The motion of a bubble of negligible viscosity, such as air, forced down a tube filled with a viscous fluid which wets the walls of the tube has become a classic of the fluid dynamical literature. The differential motion of the bubble and the fluid are determined by the thin film which surrounds the bubble, whose shape and thickness are set by the interplay between gradients in surface tension and viscous shear stresses. Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, p. 166) provided a first, clear mathematical analysis in the lubrication limit coupled with carefully constructed experimental confirmation of the thin films deposited by a bubble moving in the confining geometry of the capillary tube. Its lasting impact has been not only in the migration of bubbles, but in a host of related fluid dynamical, industrial, biological and environmental processes for which thin lubricating films on the sometimes convoluted geometries of complex microstructures, such as porous media, determine the large-scale behaviour.

Bioprosthetic heart valves create turbulent flow during early systole which might be detrimental to their durability and performance. Complex mechanisms in the unsteady and heterogeneous flow field complicate the isolation of specific instability mechanisms. We use linear stability analysis and numerical simulations of the flow in a simplified model to study mechanisms initiating the laminar–turbulent transition. The analysis of a modified Orr–Sommerfeld equation, which includes a model for fluid–structure interaction (FSI), indicates Kelvin–Helmholtz and FSI instabilities for a physiological Reynolds number regime. Two-dimensional parametrized FSI simulations confirm the growth rates and phase speeds of these instabilities. The eigenmodes associated with the observed leaflet kinematics allow for decoupled leaflet oscillations. A detailed analysis of the temporal evolution of the flow field shows that the starting vortex interacts with the aortic wall leading to a secondary vortex which moves towards the shear layer in the wake of the leaflets. This appears to be connected to the onset of the shear layer instabilities that are followed by the onset of leaflet motion leading to large-scale vortex shedding and eventually to a nonlinear breakdown of the flow. Numerical results further indicate that a narrower aorta leads to an earlier onset of the shear layer instabilities. They also suggest that the growing perturbations of the shear layer instability propagate upstream and may initiate the FSI instabilities on the valve leaflets.

In this paper, we present an analytic solution for pulse wave propagation in a flexible arterial model with tapering, physiological boundary conditions and variable wall properties (wall elasticity and thickness). The change of wall properties follows a profile that is proportional to $r^\alpha$, where $r$ represents the lumen radius and $\alpha$ is a material coefficient. The cross-sectionally averaged velocity and pressure are obtained by solving a hyperbolic system derived from the mass and momentum conservations, and they are expressed in Bessel functions of order $(4-\alpha )/(3-\alpha )$ and $1/(3-\alpha )$, respectively. The solution is successfully validated by comparing it with numerical results from three-dimensional (3-D) fluid–structure interaction simulations. Subsequently, the solution is employed to study pulse wave propagation in an arterial model, revealing that the wall properties and the physiological outlet boundary conditions, such as the resistor–capacitor–resistor (RCR) model, play a crucial role in characterizing the input impedance and reflection coefficient. At low-frequency range, the input impedance is found to be insensitive to the wall properties and is primarily determined by the RCR parameters. At high-frequency range, the input impedance oscillates around the local characteristic impedance, and the oscillation amplitude varies non-monotonically with $\alpha$. Expressions for the input impedance at both low-frequency and high-frequency limits are presented. This analytic solution is also successfully applied to model flow inside a patient-specific arterial tree, with the maximum relative errors in pressure and flow rate never exceeding $1.6\,\%$ and $9.0\,\%$ when compared with results from 3-D numerical simulations.

Characterizing the haemodynamics in intracranial aneurysms is of high interest as it impacts aneurysm growth, rupture and treatment, especially with flow-diverting stents (FDS). Flow in these geometries is known to depend on the Dean, Reynolds and Womersley numbers, $De$, $Re$, $Wo$, but is also influenced by geometrical parameters such as the sac shape or the size of the opening. Via particle image velocimetry, this parametric study aimed at evaluating the combined effects of $Re$, $De$, $Wo$ and the geometry of the aneurysmal sac on the haemodynamics before and after treatment with FDS. Eight ellipsoidal idealized aneurysm models were created with two curvatures of the parent vessel, two aspect ratios of the sac and two neck sizes. Before treatment, a single counter-rotating vortex, whose strength increases with $Re$ and $De$, as well as with the neck size and the aspect ratio, was observed in the sac for all but one geometry. After treatment with FDS, four different flow topologies were observed, depending on the geometry: no separation, separation for part of the cycle, two opposing vortices or a single counter-rotating vortex. A linear model with interaction revealed the predominant effect of $De$ and the curvature of the parent vessel on the haemodynamics before and after treatment. This work once more demonstrated the primary role of haemodynamics in the treatment of intracranial aneurysms with FDS. Future work will consider the complexity of patient-specific geometries, and their effects on both the haemodynamics in the sac and the porosity of the FDS.

We present a numerical analysis of the lateral movement and equilibrium radial positions of red blood cells (RBCs) with major diameter 8 $\mathrm {\mu }$m under a Newtonian fluid in a circular channel with 50 $\mathrm {\mu }$m diameter. Each RBC, modelled as a biconcave capsule whose membrane satisfies strain-hardening characteristics, is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$, the latter of which indicates the ratio of the fluid viscous force to the membrane elastic force. The effects of initial orientation angles and positions on the equilibrium radial position of an RBC centroid are also investigated. The numerical results show that depending on their initial orientations, RBCs have bistable flow modes, so-called rolling and tumbling motions. Most RBCs have a rolling motion. These stable modes are accompanied by different equilibrium radial positions, where tumbling RBCs are further away from the channel axis than rolling ones. The inertial migration of RBCs is achieved by alternating orientation angles, which are affected primarily by the initial orientation angles. Then the RBCs assume the aforementioned bistable modes during the migration, followed by further migration to the equilibrium radial position at much longer time periods. The power (or energy dissipation) associated with membrane deformations is introduced to quantify the state of membrane loads. The energy expenditures rely on stable flow modes, the equilibrium radial positions of RBC centroids, and the viscosity ratio between the internal and external fluids.

Two simple mathematical models of advection and diffusion of hydrogen within the retina are discussed. The work is motivated by the hydrogen clearance technique, which is used to estimate blood flow in the retina. The first model assumes that the retina consists of three, well-mixed layers with different thickness, and the second is a two-dimensional model consisting of three regions that represent the layers in the retina. Diffusion between the layers and leakage through the outer edges are considered. Solutions to the governing equations are obtained by employing Fourier series and finite difference methods for the two models, respectively. The effect of important parameters on the hydrogen concentration is examined and discussed. The results contribute to understanding the dispersal of hydrogen in the retina and in particular the effect of flow in the vascular retina. It is shown that the predominant features of the process are captured by the simpler model.

In vivo transparent vessel segmentation is important to life science research. However, this task remains very challenging because of the fuzzy edges and the barely noticeable tubular characteristics of vessels under a light microscope. In this paper, we present a new machine learning method based on blood flow characteristics to segment the global vascular structure in vivo. Specifically, the videos of blood flow in transparent vessels are used as input. We use the machine learning classifier to classify the vessel pixels through the motion features extracted from moving red blood cells and achieve vessel segmentation based on a region-growing algorithm. Moreover, we utilize the moving characteristics of blood flow to distinguish between the types of vessels, including arteries, veins, and capillaries. In the experiments, we evaluate the performance of our method on videos of zebrafish embryos. The experimental results indicate the high accuracy of vessel segmentation, with an average accuracy of 97.98%, which is much more superior than other segmentation or motion-detection algorithms. Our method has good robustness when applied to input videos with various time resolutions, with a minimum of 3.125 fps.

A mechanical theory is described for a phenomenon in the surgical procedure of resuscitative endovascular balloon occlusion of the aorta (REBOA). In this procedure a balloon is pushed into the aorta by a catheter and then inflated in order to stop haemorrhage. One of the hazards of this procedure is the tendency for the balloon to migrate away from its intended position. This work examines the mechanics of balloon anchoring and migration by analysing the effects of pressure waves, the sheet flow and solid friction in the thin gap between the walls of the aorta and balloon. A viscoelastic model is adopted for the aorta wall for pressure waves between the left ventricle and the balloon. The lubrication approximation is used for blood flow in the thin gap between the walls of the balloon and aorta. Samples of quantitative predictions are discussed on how the inflation pressure and balloon characteristics affect the balloon anchoring and migration. The crucial roles of solid friction and balloon placement are pointed out, which should help in guiding the manufacturing of balloons and their usage in the field.

The nonlinear dynamics of a two-sided collapsible channel flow is investigated by using an immersed boundary-lattice Boltzmann method. The stability of the hydrodynamic flow and collapsible channel walls is examined over a wide range of Reynolds numbers $Re$, structure-to-fluid mass ratios $M$ and external pressures $P_e$. Based on extensive simulations, we first characterise the chaotic behaviours of the collapsible channel flow and explore possible routes to chaos. We then explore the physical mechanisms responsible for the onset of self-excited oscillations. Nonlinear and rich dynamic behaviours of the collapsible system are discovered. Specifically, the system experiences a supercritical Hopf bifurcation leading to a period-1 limit cycle oscillation. The existence of chaotic behaviours of the collapsible channel walls is confirmed by a positive dominant Lyapunov exponent and a chaotic attractor in the velocity-displacement phase portrait of the mid-point of the collapsible channel wall. Chaos in the system can be reached via period-doubling and quasi-periodic bifurcations. It is also found that symmetry breaking is not a prerequisite for the onset of self-excited oscillations. However, symmetry breaking induced by mass ratio and external pressure may lead to a chaotic state. Unbalanced transmural pressure, wall inertia and shear layer instabilities in the vorticity waves contribute to the onset of self-excited oscillations of the collapsible system. The period-doubling, quasi-periodic and chaotic oscillations are closely associated with vortex pairing and merging of adjacent vortices, and interactions between the vortices on the upper and lower walls downstream of the throat.

We propose a semi-analytical solution for the advection–diffusion equation in cylindrical domains, with an aim towards extracting blood flow rates from contrast variations in a coronary computed tomography angiography image. The solution proposed in this work, in contrast with existing methods, which only consider advection, incorporates both radial velocity variation and diffusion. By means of a Galerkin approach using Bessel functions, a solution for a three-dimensional concentration field at a single time point is obtained after a Laplace transformation. This semi-analytical solution forms the basis for a novel advection–diffusion flow estimation (ADFE) method. The ADFE is derived, validated through numerical spectral-element method computations, and shown to exhibit improved accuracy against the state-of-the-art method for image-based blood flow extraction.

Flow in sidewall cerebral aneurysms can be ideally modelled as the combination of flow over a spherical cavity and flow in a curved circular pipe, two canonical flows. Flow in a curved pipe is known to depend on the Dean number $De$, combining the effects of Reynolds number $\textit {Re}$ and of the curvature along the pipe centreline, $\kappa$. Pulsatility in the flow introduces a dependence on the Womersley number $Wo$. Using stereo particle image velocimetry measurements, this study investigated the effect of these three key non-dimensional parameters, by modifying pipe curvature ($De$), flow rate ($Re$) and pulsatility frequency ($Wo$), on the flow patterns in a spherical cavity. A single counter-rotating vortex was observed in the cavity for all values of pipe curvature $\kappa$ and Reynolds number $\textit {Re}$, for both steady and pulsatile inflow conditions. Increasing the pipe curvature impacted the flow patterns in both the pipe and the cavity, by shifting the velocity profile towards the cavity opening and increasing the flow rate in to the cavity. The circulation in the cavity was found to collapse well with only the Dean number, for both steady and pulsatile inflows. For pulsatile inflow, the counter-rotating vortex was unstable and the location of its centre over time was impacted by the curvature of the pipe, as well as $\textit {Re}$ and $Wo$ in the free stream. The circulation in the cavity was higher for steady inflow than for the equivalent average Reynolds number and Dean number pulsatile inflow, with very limited impact of the Womersley number in the range studied. A second part of this study, that focuses on the changes in fluid dynamics when the intracranial aneurysm is treated with a flow-diverting stent, can be found in this issue (Barbour et al., J. Fluid Mech., vol. 915, 2021, A124).

The flow in a spherical cavity on a curved round pipe is a canonical flow that describes well the flow inside a sidewall aneurysm on an intracranial artery. Intracranial aneurysms are often treated with a flow-diverting stent (FDS), a low-porosity metal mesh that covers the entrance to the cavity, to reduce blood flow into the aneurysm sac and exclude it from mechanical stresses imposed by the blood flow. Successful treatment is highly dependent on the degree of reduction of flow inside the cavity, and the resulting altered fluid mechanics inside the aneurysm following treatment. Using stereoscopic particle image velocimetry, we characterize the fluid mechanics in a canonical configuration representative of an intracranial aneurysm treated with a FDS: a spherical cavity on the side of a curved round pipe covered with a metal mesh formed by an actual medical FDS. This porous mesh coverage is the focus of Part 2 of the paper, characterizing the effects of parent vessel $Re$, $De$ and pulsatility, $Wo$, on the fluid dynamics, compared with the canonical configuration with no impediments to flow into the cavity that is described in Part 1 (Chassagne et al., J. Fluid Mech., vol. 915, 2021, A123). Coverage with a FDS markedly reduces the flow $Re$ in the aneurysmal cavity, creating a viscous-dominated flow environment despite the parent vessel $Re>100$. Under steady flow conditions, the topology that forms inside the cavity is shown to be a function of the parent vessel $De$. At low values of $De$, flow enters the cavity at the leading edge and remains attached to the wall before exiting at the trailing edge, a novel behaviour that was not found under any conditions of the high-$Re$, unimpeded cavity flow described in Part 1. Under these conditions, flow in the cavity co-rotates with the direction of the free-stream flow, similar to Stokes flow in a cavity. As $De$ increases, the flow along the leading edge begins to separate, and the recirculation zone grows with increasing $De$, until, above $De \approx 180$, the flow inside the cavity is fully recirculating, counter-rotating with respect to the free-stream flow. Under pulsatile flow conditions, the vortex inside the cavity progresses through the same cycle – switching from attached and co-rotating with the free-stream flow at the beginning of the cycle (low velocity and positive acceleration) to separated and counter-rotating as $De$ reaches a critical value. The location of separation within the harmonic cycle is shown to be a function of both $De$ and $Wo$. The values of aneurysmal cavity $Re$ based on both the average velocity and the circulation inside the cavity are shown to increase with increasing values of $De$, while $Wo$ is shown to have little influence on the time-averaged metrics. As $De$ increases, the strength of the secondary flow in the parent vessel grows, due to the inertial instability in the curved pipe, and the flow rate entering the cavity increases. Thus, the effectiveness of FDS treatment to exclude the aneurysmal cavity from the haemodynamic stresses is compromised for aneurysms located on high-curvature arteries, i.e. vessels with high $De$, and this can be a fluid mechanics criterion to guide treatment selection.

Diabetic retinopathy (DR) is a frequent complication of diabetes mellitus and an increasingly common cause of visual impairment. Blood vessel damage occurs as the disease progresses, leading to ischemia, neovascularization, blood–retina barrier (BRB) failure and eventual blindness. Although detection and treatment strategies have improved considerably over the past years, there is room for a better understanding of the pathophysiology of the diabetic retina. Indeed, it has been increasingly realized that DR is in fact a disease of the retina’s neurovascular unit (NVU), the multi-cellular framework underlying functional hyperemia, coupling neuronal computations to blood flow. The accumulating evidence reveals that both neurochemical (synapses) and electrical (gap junctions) means of communications between retinal cells are affected at the onset of hyperglycemia, warranting a global assessment of cellular interactions and their role in DR. This is further supported by the recent data showing down-regulation of connexin 43 gap junctions along the vascular relay from capillary to feeding arteriole as one of the earliest indicators of experimental DR, with rippling consequences to the anatomical and physiological integrity of the retina. Here, recent advancements in our knowledge of mechanisms controlling the retinal neurovascular unit will be assessed, along with their implications for future treatment and diagnosis of DR.

Blood flow in the retina increases in response to light-evoked neuronal activity, ensuring that retinal neurons receive an adequate supply of oxygen and nutrients as metabolic demands vary. This response, termed “functional hyperemia,” is disrupted in diabetic retinopathy. The reduction in functional hyperemia may result in retinal hypoxia and contribute to the development of retinopathy. This review will discuss the neurovascular coupling signaling mechanisms that generate the functional hyperemia response in the retina, the changes to neurovascular coupling that occur in diabetic retinopathy, possible treatments for restoring functional hyperemia and retinal oxygen levels, and changes to functional hyperemia that occur in the diabetic brain.

The endothelial glycocalyx layer (EGL) is a brush-like layer that lines the internal surfaces of blood vessels. It is thought to serve a number of physiological functions, including as a mechanotransducer of fluid loadings to the vessel wall. However, the fragility of the EGL makes it difficult to examine experimentally, and so there is much value in theoretical models that can help to explain the dynamical behaviour of the EGL. Most previous models have employed mixture theory to mechanically describe the layer, which treats the EGL as a isotropic linearly poroelastic layer. However, there is increasing experimental evidence to suggest that the EGL has a well-defined organisational structure that might not necessarily be well captured by such mixture theory descriptions. We therefore employ homogenisation theory to incorporate into the models some of the possible EGL microstructure suggested by the current biological literature. We explore how mechanotransduction varies under the different possible EGL microstructures, which potentially has important consequences to our understanding of how structural changes to the EGL might affect a vessel’s ability to respond to hemodynamical cues. We also find that, whereas mechanotransduction through the solid components of the EGL is dominated by the fluid tractions applied at the lumen–EGL interface, the component carried through its fluid phase is most sensitive to pressure gradients within the bulk EGL. This is relevant, since it is known that the underlying endothelial cells respond differently to these two different forms of mechanical loading.

Most network models describing solute transport in the brain microcirculation use the well-mixed hypothesis and assume that radial gradients inside the blood vessels are negligible. Recent experimental data suggest that these gradients, which may result from heterogeneities in the velocity field or consumption in the tissue, may in fact be important. Here, we study the validity of the well-mixed hypothesis in network models of solute transport using theoretical and computational approaches. We focus on regimes of weak coupling where the transport problem inside the vasculature is independent of the concentration field in the tissue. In these regimes, the boundary condition between vessels and tissue can be modelled by a Robin boundary condition. For this boundary condition and for a single cylindrical capillary, we derive a one-dimensional cross-section average transport problem with dispersion and exchange coefficients capturing the effects of radial gradients. We then extend this model to a network of connected tubes and solve the problem in a complex anatomical network. By comparing with results based on the well-mixed hypothesis, we find that dispersive effects are a fundamental component of transport in transient situations with relatively rapid injections, i.e. frequencies above one Hertz. For slowly varying signals and steady states, radial gradients also significantly impact the spatial distribution of vessel/tissue exchange for molecules that easily cross the blood brain barrier. This suggests that radial gradients cannot be systematically neglected and that there is a crucial need to determine the impact of spatio-temporal heterogeneities on transport in the brain microcirculation.