A generalized analysis of perturbation patterns for the Poiseuille flow in a tube

The linearized problem on stability with respect to the three-dimensional picture of perturbations imposed on a steady flow of Newtonian viscous fluid in a pipe (the Poiseuille flow) is analyzed. The evolution in time of individual harmonics of perturbations both by angle and along axial direction is studied. A passage to quadratic functionals constructed on squares of perturbation velocity components modulus as well as derivatives with respect to the radius of these components is performed. The upper estimate of the stability parameter is obtained. It results the lower estimates of the critical Reynolds numbers in the cases of axially symmetric perturbations and two-dimensional (both axially symmetric and asymmetric) $rz$-perturbations.