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Stability analysis of Boundary Layer in Poiseuille Flow Through A Modified Orr-Sommerfeld Equation

For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive modified Orr-Sommerfeld equations that can be applied in the layer. Contrary to what one might think, we find that Squire's theorem is not applicable for the boundary layer. We find also that normalization by inertial or viscous effects leads to the same order of stability or instability. For the 2D disturbances flow ($θ=0$), we found the same critical Reynolds number for our two normalizations. This value coincides with the one we know for neutral stability of the known Orr-Sommerfeld equation. We noticed also that for all overs values of $k$ in the case $θ=0$ correspond the same values of $Re_δ$ at $c_i=0$ whatever the normalization. We therefore conclude that in the boundary layer with a 2D-disturbance, we have the same neutral stability curve whatever the normalization. We find also that for a flow with hight hydrodynamic Reynolds number, the neu- tral disturbances in the boundary layer are two-dimensional. At last, we find that transition from stability to instability or the opposite can occur according to the Reynolds number and the wave number.