Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k₂ is positive in the supercritical region (R > R_C) where k₁ is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k₁/k₂)^½. We have evaluated the coefficient k₂ and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.