Structure parameters in rotating Couette-Poiseuille channel flow (Q2277326)

The consideration of structure parameters like amplitude ones in addition to the usual Reynolds number appearing in nonlinear operational equations defined in appropriate Sobolev spaces like those related with problems in fluid mechanics, is used to determine a complete set of bifurcation equations which contain linear, quadratic and cubic terms based on a dependence of the critical characteristic value on the structure parameter when the second is considered as an amplitude one. By introducing convenient parameters the corresponding bifurcation equation is obtained including a remainder term for which an additional invariance assumption is made in order to solve the mentioned equation. This assumption is made in order to apply the implicit function theorem to guarantee the existence and uniqueness of the solution of the bifurcation equation. From this the instability under certain conditions is obtained. The above results are applied to the study of the superposition of a Poiseuille flow and a rotating Couette channel flow. The Coriolis effects are expected due to the consideration of a non-zero Rossby number. Disturbance equations are obtained by using a structure parameter related to the Rossby number and representing the superposition of the second stationary flow with respect to the former. On the Hilbert space of functions periodic in z, with zero trace with respect to y, where \(| y| =1/2\), and independent of x, it is shown that if the structure parameter \(\gamma =0\) the linear stability problem could be reduced by standard methods and the boundary value problem for the amplitudes depending on y is such that the first eigenvalue takes an absolute minimum in some value \(\sigma_ 0\), from which the necessary period with respect to z for the Hilbert space is obtained. Then in this space the linear part for \(\gamma =0\) is compact and self-adjoint, and if \(\gamma\neq 0\) it is compact but not selfadjoint. Using these facts the authors showed that all hypotheses in the general part are satisfied. Then based on the solutions of the bifurcation equation, it is proved that the addition of a small component of Poiseuille flow to a basic rotating Couette channel flow destabilizes it if some parameter related with the Reynold's and Rossby's number is greater than a critical value depending on \(\gamma\), but less than the critical eigenvalue at which the Couette flow loses stability.