**S. G. Chefranov**^{a}** and A. G. Chefranov**^{b
}

^{a}* Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevskii per. 3, Moscow, 119017 Russia
*

*e-mail: schefranov@mail.ru
*

^{b}* Eastern Mediterranean University, Famagusta, North Cyprus
*

*e-mail: Alexander.chefranov@emu.edu.tr
*

Received December 13, 2013

**Abstract**—It has been shown that the conclusion of the linear instability of the Hagen–Poiseuille flow at finite

Reynolds numbers requires the refusal of the use of the traditional “normal” form of the representation of dis-

turbances, which implies the possibility of separation of variables describing disturbances as functions of the

radial and longitudinal (along the axis of a tube) coordinates. In the absence of such separation of variables in

the developed linear theory, it has been proposed to use a modification of the Bubnov–Galerkin theory that

makes it possible to take into account the difference between the periods of the longitudinal variability for dif-

ferent radial modes preliminarily determined by the standard application of the Galerkin–Kantorovich method

to the evolution equation of extremely small axisymmetric disturbances of the tangential component of the

velocity field. It has been shown that the consideration of even two linearly interacting radial modes for the

Hagen–Poiseuille flow can provide linear instability only in the presence of the mentioned conditionally peri-

odic longitudinal variability of disturbances along the axis of the tube, when the threshold Reynolds number

Re_{th}(*p*) is very sensitive to the ratio *p* of two longitudinal periods each describing longitudinal variability for its

radial disturbance mode. In this case, the threshold Reynolds number can tend to infinity, Re_{th}(*p*) , only

at *p* = *p*_{k} = *k*, *p* = *p*_{1/k} = 1/*k*, and *p* = = [*k* + 1 ]/2, where *k*= 1, 2, 3, …. The minimum

Reynolds number Re_{th}(*p*) 448 (at which *p* 1.527) for the linear instability of the Hagen–Poiseuille flow

quantitatively corresponds to the condition of the excitation of Tollmien–Schlichting waves in the boundary

layer, where Re_{th} = 420. Similarity of the mechanisms of linear viscous dissipative instability for the Hagen–

Poiseuille flow and Tollmien–Schlichting waves has been discussed. Good quantitative agreement has been

obtained between the phase velocities of the vortex disturbances and the experimental data on the velocities of

the leading and trailing edges of turbulent “puffs” propagating along the axis of the tube.

**DOI: **10.1134/S1063776114070127

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