François Rousset

Extension of Murray's law using a non-Newtonian model of blood flow

BackgroundSo far, none of the existing methods on Murrays law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.ModelingIn the present paper, Murrays law which is applicable to an arterial bifurcation, is generalized to a non-Newtonian blood flow model (power-law model). When the vessel size reaches the capillary limitation, blood can be modeled using a non-Newtonian constitutive equation. It is assumed two different constraints in addition to the pumping power: the volume constraint or the surface constraint (related to the internal surface of the vessel). For a seek of generality, the relationships are given for an arbitrary number of daughter vessels. It is shown that for a cost function including the volume constraint, classical Murrays law remains valid (i.e. ΣRc= cste with c = 3 is verified and is independent of n, the dimensionless index in the viscosity equation; R being the radius of the vessel). On the contrary, for a cost function including the surface constraint, different values of c may be calculated depending on the value of n.ResultsWe find that c varies for blood from 2.42 to 3 depending on the constraint and the fluid properties. For the Newtonian model, the surface constraint leads to c = 2.5. The cost function (based on the surface constraint) can be related to entropy generation, by dividing it by the temperature.ConclusionIt is demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid.

Temporal Stability of Carreau Fluid Flow down an Incline

This paper deals with the temporal stability of a Carreau fluid flow down an inclined plane. As a first step, a weakly non-Newtonian behavior is considered in the limit of very long waves. It is found that the critical Reynolds number is lower for shear-thinning fluids than for Newtonian fluids, while the celerity is larger. In a second step the general case is studied numerically. Particular attention is paid to small angles of inclination for which either surface or shear modes can arise. It is shown that shear-dependency can change the nature of instability.

Wave celerity on a shear-thinning fluid film flowing down an incline

This letter presents a phenomenological model predicting the celerity of long surface waves on a non-Newtonian fluid flowing down an inclined plane. We show that, for a shear-thinning fluid, the celerity is greater than the well-known value c=2U0. The developed model points at the significant effect of the viscosity disturbance and also provides a likely explanation for the decrease in threshold for the instability.

Stability of a flow down an incline with respect to two-dimensional and three-dimensional disturbances for Newtonian and non-Newtonian fluids.

Squires theorem, which states that the two-dimensional instabilities are more dangerous than the three-dimensional instabilities, is revisited here for a flow down an incline, making use of numerical stability analysis and Squire relationships when available. For flows down inclined planes, one of these Squire relationships involves the slopes of the inclines. This means that the Reynolds number associated with a two-dimensional wave can be shown to be smaller than that for an oblique wave, but this oblique wave being obtained for a larger slope. Physically speaking, this prevents the possibility to directly compare the thresholds at a given slope. The goal of the paper is then to reach a conclusion about the predominance or not of two-dimensional instabilities at a given slope, which is of practical interest for industrial or environmental applications. For a Newtonian fluid, it is shown that, for a given slope, oblique wave instabilities are never the dominant instabilities. Both the Squire relationships and the particular variations of the two-dimensional wave critical curve with regard to the inclination angle are involved in the proof of this result. For a generalized Newtonian fluid, a similar result can only be obtained for a reduced stability problem where some term connected to the perturbation of viscosity is neglected. For the general stability problem, however, no Squire relationships can be derived and the numerical stability results show that the thresholds for oblique waves can be smaller than the thresholds for two-dimensional waves at a given slope, particularly for large obliquity angles and strong shear-thinning behaviors. The conclusion is then completely different in that case: the dominant instability for a generalized Newtonian fluid flowing down an inclined plane with a given slope can be three dimensional.

Structure Development of Biodegradable Polymers: Crystallization of PLA

Poly-(lactic acid) or PLA is a biodegradable polymer produced from renewable resources. Recently new polymerization routes have been discovered which allows increasing the produced quantity. Hence, PLA becomes of great interest to lessen the dependence on petroleum-based plastics. Due to its good mechanical properties, PLA is a potential substitute to some usual polymers such as PET. Nevertheless the kinetics of crystallization is relatively slow which can be an inconvenient in polymer processing. Thermomechanical history experienced by the polymer during processing affects drastically its relative crystallinity. For example, the flow is known to enhance the crystallization kinetics. Nevertheless, only a few studies were found in the literature about the crystallization of PLA under flow conditions. In the present work we investigate the crystallization of PLA under quiescent and flow conditions. A combination of DSC, rheological and optical measurements is used to identify the crystallization kinetic parameters. Thermal and flow-induced crystallization are then simulated using two sets of Schneider’s differential equations [1] based on a previously developed model Zinet & al [2]. Experimental results are analyzed and compared to the numerical model. New features about the influence of thermal and flow conditions on the crystallization of PLA are discussed.