Investigation of the relationship between fractal and hydraulic properties of porous structures of the upper respiratory tract of some Arctic animals

The respiratory ducts of animals and humans are presented by curved tubes with complex geometries. The open areas in such structures are filled with moving air governed by a pressure drop between the inlet and outlet of the duct. The complex structures formed by thin walls and warmed by constant blood flow at the body temperatures T=36-39 C serve for fast and efficient warming of the inhaled air to the body temperature and its moistening up to 100% humidity. The Arctic animals possess the most efficient nasal ducts allowing the heating of the inhaled air from T=-30-60C to T=38-39 C during the duct with the length L=8-15 only. The detailed geometry of the nasal ducts of some Arctic animal has been studied on the computed tomograms (CT) scans of the heads of the animals found in the open databases and published in literature. The highly porous structures on some slices are formed by fractal-like divisions of the walls protruded into the nasal lumen. Since the fractal structures are characterized by their fractal dimensions D, the relationships between the hydrodynamic properties and fractal dimensions of the porous structures of the upper respiratory tract of some Arctic animals has been studied. The dimensions D of the cross sections of the tract have been calculated by the counting box method. The porosities of the samples, the tortuosity of the pores, and the equivalent hydraulic diameter Dh of the channel have been calculated. Sierpinski fractals of various types have been used as models of porous structures, for which the above listed parameters, as well as hydraulic resistance to a stationary flow, have also been computed. A number of statistical dependencies between the calculated parameters were revealed, but the absence of their correlations with D was shown. It was obtained, the structures with different porosities and hydraulic resistance Dh can have the same values ​​of D. Therefore, the choice of an adequate model based on only D value introduces significant errors in the calculations of air heating along the upper respiratory tract. The statistical dependences inherent in the natural samples studied can be obtained only on the basis of multifractal models in which the number and shape of the channels, as well as the scale of their decrease, change in a certain way at each generation.