Abstract
The use of porous ducts to deliver water to agricultural fields is an old technique which helps saving water and prevents ground erosion. Designing porous duct is not as a simple task as it looks and apparently has never been the subject of mathematical research. Here the problem is addressed making use of a double rescaling of space and velocity variables, which allows the derivation of the governing equations starting from the study of the classical Navier Stokes equations in a pipe. Such equations are then solved obtaining results of practical interest in design of irrigation pipes, both for low discharge pipes (small plants) and for high discharge pipes (large plants).
Highlights
A widely used irrigation technique consists in delivering water by letting it filtrate through pipes laid down or suspended over the ground
In small plants pipes made of a permeable and flexible material and having length of the order of 100 m and radius of the order of 1 cm are laid on the ground and connected to a reservoir whose capacity is of the order of 1 m3 located in an elevated position so that the driving pressure is provided by gravity
In large plants irrigation pipes are suspended over the ground and have lengths of the order of 1 Km
Summary
A widely used irrigation technique consists in delivering water by letting it filtrate through pipes laid down or suspended over the ground. The key point in designing an irrigation plant of both kinds is to regulate the total discharge, and to obtain, at the far end of the pipe, a water delivery rate similar to the one close to the inlet. The main difference between our approach and the quoted studies is that, in place of heuristic assumptions, we just apply the fundamental laws of dynamics of fluids in a pipe (Navier Stokes equations) and through porous media (Darcy’s law). This is precisely the advantage of the upscaling procedure described above, which gives a sound rigorous basis to the theory. The target we have in mind is to look for the physical and geometrical parameters which produce a prescribed discharge, while fulfilling additional constraints compatible with an effective operation of the plant
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