Rodolfo Repetto

Channel bifurcation in braided rivers: Equilibrium configurations and stability

[1] We investigate the equilibrium configurations and the stability of river bifurcations in gravel braided networks. Within the context of a one-dimensional approach, the nodal point conditions play a crucial rule, as pointed out by Wang et al. [1995] who propose an empirical relationship relating water and sediment flow rates into the downstream branches. In the present paper, an alternative formulation of nodal point conditions is proposed based on a quasi two-dimensional approach. The results show that, if the Shields parameter of the upstream channel is large enough, the system only admits of one solution with both branches open, which is invariably stable. As the Shields parameter of the upstream channel decreases, two further stable solutions appear characterized by a different partition of water discharge into the downstream branches: in this case, the previous solution becomes unstable. Theoretical findings are confirmed by the numerical solution of the nonlinear one-dimensional equations.

Free bars in rivers

In the paper we review some recent work on the mechanics of formation and development of river bars. The emphasis is placed on the instability process which leads to the spontaneous development of bars in almost straight reaches of alluvial rivers. A three dimensional formulation of the problem is presented along with a discussion on the relevant closure relationships. Results of linear and non linear theories for free bars under bedload dominated conditions are summarised. Furthermore, account is given on the effect on bar instability induced by suspended load, grain sorting and width variations. Some as yet unpublished results are also presented.

Planimetric instability of channels with variable width

We study the steady three-dimensional flow field and bed topography in a channel with sinusoidally varying width, under the assumptions of small-amplitude width variations and sufficiently wide channel to neglect nonlinear effects and sidewall effects. The aim of the work is to investigate the role of width variations in producing channel bifurcation in braided rivers. We infer incipient bifurcation in cases where the growth of a central bar leads to planimetric instability of the channel, i.e. when the given infinitesimal width perturbation is enhanced. Results of the three-dimensional model suggest that the equilibrium bottom profile mainly consists of a purely longitudinal component, uniformly distributed over the cross-section, which induces deposition at the wide section and scour at the constriction, and of a transverse component in the form of a central bar (wide sections) and scour (constrictions), with longitudinal wavelength equal to that of width variations. A comparison between the results of the three-dimensional model and those obtained by means of a two-dimensional depth-averaged approach shows that the transverse component is mainly related to three-dimensional effects. Theoretical findings display a satisfactory agreement with results of flume experiments. Transverse variations are responsible for the planimetric instability of the channel; we find that in the range of values of Shields stress typical of braided rivers, the incipient bifurcation is enhanced as the width ratio of the channel increases.

Experimental investigation of vitreous humour motion within a human eye model

We present an experimental study of the vitreous motion induced by saccadic eye movements. A magnified model of the vitreous chamber has been employed, consisting of a spherical cavity carved in a perspex cylindrical container, which is able to rotate with a prescribed time law. Care has been taken to correctly reproduce real saccadic eye movements. The spherical cavity is filled with glycerol and the flow field is measured on the equatorial plane orthogonal to the axis of rotation, through the PIV technique. Visualizations of the fully three-dimensional flow suggest that it essentially occurs on planes perpendicular to the axis of rotation, the motion orthogonal to such planes being smaller by three to four orders of magnitude. Theoretical results, based on a simplified solution, are in very good agreement with the experimental findings. The maximum value of the shear stress at the wall, which is thought to play a possibly important role in the pathogenesis of retinal detachment, does not significantly depend on the amplitude of saccadic movements. This suggests that relatively small eye rotations, being much more frequent than large movements, are mainly responsible for vitreous stresses on the retina. Results also illustrate the dependence of the maximum shear stress at the wall from the vitreous viscosity.

Mathematical Modeling of the Circulation in the Liver Lobule

In this paper, we develop a mathematical model of blood circulation in the liver lobule. We aim to find the pressure and flux distributions within a liver lobule. We also investigate the effects of changes in pressure that occur following a resection of part of the liver, which often leads to high pressure in the portal vein. The liver can be divided into functional units called lobules. Each lobule has a hexagonal cross-section, and we assume that its longitudinal extent is large compared with its width. We consider an infinite lattice of identical lobules and study the two-dimensional flow in the hexagonal cross-sections. We model the sinusoidal space as a porous medium, with blood entering from the portal tracts (located at each of the vertices of the cross-section of the lobule) and exiting via the centrilobular vein (located in the center of the cross-section). We first develop and solve an idealized mathematical model, treating the porous medium as rigid and isotropic and blood as a Newtonian fluid. The pressure drop across the lobule and the flux of blood through the lobule are proportional to one another. In spite of its simplicity, the model gives insight into the real pressure and velocity distribution in the lobule. We then consider three modifications of the model that are designed to make it more realistic. In the first modification, we account for the fact that the sinusoids tend to be preferentially aligned in the direction of the centrilobular vein by considering an anisotropic porous medium. In the second, we account more accurately for the true behavior of the blood by using a shear-thinning model. We show that both these modifications have a small quantitative effect on the behavior but no qualitative effect. The motivation for the final modification is to understand what happens either after a partial resection of the liver or after an implantation of a liver of small size. In these cases, the pressure is observed to rise significantly, which could cause deformation of the tissue. We show that including the effects of tissue compliance in the model means that the total blood flow increases more than linearly as the pressure rises.

Eye rotation induced dynamics of a Newtonian fluid within the vitreous cavity: the effect of the chamber shape.

The dynamics of the vitreous body induced by eye rotations is studied experimentally. In particular, we consider the case in which the vitreous cavity is filled by a Newtonian fluid, either because the vitreous is liquefied or because it has been replaced, after vitrectomy, by a viscous fluid. We employ a rigid Perspex container which models, in a magnified scale, the vitreous cavity of the human eye. The shape of the cavity closely resembles that of the real vitreous chamber; in particular, the anterior part of the container is concave in order to model the presence of the eye lens. The container is filled with glycerol and is mounted on the shaft of a computer-controlled motor which rotates according to a periodic time law. PIV (particle image velocimetry) measurements are taken on the equatorial plane orthogonal to the axis of rotation. The experimental measurements show that the velocity field is strongly influenced by the deformed geometry of the domain. In particular, the formation of a vortex in the vicinity of the lens, which migrates in time towards the core of the domain, is invariably observed. The vortex path is tracked in time by means of a vortex identification technique and it is found that it is significantly influenced by the Womersley number of the flow. Particle trajectories are computed from the PIV measurements. Particles initially located at different positions on the equatorial horizontal plane (perpendicular to the axis of rotation) tend to concentrate in narrow regions adjacent to the lens, thus suggesting the existence, in such regions, of a vertical fluid ejection. Such a strong flow three-dimensionality, which is essentially induced by the irregular shape of the domain, may play a significant role in the mixing processes taking place inside the eye globe. The tangential stresses acting on the rigid boundary of the domain are also computed from the experimental measurements showing that regions subject to particularly intense stresses exist along the boundary close to the lens.

A 3D porous media liver lobule model: the importance of vascular septa and anisotropic permeability for homogeneous perfusion

The hepatic blood circulation is complex, particularly at the microcirculatory level. Previously, 2D liver lobule models using porous media and a 3D model using real sinusoidal geometries have been developed. We extended these models to investigate the role of vascular septa (VS) and anisotropic permeability. The lobule was modelled as a hexagonal prism (with or without VS) and the tissue was treated as a porous medium (isotropic or anisotropic permeability). Models were solved using computational fluid dynamics. VS inclusion resulted in more spatially homogeneous perfusion. Anisotropic permeability resulted in a larger axial velocity component than isotropic permeability. A parameter study revealed that results are most sensitive to the lobule size and radial pressure drop. Our model provides insight into hepatic microhaemodynamics, and suggests that inclusion of VS in the model leads to perfusion patterns that are likely to reflect physiological reality. The model has potential for applications to unphysiological and pathological conditions.

Steady streaming within a periodically rotating sphere

We consider the flow in a spherical chamber undergoing periodic torsional oscillations about an axis through its centre, and analyse it both theoretically and experimentally. We calculate the flow in the limit of small-amplitude oscillations in the form of a series expansion in powers of the amplitude, finding that at second order, a steady streaming flow develops consisting of two toroidal cells. This streaming behaviour is also observed in our experiments. We find good quantitative agreement between theory and experiments, and we discuss the dependence of the steady streaming behaviour as both the oscillation frequency and amplitude are varied.

Ultrasound imaging velocimetry of the human vitreous

Knowledge of vitreous motion in response to saccades is a prerequisite for understanding vitreous rheology. Purpose of present paper is to introduce Ultrasound Image Velocimetry of the human eye, measure scleral and vitreous velocity fields and test the reproducibility of the proposed technique. Twelve patients with varying diagnosis underwent Ocular Dynamic Ultrasound; scleral angular velocity (V(S)) was measured by 2 different operators and reproducibility calculated. Squared velocity of the vitreous (E), which is representative of kinetic energy per unit mass, was computed from velocity. The time evolution of the energy of the vitreous was described by its spatial average (E(S)), whereas spatial distribution was described by its time average (E(T)). Peak and average E(S), the ratio K(p) of the peak of the spatially averaged kinetic energy per unit mass to the maximum squared scleral angular velocity, vitreous motion onset time (T(O)) and vitreous motion decay time (T(D)) were also defined. Inter-operator reproducibility coefficient was 0.043 and correlation between operators was significant. V(S), peak and average E(S), K(p) ratio and T(D) differed among patients but not among operators. V(S) correlated with E(S) and T(D). E(S) and T(D) but not V(S), were significantly different in patients with Posterior Vitreous Detachment. Patients with retinal detachment showed significantly higher V(S) and E(S). K(p) was inversely correlated to age and refraction. Measures proved accurate and reproducible. E is related to V(S), retinal traction and mechanical stimulation. Identified variables varied with age, refraction pathologic conditions.

Traction on the retina induced by saccadic eye movements in the presence of posterior vitreous detachment

Posterior vitreous detachment is a fairly common condition in elderly people. Tractions exerted by the detached vitreous on the retina may result in retinal tears and detachments. We studied how these tractions can arise from saccadic eye movements. Numerical simulations have been performed on a two-dimensional model of the vitreous chamber within a rigid spherical sclera, subjected to prescribed finite-amplitude rotations about a vertical axis. The vitreous chamber was assumed to be split into two regions: one occupied by the detached vitreous, modeled as an elastic viscous solid, and the other occupied by the separated liquefied vitreous, modeled as a Newtonian fluid. At the interface between the two phases, we also considered the presence of the vitreous cortex, modeled as an elastic membrane. We tested several different configurations of the interface. In all cases, we found that eye rotations generate large tractions on the retina close to the attachment points of the membrane. Comparing them, we identified configurations of the vitreous detachment that exhibit higher tractions. We also investigated how the response to saccadic movements depends on some physical parameters, in particular on the rheological properties of the solid phase and the membrane. The numerical simulations show that the generated tractions may be of the same order of magnitude as the adhesive force between the retina and the pigment epithelium. Therefore, the model provides a sound physical justification for the hypothesis that saccadic movements, in the presence of posterior vitreous detachment, could be responsible for high tractions on the retina, which may trigger retinal tear formation.