Biorthogonal stretching of an elastic membrane beneath a uniformly rotating fluid

The flow generated by a biorthogonally stretched membrane below a steadily rotating flow at infinity is examined. The flow's velocity field is shown to be an exact, self-similar, solution of the fully three-dimensional Navier-Stokes equations with the solution governed by a set of four ordinary differential equations. It is demonstrated that dual solutions exist when the membrane is stretched in both directions (except in the radially symmetric case), as well as for a range of parameters where the membrane is stretched in one direction and allowed to shrink in the other. For stretching rates close to the radially stretched symmetric case, four solutions exist, including one which has a large wall-jet velocity profile close to the membrane. The linear stability of each solution is also examined, and it is found that only a single solution is stable (where one exists) for a given stretching and rotation rate