Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

Blood flow multiscale phenomena

The cardiovascular disease is one of most frequent cause deaths in modern society. The objective of this work is analyse the effect of dynamic vascular geometry (curvature, torsion, bifurcation) and pulsatile blood nature on secondary flow, wall shear stress and platelet deposition. The problem was examined as multi-scale physical phenomena using perturbation analysis and numerical modelling. The secondary flow determined as influence pulsatile pressure, vascular tube time-dependent bending and torsion on the main axial flow. Bifurcation and branching phenomena are analysed experimentally through, blood-like fluid pulsatile flow across elastic rubber-like Y-model model. The problem complex geometry near branching in platelet deposit modelling is resolved numerically as Falker-Skan flow.     

Flow of a newtonian fluid through a permeable tube: The application to the proximal renal tubule

Creeping flow of a Newtonian fluid through a rigid permeable tube is considered and the transmural seepage is assumed to obey Darcy's law. Closed-form solutions for the pressure and velocity fields are presented and equations describing the axial variation of the mean cross-sectional pressure, the axial volumetric flow and the transmural fluid flux are derived. Approximate solutions for small seepage rates are given and are applied to the flow in the proximal renal tubule. Probable values for the epithelium permeability and the intraluminal hydrostatic pressure drop are obtained.     

The "black hole" phenomenon in ultrasonic backscattering measurement under pulsatile flow with porcine whole blood in a rigid tube

The "black hole" phenomenon was further investigated with porcine whole blood under pulsatile flow conditions in a straight rigid tube 120 cm long and of 0.95 cm diameter. A modified Aloka 280 commercial scanner with a 7.5 MHz linear array was used to collect the radio frequency (RF) signal of backscattering echoes from the blood inside the tube. The transducer was located downstream from the entrance and parallel to the longitudinal direction of the tube. The experimental results showed that higher hematocrits enhanced the black hole phenomenon, leading to a more apparent and larger diameter black hole. The black hole was not apparent at hematocrits below 23%. The highest hematocrit used in the experiment was 60%. Beat rates of 20, 40 and 60 beats per minute (bpm) were used, and the black hole became weaker in amplitude and smaller in diameter when the peak flow velocity was increased at each beat rate. These results are consistent with the suggestion in previous work that the black hole arises from insufficient aggregation of red blood cells (RBCs) at the center of the tube because of the low shear rate. At 20 and 40 bpm, the peak flow velocity ranges were 10 approximately 25 cm/s and 18 approximately 27 cm/s, respectively. The black hole was very clear at the minimal peak flow velocity but almost disappeared at the maximal velocities for each beat rate. At 60 bpm, experiments were only performed at one peak flow velocity of 31 cm/s and the black hole was clear. The results showed that the black hole was more pronounced at higher beat rates when the peak velocity was the same. This phenomenon cannot be explained by previous hypotheses. Acceleration seems to be the only flow parameter that varies at different beat rates when peak velocities are the same. Therefore, the influence of acceleration on the structural organization and orientation of RBC rouleaux might be another factor involved in the formation of the black hole in addition to the shear rate. As the entrance length was changed from 110 to 15 diameters (D) in seven steps at the hematocrit of 60%, it was found that a position farther downstream yielded a black hole with a greater contrast relative to the surrounding region, while the backscattering power at the central hypoechoic zone did not increase with increasing entrance length.     

Effect of geometric variations on pressure loss for a model bifurcation of the human lung airway

Characteristics of pressure loss (ΔP) in human lung airways were numerically investigated using a realistic model bifurcation. Flow equations were numerically solved for the steady inspiratory condition with the tube length, the branching angle and flow velocity being varied over a wide range. In general, the ΔP coefficient K showed a power-law dependence on Reynolds number (Re) and length-to-diameter ratio with a different exponent for Re≥100 than for Re<100. The effect of different branching angles on pressure loss was very weak in the smooth-branching airways.     

Influence of model boundary conditions on blood flow patterns in a patient specific stenotic right coronary artery

Background

In literature, the effect of the inflow boundary condition was investigated by examining the impact of the waveform and the shape of the spatial profile of the inlet velocity on the cardiac hemodynamics. However, not much work has been reported on comparing the effect of the different combinations of the inlet/outlet boundary conditions on the quantification of the pressure field and flow distribution patterns in stenotic right coronary arteries.

Method

Non-Newtonian models were used to simulate blood flow in a patient-specific stenotic right coronary artery and investigate the influence of different boundary conditions on the phasic variation and the spatial distribution patterns of blood flow. The 3D geometry of a diseased artery segment was reconstructed from a series of IVUS slices. Five different combinations of the inlet and the outlet boundary conditions were tested and compared.

Results

The temporal distribution patterns and the magnitudes of the velocity, the wall shear stress (WSS), the pressure, the pressure drop (PD), and the spatial gradient of wall pressure (WPG) were different when boundary conditions were imposed using different pressure/velocity combinations at inlet/outlet. The maximum velocity magnitude in a cardiac cycle at the center of the inlet from models with imposed inlet pressure conditions was about 29% lower than that from models using fully developed inlet velocity data. Due to the fact that models with imposed pressure conditions led to blunt velocity profile, the maximum wall shear stress at inlet in a cardiac cycle from models with imposed inlet pressure conditions was about 29% higher than that from models with imposed inlet velocity boundary conditions. When the inlet boundary was imposed by a velocity waveform, the models with different outlet boundary conditions resulted in different temporal distribution patterns and magnitudes of the phasic variation of pressure. On the other hand, the type of different boundary conditions imposed at the inlet and the outlet did not have significant effect on the spatial distribution patterns of the PD, the WPG and the WSS on the lumen surface, regarding the locations of the maximum and the minimum of each quantity.

Conclusions

The observations from this study indicated that the ways how pressure and velocity boundary conditions are imposed in computational models have considerable impact on flow velocity and shear stress predictions. Accuracy of in vivo measurements of blood pressure and velocity is of great importance for reliable model predictions.

     

Analysis of the Velocity Distribution in Partially-Filled Circular Pipe Employing the Principle of Maximum Entropy

The flow velocity distribution in partially-filled circular pipe was investigated in this paper. The velocity profile is different from full-filled pipe flow, since the flow is driven by gravity, not by pressure. The research findings show that the position of maximum flow is below the water surface, and varies with the water depth. In the region of near tube wall, the fluid velocity is mainly influenced by the friction of the wall and the pipe bottom slope, and the variation of velocity is similar to full-filled pipe. But near the free water surface, the velocity distribution is mainly affected by the contractive tube wall and the secondary flow, and the variation of the velocity is relatively small. Literature retrieval results show relatively less research has been shown on the practical expression to describe the velocity distribution of partially-filled circular pipe. An expression of two-dimensional (2D) velocity distribution in partially-filled circular pipe flow was derived based on the principle of maximum entropy (POME). Different entropies were compared according to fluid knowledge, and non-extensive entropy was chosen. A new cumulative distribution function (CDF) of partially-filled circular pipe velocity in terms of flow depth was hypothesized. Combined with the CDF hypothesis, the 2D velocity distribution was derived, and the position of maximum velocity distribution was analyzed. The experimental results show that the estimated velocity values based on the principle of maximum Tsallis wavelet entropy are in good agreement with measured values.     

Hydrodynamics of arterial branching--the effect of arterial branching on distal blood supply

A study was conducted to investigate the hydrodynamics of branching flow in relation to the blood supply to the basal part of the brain. A series of measurements of the branching loss-coefficients under laminar steady flow were conducted using model branches with various geometries, and the effect of branching on blood supply to distal areas was described using a lumped-parameter model of the vascular structure. It was revealed that in the blood circulation, branching loss is important where a small artery divides off with a large branching angle from a large trunk. It was also indicated that the effect of such branching on the distal blood supply might become more significant when the peripheral resistance is reduced, thereby increasing the blood velocity in the trunk.     

Numerical simulation of sinusoidal flow in a straight elastic tube: effects of phase angles

Numerical simulations of flow in straight elastic (moving wall) tubes subjected to a sinusoidal pressure gradient were performed for conditions prevailing in large and medium sized arteries. The effects of varying the phase angle between the pressure gradient and the tube radius, the amplitude of wall motion, and the unsteadiness parameter (alpha) on flow rate and wall shear stress were investigated. Mean and peak flow rates and shear stresses were found to be strongly affected by the phase angle between the pressure gradient and the tube radius with greater sensitivity at higher diameter variation and higher alpha. In large artery simulations (alpha = 12), means flow rate was found to be 60% higher and peak flow rate to be 73% higher than corresponding rigid tube values for certain phase angles, while a threefold increase in mean wall shear stress and sevenfold increase in peak wall shear stress were observed in a sensitive phase angle range. Significant reversal in the wall shear stress direction occurred in the sensitive phase angle range even when there was negligible flow rate reversal. All effects were greatly diminished in simulations of medium sized vessels (alpha = 4). Some experimental evidence to support the predictions of a strong effect of phase angle on wall shear stress in large vessels is presented. Finally, physiological implications of the present work are discussed from a basis of aortic input impedance data, and a physical explanation for the extreme sensitivity of the flow field to small amplitude wall motion at high alpha is given.     

A numerical calculation of flow in a curved tube model of the left main coronary artery

The flow pattern in the left main coronary artery has been calculated using an idealized geometry and by numerically solving the full Navier-Stokes equations for a Newtonian fluid. Two different forms for the entrance velocity profile were used, one a time-varying, flat profile and the other a time-varying, less flat velocity profile. The results obtained demonstrate the presence of secondary motions for conditions simulating flow in the left main coronary artery, with maximum secondary flow velocities being on the order of three to four percent of the maximum axial velocity. This secondary flow phenomenon has an important influence on the wall shear stress distribution, in spite of the fact that there is virtually no alteration in the axial velocity profile. The maximum ratio of the outer wall shear stress to that on the inner wall is 1.4 at a Reynolds number of Re = 270, and it increases with increasing Reynolds number, reaching a value of 1.7 at Re = 810. Although there are significant differences in the results in the immediate vicinity of the inlet for the two different forms of the entrance velocity profile used, this difference does not persist far into the tube. Independent of the choice of the entrance velocity profile, it appears that there will be significant secondary flow effects on the wall shear stress.     

On connecting large vessels to small. The meaning of Murray's law

A large part of the branching vasculature of the mammalian circulatory and respiratory systems obeys Murray's law, which states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughters. Where this law is obeyed, a functional relationship exists between vessel radius and volumetric flow, average linear velocity of flow, velocity profile, vessel-wall shear stress, Reynolds number, and pressure gradient in individual vessels. In homogeneous, full-flow sets of vessels, a relation is also established between vessel radius and the conductance, resistance, and cross- sectional area of a full-flow set.     

Spatial and temporal variation of secondary flow during oscillatory flow in model human central airways

Axial and secondary velocity profiles were measured in a model human central airway to clarify the oscillatory flow structure during high-frequency oscillation. We used a rigid model of human airways consisting of asymmetrical bifurcations up to third generation. Velocities in each branch of the bifurcations were measured by two-color laser-Doppler velocimeter. The secondary velocity magnitudes and the deflection of axial velocity were dependent not only on the branching angle and curvature ratio of each bifurcation, but also strongly depended on the shape of the path generated by the cascade of branches. Secondary flow velocities were higher in the left bronchus than in the right bronchus. This spatial variation of secondary flow was well correlated with differing gas transport rates between the left and right main bronchus.     

A problem in the mathematical biophysics of blood circulation: II. Relation between pressure and flow of a viscous fluid in an elastic distensible tube

In an elastic distensible tube, like a blood vessel, the radius is determined by the equality of the hydrostatic pressure and the elastic forces. If a viscous fluid flows through such a tube, there is a pressure drop along the line of flow. This results in a variation of the radius of the tube along the axis. An approximate expression, valid within a limited range of values, is derived for the radius of the tube as a function of the distance along the axis. Another approximate expression is derived for the relation between pressure drop and total flow in such a case. For sufficiently high rates of flow the pressure drop does not vary linearly with the flow, as in the usual poiseuille's law, but more rapidly.     

An apparatus to study the response of cultured endothelium to shear stress

An apparatus which has been developed to study the response of cultured endothelial cells to a wide range of shear stress levels is described. Controlled laminar flow through a rectangular tube was used to generate fluid shear stress over a cell-lined coverslip comprising part of one wall of the tube. A finite element method was used to calculate shear stresses corresponding to cell position on the coverslip. Validity of the finite element analysis was demonstrated first by its ability to generate correctly velocity profiles and wall shear stresses for laminar flow in the entrance region between infinitely wide parallel plates (two-dimensional flow). The computer analysis also correctly predicted values for pressure difference between two points in the test region of the apparatus for the range of flow rates used in these experiments. These predictions thus supported the use of such an analysis for three-dimensional flow. This apparatus has been used in a series of experiments to confirm its utility for testing applications. In these studies, endothelial cells were exposed to shear stresses of 60 and 128 dynes/cm2. After 12 hr at 60 dynes/cm2, cells became aligned with their longitudinal axes parallel to the direction of flow. In contrast, cells exposed to 128 dynes/cm2 required 36 hr to achieve a similar reorientation. Interestingly, after 6 hr at 128 dynes/cm2, specimens passed through an intermediate phase in which cells were aligned perpendicular to flow direction. Because of its ease and use and the provided documentation of wall shear stress, this flow chamber should prove to be a valuable tool in endothelial research related to atherosclerosis.     

Velocity field of pulsatile flow in a porous tube

This paper describes velocity fields for fully developed periodic laminar flow in a rigid tube with a porous wall. We obtained an analytical solution of the flow by the linear approximation of the Navier-Stokes equation. Unlike the previous works with a constant seepage rate along the axis, we used a wall velocity which contained hydraulic permeation constant Lp. The axial velocity profile shows a local maximum velocity near the wall at a large Womersley number alpha. This suggests that concentration polarization in porous tubular membrane may be reduced at high frequencies if a membrane device is operated under pulsatile flow conditions. The magnitude of wall permeation velocity decreases linearly along the tube axis because the damping of the pressure difference between the inside and the outside of the tube is very small.     

Purification of exhaust air containing organic pollutants in a trickle-bed bioreactor

Summary This paper reports experiments on the purification of exhaust air containing organic pollutants by a new biological process using a trickle-bed reactor. Pollutant-specific microorganisms in high concentration were fixed to a suitable bed. The absorption and conversion of propionaldehyde as a model pollutant was measured by systematic variation of the gas and liquid flow rates in the reaction system. At a space velocity of 1000 h^–1, it was possible to achieve conversion rates of between 68 and 96%, depending on the trickling density. The degradation capacity of the biological trickle bed is over 500 g propionaldehyde/m³ of reactor per hour. By using a tube bundle (honeycomb tube), it was possible to ensure continuous operation of the reactor with reduced conversion and pressure loss.Dedicated to Professor Dr. Dr. D. Behrens on the occasion of 65th birthday     

Mass transfer in an airlift reactor with a net draft tube.

Gas-liquid mass transfer in an airlift reactor with net draft tube is investigated. The effects of both the ratio of draft tube to reactor diameter and the reactor pressure on oxygen transfer are considered. The value of the volumetric mass transfer coefficient, kLa, increases with a decreasing diameter ratio at higher air flow rates. The correlation of volumetric mass transfer coefficient with respect to the true superficial air velocity under different reactor pressures is determined. The kLa value decreases with increasing reactor pressure.     

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.     

Peristaltic flow of a couple stress fluid in an asymmetric channel

This paper presents an analysis of the peristaltic flow of a couple stress fluid in an asymmetric channel. The asymmetric nature of the flow is introduced through the peristaltic waves of different amplitudes and phases on the channel walls. Mathematical modelling corresponding to a two-dimensional flow has been carried out. The flow analysis is presented under long wavelength and low Reynolds number approximations. Closed form solutions for the axial velocity, stream function and the axial pressure gradient are given. Numerical computations have been carried out for the pressure rise per wavelength, friction forces and trapping. It is noted that there is a decrease in the pressure when the couple stress fluid parameter increases. The variation of the couple stress fluid parameter with the size of the trapped bolus is also similar to that of pressure. Furthermore, the friction force on the lower channel wall is greater than that on the upper channel wall.     

Numerical Simulation of Blood Flow Through Microvascular Capillary Networks

A numerical method is implemented for computing blood flow through a branching microvascular capillary network. The simulations follow the motion of individual red blood cells as they enter the network from an arterial entrance point with a specified tube hematocrit, while simultaneously updating the nodal capillary pressures. Poiseuille’s law is used to describe flow in the capillary segments with an effective viscosity that depends on the number of cells residing inside each segment. The relative apparent viscosity is available from previous computational studies of individual red blood cell motion. Simulations are performed for a tree-like capillary network consisting of bifurcating segments. The results reveal that the probability of directional cell motion at a bifurcation (phase separation) may have an important effect on the statistical measures of the cell residence time and scattering of the tube hematocrit across the network. Blood cells act as regulators of the flow rate through the network branches by increasing the effective viscosity when the flow rate is high and decreasing the effective viscosity when the flow rate is low. Comparison with simulations based on conventional models of blood flow regarded as a continuum indicates that the latter underestimates the variance of the hematocrit across the vascular tree.