Eigenvalue bounds for the Orr-Sommerfeld equation and their relevance to the existence of backward wave motion

Theoretical estimates of the phase velocity c(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 Poi seuille flow or nearly parallel viscous shear flows in straight channels wi th velocity U(z) (= 1- z(2), z is an element of[- 1, + 1] for plane Poiseui lle flow), leave open the possibility that these phase velocities lie outsi de the range U-min < c(r) < U-max but not a single experimental or numerica l investigation, concerned with unstable waves in the context of flows with (d(2)U/dz(2))(max) less than or equal to 0, has supported such a possibili ty as yet. U-min,U-max and (d(2)U/dz(2))(max) are, respectively, the minimu m value of U(z), the maximum value of U(z), and the maximum value of (d(2)U /dz(2)) for Z is an element of[- 1, + 1]. This gap between the theory on on e hand and experiment and computation on the other has remained unexplained ever since Joseph [3] derived these estimates, first in 1968, and has even led to the speculation of a negative phase velocity in plane Poiseuille ho w (i.e., c(r) < U-min = 0) and hence the possibility of a "backward" wave a s in Jeffrey-Hamel flow in a diverging channel with backflow [1]. A simple mathematical proof of the nonexistence of such a possibility is given herei n by showing that if (d(2)U/dz(2))(max) less than or equal to 0 and (d(4)U/ dz(4))(min) greater than or equal to 0 for z is an element of[- 1, + 1], th en the phase velocity c(r) of an arbitrary unstable wave must satisfy the i nequality U-min < c(r) < U-max, (d(4)U/dz(4))(min) is the minimum value of (d(4)U/dz(4)) for z is an element of[- 1, + 1], and therefore c(r) cannot b e negative when U-min = 0. Another result that provides valuable insight in to the general modal structure of the problem of instability of the above c lass of flows with U-min greater than or equal to 0 (e.g., plane Poiseuille flow) is that all standing waves, that is, modes for which c(r) = 0, are s table.