Comparison of pressure losses in steady non-Newtonian flow through experimental tapered and cylindrical arterial prostheses

The use of an arterial prosthesis with a tapered lumen has several important advantages; for example, improved stability of flow, increased wall shear and better matching of its size with that of the host vessel. Tapering may, however, lead to increased energy losses, particularly if the angle of taper is large and the flow is high. This study is concerned with the determination of pressure drop for steady and laminar converging flow through rigid wall models of tapered arterial grafts. The angles of taper examined ranged from 0.5° to 1.0°. Aqueous solutions of polyacrylamide, with non-Newtonian viscous properties similar to those of blood, were used. The pressure drops across the tapered tubes were measured and the data were measured and the data were related to the pressure loss in cylindrical tubes of equivalent dimensions. Expressions for the ratio of the pressure drop in a tapered tube to that in a cylindrical tube for steady flow of a power law fluid were derived; there was good agreement between the predicted and the measured pressure drop ratios over a wide range of flows. The results of this study may be applied to the design of tapered arterial grafts. The pressure losses to be expected in tapered bypass grafts having various dimensions can easily be computed.     

Steady flow of couple stress fluid through tubes of slowly varying cross-sections--application to blood flows

Steady laminar flow of a non-Newtonian fluid based on couple stress fluid theory, through narrow tubes of varying cross-sections has been studied theoretically. Asymptotic solutions are obtained for the basic equations and the expressions for the velocity field and the wall shear stress are derived for a general cross-section. Computation and discussions are carried out for the geometries which occur in the context of physiological flows or in particular blood flows. The tapered tubes and constricted tubes are of special importance. It is observed that increase in certain parameters results in erratic flow behaviour proximal to the constricted areas which is further enhanced by the increase in the geometric parameters. This elucidates the implications of the flow in the development of vascular lesions.     

Pressure losses in non-Newtonian flow through rigid wall tapered tubes

Pressure drop and pressure gradient were measured in steady Newtonian and non-Newtonian flow through tapered tubes having angles of taper, alpha, between 0.5 degree and 1.25 degrees. Aqueous solutions of polyacrylamide, characterized as power law fluids, were used for the non-Newtonian flow measurements. These solutions had power law parameters similar in magnitude to those of blood. The pressure drop-flow rate data compared well with the predictions of a semi-empirical flow model over a large range of flow rates (Re alpha up to 10 for Newtonian flow and 5.7 for non-Newtonian flow). The pressure gradient increased with distance, z, into the taper as the radius decreased. The linear relationship between pressure gradient and z, derived by Oka (Biorheology, 10, 207-212, 1973) was found to be valid only when alpha z was small. For the tapered tubes examined here, agreement was confined to the region near the inlet. If higher order terms in alpha z were taken into account the agreement was extended further into the taper. However, under higher flow conditions, when the inertial losses are not negligible, the semi-empirical model provides much better estimates of pressure gradient.     

Flow through a Collapsible Tube: Experimental Analysis and Mathematical Model

Flow through thin-wall axisymmetric tubes has long been of interest to physiologists. Analysis is complicated by the fact that such tubes will collapse when the transmural pressure (internal minus external pressure) is near zero. Because of the absence of any body of related knowledge in other sciences or engineering, previous workers have directed their efforts towards experimental studies of flow in collapsible tubes. More recently, some attention has been given towards analytical studies. Results of an extensive series of experiments show that the significant system parameter is transmural pressure. The cross-sectional area of the tube depends upon the transmural pressure, and changes in cross-section in turn affect the flow geometry. Based on experimental studies, a lumped parameter system model is proposed for the collapsible tube. The mathematical model is simulated on a hybrid computer. Experimental data were used to define the functional relationship between cross-sectional area and transmural pressure as well as the relation between the energy loss coefficient and cross-sectional area. Computer results confirm the validity of the model for both steady and transient flow conditions.     

Casson fluid model for pulsatile flow of blood under periodic body acceleration

Pulsatile flow of a Casson fluid under the influence of a periodic body acceleration has been studied in this paper. An implicit finite difference numerical procedure has been used to analyze the flow. Applicability of this method has been checked by comparing the obtained results with the analytical solution for Newtonian flow and explicit scheme solution. The agreement between the implicit and explicit scheme solutions and the analytical solution is good (error less than 1%). Flow variables have been computed at three locations in cardiovascular system (wide (femoral) and narrow (arteriole and coronary) tubes). Effects of yield stress, tube radius and pressure gradient combined, body acceleration amplitude and frequency etc., on flow have been studied. The following observations have been made: (i) Initial transient time It changes with yield stress in narrow tubes are insignificant, whereas in wide tubes It decreases with yield stress; (ii) The axial velocity and fluid acceleration variations with yield stress are uniform (changes only quantitatively, profiles shape remain same) in narrow tubes, whereas in wide tubes these variations are non-uniform (profiles change qualitatively as well as quantitatively); (iii) Yield stress effects on wall shear amplitude are insignificant in narrow tubes (congruent to 0.3% in arteriole and congruent to 6% in femoral); and (iv) For Newtonian fluid, mean flow rate does not change with body acceleration amplitude a0 and frequency fb but it increases (decreases) with a0(fb) for Casson fluid.     

Convection-diffusion interaction for oscillatory flow in a tapered tube

Transport of soluble material is analyzed for volume-cycle oscillatory flow in a tapered tube. The equations of motion are solved using a regular perturbation method for small taper angle and order unity amplitude over a range of the Womersley parameter. The transport equation is also solved by a regular perturbation method where uniform end concentrations and no wall flux are assumed. The time-averaged axial transport of solute is calculated for several tapered tubes. There is substantial modification of transport compared to the straight tube case and the results are interpreted with respect to pulmonary gas exchange.     

Pulse velocities in cylindrical,tapered and curved anisotropic elastic arteries

Modified Moens-Korteweg formulae have been developed for the pulse velocities in anisotropic elastic arteries of varying geometries. In particular, cylindrical straight, tapered and curved tubes have been considered. Numerical results indicate that the cylindrical tube formula can be applied in all cases and that the circumferential elastic modulus is the dominant elastic parameter.     

The influence of suspending phase viscosity on the passage of red blood cells through capillary-size micropores

Much attention has been paid to the study of blood flow in long, narrow tubes. While the influence of tube diameter and driving pressure have been examined in detail, the influence of suspending phase viscosity has generally been assumed only to affect the blood viscosity in a linearly proportional manner, hence the practice of normalizing apparent blood viscosity values by the suspending phase viscosity to give a relative viscosity (e.g., Pries et al., 1992). While this assumption is probably valid for long tubes, it apparently does not hold for blood flow in short tubes (and by extension also for flow in short or branching capillary segments in vivo) in which RBC deformation plays a more significant role. In this paper we present a series of experiments using the Cell Transit Analyzer (CTA) in which the influence of driving pressure and suspending phase viscosity on RBC passage through short, narrow tubes has been systematically evaluated. Over the range studied (1 to 10 cm water), the influence of driving pressure was found to be unremarkable, in that RBC velocity scaled directly and linearly with pressure. This finding is consistent with previous studies. However, a distinct intercept was observed in the linear relationship between RBC pore transit time and suspending phase viscosity, which presumably arises as a consequence of RBC deformation either at the pore entrance or within the pore. Two simple mathematical models for the suspending phase-viscosity/transit-time relationship were considered. The results show that making CTA measurements over a range of suspending medium viscosities is a simple and practical way to obtain additional information about RBC mechanical properties.     

Platelet near-wall excess in porcine whole blood in artery-sized tubes under steady and pulsatile flow conditions

Platelet margination (enhanced platelet concentration in the near wall region of a blood vessel) has been well documented in small vessels. In artery-sized vessels margination has only been demonstrated in one study, using ghost cell suspensions and under relatively non-physiologic conditions of steady flow and 50 cm development length. Local sampling experiments were performed to confirm platelet margination in artery-sized stainless steel tubes, for a typical anatomical length and under pulsatile flow, using fresh EDTA-anticoagulated porcine whole blood (N=21). Experiments were designed using three-dimensional Computational Fluid Dynamics (CFD) to model the sample region with greater fidelity. Steady flow experiments in 50 cm long tubes verify published laser Doppler measurements of platelet margination in 3 mm ID tubes at normal arterial shear rate (500 s(-1). Margination persists under pulsatile flow conditions (63.8 pulses/min), but in steady flow at length of 10 cm, margination is reduced. Platelet margination ratio (the ratio of the platelet concentration near the wall to bulk average platelet count) ranges from 1.21 to 2.48. No significant effects of calculated sampling thickness (20 microm and 50 microm) or pulsatility were detected. Hematocrit margination ratio is 0.68 to 0.90. Two model platelet concentration profiles are fit to the experimental results.     

A mathematical note on the fahraeuslindqvist effect in power law fluid

An axisymmetric flow of a power law fluid through circular tubes under constant pressure gradient with the flow parameters varying radially is analyzed theoretically. The main finding is that for the Fahraeus-Lindqvist (F-L) effect to occur, it is necessary to have at least one of the parametersK (consistency) andn (index) as a discontinuous function ofr in the absence of wall slip; and with slip condition the parameters could be continuous functions ofr under specific conditions. In both the cases the existence of more than one discontinuity cannot be ruled out. The results obtained are consistent with experimental findings of blood flow through narrow tubes.     

Two-equation turbulence modeling of pulsatile flow in a stenosed tube

The study of pulsatile flow in stenosed vessels is of particular importance because of its significance in relation to blood flow in human pathophysiology. To date, however, there have been few comprehensive publications detailing systematic numerical simulations of turbulent pulsatile flow through stenotic tubes evaluated against comparable experiments. In this paper, two-equation turbulence modeling has been explored for sinusoidally pulsatile flow in 75% and 90% area reduction stenosed vessels, which undergoes a transition from laminar to turbulent flow as well as relaminarization. Wilcox's standard k-omega model and a transitional variant of the same model are employed for the numerical simulations. Steady flow through the stenosed tubes was considered first to establish the grid resolution and the correct inlet conditions on the basis of comprehensive comparisons of the detailed velocity and turbulence fields to experimental data. Inlet conditions based on Womersley flow were imposed at the inlet for all pulsatile cases and the results were compared to experimental data from the literature. In general, the transitional version of the k-omega model is shown to give a better overall representation of both steady and pulsatile flow. The standard model consistently over predicts turbulence at and downstream of the stenosis, which leads to premature recovery of the flow. While the transitional model often under-predicts the magnitude of the turbulence, the trends are well-described and the velocity field is superior to that predicted using the standard model. On the basis of this study, there appears to be some promise for simulating physiological pulsatile flows using a relatively simple two-equation turbulence model.     

Liquid film bioreactors for cell mass production

In order to make heat and mass transfer processes in bioreactors more intensive, the turbulent liquid film flow which is proposed for realization in multitubular reactors has been developed. Experiments have been carried out in tubes with smooth and rough surfaces at cocurrent gas and liquid film downward flows with a view to determine the values of the mass transfer coefficients in turbulent liquid hold-up in contact tubes. The liquid viscosity and REYNOLDS number are changing in a wide range (from 2000 to 60000). The results show that the maximum values of the mass transfer coefficients and an acceptable hold-up can be obtained in tubes with rough surfaces at high REYNOLDS numbers of the liquid film. The results of culturing yeasts on hydrolyzate in a laboratory-size liquid film fermenter are also presented.     

Non-Newtonian flow patterns associated with an arterial stenosis.

A non-Newtonian constitutive equation for blood has been introduced in this paper. Using this equation, blood flow attributes such as velocity profiles, flowrate, pressure gradient, and wall shear stress in both straight and stenotic (constricted) tubes have been examined. Results showed that compared with Newtonian flow at the same flowrate, the non-Newtonian normally features larger pressure gradient, higher wall shear stress, and different velocity profile, especially in stenotic tube. In addition, the non-Newtonian stenotic flow appears to be more stable than Newtonian flow.     

Reliable genotyping of samples with very low DNA quantities using PCR.

Our purpose was to identify an experimental procedure using PCR that provides a reliable genotype at a microsatellite locus using only a few picograms of template DNA. Under these circumstances, it is possible (i) that one allele of a heterozygous individual will not be detected and (ii) that PCR-generated alleles or 'false alleles' will arise. A mathematical model has been developed to account for stochastic events when pipetting template DNA in a very dilute DNA extract and computer simulations have been performed. Laboratory experiments were also carried out using DNA extracted from a bear feces sample to determine if experimental results correlate with the mathematical model. The results of 150 typing experiments are consistent with the proposed model. Based on this model and the level of observed false alleles, an experimental procedure using the multiple tubes approach is proposed to obtain reliable genotypes with a confidence level of 99%. This multiple tubes procedure should be systematically used when genotyping nuclear loci of ancient or forensic samples, museum specimens and hair or feces of free ranging animals.     

Use of a tapered fluidized bed as a Continuous bioreactor

Reactor systems based on tapered fluidized beds are being developed for aqueous bioprocesses in which adhering microorganisms or immobilized active biological fractions are used. The use of a fluidized bed prevents biomass buildup, accommodates particulates in the feed stream, is compatible with gas sparging, and allows easy removal or addition of the active materials. The tapered reactor tends to stabilize the fluidized bed, thus allowing a much wider range of operating conditions. Preliminary experimental results and an empirical mathematical model of the tapered bed indicate that bed stability is associated with a decreasing velocity and void-fraction profile up the bed and the pressure drop across the bed decreases with increasing flow rates. The tapered fluidized bed bioreactor is being evaluated for use in the enzymatic production of hydrogen, microbiological denitrification, and microbiological degradation of coal conversion aqueous waste streams. The enzyme catalyzed conversion of lactose to glucose and galactose was used in the evaluation of the reactor concept.     

The flow of a viscous liquid in a converging tube

The problem of the viscous flow of an incompressible Newtonian liquid in a converging tapered tube has been solved in spherical polar coordinates. The method of the solution involves the Stokes' stream function and a technique introduced by Stokes in the study of a sphere oscillating in a fluid. The theory for the flow in a rigid tube includes: (1) the pulsatile flow with both radial and angular velocity components; (2) the steady state flow with both radial and angular velocity components and (3) the very slow steady state flow with only a radial velocity component present. For a tapered elastic tube, the velocity of the propagated pulse wave is determined. The solution given is in terms of the elastic constants of the system and the coordinates for this type of geometry. The pulse velocity is then related to the velocity in an elastic cylindrical tube with the necessary correction terms to account for the tapered tube. Supported in part by the American Heart Association (No. 62F4EG). This work was done during the tenure of an Established Investigatorship of the American Heart Association.     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.     

Effect of needle geometry on flow rate and cell damage in the dispensing-based biofabrication process

Biodispensing techniques have been widely applied in biofabrication processes to deliver cell suspensions and biomaterials to create cell-seeded constructs. Under identical operating conditions,two types of dispensing needles—tapered and cylindrical—can result in different flow rates of material and different cell damage percent induced by the mechanical forces. In this work, mathematical models of both flow rate and cell damage percent in biodispensing systems using tapered and cylindrical needles, respectively, were developed, and experiments were carried out to verify the effectiveness of the developed models. Both simulations and experiments show tapered needles produce much higher flow rates under the same pressure conditions than cylindrical needles. Use of a lower pressure in a tapered needle can therefore achieve the same flow rate as that in a cylindrical needle. At equivalent flow rates, cell damage in a tapered needle is lower than that in a cylindrical one. Both Schwann cells and 3T3 fibroblasts, which have been widely used in tissue engineering, were used to validate the cell damage models. Application of the developed models to specify the influence of process parameters, including needle geometry and air pressure, on the flow rate and cell damage percent represents a significant advance for biofabrication processes.The models can be used to optimize process parameters to preserve cell viability and achieve the desired cell distribution in dispensing-based biofabrication.     

Hemodynamic effects of red blood cell aggregation

The influence of red blood cell (RBC) aggregation on blood flow in vivo has been under debate since early 1900's, yet a full understanding has still has not been reached. Enhanced RBC aggregation is well known to increase blood viscosity measured in rotational viscometers. However, it has been demonstrated that RBC aggregation may decrease flow resistance in cylindrical tubes, due to the formation of a cell-poor zone near the tube wall which results from the enhanced central accumulation of RBC. There is also extensive discussion regarding the effects of RBC aggregation on in vivo blood flow resistance. Several groups have reported increased microcirculatory flow resistance with enhanced RBC aggregation in experiments that utilized intravital microscopy. Alternatively, whole organ studies revealed that flow resistance may be significantly decreased if RBC aggregation is enhanced. Recently, new techniques have been developed to achieve well-controlled, graded alterations in RBC aggregation without influencing suspending phase properties. Studies using this technique revealed that the effects of RBC aggregation are determined by the degree of aggregation changes, and that this relationship can be explained by different hemodynamic mechanisms.     

Phloem ultrastructure and pressure flow: Sieve-Element-Occlusion-Related agglomerations do not affect translocation

Since the first ultrastructural investigations of sieve tubes in the early 1960s, their structure has been a matter of debate. Because sieve tube structure defines frictional interactions in the tube system, the presence of P protein obstructions shown in many transmission electron micrographs led to a discussion about the mode of phloem transport. At present, it is generally agreed that P protein agglomerations are preparation artifacts due to injury, the lumen of sieve tubes is free of obstructions, and phloem flow is driven by an osmotically generated pressure differential according to Münch's classical hypothesis. Here, we show that the phloem contains a distinctive network of protein filaments. Stable transgenic lines expressing Arabidopsis thaliana Sieve-Element-Occlusion-Related1 (SEOR1)-yellow fluorescent protein fusions show that At SEOR1 meshworks at the margins and clots in the lumen are a general feature of living sieve tubes. Live imaging of phloem flow and flow velocity measurements in individual tubes indicate that At SEOR1 agglomerations do not markedly affect or alter flow. A transmission electron microscopy preparation protocol has been generated showing sieve tube ultrastructure of unprecedented quality. A reconstruction of sieve tube ultrastructure served as basis for tube resistance calculations. The impact of agglomerations on phloem flow is discussed.