On the stability of Poiseuille-Couette flow: a bifurcation from infinity

The linear and weakly nonlinear stability of Poiseuille–Couette flow is considered for various values of the relative wall velocity 2 u w , An account is given first of the asymptotic upper and lower branches of the linear neutral curve ( s ), followed by their disappearance, as u w is increased. Two main (and one minor) neutral curves are found to exist for smaller O (1) (or lesser) values of u w , then one for moderate O (1) values of u w , and none for larger O (1) values of u w . The cut-off velocity at which each main neutral curve disappears is determined, and in each case the whole neutral curve for u w just below the cut-off value is determined in closed form. Secondly, weakly nonlinear solutions are found to bifurcate subcritically from the neutral curve for u w just below cut-off, but to ‘bifurcate from infinity’ just above cut-off. This identifies a minimum threshold amplitude at the entry to the regime where no linear neutral curve exists.