Blood flow in small curved tubes

Blood flow in small curved tubes is modeled by the two-fluid model where a relatively cell-free fluid layer envelops a fluid core of higher viscosity. The parameters in the model are successfully curve fitted to experimental data for straight tubes. The curved tube equations are then solved by perturbation theory. It was found that curvature in general lowers the tube resistance, but increases the shear stress near the inside wall.     

Collective swimming and the dynamics of bacterial turbulence

To swim, a bacterium pushes against the fluid within which it is immersed, generating fluid flow that dies off on a length scale comparable to the size of the bacterium. However, in dense colonies of bacteria, the bacteria are close enough that flow generated by swimming is substantial. For these cases, complex flows can arise due to the interaction and feedback between the bacteria and the fluid. Recent experiments on dense populations of swimming Bacillus subtilis have revealed a volume fraction-dependent transition from random swimming to transient jet and vortex patterns in the bacteria/fluid mixture. The fluid motions that are observed are reminiscent of flows that are observed around translating objects at moderate to high Reynolds numbers. In this work, I present a two-phase model for the bacterial/fluid mixture. The model explains turbulent flows in terms of the dipole stress that the bacteria exert on the fluid, entropic elasticity due to the rod shape of each bacterium, and the torque on the bacteria due to fluid gradients. Solving the equations in two dimensions using realistic parameters, the model reproduces empirically observed velocity fields. Dimensional analysis provides scaling relations for the dependence of the characteristic scales on the model parameters.     

Oscillatory flow of blood in a stenosed artery

In order to understand the abnormal flow conditions of blood in a locally constricted blood vessel, the analytical results are obtained for the oscillatory flow of blood which behaves as a Newtonian fluid. It is here assumed that the surface roughness is cosine-shaped and the maximum height of the roughness is very small compared with the radius of the unconstricted tube. Numerical solutions are presented for the instantaneous flow rate, resistive impedance, wall shear stress and phase lag.     

Oscillatory flow in thin-walled curved elastic tubes—Summary

It is relevant to study the mechanics of oscillatory flow of a viscous fluid in curved elastic tubes, as a step toward understanding the flow of blood at curvature sites in the major arteries. A numericla analysis is carried out for oscillatory flow of a viscous incompressible fluid through thin-walled elastic tubes with small curvature. Parameters in the computation are chosen to correspond to physiological flows at arterial sites of small curvature. This author was Dr. Anthony F. Bartholomay's first post-doctoral research fellow at Harvard University in 1963.     

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.     

Viscous flow simulation in a stenosis model using discrete particle dynamics: a comparison between DPD and CFD

Flow and stresses induced by blood flow acting on the blood cellular constituents can be represented to a certain extent by a continuum mechanics approach down to the order of the?μm level. However, the molecular effects of, e.g., adhesion/aggregation bonds of blood clotting can be on the order of nm. The coupling of the disparate length and timescales between such molecular levels and macroscopic transport represents a major computational challenge. To address this challenge, a multiscale numerical approach based on discrete particle dynamics (DPD) methodology derived from molecular dynamics (MD) principles is proposed. The feasibility of the approach was firstly tested for its ability to simulate viscous flow conditions. Simulations were conducted in low Reynolds numbers flows (Re = 25–33) through constricted tubes representing blood vessels with various degrees of stenosis. Multiple discrete particles interacting with each other were simulated, with 1.24–1.36 million particles representing the flow domain and 0.4 million particles representing the vessel wall. The computation was carried out on the massive parallel supercomputer NY BlueGene/L employing NAMD-a parallel MD package for high performance computing (HPC). Typical recirculation zones were formed distal to the stenoses. The velocity profiles and recirculation zones were in excellent agreement with computational fluid dynamics (CFD) 3D Navier–Stokes viscous fluid flow simulations and with classic numerical and experimental results by YC Fung in constricted tubes. This feasibility analysis demonstrates the potential of a methodology that widely departs from a continuum approach to simulate multiscale phenomena such as flow induced blood clotting.     

Flutter in collapsible tubes: a theoretical model of wheezes

A mathematical analysis of flow through a flexible channel is examined as a model of flow-induced flutter oscillations that pertain to the production of wheezing breath sounds. The model provides predictions for the critical fluid speed that will initiate flutter waves of the wall, as well as their frequency and wavelength. The mathematical results are separated into linear theory (small oscillations) and nonlinear theory (larger oscillations). Linear theory determines the onset of the flutter, whereas nonlinear theory determines the relationships between the fluid speed and both the wave amplitudes and frequencies. The linear theory predictions correlate well with data taken at the onset of flutter and flow limitation during experiments of airflow in thick-walled collapsible tubes. The nonlinear theory predictions correlate well with data taken as these flows are forced to higher velocities while keeping the flow rate constant. Particular ranges of the parameters are selected to investigate and discuss the applications to airway flows. According to this theory, the mechanism of generation of wheezes is based in the interactions of fluid forces and friction and wall elastic-restoring forces and damping. In particular, a phase delay between the fluid pressure and wall motion is necessary. The wave speed theory of flow limitation is discussed with respect to the specific data and the flutter model.     

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.     

Pulsatile flow in a constricted channel.

A nonuniform channel is used as a simple model of a constricted arterial vessel. Flow patterns have been calculated for pulsatile flow with both sinusoidal and nonsinusoidal flow rates for a range of Reynolds number, Re, and Strouhal number, St. The results show that even for relatively low frequency flows a strong vortex wave will be generated with a complex wall shear stress distribution and peak values much greater than those found in steady or unsteady parallel flow. The vortex wave increases in strength with increasing Re and St, with its total length and wavelength independent of Re but inversely proportional to St. The form of the imposed flow rate is found to have an important effect on the flow and the shear stress distribution.     

The decomposition of apparent stresses in disturbed pulsatile flow in the presence of large scale organized structures

Flow disturbance phenomena that occur in unsteady-in-the-mean flows (i.e. pulsatile or oscillating) at moderate Reynolds numbers are analyzed in both the time domain and the frequency domain. The analysis utilizes variable decomposition into a time-varying underlying waveform and flow disturbances which are composed of large scale organized structures and random fluctuations. A practical technique which incorporates time domain phase conditioning, trend removal, and frequency domain matched filtering, is presented and examined using simulated data of known statistical behavior. The applicability of the method is shown by the decomposition of the simulated data and the technique is then applied to experimental data obtained in pulsatile flow through a constricted tube by means of a laser Doppler anemometer. The cross-sectional area reduction at the constriction throat was 90%. The Womersley parameter in the experiments was 5.3 and the Reynolds number based on the average flow rate per cycle was 300 with a minimum/maximum value of 55/600 based on the instantaneous flow rate. Measurements were taken in the flow region downstream of the constriction throat which included several interesting flow disturbance phenomena. The results of the decomposed flow phenomena demonstrate the significant role of large scale organized structures in such flows. This is particularly important when analyzing blood flow in the large arteries in the presence of severe stenosis or behind prosthetic devices in an attempt to estimate the 'turbulent' stress which act on cellular elements. Estimation of the apparent stress tensor is of importance in an effort to elucidate the mechanical factors which influence the durability of red blood cells under abnormal conditions.     

Sheared bioconvection in a horizontal tube

The recent interest in using microorganisms for biofuels is motivation enough to study bioconvection and cell dispersion in tubes subject to imposed flow. To optimize light and nutrient uptake, many microorganisms swim in directions biased by environmental cues (e.g. phototaxis in algae and chemotaxis in bacteria). Such taxes inevitably lead to accumulations of cells, which, as many microorganisms have a density different to the fluid, can induce hydrodynamic instabilites. The large-scale fluid flow and spectacular patterns that arise are termed bioconvection. However, the extent to which bioconvection is affected or suppressed by an imposed fluid flow and how bioconvection influences the mean flow profile and cell transport are open questions. This experimental study is the first to address these issues by quantifying the patterns due to suspensions of the gravitactic and gyrotactic green biflagellate alga Chlamydomonas in horizontal tubes subject to an imposed flow. With no flow, the dependence of the dominant pattern wavelength at pattern onset on cell concentration is established for three different tube diameters. For small imposed flows, the vertical plumes of cells are observed merely to bow in the direction of flow. For sufficiently high flow rates, the plumes progressively fragment into piecewise linear diagonal plumes, unexpectedly inclined at constant angles and translating at fixed speeds. The pattern wavelength generally grows with flow rate, with transitions at critical rates that depend on concentration. Even at high imposed flow rates, bioconvection is not wholly suppressed and perturbs the flow field.     

Mass transfer to fluids flowing through rotating nonaligned straight tubes

Relatively inefficient heat/mass transfer is characteristic of tubular devices if the Reynolds number is low. One method of improving the heat/mass transfer efficiency of such devices is by inducing transverse laminar secondary circulations that are superimposed on the primary flow field; the resulting transverse velocity components lead to fluid mixing and hence augmented mass transfer in the tube lumen. The present work is a theoretical and experimental investigation of the enhanced transport in rotating, nonaligned, straight tubes, a method of transport enhancement that utilizes Coriolis acceleration to create transverse fluid mixing. This technique couples the transport advantages of coiled tubes with the design advantages of straight tubes. The overall mass balance equation is numerically solved for transfer into fluids flowing steadily through rotating nonaligned straight tubes. This solution, for small Coriolis disturbances, incorporates a third order perturbation solution for the primary and secondary flow fields. For sufficiently small Coriolis disturbances the bulk concentration increase is found to be uniquely determined by the value of a single similarity parameter. As the Coriolis disturbance is increased, however, two additional parameters are required to accurately characterize the mass transfer. In general, increasing the Coriolis accelerations results in an increase in mass transfer. There are solution regimens, however, in which increasing this acceleration can lead to a decrease in mass transfer efficiency. This interesting phenomena, which has important design implications, appears to result from velocity-weighting effects on the exiting sample. Experiments, involving the measurement of oxygen transferred into water and blood, produced data that agree with the theoretical predictions.     

Effects of helium-oxygen mixtures on endotracheal tubes: an in vitro study

QUESTION: To determine flow pattern and critical Reynolds numbers in endotracheal tubes submitted to different helium-oxygen mixtures under laboratory conditions. MATERIALS AND METHODS: Flow-pressure relationships were performed for seven endotracheal tubes (rectilinear position, entry length applied) with distal end open to atmosphere (predicted internal diameters: 6-9 mm). Nine helium-oxygen mixtures were tested, with FIHe varying from zero to 0.78 (increment: 10%). Nine flows were tested, with rates varying from 0.25 to 1.60 l s(-1) (increment: 0.15 l s(-1)). Gas flow resistance was calculated, and for each endotracheal tube, a Moody diagram was realised. Flow regime and critical Reynolds numbers were then determined (fully established laminar, nonestablished laminar, smooth turbulent, or rough). RESULTS: Even low concentration of helium in inspiratory mixture reduces endotracheal tubes resistance. Effect is maximal for high flows, small tube and high FIHe. Critical Reynolds numbers are inversely correlated to tube diameter. ANSWER: Under laboratory conditions, flow pattern in endotracheal tubes varies from fully established laminar to rough. Knowledge of the critical Reynolds numbers allows correct application of fluid mechanic formula when studying tube or gaseous mixture effects on respiratory mechanisms.     

The effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes

An analytical solution for pulsatile flow of a generalized Maxwell fluid in straight rigid tubes, with and without axial vessel motion, has been used to calculate the effect of blood viscoelasticity on velocity profiles and shear stress in flows representative of those in the large arteries. Measured bulk flow rate Q waveforms were used as starting points in the calculations for the aorta and femoral arteries, from which axial pressure gradient ▿P waves were derived that would reproduce the starting Q waves for viscoelastic flow. The ▿P waves were then used to calculate velocity profiles for both viscoelastic and purely viscous flow. For the coronary artery, published ▿P and axial vessel acceleration waveforms were used in a similar procedure to determine the separate and combined influences of viscoelasticity and vessel motion.Differences in local velocities, comparing viscous flow to viscoelastic flow, were in all cases less than about 2% of the peak local velocity. Differences in peak wall shear stress were less than about 3%.In the coronary artery, wall shear stress differences between viscous and viscoelastic flow were small, regardless of whether axial vessel motion was included. The shape of the wall shear stress waveform and its difference, however, changed dramatically between the stationary and moving vessel cases. The peaks in wall shear stress difference corresponded with large temporal gradients in the combined driving force for the flow.     

The impedance of curved artery models

The impedance (pressure drop/flow rate) of four curved artery models has been determined experimentally for steady and periodic flows simulating conditions in the aortic arch. Steady flow results indicate that very short entry lengths are required for flow development in curved artery models, and impedance is elevated above straight tube values by a factor of 3-4 for mean flow conditions in the aortic arch. Results for periodic flow with a nonzero mean show a significant elevation of mean flow impedance relative to values for steady flow at the mean flow rate--a factor of 2-3 for aortic arch flow conditions. The impedance of the first harmonic of periodic flows follows straight tube theory at high values of the unsteadiness parameter in agreement with available theory for curved tubes. The implications of the impedance measurements for wall shear stress in the aortic arch are discussed.     

Flow parameters associated with hydrodynamic cell injury

Two flow parameters are proposed for the analysis of flows that have potential to damage animal cells. They are the state of stress (characterized by the second invariant of the stress tensor) and the flow classification parameter R(D) (which is related to the possibility of stress relaxation). We consider the flow that occurs when a 1.7-mm bubble collapses at a liquid interface. Using these two parameters, we show the regions in which the flow is strong in terms of high hydrodynamic stresses and elongation characteristics. (c) 1994 John Wiley & Sons, Inc.     

Numerical analysis of flow in an elastic artery model.

Oscillatory and pulsatile flows of Newtonian fluids in straight elastic tubes are simulated numerically with the aid of Ling and Atabek's "local flow" assumption for the nonlinear convective acceleration terms. For the first time, a theoretical assessment of the local flow assumption is presented, and the range of validity of the assumption is estimated by comparison with perturbation solutions of the complete flow problem. Subsequent simulations with the local flow model indicate that the flow field and associated wall shear stress are extremely sensitive to the phase angle between oscillatory pressure and flow waves (impedance phase angle). This phase angle, which is a measure of the wave reflection present in the system, is known to be altered by arterial disease (e.g., hypertension) and vasoactive drugs. Thus, the paper elucidates a mechanism by which subtle changes in systemic hemodynamics (i.e., phase angles) can markedly influence local wall shear stress values.     

Micromechanically based poroelastic modeling of fluid flow in Haversian bone

To explore the hypothesis that load-induced fluid flow in bone is a mechano-transduction mechanism in bone adaptation, unit cell micro-mechanical techniques are used to relate the microstructure of Haversian cortical bone to its effective poroelastic properties. Computational poroelastic models are then applied to compute in vitro Haversian fluid flows in a prismatic specimen of cortical bone during harmonic bending excitations over the frequency range of 10(0) to 10(6) Hz. At each frequency considered, the steady state harmonic response of the poroelastic bone specimen is computed using complex frequency-domain finite element analysis. At the higher frequencies considered, the breakdown of Poisueille flow in Haversian canals is modeled by introduction of a complex fluid viscosity. Peak bone fluid pressures are found to increase linearly with loading frequency in proportion to peak bone stress up to frequencies of approximately 10 kHz. Haversian fluid shear stresses are found to increase linearly with excitation frequency and loading magnitude up until the breakdown of Poisueille flow. Tan delta values associated with the energy dissipated by load-induced fluid flow are also compared with values measured experimentally in a concurrent broadband spectral analysis of bone. The computational models indicate that fluid shear stresses and fluid pressures in the Haversian system could, under physiologically realistic loading, easily reach the level of a few Pascals, which have been shown in other works to elicit cell responses in vitro.     

Characterization of erythrocyte membrane tension for hemolysis prediction in complex flows

Hemolysis is a persistent issue with blood-contacting devices. Many experimental and theoretical contributions over the last few decades have increased insight into the mechanisms of hemolysis in both laminar and turbulent flows, with the ultimate goal of developing a comprehensive, mechanistic hemolysis model. Many models assume that hemolysis scales with a resultant, scalar stress representing all components of the fluid stress tensor. This study critically evaluates this scalar stress hypothesis by calculating the response of the red blood cell membrane to different types of fluid stress (laminar shear and extension, and three turbulent shear and extension cases), each with the same scalar stress. It was found that even though the scalar stress is the same for all cases, membrane tension varied by up to three orders of magnitude. In addition, extensional flow causes constant tension, while tank-treading in shear flow causes periodic tension, with tank-treading frequency varying by three orders of magnitude among the cases. For turbulent flow, tension also depends on eddy size. It is concluded, therefore, that scalar stress alone is inadequate for scaling hemolysis. Fundamental investigations are needed to establish a new index of the fluid stress tensor that provides reliable hemolysis prediction across the wide range of complex flows that occur in cardiovascular devices.     

Fluid-induced osteolysis: modelling and experiments

A model to calculate bone resorption driven by fluid flow at the bone-soft tissue interface is developed and used as a basis for computer calculations, which are compared to experiments where bone is subjected to fluid flow in a rat model. Previous models for bone remodelling calculations have been based on the state of stress, strain or energy density of the bone tissue as the stimulus for remodelling. We believe that there is experimental support for an additional pathway where an increase in the amount of the cells directly involved in bone removal, the osteoclasts, is caused by fluid pressure, flow velocity or other parameters related to fluid flow at the bone-soft tissue interface, resulting in bone resorption.