On reducing Howard's semicircle for homogeneous shear flows

On stability of steady-state plane–parallel shearing flows in a homogeneous in density ideal incompressible fluid

“…Unfortunately, results obtained with this method often have a conditional character because they are valid only for perturbations in the form of normal modes [3]. Moreover, while using integral relations, we actually deal not with all possible perturbations, but with perturbations which satisfy some additional requirements on the differential operator in the Rayleigh equation [4][5][6][7][8]10].…”

“…Stability for stationary plane-parallel shearing flows of a homogeneous in density ideal incompressible fluid in a gap between two immovable impermeable solid parallel infinite walls with respect to small plane perturbations is the most studied by the spectral method with the use of integral relations [4][5][6][7][8][9][10]. Unfortunately, results obtained with this method often have a conditional character because they are valid only for perturbations in the form of normal modes [3].…”

“…Hence, the inequality (4.8), by its existence, automatically converts both last sufficient conditions for stability of the steady-state plane-parallel shearing flows for a homogeneous in density inviscid incompressible fluid in the layer between two immovable impermeable solid parallel unbounded plates with respect to small plane perturbations in the form of normal modes which meet additional requirements on the differential operator in the Rayleigh equation [1][2][3][4][5][6][7][8]10].…”

On the Stability of Three-Dimensional Disturbances in Stratified Flow with Lateral and Vertical Shear

“…Howard's theorem was extended by Kochar and Jain [3], who proved that the complex wave velocity for any unstable mode lies within a semi-ellipse whose major axis coincides with the diameter of Howard's semi-circle, while its minor axis depends on the stratification. Banerjee et al [4] found a more limiting version of Howard's theorem by restricting attention to homogeneous flow. Pedloski [5,6] proved Howard's theorem for flow with a base rotation, which was later improved by Kanwar and Sinha [7].…”

An improved instability region for the extended Rayleigh problem of hydrodynamic stability

“…In this paper, we obtained an improved parabolic instability region. Banerjee et al (1988) has proved the parabolic instability region for the Rayleigh problem. This was taken deeper to extended Rayleigh problem by Subbiah and Ganesh (2007).…”