Abstract
Energetically optimal nonstationary mode of flow along tube with constant and time-varying radius
Highlights
The obtained the equations of hydrodynamics the conclusions may be used for development of the hydrodynamic basis of model- parameters characterizing the fluid ling the energy-optimal blood flow realized in the cardiovascular system in norm. boundaries and its volume are supposed to be temporary independent
Since the times of Leonardo da Vin- of such an efficiency cannot be es- simplest modification, which, for ci there has been an issue of extreme- tablished when modelling the blood the present, takes into account only a ly high energetic efficiency of blood flow process along the vessels using synchronous change in time of radicirculation in the cardiovascular sys- the traditional Poiseuille law [1]. us of the tube along its entire length
Tem (CVS) observed under its normal This law relates the velocity of fluid Besides, obtained is a generalization of the well-known model of optimal branching tube [5] for the case of a new non-stationary solution of hydrodynamic equations
Summary
Let us consider the relevant modification of the Navier–Stokes equations and the continuity equations in order to model a nonstationary flow mode in the time-varying radius tube. Let us consider the solution of the following variational problem in conditional extremum (type of isoperimetric problem) when it is necessary to define the form of functions s, for which the work functional reaches its extremum (minimum) in fixed fluid volume g = g0 = const in (14) pumped within the period of extension and further reduction to the initial tube radius s(y) = 1. In solutions (19) and (20) at b > 0 (that corresponds to an initial reduction and only afterwards an extension of the tube up to initial radius) possible is the realization of energetically advantageous fluid flow mode, and an execution of condition (23) is necessary for it.
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