Energy-optimal time-dependent regimes of viscous incompressible fluid flow

The hydrodynamic equations of a viscous incompressible fluid are modified for axisymmetric flows in a pipe of time-varying radius. A new exact time-dependent solution of these equations which generalizes the well-known classic steady-state Hagen–Poiseuille solution for flow in a pipe of constant radius (independent of time) is obtained. It is shown that the law of time variation in the pipe radius can be determined from the condition of the minimum work done to pump a given fluid volume through such a pipe during the radius variation cycle period. A generalization of the optimal branching pipeline in which, instead of the Poiseuille law, its modification based on the use of the exact solution corresponding to the time-dependent M-shaped regime is employed is suggested. It is shown that the hydraulic resistance can be reduced over a certain range of the parameters of the time-dependent flow regime as compared with the steady-state pipe flow regime. The conclusion obtained can be used for the development of the hydrodynamic basis for simulating the optimal hydrodynamic blood flow regime.