Tradeoffs between energy efficiency and mechanical response in fluid flow networks

Optimization of transport networks is a ubiquitous problem that can be found in a variety of natural and artificial systems. In the case of systems such as the animal vasculature, the transport of fluids is not only hindered by the inherent resistance to flow but also kept in a dynamic state by the pulsatile nature of the heart and elastic properties of the vessel walls. By linearizing the Navier-Stokes equation, we show that while this imparted pulsatility necessarily increases the dissipation of energy caused by the resistance, the vessel elasticity helps to reduce overall dissipation by attenuating the amplitude of the pulsatile components of the flow. However, we find that this reduction in energy loss comes at the price of increasing the time required to respond to changes in the flow boundary conditions. Dissipation and response time are found to obey a simple power law scaling relation in single vessels as well as hierarchically structured networks with relatively few loops.