabilities are always present in zero gravity environments with a growth rate that increases as the film thickness decreases due to evaporation. Our equation for growth rate of long wave instabilities may be used as an engineering design tool to confine operating parameters of zero gravity heat transfer equipment, that include or harness phase change, to safe limits.

Robert W. Kolkka

Papers

Spurt phenomena of the Johnson-Segalman fluid and related models

On the convected linear stability of a viscoelastic oldroyd B fluid heated from below

Phase space analysis of the spurt phenomenon for the giesekus viscoelastic fluid model

The Effect of Gravity on the Stability of an Evaporating Liquid Film

Reduction of anoxia through myoglobin-facilitated diffusion of oxygen.