Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem

Abstract

Theoretical estimates of the phase velocityC r of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr-Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow (U(z)=1−z 2,−1≤z≤+1), leave open the possibility of these phase velocities lying outside the rangeU min<C r <U max, but not a single experimental or numerical investigation in this regard, which are concerned with unstable or marginally stable waves has supported such a possibility as yet,U min andU max being respectively the minimum and the maximum value ofU(z) forz∈[−1, +1]. This gap between the theory on one side and the experiment and computation on the other has remained unexplained ever since Joseph derived these estimates, first, in 1968, and has even led to the speculation of a negative phase velocity (or rather,C r <U min=0) and hence the possibility of a ‘backward’ wave as in the case of the Jeffery-Hamel flow in a diverging channel with back flow ([1]). A simple mathematical proof of the non-existence of such a possibility is given herein by showing that the phase velocityC r of an arbitrary unstable or marginally stable wave must satisfy the inequalityU min<C r <U max. It follows as a consequence stated here in this explicit form for the first time to the best of our knowledge, that ‘overstability’ and not the ‘principle of exchange of stabilities’ is valid for the problem of plane Poiseuille flow.