On the eigenvalue problems of Poiseuille flows in a circular pipe

In this work we introduce a novel formulation of Sexl's equations obtained by expanding three dimensional infinitesimal disturbances about the axisymmetric Hagen-Poiseuille flow in a pipe of circular cross section. The formulation is based on a representation theorem of solenoidal vector fields due to Schmitt and von Wahl [53]. We use the new formulation to prove that the linear nonaxisymmetric eigenvalue problem associated with Hagen-Poiseuille flow has infinitely many eigenfunctions which form a complete set. The method of proof includes the DiPrima-Habetler completeness theorem [14]. A nontrivial transformation that links the new formulation with the formulation used by Salwen et al [50] is presented.; The relationship between axisymmetric eigenfunctions associated with Hagen-Poiseuille flow and their counterparts associated with parabolic Poiseuille flow is explored by means of a numerical study. We use a modified Chebyshev tau numerical scheme to compute eigenvalues and eigenfunctions of the two problems. The parabolic Poiseuille flow is easier to deal with because of the absence of a singularity in the differential equations. An exact solution to the axisymmetric parabolic Poiseuille flow problem is presented for the first time.