Linear instability of plane Couette and Poiseuille flows

Abstract

It is shown that linear instability of plane Couette flow can take place even at finite Reynolds numbers Re > Re_th ≈ 139, which agrees with the experimental value of Re_th ≈ 150 ± 5 [16, 17]. This new result of the linear theory of hydrodynamic stability is obtained by abandoning traditional assumption of the longitudinal periodicity of disturbances in the flow direction. It is established that previous notions about linear stability of this flow at arbitrarily large Reynolds numbers relied directly upon the assumed separation of spatial variables of the field of disturbances and their longitudinal periodicity in the linear theory. By also abandoning these assumptions for plane Poiseuille flow, a new threshold Reynolds number Re_th ≈ 1035 is obtained, which agrees to within 4% with experiment—in contrast to 500% discrepancy for the previous estimate of Re_th ≈ 5772 obtained in the framework of the linear theory under assumption of the “normal” shape of disturbances [2].