Note on Discrepancies between Two Theories on the Stability of Plane Poiseuille Flow

Abstract

Existing theories of the hydrodynamical stability of plane Poiseuille flow are divided into two groups by their quite different conclusions. Results attained by Lin and other earlier authors are that this flow is unstable for sufficiently large Reynolds numbers. On the other hand, Pekeris has obtained different results that the flow is stable for any value of the Reynolds number and that, in addition to the ordinary disturbances discussed by earlier authors, there exists another class of perturbation which is characterized by c_r→1 as αR→∞.
In this paper, a critical survey for these conflicting results is presented. It is shown that Pekeris’s first result arises essentially from the poor convergency of his series expansion of c, namely c=c₀+α²c₁+…, and that if this procedure is avoided, the existence of Lin’s stability limit is asymptotically confirmed. It is also found that Pekeris’s second prediction is erroneous.