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.     

Microcontinuum model for pulsatile blood flow through a stenosed tube

The effects of polar nature of blood and pulsatility on flow through a stenosed tube have been analysed by assuming blood as a micropolar fluid. Linearized solutions of basic equations are obtained through consecutive applications of finite Hankel and Laplace transforms. The analytical expressions for axial and particle angular velocities, wall shear stress, resistance to flow and apparent viscosity have been obtained. The axial velocity profiles for Newtonian and micropolar fluids have been compared. The interesting observation of this analysis is velocity, in certain parts of cycle, for micropolar fluid is higher than Newtonain fluid. Variation of apparent viscosity eta a with tube radius shows both inverse Fahraeus-Lindqvist and Fahraeus-Lindqvist effects. Finally, the resistance to flow and wall shear stress for normal and diseased blood have been computed and compared.     

Plug Effect of Erythrocytes in Capillary Blood Vessels

As an idealized problem of the motion of blood in small capillary blood vessels, the low Reynolds number flow of plasma (a newtonian fluid) in a circular cylindrical tube involving a series of circular disks is studied. It is assumed in this study that the suspended disks are equally spaced along the axis of the tube, and that their centers remain on the axis of the tube and that their faces are perpendicular to the tube axis. The inertial force of the fluid due to the convective acceleration is neglected on the basis of the smallness of the Reynolds number. The solution of the problem is derived for a quasi-steady flow involving infinitesimally thin disks. The numerical calculation is carried out for a set of different combinations of the interdisk distance and the ratio of the disk radius to the tube radius. The ratio of the velocity of the disk to the average velocity of the fluid is calculated. The different rates of transport of red blood cells and of plasma in capillary blood vessels are discussed. The average pressure gradient along the axis of the tube is computed, and the dependence of the effective viscosity of the blood on the hematocrit and the diameter of the capillary vessel is discussed.     

Spatial velocity profile in mouse embryonic aorta and Doppler-derived volumetric flow: a preliminary model

Characterizing embryonic circulatory physiology requires accurate cardiac output and flow data. Despite recent applications of high-frequency ultrasound Doppler to the study of embryonic circulation, current Doppler analysis of volumetric flow is relatively crude. To improve Doppler derivation of volumetric flow, we sought a preliminary model of the spatial velocity profile in the mouse embryonic dorsal aorta using ultrasound biomicroscopy (UBM)-Doppler data. Embryonic hematocrit is 0.05-0.10 so rheologic properties must be insignificant. Low Reynolds numbers (<500) and Womersley parameters (<0.76) suggest laminar flow. UBM demonstrated a circular dorsal aortic cross section with no significant tapering. Low Dean numbers (<100) suggest the presence of minimal skewing of the spatial velocity profile. The inlet length allows for fully developed flow. There is no apparent aortic wall pulsatility. Extrapolation of prior studies to these vessel diameters (300-350 microm) and flow velocities (~50-200 mm/s) suggests parabolic spatial velocity profiles. Therefore, mouse embryonic dorsal aortic blood flow may correspond to Poiseuille flow in a straight rigid tube with parabolic spatial velocity profiles. As a first approximation, these results are an important step toward precise in utero ultrasound characterization of blood flow within the developing mammalian circulation.     

The surface-tension-driven flow of blood from a droplet into a capillary tube

In tissue, medical, or dental engineering, when blood comes into contact with a new artificial material, the flow may be influenced by surface tension between the blood and the surface of the material. The effect of surface tension on the flow of blood is significant, especially in microscale. The leading edge of the flowing blood is the triple point where the blood, the material surface, and a stationary gas orfluid meet. The movement of the triple point, i.e., the advancing front of the flow, is driven by surface tension, resisted by viscous shear stress, and balanced by the inertial force (-mass x acceleration). In this article, the dynamics is illustrated in detail in the case of blood flowing into a capillary tube by contact. The capillar, tube draws the blood into it. It is shown theoretically that initially the flow of blood in the capillary has a large acceleration, followed by a relatively large deceleration over the next short period of time, then the acceleration becomes small and oscillatory. The velocity history appears impulsive at first, then slows down. The history of the length of blood column appears smooth after integration. Existing solutions of the Navier-Stokes equation permit the analysis of simpler cases. Further fluid mechanics development is needed to meet the practical needs of bioengineering. The importance of experimental study of surface tension and contact angle over a biological surface or a man-made material as a future direction of research is pointed out.     

Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals.     

Shear-layer detection in poststenotic flow by spectrum analysis of Doppler signals

Spectrum analysis of the Doppler signals was performed 0.5 tube diameters downstream from an axisymmetric constriction with an area reduction of 80 percent in steady flow at a jet Reynolds number of 2840. Both pulsed and continuous wave (CW) Doppler spectra showed significant reverse flow components in the separated flow. The pulsed Doppler spectra exhibited sudden changes when the sample volume crossed the shear layer between the center jet and the separated flow. A power spectrum equation was theoretically derived from continuity of flow to define the Doppler shift frequency for the shear layer velocity. The CW Doppler spectrum showed a minimum spectrum density at a frequency which equalled the shear layer Doppler shift frequency derived from the equation. The pulsed spectra exhibited the sudden changes at the same frequency as well.     

Stenotic effects in a tube of elliptic cross-section at low reynolds numbers

With an objective to understanding arteriosclerosis, the blood flow in a cylindrical tube with local constriction is analysed. The cross-section of the tube is an ellipse, the axes of which are in an arbitrary position with respect to the axis of the tube. Blood is taken to be a Newtonian and homogeneous fluid. The cross-sectional area varies slowly with the longitudinal distance and the area change is so adjusted to take account of stenosis. The transverse velocity field and the effects of inertia on the primary velocity and pressure distribution are calculated to a first order in the relevant small parameter and effects of asymmetry on the wall shear stress and impedance are presented.     

Turbulence modeling in three-dimensional stenosed arterial bifurcations

Under normal healthy conditions, blood flow in the carotid artery bifurcation is laminar. However, in the presence of a stenosis, the flow can become turbulent at the higher Reynolds numbers during systole. There is growing consensus that the transitional k-omega model is the best suited Reynolds averaged turbulence model for such flows. Further confirmation of this opinion is presented here by a comparison with the RNG k-epsilon model for the flow through a straight, nonbifurcating tube. Unlike similar validation studies elsewhere, no assumptions are made about the inlet profile since the full length of the experimental tube is simulated. Additionally, variations in the inflow turbulence quantities are shown to have no noticeable affect on downstream turbulence intensity, turbulent viscosity, or velocity in the k-epsilon model, whereas the velocity profiles in the transitional k-omega model show some differences due to large variations in the downstream turbulence quantities. Following this validation study, the transitional k-omega model is applied in a three-dimensional parametrically defined computer model of the carotid artery bifurcation in which the sinus bulb is manipulated to produce mild, moderate, and severe stenosis. The parametric geometry definition facilitates a powerful means for investigating the effect of local shape variation while keeping the global shape fixed. While turbulence levels are generally low in all cases considered, the mild stenosis model produces higher levels of turbulent viscosity and this is linked to relatively high values of turbulent kinetic energy and low values of the specific dissipation rate. The severe stenosis model displays stronger recirculation in the flow field with higher values of vorticity, helicity, and negative wall shear stress. The mild and moderate stenosis configurations produce similar lower levels of vorticity and helicity.     

On the propagation of a disturbance through a viscous liquid flowing in a distensible tube of appreciable mass

The velocity of propagation of a disturbance wave in a liquid flowing in a distensible tube is computed. The mathematical model is more general than those used in previous analyses: the tube wall properties are realistic; the convective part of the axial inertia forces is taken into account; radial inertia forces of both the fluid and tube wall are present; viscous stresses are present. Four parameters influencing the velocity of propagation are obtained and discussed. Curves are plotted illustrating the effects of the parameters. Contrary to the results of previous analyses, viscous effects are shown to be appreciable in blood flow. It is also shown that radial inertia effects can be important in laboratory set-ups. The material presented in this paper was adapted from the Ph.D. thesis written by the author at Harvard University.     

Numerical simulation for the propagation of nonlinear pulsatile waves in arteries.

The behavior of nonlinear pulsatile flow of incompressible blood contained in an elastic tube is examined. The theory takes into account the nonlinear convective terms of the Navier-Stokes equations. The motion of the arterial wall is characterized by a set of linearized differential equations. The region bounded by the flexible arterial wall is mapped into a fixed area in which numerical discretization takes place. The finite element method (Galerkin weighted residual approach) is used for the solution of this nonlinear system. The results obtained are pressure distribution, velocity profile, flow rate and wall displacements along the elastic tube (20 cm long).     

A couple stress model of blood flow in the microcirculation

A simple mathematical model depicting blood flow in the capillary is developed with an emphasis on the permeability property of the blood vessel based on Starling's hypothesis. In this study the effect of inertia has been neglected in comparison with the viscosity on the basis of the smallness of the Reynolds number of the flow in the capillary. The capillary blood vessel is approximated by a circular cylindrical tube with a permeable wall. The blood is represented by a couple stress fluid. With such an ideal model the velocity and pressure fields are determined. It is shown that an increase in the couple stress parameter increases the resistance to the flow and thereby decreases the volume rate flow. A comparison of the results with those of the Newtonian case has also been made.     

Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals. This research was partially supported by the National Science Foundation; it was done in part by D. K. Whirlow in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Carnegie Institute of Technology.     

Modeling of arterial stenosis and its applications to blood diseases

Blood flow in a stenosed tube has been modeled in the present studies. Blood flow is assumed to be represented by a couple stress fluid. Flow parameters such as velocity, resistance to flow, and shear stress distribution have been computed for different suspension concentrations (haematocrit), and for the blood diseases; polycythemia, plasma cell dyscrasias, and for Hb SS (sickle cell). The results have been compared with the case of normal blood and for other theoretical models. The importance of size effects in blood flow studies has been highlighted.     

Experimental assessment of wall shear flow in models

The blood flow immediately adjacent to the wall of a blood vessel or an artificial surface is of great interest. This flow defines the shear stress at the wall and is known to have a great physiological importance. The use of models is a viable method to investigate this flow. However, even in models the shear stress at the wall is difficult to assess. A new optical method is based on transparent models and uses particles in the model fluid, which are only visible near the wall. This is achieved with a model fluid having a defined opacity. This fluid obscures particles in the center of the models, but permits the observation and recording of particles close to the wall. The method has been applied for Hagen-Poiseuille flow and for the likewise well researched flow in a tube with a sudden expansion.     

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.     

The flow field downstream of an oscillating collapsed tube

A two-component laser Doppler anemometer was used to determine the velocity of aqueous flow in the region from 0.25 to 2.5 diameters downstream of a collapsible tube while the tube was executing vigorous repetitive flow-induced oscillations. The Reynolds number for the time-averaged flow was 10,750. A simultaneous measurement of the pressure at the downstream end of the tube was used to align all the results in time at sixty locations in each of the two principal planes defined by the axes of collapse of the flexible tube upstream. The raw data of seed-particle velocity were used to create a periodic waveform for each measured velocity component at each location by least-squares fitting of a Fourier series. The results are presented as both velocity vectors and interpolated contours, for each of ten salient instants during the cycle of oscillation. In the plane of the collapse major axis, the dominant feature is the jet which emerges from each of the two tube lobes when it collapses, but transient retrograde flow is observed on both the central and lateral edges of this jet. In the orthogonal, minor-axis plane, the dominant feature is the retrograde flow, which during part of the cycle extends over the whole plane. All these features are essentially confined to the first 1.5 diameters of the rigid pipe downstream of the flexible tube. These data map the temporal and spatial extent of the highly three-dimensional reversing flow just downstream of an oscillating collapsed tube.     

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.     

Pulsatile flow of blood through a stenosed tube: effect of periodic body acceleration and a magnetic field

A proper understanding of the interactions of body acceleration and a magnetic field with blood flow could be useful in the diagnosis and treatment of some health problems. In the work reported in this paper we studied the pulsatile flow of blood through stenosed arteries, including the effects of body acceleration and a magnetic field. Blood is regarded as an electrically conducting, incompressible, couple-stress fluid in the presence of a magnetic field along the radius of the tube. The effects of the body acceleration and the magnetic field on the axial velocity, flow rate, and fluid acceleration were obtained analytically by use of the Hankel transform and the Laplace transform. Velocity variations under different conditions are shown graphically. The results have been compared with those from other theoretical models, and are in good agreement. Finally, our mathematical model gives a simple velocity expression for blood flow so it will help not only in the field of physiological fluid dynamics but will also help medical practitioners with elementary knowledge of mathematics.