Peristaltic transport of blood: Casson model--II

The problem of peristaltic transport of blood in a uniform and non-uniform tube has been investigated, under zero Reynolds number and long wavelength approximation. Blood is represented by a two-layered fluid model consisting of a central layer of suspension of all erythrocytes, etc., assumed to be a Casson fluid, and a peripheral layer of plasma as a Newtonian fluid. A comparison of results with those without peripheral layer shows that the magnitude of the pressure rise, under a given set of conditions is smaller in the case of model with peripheral layer. It is found that, for a given flow rate, the pressure rise decreases as the viscosity of the peripheral layer decreases, and for a given non zero pressure drop, the flow rate increases as the viscosity of the peripheral layer decreases. However, the flow is independent of the presence of the peripheral layer, for zero pressure rise. Further, the pressure rise in the case of non-uniform geometry is found much smaller than the corresponding value in the uniform geometry.     

Peristaltic transport of multilayered power-law fluids with distinct viscosities: a mathematical model for intestinal flows

This paper is concerned with the theoretical study of two-dimensional peristaltic flow of power-law fluids in three layers with different viscosities. The analysis is carried out under low Reynolds number and long wavelength approximations. The shapes of the interfaces are described by a system of non-linear algebraic equations which are solved numerically as streamlines. It is found that the non-uniformity in the intermediate and peripheral layers diminishes when the viscosity of the intermediate layer is increased and that of the outermost layer is kept unaltered for both the pseudo-plastic and dilatant fluids. Similar are the observations when the viscosity of the outermost layer is increased and that of the intermediate layer is kept fixed. The flow rate increases with the viscosities of the peripheral and the intermediate layers but the viscosity of the outermost layer is more effective. However, the knowledge of the effect of the viscosity of the intermediate layer facilitates us to achieve the required flow rate without disturbing the outermost layer. An increase in the flow behaviour index too favours larger flow rates. The trapping limits increase with the viscosity of the intermediate layer but decrease with the viscosity of the outermost layer and the flow behaviour index. Thus, a medicinal intervention that creates a more viscous intermediate layer and reduces pseudo plasticity may reduce constipation.     

A Mathematical Study on Three Layered Oscillatory Blood Flow Through Stenosed Arteries

A mathematical model is constructed to examine the characteristics of three layered blood flow through the oscillatory cylindrical tube (stenosed arteries).The proposed model basically consists three layers of blood (viscous fluids with different viscosities) named as core layer (red blood cells),intermediate layer (platelets/white blood cells) and peripheral layer (plasma).The analysis was restricted to propagation of small-amplitude harmonic waves,generated due to blood flow whose wave length is larger compared to the radius of the arterial segment.The impacts of viscosity of fluid in peripheral layer and intermediate layer on the interfaces,average flow rate,mechanical efficiency,trapping and reflux are discussed with the help of numerical and computational results.This model is the generalized form of the preceding models.On the basis of present discussion,it is found that the size of intermediate and peripheral layers reduces in expanded region and enhances in contracted region with the increasing viscosity of fluid in peripheral layer,whereas,opposite effect is observed for viscosity of fluid in intermediate layer.Final conclusion is that the average flow rate and mechanical efficiency increase with the increasing viscosity of fluid in both layers,however,the effects of the viscosity of fluid in both layers on trapping and reflux are opposite to each other.     

Peripheral-layer viscosity and microstructural effects on the capillary-tissue fluid exchange

A theoretical investigation of capillary-tissue fluid exchange has been studied including the characteristics and influence of the boundaries and media through which the fluid flows. Filtration from a cylindrical capillary into the concentrically surrounding tissue space and flow from a capillary into the tissue across a thin membrane are analyzed in detail. In has been observed that the filtration efficiency of the functional unit decreases as the viscosity of the peripheral layer increases. Contrary to the results of Apelblat [17], the slip velocity at the porous boundary plays a dominant role in filtration efficiency. It has also been noticed that the filtration efficiency decreases as the slip velocity at the porous boundary increases.     

Peristaltic transport in a channel with a porous peripheral layer: model of a flow in gastrointestinal tract

Peristaltic transport in a two dimensional channel, filled with a porous medium in the peripheral region and a Newtonian fluid in the core region, is studied under the assumptions of long wavelength and low Reynolds number. The fluid flow is investigated in the waveframe of reference moving with the velocity of the peristaltic wave. Brinkman extended Darcy equation is utilized to model the flow in the porous layer. The interface is determined as a part of the solution using the conservation of mass in both the porous and fluid regions independently. A shear-stress jump boundary condition is used at the interface. The physical quantities of importance in peristaltic transport like pumping, trapping, reflux and axial velocity are discussed for various parameters of interest governing the flow like Darcy number, porosity, permeability, effective viscosity etc. It is observed that the peristalsis works as a pump against greater pressure in two-layered model with a porous medium compared with a viscous fluid in the peripheral layer. Increasing Darcy number Da decreases the pumping and increasing shear stress jump constant beta results in increasing the pumping. The limits on the time averaged flux Q for trapping in the core layer are obtained. The discussion on pumping, trapping and reflux may be helpful in understanding some of the fluid dynamic aspects of the transport of chyme in gastrointestinal tract.     

Analytical investigation of cilia induced mucous flow

This is an analytic study of mucous flow caused by ciliary motion. The computed flow data may be compared with that already found exprimentally. The effects of mucous density, viscosity and layer depth on flow phenomena are investigated. The effects of cilia diameter, length, spacing and oscillation frequency are determined from the equations governing the flow of the mucous blanket. A pertinent finding of the analysis is that the mucous flow in the airway tubes can satisfy physical constraints only through the assumption of a variable viscosity in the covering mucous blanket. The mucous viscosity must increase considerably from the low value at the cillium layer to a much higher value at the air-mucus interface.     

Microvascular blood viscosity in vivo and the endothelial surface layer

The apparent viscosity of blood in glass tubes declines with decreasing diameter (F?hraeus-Lindqvist effect) and exhibits a distinctive minimum at 6-7 microm. However, flow resistance in vivo in small vessels is substantially higher than predicted by in vitro viscosity data. The presence of a thick endothelial surface layer (ESL) has been proposed as the primary cause for this discrepancy. Here, a physical model is proposed for microvascular flow resistance as a function of vessel diameter and hematocrit in vivo; it combines in vitro blood viscosity with effects of a diameter-dependent ESL. The model was developed on the basis of flow distributions observed in three microvascular networks in the rat mesentery with 392, 546, and 383 vessel segments, for which vessel diameters, network architecture, flow velocity, and hematocrit were determined by intravital microscopy. A previously described hemodynamic simulation was used to predict the distributions of flow and hematocrit from the assumed model for effective blood viscosity. The dependence of ESL thickness on vessel diameter was estimated by minimizing deviations of predicted values for velocities, flow directions, and hematocrits from measured data. Optimal results were obtained with a layer thickness of approximately 0.8-1 microm for 10- to 40-microm-diameter vessels and declined strongly for smaller diameters, with an additional hematocrit-dependent impact on flow resistance exhibiting a maximum for approximately 10-microm-diameter vessels. These results show that flow resistance in vivo can be explained by in vitro blood viscosity and the presence of an ESL and indicate the rheologically effective thickness of the ESL in microvessels.     

A semi-empirical model of apparent blood viscosity as a function of vessel diameter and discharge hematocrit

A semi-empirical model is developed to describe the dependence of apparent viscosity of blood on vessel diameter (2.7 to 500 microns) and vessel discharge hematocrit (5% to 60%). The blood flow is modeled as a cell-rich core and a cell-free marginal layer in the larger vessels and an axial-train in the smaller vessels. Laminar (Poiseuille) flow is assumed in all cases. An equation is derived in which apparent viscosity is a function of vessel diameter, core viscosity, and width of marginal layer. This is then complemented by empirical equations in which core viscosity varies exponentially with discharge hematocrit while the width of marginal layer varies linearly with discharge hematocrit. The model correlates well with several sets of experimental data and behaves according to the Fahraeus-Lindqvist effect. Predicted apparent viscosity tends to the expected finite value for large vessel diameters. Dependence of apparent viscosity on vessel diameter is realistically smooth in the whole diameter range.     

A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall

A two-phase model for the flow of blood in narrow tubes is described. The model consists of a central core of suspended erythrocytes and a cell-free layer surrounding the core. It is assumed that the viscosity in the cell-free layer differs from that of plasma as a result of additional dissipation of energy near the wall caused by the red blood cell motion near the cell-free layer. A consistent system of nonlinear equations is solved numerically to estimate: (i) the effective dimensionless viscosity in the cell-free layer (beta), (ii) thickness of the cell-free layer (1-lambda) and (iii) core hematocrit (H(c)). We have taken the variation of apparent viscosity (mu(app)) and tube hematocrit with the tube diameter (D) and the discharge hematocrit (H(D)) from in vitro experimental studies [16]. The thickness of the cell-free layer computed from the model is found to be in agreement with the observations [3,21]. Sensitivity analysis has been carried out to study the behavior of the parameters 1-lambda, beta, H(c), B (bluntness of the velocity profile) and mu(app) with the variation of D and H(D).     

Reduction of cutaneous blood flow in cold fingers

Blood flow to fingers is reduced during cold exposure. This is generally attributed to vasoconstriction. We tested the hypothesis that increased blood viscosity, not vasoconstriction, accounts for reductions of cutaneous flow after fingers cool. Blood viscosity was higher at 10 degree C than at 27 degree C and independent of hematocrit at low shear rates. The increase of finger vascular resistance may be due to increased vascular hindrance early in cold exposure (< 15 min) and is more likely due to increased viscosity after 20-30 min, a factor that may dominate the peripheral microcirculaton during prolonged cold exposure.     

Effects of sedimentation of small red blood cell aggregates on blood flow in narrow horizontal tubes

The flow properties of aggregating red cell suspensions flowing at low flow rates through horizontal tubes are analyzed using a theoretical model. The effects of sedimentation of small aggregates, which will be formed at comparatively high flow rates, on the relative apparent viscosity are considered. In the case in which a large number of small aggregates are formed in a suspension flowing through a horizontal tube, it seems that red cells are transported as a concentrated suspension through the bottom part of the tube because of sedimentation of aggregates. A two-layer flow model is used for the distribution of red cells. It consists of plasma in the upper part and a concentrated red cell suspension in the bottom part of the tube divided by a smooth and horizontal interface. It is assumed that the suspension is a Newtonian fluid whose viscosity increases exponentially with hematocrit. The velocity distribution, the relative apparent viscosity and the flux of red cells are calculated as functions of width of plasma layer for a different discharge hematocrit. The theoretical results are compared with the results obtained from experimental data. The relative apparent viscosity increases rapidly with an increasing degree of sedimentation over a wide range of plasma layer widths.     

Two-layered blood flow in stenosed tubes for different diseases

A two-fluid model for blood flow through a stenosed tube has been developed. The model consists of a core (suspension of RBCs) and peripheral plasma layer. The core is assumed to be represented by a polar fluid and the plasma layer by a Newtonian fluid. The flow is assumed to be steady and laminar, and the fluids incompressible. The flow variables are computed for normal blood and for the cases of polycythemia, plasma cell dyscrasias and for Hb SS diseases. Resistance to flow has been computed for different stenosis length and for different stenosis height. Shear stress distribution along the axial distance has been computed for different stenosis height. The impact of size effects (particle size to tube diameter) on blood diseases is discussed.     

Effects of aggregation on the flow properties of red blood cell suspensions in narrow vertical tubes

The flow properties of aggregating red cell suspensions flowing at low rates through vertical tubes with diameters from 30 microns to 150 microns are analyzed using a theoretical model. Unidirectional flow is assumed, and the distributions of velocity and red cell concentration are assumed to be axisymmetric. A three-layer approximation is used for the distribution of red cells, with a cylindrical central core of aggregated red cells moving with uniform velocity, a cell-free marginal layer near the tube wall, and an annular region located between the core and the marginal layer containing suspended non-aggregating red cells. This suspension is assumed to behave approximately as a Newtonian fluid whose viscosity increases exponentially with red cell concentration. Physical arguments concerning the mechanics of red cell attachment to, and detachment from the aggregated core lead to a kinetic equation for core formation. From this kinetic equation and the equation for conservation of red cell volume flux, a relationship between core radius and pressure gradient is obtained. Then the relative viscosity is calculated as a function of pseudo-shear rate. At low flow rates, it is shown that the relative viscosity decreases with decreasing flow and that the dependence of relative viscosity on shear rates is more pronounced in larger tubes. It is also found that the relative viscosity decreases with increasing aggregation tendency of suspension. These theoretical predictions are in good qualitative and quantitative agreement with experimental results.     

Multiphase hemodynamic simulation of pulsatile flow in a coronary artery

A multiphase transient non-Newtonian three-dimensional (3-D) computational fluid dynamics (CFD) simulation has been performed for pulsatile hemodynamics in an idealized curved section of a human coronary artery. We present the first prediction, to the authors' knowledge, of particulate buildup on the inside curvature using the multiphase theory of dense suspension hemodynamics. In this study, the particulates are red blood cells (RBCs). The location of RBC buildup on the inside curvature correlates with lower wall shear stress (WSS) relative to the outside curvature. These predictions provide insight into how blood-borne particulates interact with artery walls and hence, have relevance for understanding atherogenesis since clinical observations show that atherosclerotic plaques generally form on the inside curvatures of arteries. The buildup of RBCs on the inside curvature is driven by the secondary flow and higher residence times. The higher viscosity in the central portion of the curved vessel tends to block their flow, causing them to migrate preferentially through the boundary layer. The reason for this is the nearly neutrally buoyant nature of the dense two-phase hemodynamic flow. The two-phase non-Newtonian viscosity model predicts greater shear thinning than the single-phase non-Newtonian model. Consequently, the secondary flow induced in the curvature is weaker. The waveforms for computed hemodynamic parameters, such as hematocrit, WSS, and viscosity, follow the prescribed inlet velocity waveforms. The lower oscillatory WSS produced on the inside curvature has implications for understanding thickening of the intimal layer.     

The clinical significance of whole blood viscosity in (cardio)vascular medicine

Whole blood is a non-Newtonian fluid, which means that its viscosity depends on shear rate. At low shear, blood cells aggregate, which induces a sharp increase in viscosity, whereas at higher shear blood cells disaggregate, deform and align in the direction of flow. Other important determinants of blood viscosity are the haematocrit, the presence of macro-molecules in the medium, temperature and, especially at high shear, the deformability of red blood cells. At the sites of severe atherosclerotic obstructions or at vasospastic locations, when change of vessel diameter is limited, blood viscosity contributes to stenotic resistance thereby jeopardising tissue perfusion. However, blood viscosity plays its most important role in the microcirculation where it contributes significantly to peripheral resistance and may cause sludging in the postcapillary venules. Apart from the direct haemodynamic significance, an increase in blood viscosity at low shear by red blood cell aggregation is also associated with increased thrombotic risk, as has been demonstrated in atrial fibrillation. Furthermore, as increased red blood cell aggregation is a reflection of inflammation, hyperviscosity has been shown to be a marker of inflammatory activity. Thus, because of its potential role in haemodynamics, thrombosis and inflammation, determination of whole blood viscosity could provide useful information for diagnostics and therapy of (cardio)vascular disease.     

Correlation of blood rheology with vascular resistance in critically ill patients.

Blood rheologic measurements together with peripheral resistance determinations in vivo were made in 27 critically ill patients. Eighteen of these patients (group I) suffered from violent trauma or operative injury and the other 9 (group II) were patients with generalized sepsis. As a result of fluid therapy all patients underwent hemodilution, resulting in a decrease in blood viscosity. This drop in blood viscosity was counteracted to some extent by an increased plasma viscosity due to elevated fibrinogen levels and a decreased red cell deformability associated with massive transfusions of stored blood. The correlation of vivo hemodynamics with blood rheological data made it possible to separate the relative roles of vascular dimensions and blood viscosity in affecting the total peripheral resistance. This approach permitted us to distinguish varying degrees of vasoconstriction in nonseptic patients in low flow states (group I) and varying degrees of vasodilation in septic patients (group II). This type of analysis serves to elucidate the pathophysiology of hemodynamic alterations in disease and provides a rational basis for devising an effective therapeutic program.     

Steady-state viscosity of the liquid two-phase disperse system water-casein-sodium alginate

The steady-state viscosity (η^*) of the liquid two-phase disperse system water-casein-sodium alginate of varying composition, with a disperse particle diameter of about 30 μm, has been investigated in the shear rate range . The flow curves obtained are similar in shape. They are invariant in composition and can be fitted to the flow curve equation where η^*₀ is the Newtonian viscosity obtained from data at low shear stresses and k is an empirical parameter. Both η^*₀ and k depend on the composition of the two-phase system.The experimental dependence of the viscosity on the composition follows a logarithmic additivity law at high shear stresses, but at low shear stresses the observed viscosities are lower than would be predicted by this law. The experimental dependence of the ratio (the mean activation volume) on the composition is also not in agreement with the additivity law. These special features of the steady-state flow of the investigated disperse system are explained by the presence of a low-concentration interphase layer having a lower viscosity and less pronounced non-Newtonian behaviour than the disperse and continuous phases. This interphase layer exists in other two-phase systems which consist of a solvent and two incompatible polymers. For this reason, the rheological behaviour of the system investigated may be expected to be common for two-phase systems of this type, particularly protein-polysaccharide mixtures in water.     

Theory for flow of Casson and Herschel-Bulkley fluids in cone-plate viscometers

Mathematical models for blood flow in cone-plate viscometer have been considered, by assuming blood as a Casson/Herschel-Bulkley fluid. Three different cases have been analyzed (i) when there is no shearing, (ii) partial shearing and (iii) full shearing. The relationships between the angular velocity and torque have been obtained for the above three cases. By assuming total shearing, the analytical expression for apparent viscosity has been obtained. Variation of apparent viscosity with yield stress, angular velocity, Casson co-efficient of viscosity, consistency index and flow behaviour index has been computed. It is observed that as the angular velocity increases, the apparent viscosity decreases for both fluids. Further, it is found that as the cone angle increases, the apparent viscosity increases. This behaviour of apparent viscosity in cone-plate viscometer is interesting and unexpected and is being reported first time.     

Reduced erythrocyte deformability in Krushinsky-Molodkina rats with acute hemorrhagic stroke

Acute hemorrhagic stroke in Krushinsky-Molodkina rats was used to assess the ability of erythrocytes to change their shape in a shear flow. Membrane rigidity and internal viscosity of erythrocytes were measured by laser diffractometry (i.e., obtaining a diffraction pattern from a thin layer of an erythrocyte suspension in a shear flow followed by computer processing of the image). The results testify to reduced deformability of erythrocytes under hemorrhagic stroke.     

The physiologic reserve in oxygen carrying capacity: studies in experimental hemodilution

The mechanisms by which the body attempts to avoid tissue hypoxia when total body oxygen delivery is compromised during acute anemia are reviewed. When the hematocrit is reduced by isovolemic hemodilution the compensatory adjustments include an increase in cardiac output, redistribution of blood flow to some tissues, and an increase in the whole body oxygen extraction ratio. These responses permit whole body oxygen uptake to be maintained until the hematocrit has been lowered to about 10%. Several factors are discussed which contribute to the increase in cardiac output during acute anemia including the reduction in blood viscosity, sympathetic innervation of the heart, and increased venomotor tone. The latter has been shown to be dependent on intact aortic chemoreceptors. With respect to peripheral vascular responses, the rise in coronary and cerebral blood flows which occur following hemodilution is proportionally greater than the increase in cardiac output while the opposite is true for kidney, liver, spleen, and intestine. Skeletal muscle does not contribute to a redistribution of blood flow to more vital areas during acute anemia despite its relatively large anaerobic capacity. Overall, peripheral compensatory adjustments result in an increased oxygen extraction ratio during acute anemia which reflects a better matching of the limited oxygen supply to tissue oxygen demands. However, some areas such as muscle are relatively overperfused which limits an even more efficient utilization of the reduced oxygen supply. Studies of the response of the microcirculation and the extent to which sympathetic vascular controls are involved in peripheral blood flow regulation are necessary to further appreciate the complex pattern of physiological responses which help ensure survival of the organism during acute anemia.