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

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.