A general treatment of stability is considered for an isentropic flow equilibrium against three-dimensional incompressible perturbations by taking into account the difference in the orientations of the system rotation and flow vorticity. It is shown that the aforementioned orientation difference can indeed generate a coupling that drives instabilities at the expense of the rotational energy. Two types of instability are identified, with one growing algebraically and the other growing exponentially; the parameter regimes for both instabilities are also located. The algebraically growing modes are destabilized more easily than the exponentially growing modes; for example, the former can be unstable when the angle between the rotation axis and the vorticity is beyond 70 degrees.5, whereas the latter becomes unstable when this angle is greater than 90 degrees. In addition, we find that even in the limit of small vorticity, the system may still be unstable algebraically at a considerable strength, in contrast to the case of exact zero vorticity, which is absolutely stable. This finding indicates the existence of structural instability for a rotating fluid. The present analysis is applied also to examination of the problem of shear mixing interior of an accreting white dwarf in the context of nova explosions. In order for the nuclear fuels to be blended deep inside the star and make the explosion, the high angular momentum accreted materials combined with the stellar materials should undergo shear flow instabilities. We find that the shear flow instabilities happen when the disk rotation axis is off by more than 90 degrees from the star rotation axis. The instability has in general an exponential growth, on a timescale much shorter than that of the runaway nuclear burning.
