[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 onedimensional equations.

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

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

[1] In a recent paper, Bolla Pittaluga et al. proposed a physically based formulation for the nodal point conditions to be adopted at a channel bifurcation in the context of a onedimensional approach. They employed such conditions to study the equilibrium configurations of a simple bifurcation and their stability, assuming erodible bed and fixed banks. With the present work we extend the model proposed by Bolla Pittaluga et al. to the case of channels with erodible banks, i.e., channels which may adapt their width to the actual flow conditions. Such an extension is of a particular interest for river bifurcations in gravel bed braided streams, in which bed evolution and bank erosion processes occur over comparable timescales. Moreover, to study the morphological evolution of a bifurcation, we employ a different approach with respect to Bolla Pittaluga et al. and introduce a ''local analysis'' which does not account for the influence exerted on channel morphology by downstream conditions on longer timescales. For values of the controlling parameters typical of gravel bed braided streams, the model shows that the stable equilibrium solutions of a bifurcation are invariably characterized by a strongly unbalanced partition of water and sediment discharges in the two branches. Numerical simulations, based on a simplified model of the bifurcation evolution, allow us to investigate the role of the mechanism of generation of a bifurcation on its final equilibrium configurations. It is shown that bifurcations which form through the incision of a new, initially narrow, channel may lead to significantly different equilibrium configurations with respect to bifurcations which generate from a central deposition in a wide channel. In spite of its simplicity the model seems to retain the most relevant effects which govern the behavior of gravel bed channel bifurcations.

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[1] In the present work we investigate the interaction between migrating alternate bars and the dynamics of river bifurcations. Laboratory experiments are carried out to study a Y-shaped bifurcation with fixed banks and erodible bed composed of well-sorted sand. The problem is also analyzed by developing a theoretical, one-dimensional model. Results show the occurrence of regular fluctuations in the discharge distribution at the bifurcation node, which are strictly related to bar migration. The effectiveness of bars in conditioning the bifurcation behavior increases with bar amplitude and decreases with bar migration speed. Four qualitatively different behaviors of the system are observed as the controlling parameters of the flow are varied within a range significant for gravel bed rivers. The theoretical predictions are in good qualitative agreement with the experimental observations.

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.

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.