Robert W. Kolkka

Spurt phenomena of the Johnson-Segalman fluid and related models

Abstract 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.

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

Abstract 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].

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

Abstract 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.

Reduction of anoxia through myoglobin-facilitated diffusion of oxygen.

At relatively low perfusion rates, anoxic regions may occur in tissue even though oxygen remains in the blood as it leaves the capillary at the venous end. In this paper a mathematical theory of facilitated diffusion is developed and used to determine the extent to which myoglobin increases the removal of oxygen from blood and aids in the reduction or elimination of regions of anoxia.

A mathematical analysis of carrier-facilitated diffusion

Abstract Carrier-facilitated diffusion in a one dimensional slab is analyzed. The method of matched asymptotic expansion is used to obtain solutions in both the near-equilibrium and the small facilitation limit. Consumption of the diffusing species within the slab is included in the analysis, so that the solutions are applicable to situations of physiological interest.

Singular Perturbations of Bifurcations with Multiple Independent Bifurcation Parameters

There are several nonlinear bifurcation problems which involve multiple bifurcation parameters (“specified inputs”). Often there is a corresponding “perturbed” problem associated with the bifurcation problem which models imperfections. Instead of bifurcation, the perturbed problem exhibits a smooth but rapid transition in the critical range of the bifurcation parameter. A new generalization of the single bifurcation parameter method of Matkowsky and Reiss, SPB (“Singular Perturbations of Bifurcations”) [SIAM J. Appl. Math., 33 (1977), pp. 230–255] is employed to solve the problem of the slightly crooked rotating elastica (shaft) subject to a dead load applied at the ends.

Stability of Liquid Films

Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. The present work examines, through the solution of a onesided evolution equation as an initial value problem with periodic boundary conditions, the various mechanisms that affect the stability of liquid films. The numerical program employed to solve the non-linear evolution equation is validated by comparing the results produced with previously published data. The wavenumber associated with various destabilizing mechanisms is extracted. The effect of pinned boundary conditions versus periodic boundary conditions will be discussed.© 2010 ASME