Linear Stability of Plane Poiseuille Flow in the Sense of Lyapunov

In this paper, we present a linear stability analysis formulation for a plane Poiseuille flow developed in a continuous time domain. Contrary to a conventional approach based on an eigenvalue analysis, which can only proof stability with respect to certain solutions that are assumed to be time harmonics modulated by an exponentially growing or decaying amplitude, the presented methodology does not make any assumptions on a solution form. By analyzing all time-varying solutions and not only the ones restricted to a specific functional form, the developed stability test provides a stronger condition with regard to the system stability. Stability analysis is performed by first casting the corresponding linearized partial differential equation into a partial integral equation (PIE) form, and subsequently employing a linear partial inequality (LPI) stability test, which searches for a corresponding Lyapunov function parameterized through polynomial expansions to prove or disprove stability. Stability results of the continuous-time formulation for the plane Poiseuille flow are compared with a traditional eigenvalue-based analysis, demonstrating that the developed methodology represents a stricter condition on stability.