Short-time evolution of pipe Poiseuille flow

TL;DR

This paper proves that pipe Poiseuille flow with a parabolic profile decays exponentially over time when subjected to small disturbances, highlighting the dominance of viscous diffusion in the flow's linear regime.

Contribution

It provides a rigorous proof of exponential decay for infinitesimal disturbances in pipe flow and characterizes the eigenvalue spectrum, including asymptotic formulas and numerical validation.

Findings

Flow disturbances decay exponentially over time.

Eigenvalue spectrum contains infinitely many discrete modes.

Asymptotic formulas match numerical computations.

Abstract

In the present paper we prove that the pipe Poiseuille flow of parabolic velocity profile attenuates exponentially in time with respect to three dimensional infinitesimal disturbances at all finite wave numbers and Reynolds numbers for given azimuthal periodicity if the equations of motion are linearized. The spectra of the eigenvalue are shown to consist of infinitely many discrete eigen-modes. Results of asymptotic analysis, expressed in simple algebraic formulas and functional relations, are given. Good comparison has been found in the approximations and numerical computations. The present results are best interpreted as a description of the pipe flow regime, where the linear diffusion due to viscosity dominates.