Joe D. Goddard

An inverse for the Jaumann derivative and some applications to the rheology of viscoelastic fluids

SummaryBy using a generalization of the matrizant of matrix calculus, it is shown how one can construct formally an inverse, or integral, for the well-knownJaumann derivative of continuum mechanics. Some applications to fluid rheology are then considered. First, it is shown that this integral provides, via theBoltzmann super-position principle, a generalization of Oldroyds quasi-linear fluid model, which is related to the molecular model ofBueche. Explicit expressions for the stresses arising in a general laminar shear flow are then derived for this model. Secondly, it is indicated how the operation can be used with rheological equations which are nonlinear in the deformation-rate, but quasi-linear in stress, to solve explicitly for the stress in terms of kinematic quantities. As an example, a rheological equation for suspensions of viscoelastic spheres in aNewtonian fluid is treated.

An elastohydrodynamic theory for the rheology of concentrated suspensions of deformable particles

Abstract The equations which govern thin films of a Newtonian liquid confined between deformable solid surfaces are applied to the regions of near contact in a concentrated suspension of deformable particles. For the case of slightly deformable elastic particles, one obtains the socalled “elastohydrodynamic” equations of lubrication theory. The appropriate asymptotic solution of these equations yields estimates for the viscosity, of a form proposed earlier by Frankel and Acrivos [1] for rigid particles, as well as a relaxation time for a suspension of near spheres. The present method, which goes beyond the dissipation calculation of Frankel and Acrivos to a derivation of the full stress tensor, yields the same form of dependence of viscosity on particle concentration. However, there is an as yet unexplained difference between the methods in the value of a numerical coefficient determined by the assumed packing of the spheres. While further work is needed on the kinetic theory for fluid suspensions, the methods employed here for the derivation of the stress tensor should have direct utility for certain solid dispersions, where it is possible to specify a priori the particle-packing in the system.

Tensile stress contribution of flow-oriented slender particles in non-newtonian fluids

Abstract A formula is derived for the intrinsic stress contribution from closely-spaced, rod-like rigid particles suspended in a non-Newtonian liquid subject to a simple extensional flow. The present results indicate that, compared to the Newtonian-fluid result given previously by Batchelor [1], a much smaller particle-stress effect may occur in non-Newtonian fluid, owing to shear thinning and possible tensile stiffening in the fluid itself. While unsubstantiated by any critical experimental test, this prediction appears to agree qualitatively with some recent experimental observations of Charrier and Rieger [29] on glass fibers in polymer melts. Further work is suggested for the relevant slender-body theory, particularly the far-field body influence in non-linear fluids.

AN EX VIVO METHOD FOR THE EVALUATION OF BIOMATERIALS IN CONTACT WITH BLOOD

The use of extracoporeal (ex vivo) shunts for studying thrombus formation was given impetus by a number of studies from J. F. Mustard’s 1aboratory.lS In these studies the essential problems of this method were recognized and the requirements for a reproducible system were delineated. The basic features are: 1) uniform blood flow, 2 ) a chamber that represents preferential region for thrombus formation, and 3 ) access to the thrombus for analysis. Murphy et a1.l tried a number of flow chamber designs and finally settled on the use of a model bifurcation for their studies,2 and other investigators s , have evaluated alternative flow chamber configurations. This type of ex vivo test with recirculation of blood should be distinguished from other ex vivo procedures in which animals are bled through a test chamber.5, In the latter cases chronic shunts are not used, which simplifies the procedure, but the loss of blood usually limits the duration of the experiment to less than one hour. To develop an ex vivo method for the evaluation of the thrombogenicity of materials in contact with blood, a system was devised that exposes a test chamber to a dog’s systemic circulation by means of a chronic arteriovenous shunt. The general aspects of this approach are shown in the flow diagram in FIGURE 1 and the schematic FIGURE 2. Male dogs were selected after screening to ensure that they were normal with respect to weight and blood coagulation values. An arteriovenous shunt was implanted in accepted animals, and after surgery, coagulation factors and other physiological parameters were evaluated periodically to monitor the approach of the dog’s coagulation system toward steady state. After about one week, labeled fibrinogen and platelets were introduced systemically, and after 24-48 hours the chamber containing the test material was introduced into the bypass system. The rate of thrombus formation was determined by monitoring the buildup of radioactive fibrinogen and platelets in the test chamber. Blood flow through the chamber was measured with an electromagnetic or Doppler flow meter and controlled at a constant value. After a predetermined time the chamber was disconnected from the shunt, and any thrombus formed was removed and quantitated in terms of clot weight, moisture, fibrinogen, red cell, and platelet content. These values are used as indices of thrombus formation on a given biomaterial under defined flow conditions.

Carrier-mediated heat transport

Abstract-It is shown how an asymptotic, boundary-layer method, developed specifically for application to facilitated mass transport, can also be applied to the corresponding problem of facilitated enthalpy transport. The method, which merely entails an implicit algebraic computation, is applied to the (hydrodynamic) film-theory model for heat transfer between a hot or cold wall and a reversibly dissociating gas mixture, 2N0 &2NO + 02, which has previously been treated by Brian and Bodman[l]. Good agreement is found with their approximate analytical formula over the parameter range of their computations. However, for more extreme states of dissociation and larger overall temperature differences, the present results differ from those given by their approximate formula, especially for the cooling (cold wall) problem. INTRODUCTION The conduction of heat in chemically reactive sys- tems is by no means a novel topic in the general area of diffusional transport phenomena. It has been the subject of several previous works, which are exemplified most notably by those of Hirshfelder[2], Brian and Reid[3], Brian and Bodman[l], Bodman et al. [4], Fan and Mason[5], and co-workers. One of the principal objectives of such studies has been to predict theoretically the reactive enhancement of steady-state heat trans- fer through a chemical mixture due to reversible thermal dissociation and recombination accom- panied by molecular diffusion of various chemical species. If, in such systems, say reacting gas mix- tures, the intrinsic rates of the chemical reactions are, in some appropriate sense, small compared to diffusional transport rates then one obtains the so-called ‘frozen’ (conductivity) regime, where reaction exerts a negligible influence on energy transport. At the opposite extreme, where reaction rates are in effect infinitely rapid, one obtains the (reaction) ‘equilibrium’ regime and, usually, max- imum enhancement of heat flux. In either of these asymptotic regimes one is generally able to perform fairly direct computations of the heat flux through a mixture from given trans- port and equilibrium properties; whereas, other- wise, one has to solve a set of non-linear differential equations, involving diffusion and reaction rates and governing spatial temperature and composition fields. While such boundary-value problems are in principle straightforward, albeit not analytically tractable, and actual numerical computation can be difficult and time-consuming, even with advanced computer-executed numerical schemes. These practical difficulties are traceable directly to a boundary layer structure that develops whenever reaction rates are moderately to highly rapid. This boundary-layer nature of the near- equilibrium regime was anticipated in the early works of Hirshfelder[2], but, up to the present, no general mathematical techniques have apparently been proposed to cope with the problem in this particular physical context. However, in very re- cent times there has been a good deal of effort devoted to the treatment of problems closely akin to the above in the related area of facilitated mass transport. Although there had been a previous long- standing interest in the reactive enhancement of mass transport, such problems have received re- newed attention because of their possible relevance to carrier-mediated biological transport in mem- branes and liquid films. Thus, Goddard

An Analysis of the Competitive Diffusion of O2 and Co through Hemoglobin Solutions

The subject of carrier-mediated or “facilitated” transport has received widespread attention in the chemical and biological literature, especially in conjunction with diffusion across thin films or membranes. One of the simpler conceptual models for the phenomenon is based on the postulate of ordinary molecular diffusion coupled with reversible homogeneous chemical reactions. The diffu-sional flux of a transferred chemical species is thereby “mediated” or altered by reversible combination with certain mobile “carrier” species which are indigenous to the film or membrane. Any such species, which is physically restricted to the carrier-mediated transport system, will be designated here as a “non-volatile” or “non-transferred” species.

A note on a statistical-mechanical treatment of activation-limited surface diffusion

A brief review is given of a class of simple statistical-mechanical models for surface diffusion, applicable to the limiting case where diffusional “hopping” is controlled by the thermal activation rate of the adsorbed particle. A theoretical result of Reyes, for the pre-exponential or “frequency” factor, is discussed and compared briefly to experiment.AbstractПриводится краткий обзор класса простых статистическо-механических моделей поверхностной диффузии, примени-мых в предельных случаях, когда диффузионные “прыжки” контролируются скоростью термической активации адсорбированных частиц. Теоретические результаты Рейса отно-сительно предэкспоненциального множителя или “частот-ного” фактора обсуждаются и сравниваются с экспериментальными значениями.