Bounds on the phase velocity in the linear instability of viscous shear flow problem in the β - plane

Parallel shear flows problem is a classical hydrodynamic instability problem and continues to attract attention of researchers [1,2,3,5]. Kuo [8] considered slightly general case of a homogeneous, inviscid, parallel shear flow problem in the β -plane. He obtained the linearized perturbation equation for this problem and derived an extension of the celebrated Rayleigh’s [9] inflexion point criterion for inviscid, homogeneous parallel shear flows. Fjortoft’s [4] instability criterion for inviscid, homogeneous parallel shear flows has also been extended for such barotropic flows (see [6]). Hickernel reviewed various results obtained for this problem and obtained new upper bound on the growth rate of the temporally growing modes of the linearized equations of motion for this problem. Joseph [7] obtained bounds for the eigenvalues of the Orr–Sommerfeld’s equation. Derivation of bounds is of importance because exact solution of the problem is not obtainable in closed form. In the present paper, we extend the results of Joseph [7] to the problem of viscous parallel shear flow problem in the β -plane and also obtain a sufficient condition for stability. The governing equation for Kuo’s [8] problem for the linear stability of an incompressible, viscous, parallel shear flow in the β -plane is given by (cf. [6])