New Linear Theory of Hydrodynamic Instability of the Hagen-Poiseuille Flow

TL;DR

This paper introduces a new linear instability condition for Hagen-Poiseuille flow, showing that certain quasi-periodic disturbances can cause instability at Reynolds numbers above 124, challenging previous stability assumptions.

Contribution

It presents a novel instability criterion involving quasi-periodic disturbances with two radial modes, differing from traditional periodic disturbance analysis.

Findings

Instability occurs at Re > 124 for specific quasi-periodic disturbances.

The spatial structure involves two non-commensurable longitudinal periods with a ratio near the golden ratio.

Traditional pure periodic disturbances do not induce instability at any Reynolds number.

Abstract

New condition Re>Re_th_min=124 of linear (exponential) instability of the Hagen-Poisseuille (HP) with respect to extremely small by magnitude axially-symmetric disturbances of the tangential component of the velocity field is obtained. For this, disturbances must necessarily have quasi-periodic longitude variability (not representable as a Fourier series or integral) along the pipe axis that complies with experimental data and differs from the usually considered idealized case of pure periodic disturbances for which HP flow is stable for arbitrary large Reynolds numbers Re. Obtained minimal threshold Reynolds number is related to the spatial structure of disturbances (having two radial modes with non-commensurable longitudinal periods) in which irrational value p=1.58 of the ratio of the two longitudinal periods is close to the value of the "golden ratio" equal to 1.618.