Mathematical model for a Herschel-Bulkley fluid flow in an elastic tube

The constitution of blood demands a yield stress fluid model, and among the available yield stress fluid models for blood flow, the Herschel-Bulkley model is preferred (because Bingham, Power-law and Newtonian models are its special cases). The Herschel-Bulkley fluid model has two parameters, namely the yield stress and the power law index. The expressions for velocity, plug flow velocity, wall shear stress, and the flux flow rate are derived. The flux is determined as a function of inlet, outlet and external pressures, yield stress, and the elastic property of the tube. Further when the power-law index n = 1 and the yield stress τ ₀ → 0, our results agree well with those of Rubinow and Keller [J. Theor. Biol. 35, 299 (1972)]. Furthermore, it is observed that, the yield stress and the elastic parameters (t ₁ and t ₂) have strong effects on the flux of the non-Newtonian fluid flow in the elastic tube. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Asymmetry of the wall shear stress in a stimulated ascendant flow of a liquid with monodisperse bubbles

Results of an experimental investigation of the two-phase wall shear stress averaged over the tube perimeter and the pulsation of wall shear stress in a stimulated ascendant flow with monodisperse bubbles with an average diameter of 1.2 and 2.2 mm are presented. Regimes with various hydrodynamic parameters such as high shear stress on the wall, low and negative wall shear stress, a high level of shear stress pulsation on the wall, and possible decrease in this level of pulsation are found. An increase in the void gas fraction results in a monotonic increase of perturbation of the single-phase flow. The dependences of the ratio of two-phase and single-phase wall shear stresses for two average bubble diameters seem to be qualitatively similar. The analysis of data revealed a complex dependence of the shear stress pulsation on the bubble diameter. The averaged flow characteristics quantitatively change upon the decrease in the bubble diameter. A further decrease in the average bubble diameter at the same void gas fraction will probably increase the heat-and mass-transfer characteristics of the flow. This is an issue for the futures study.     

Influence of bubbly clusters on the characteristics of the two-phase gas-liquid flow in a flat channel

Results of experimental investigation of a bubbly gas-liquid flow in horizontal and weakly inclined (from −20° to +20°) flat channel are presented. These measurements were carried out within the 0.2–1 m/s range of superficial velocities and volumetric gas flow rate ratio of up to 0.2. The hydrodynamic structure was measured by the electrochemical method with application of wall shear stress and conductivity microprobes. During the experiments signals of shear stress on the upper channel wall and local gas flow rate ratio were recorded completely. After numerical treatment of recorded signals the profiles of local gas flow rate ratio were obtained, average shear stress and its relative mean square pulsations on the upper channel wall were determined. It is shown that under the studied regimes the bubbles are grouped into clusters, and the bubbly flow is presented by alternation of bubbly clusters and single-phase liquid with separate bubbles and without them. Average wall shear stress and absolute shear stress pulsations in the range of bubbly clusters and beyond them were determined. Histograms of probability density distribution were obtained for the wall shear stress on the upper wall. It is shown that average shear stress and absolute pulsations in clusters are significantly higher than those in the flow zone free from bubbles. The work was financially supported by the Russian Foundation for Basic Research (No. 07-08-00405a).     

Shear Stress in a Couette Flow of Liquid-Particle Suspensions

The mechanisms of momentum transfer and shear stress of liquid-particle suspensions in two-dimensional Couette flow are studied using direct numerical simulation by lattice-Boltzmann techniques. The results obtained display complex flow phenomena that arise from the two-phase nature of the fluid including a nonlinear velocity profile, layering of particles, and apparent slip near the solid walls. The general rheological behaviour of the suspension is dilatant. A detailed study of the various momentum transfer mechanisms that contribute to the total shear stress indicates that the observed shear thickening is related to enhanced relative solid phase stress for increasing shear rates.     

Simulation of blood flow using extended Boltzmann kinetic approach

Lattice Boltzmann (LB) simulations are conducted to obtain the detailed hydrodynamics in a variety of blood vessel setups, including a prototype stented channel and four human coronary artery geometries based on the images obtained from real patients. For a model of stented flow involving an S-shape stent, a pulsatile flow rate is applied as the inlet boundary condition, and the time- and space-dependent flow field is computed. The LB simulation is found to reproduce the analytical solutions for the velocity profiles and wall shear stress distributions for the pulsatile channel flow. For the coronary arteries, the distributions of wall shear stress, which is important for clinical diagnostic purposes, are in good agreement with the conventional CFD predictions.     

Lattice Boltzmann method simulating hemodynamics in the three-dimensional stenosed and recanalized human carotid bifurcations

By using the lattice Boltzmann method(LBM)pulsatile blood flows were simulated in three-dimensional moderate stenosed and recanalized carotid bifurcations to understand local hemodynamics and its relevance in arterial atherosclerosis formation and progression.The helical flow patterns,secondary flow and wall dynamical pressure spatiotemporal distributions were investigated,which leads to the disturbed shear forces in the carotid artery bifurcations.The wall shear stress distributions indicated by time-averaged wall shear stress(TAWSS),oscillatory shear index(OSI),and the relative residence time(RRT)in a cardiac cycle revealed the regions where atherosclerotic plaques are prone to form,extend or rupture.This study also illustrates the point that locally disturbed flow may be considered as an indicator for early atherosclerosis diagnosis.Additionally the present work demonstrates the robust and highly efficient advantages of the LBM for the hemodynamics study of the human blood vessel system.     

Heat transfer and shear stress in a gas-liquid flow in an inclined flat channel

Results of experimental investigation of a bubbly gas-liquid flow in an inclined flat channel are presented. Themeasurements were carried out in the range of superficial liquid velocities of 0.3–1.1 m/s and with different values of the volumetric gas flow rate ratio. The hydrodynamic structure wasmeasured bymeans of an electrochemical method using miniature shear stress probes. Values of average shear stress and heat transfer coefficient for different orientation of the channel were found. It is shown that in a bubbly gas-liquid flow the shear stress and heat transfer depend substantially on the channel inclination angle.     

Numerical simulation of flow in a screw-type blood pump

This study presents a numerical investigation of the flow field in a screw pump designed to circulate biological fluid such as blood. A simplified channel flow model is used to allow analysis using a Cartesian set of coordinates. Finite analytic (FA) numerical simulation of the flow field inside the channel was performed to study the influence of Reynolds number and pressure gradient on velocity distribution and shear stresses across the channel cross-section. Simulation results were used to predict flow rates, circulatory flow and the shear stresses, which are known to be related to the level of red blood cell damage (hemolysis) caused by the pump. The study shows that high shear levels are confined to small regions within the channel cross-section, but the circulatory nature of the flow causes an increased percentage of blood elements to pass through the high shear regions, and increases the likelihood of cell damage.     

Stretching of polymers in a random three-dimensional flow

Behavior of a dilute polymer solution in a random three-dimensional flow with an average shear is studied experimentally. Polymer contribution to the shear stress is found to be more than 2 orders of magnitude higher than in a laminar shear flow. The results indicate that the polymer molecules are strongly stretched by the random motion of the fluid.     

Pulsatile Flow of Blood through a Porous Mediumunder Periodic Body Acceleration

Pulsatile flow of blood through a porous medium has been studied studied underthe influence of body acceleration. With the help of Laplace and finite Hankeltransforms, analytic expressions for axial velocity, fluid acceleration, flow rate,and shear stress have been obtained.     

Experimental study of the Taylor bubbles shear stress in an upward flow in a vertical tube

The modification of electrodiffusional method of the wall shear stress measurements is applied for registration of the Taylor bubble shear stress in an upward liquid flow. Time realization of shear is considered as a structure frozen into the flow, which moves together with a bubble. Experiments were carried out in laminar and transitional liquid flows. The wall shear stress in the liquid film around bubble averaged over the tube perimeter is presented for different flow Reynolds numbers and different lengths of the bubble.     

Sample-to-sample torque fluctuations in a system of coaxial randomly charged surfaces

The stress propagation in a concentrated attractive colloidal suspension under shear is studied using numerical simulations. The spatial correlations of the intercolloidal stress field are studied and an inertia-like tensor is defined in order to characterize the anisotropic nature of the stress field. It is shown that the colloids remain in a liquid order, the intercolloidal stress is strongly anisotropic. A transition under flow is observed: during a transient regime at low deformation, the stress propagates along the compression direction of the shear, whereas at larger deformations, the stress is organized into layers parallel to the (flow, vorticity) plane.     

Surface roughness effects on the turbulence statistics in a low Reynolds number channel flow

A high-resolution particle image velocimetry was used to characterize a low Reynolds number turbulent flow in a channel. Experiments were conducted over a sand grain-coated surface of large relative roughness, and the results were compared with measurements over a smooth surface. The roughness perturbation significantly modified the outer layer. Even though the streamwise Reynolds stress shows less sensitivity in the outer layer to the boundary condition, significant enhancements were observed in the wall-normal Reynolds stress and the Reynolds shear stress. These modifications were considered as footprints of the larger-scale eddies transporting intense wall-normal motions away from the rough wall. A quadrant decomposition shows that strong and more frequent ejections are responsible for the larger values of the mean Reynolds shear stress over the rough wall. The results also indicate that spanwise vortex cores with mean vorticity of the same sign as the mean shear are the dominant smaller-scale vortical structures over the smooth and rough walls. A linear stochastic estimation-based analysis shows that the average larger-scale structure associated with these vortices is a shear layer that strongly connects the outer layer flow to the near-wall flow. A proper orthogonal decomposition of the flow suggests that the large-scale eddy is more energetic for the rough wall, and contributes more significantly to the resolved turbulent kinetic energy and the Reynolds shear stress than the smooth wall.     

Temporal oscillations of the shear stress and scattered light in a shear-banding-shear-thickening micellar solution

The results of optical and rheological experiments performed on a viscoelastic solution (cetyltrimethylammonium bromide + sodium salicylate in water) are reported. The flow curve has a horizontal plateau extending between two critical shear rates characteristic of heterogeneous flows formed by two layers of fluid with different viscosities. These two bands which also have different optical anisotropy are clearly seen by direct observation in polarized light. At the end of the plateau, apparent shear thickening is observed in a narrow range of shear rates; in phase oscillations of the shear stress and of the first normal stress difference are recorded in a shearing device operating under controlled strain. The direct observation of the annular gap of a Couette cell in a direction perpendicular to a plane containing the vorticity shows that the turbidity of the whole sample also undergoes time dependent variations with the same period as the shear stress. However no banding is observed during the oscillations and the flow remains homogeneous.     

Shear Flow of a Granular Material

The shear flow of a granular material between parallel plates is treated by means of the Boltzmann equation with pseudo-Maxwellian grains. The moments for reverse reflection boundary conditions are found explicitly. The shearing stress is found to depend quadratically on the shear rate.     

Azimuthal instability of the interface in a shear banded flow by direct visual observation

The stability of the shear banded flow of a Maxwellian fluid is studied from an experimental point of view using rheology and flow visualization with polarized light. We show that the one-layer homogeneous flow cannot sustain shear rates corresponding to the end of the stress plateau. The high shear rate branch is not found and the shear stress oscillates at the end of the plateau. An azimuthal instability appears: the shear induced band becomes unstable and the interface between the two bands undulates in time and space with a period τ, a wavelength λ and a wave vector k parallel to the direction of the tangential velocity.     

Simulation of Blood Flow at Vessel Bifurcation by Lattice Boltzmann Method

The application of the lattice Boltzmann method to the large vessel bifurcation blood flow is investigated in a wide range of Reynolds numbers. The velocity, shear stress and pressure distributions at the bifurcation are presented in detail The flow separation zones revealed with increase of Reynolds number are located in the areas of the daughter branches distal to the outer corners of the bifurcation where some deposition of particular blood components might occur to form arteriosclerosis. The results also demonstrate that the lattice Boltzmann method is adaptive to simulating the flow in larger vessels under a high Reynolds number.     

Shear-induced stress relaxation in a two-dimensional wet foam

We report on experimental measurements of the flow behavior of a wet, two-dimensional foam under conditions of slow, steady shear. The initial response of the foam is elastic. Above the yield strain, the foam begins to flow. The flow consists of irregular intervals of elastic stretch followed by sudden reductions of the stress, i.e., stress drops. We report on the distribution of the stress drops as a function of the applied shear rate. We also comment on our results in the context of various two-dimensional models of foams.     

Oscillating viscosity in a lyotropic lamellar phase under shear flow

We report some time-dependent behavior of lyotropic lamellar phase under shear flow. At fixed stress, near a layering instability, the system presents an oscillating shear rate. We build up a new stress versus shear rate diagram that includes temporal behavior. This diagram is made of two distinct branches of stationary states which correspond, respectively, to disordered and ordered multilamellar vesicle phases. When increasing the shear stress, prior to the transition to the ordered structural state, sustained oscillations of the viscosity are recorded. They correspond to periodic structural change of the entire sample between a disordered and a ordered state of multilamellar vesicles.     

Effect of elasticity of a polymer solution on the Hele-Shaw flow

In this study, the Hele-Shaw cell is used to examine the effect offluid elasticity on the flow patterns of two-dimensional potential flow. Flows around a circular cylinder, a square cylinder and flows through abruptly converging-diverging channels (slits) with different throat lengths are tested for water and 0.2 wt % polyacrylamide aqueous solution (PAA-solution). The viscosity of the latter is well modeled by the power law, and the first normal stress difference in the steady shear flow is around ten times higher than the shear stress. Although the PAA-solution is highly shear-thinning, the flows of PAA-solution well reproduce the two-dimensional potential flow patterns that correspond to the respective flow configurations when the flow rate is very low. The potential flow patterns ofPAA solution are disturbed in the opposite way of inertia effect observed for water. The streamlines near the upstream stagnation point of cylinders are shifted upstream separating from the cylinder surface when the flow rate is higher, while streamlines in the wake approach closer to the downstream stagnation point. Streamlines offlow through the slit at flow rates higher than the potential flow region show that a pair ofvortices is formed upstream the slit entrance, while the streamlines remain attached to the downstream wall after passing the slit.