Passivity of plane Poiseuille flow

Abstract

This paper considers the passivity of plane Poiseuille flow, which is the incompressible flow observed between two parallel plates that are assumed to be of infinite extent. A model in which the flow is considered as the feedback connection between a linear time-invariant system and a static, memoryless nonlinear system is used. It is well known that the nonlinearity of plane Poiseuille flow is not only passive but lossless, which means that it does not generate or consume energy and the only effect of the nonlinearity is to move energy from one flow mode to another. However, little has been addressed about the passivity of the entire flow system. The primary aim of this paper is to find a subcritical Reynolds number below which the linearized flow is always strictly passive, which means that the origin of the full nonlinear system is globally asymptotically stable when the Reynolds number does not exceed the subcritical number. Our results show that the subcritical Reynolds number obtained by the passivity approach is equal to the energy Reynolds number, which is derived by the classical energy approach. This result indicates that the passivity approach is closely related to the energy approach and is a valuable tool to study the stability of fluid flows.