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In the linear theory of hydrodynamic stability up to now there exist examplesof flows for which there is full quantitative distinction, as for cylindricalHagen-Poiseuille (HP) flow in a pipe with round section, between theoryconclusions and experimental data on the threshold Reynolds number Reth. In thepresent work, we show that to get a conclusion of linear instability of the HPflow for finite Reynolds numbers Re, it is necessary to abandon the use oftraditional 'normal' form of disturbances which assumes an opportunity ofseparation of variables describing disturbances variability depending on radialand longitudinal (along the pipe axis) coordinates. In the result of theabsence of such variables separation, in the suggested linear theory, it isproposed to use Bubnov-Galerkin's approximation method modification that givesan opportunity to account longitudinal variability periods distinctions fordifferent radial modes defined a priori in the result of standardGalerkin-Cantorovich's method to the equation of evolution of extremely smallaxially symmetric velocity field tangential component disturbances. We foundthat when considering even two linearly interacting radial modes for the HPflow, linear instability is possible only when there exists mentioned aboveconditionally periodic longitudinal along the pipe axis disturbance variabilitywhen Re_th(p) very sensitively depends on the ratio p of two longitudinalperiods each of which describes longitudinal variability for its own radialmode only. Obtained for the HP flow linear instability realization minimalvalue Re_th=448 (when p=1.527) quantitatively agrees with the Tolmin-Shlihtingwaves in the boundary layer emergence, where also Re_th=420. We getquantitative agreeing of the phase velocity values of the considereddisturbances with experimental data on the fronts of the turbulent 'puffs'spreading in the pipe.

Hagen-Poiseuille Flow Linear Stability Paradox Resolving and Viscous Dissipative Mechanism of the Turbulence Emergence in the Boundary Layer