Modelling Complex Flows in Porous Media by Means of Upscaling Procedures

We review a series of problems arising in the field of flows through porous media and that are highly nontrivial either because of the presence of mass exchange between the fluid and the porous matrix (or other concurrent phenomena of physical or chemical nature), or because of a particularly complex structure of the medium. In all these cases there is a small parameter $\varepsilon$, representing the ratio between the microscopic and the macroscopic space scale. Our attention is focussed on a modelling technique (upscaling) which start from the governing equations written at the pore scale, introduces an expansion in power series of $\varepsilon$of all the relevant quantities and eventually leads to the formulation of the macroscopic governing equations at the various orders in $\varepsilon$ by a matching procedure, followed by suitable averaging. Two problems will be analyzed with some detail: soil erosion and the dynamics of water ultrafiltration devices. Moreover other problems will be occasionally discussed and open questions will be proposed

Designing irrigation pipes

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Mathematical Models for Some Aspects of Blood Microcirculation

Blood rheology is a challenging subject owing to the fact that blood is a mixture of a fluid (plasma) and of cells, among which red blood cells make about 50% of the total volume. It is precisely this circumstance that originates the peculiar behavior of blood flow in small vessels (i.e., roughly speaking, vessel with a diameter less than half a millimeter). In this class we find arterioles, venules, and capillaries. The phenomena taking place in microcirculation are very important in supporting life. Everybody knows the importance of blood filtration in kidneys, but other phenomena, of not less importance, are known only to a small class of physicians. Overviewing such subjects reveals the fascinating complexity of microcirculation

Flow stability in a wide vaneless diffuser

Abstract This work is concerned with the theoretical aspects of flow stability in a two dimensional vaneless diffuser. Specifically, the appearance of self-excited oscillations, also referred to as rotating stall, is investigated considering a two-dimensional inviscid flow in an annulus. We consider a linear perturbation method, taking as basic flow the steady potential velocity field whose radial and tangential components are inversely proportional to the radial coordinate. We show that such flow may become unstable to small two-dimensional perturbations provided that the ratio between the inlet tangential velocity and the radial one is sufficiently large and a certain amount of vorticity is injected in the flow field. Such an instability is purely kinematical, i.e. it does not involve any boundary layer effects, contrary to the classical hypothesis which ascribes the instability to a peculiar boundary layers interaction