Theory Of Orthogonal Flow In Elastic Chambers And Bio-bags
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This study presents a theory of the orthogonal flow of Newtonian fluid in an elastic chamber. The elastic chamber is a space bounded by a movable material boundary S that has a single orifice and is pliable with respect to normal forces. The characteristic feature of this flow is that the fluid motion is perpendicular to moving S and S to the one-parameter level surfaces Si inside the chamber. The orthogonal flow domain is confined to such parts of the chamber interior V, for which the level function exists defining the system of level surfaces Si, by means of the equation (x,y,z)= i with the set of constant parameters i, i=1,2…,. This flow property enables us to define the three-component velocity field a(u,v,w) by the single vector q normal to Si for whole V. By means of vector algebra and the Navier-Stokes description of the shear stress tensor, a mathematical formulation of the mass and momentum conservation laws for orthogonal flow has been derived as the main result of the theory. The certainty and the uncertainty of the direction field of flow were used as the determining factors in this process. A constitutive form of the level function (x,y,z) as the tool for obtaining the system of level surfaces Si for chambers of real geometry has been proposed and applied for the sphere and slanted ellipsoid chambers. The study of liquor dynamics effects on hydrocephalus development in the system of brain ventricles inspired this work. It is assumed that the orthogonal flow theory can effectively simplify the mathematical description of a certain kind of fluid flow in elastic chambers as well as in some cavities and bags of live organisms. Keywords: single-orifice elastic chamber, orthogonal flow, directional cosines, directional field of flow, level function, level surface system, stream lines, intrinsic derivatives, constitutive form of level function, Navier-Stokes equations.
single-orifice elastic chamber, orthogonal flow, directional cosines, directional field of flow, level function, level surface system, stream lines, intrinsic derivatives, constitutive form of level function, Navier-Stokes equations