Energetically optimal nonstationary mode of flow along tube with constant and time-varying radius

Derived is a new modification of hydrodynamic equations of viscous incompressible fluid flowing along the tube with radius changing in time. Obtained are exact non-stationary solutions of these equations generalizing a well-known classic stationary solution for Hagen-Poiseuille flow in the tube with radius constant in time. It is demonstrated that the law of changing the tube radius in time may be determined basing on the condition of minimality of the work expended for flowing the set fluid volume along such a tube during the period of radius change cycle. Obtained is the solution of the corresponding variational (isoperimetric) problem on conditional extremum determining the limits to dimensionless quantity of the cycle duration set by the specified dimensionless value of the flowed fluid volume. Identified is the generalization of well-known model of optimal branching pipeline in which the Poiseuille law modification is used for a new exact non-stationary solution of hydrodynamic equation instead of the law itself. It is demonstrated that the energetically favorable non-stationary modes with negative hydraulic resistance are permissible in certain conditions. The obtained conclusions may be used for development of the hydrodynamic basis of modelling the energy-optimal blood flow realized in the cardiovascular system in norm.