Linear stability analysis of plane Poiseuille flow of a De-Kee-Turcotte fluid

In this paper, we study the linear stability of a planar Poiseuille flow of a De-Kee-Turcotte fluid. After a suitable scaling of the governing equations, we explicitly determine the base one-dimensional (1D) flow by exploiting the Lambert function. We show that the problem admits two steady-state solutions, one for each branch of the Lambert function. We perturb the base flow with a disturbance in the form of longitudinal wave of unknown amplitude. We derive the modified Orr-Sommerfeld equation for the second component of the perturbed velocity. The relative eigenvalue problem is solved through a spectral collocation scheme based on Chebyshev polynomials. We prove that the solution corresponding to the secondary branch of the Lambert function is unconditionally unstable, whereas the one relative to the principal branch may exhibit two critical Reynolds numbers: one marking the onset of instability and the other its cessation.