Context :

Aims : We examine the interactions of various instabilities in rotating stars , which usually are considered as independent .

Methods : An analytical study of the problem is performed , account is given to radiative losses , \mu –gradients and horizontal turbulence .

Results : The diffusion coefficient for an ensemble of instabilities is not given by the sum of the specific coefficients for each instability , but by the solution of a general equation .
We find that thermohaline mixing is possible in low-mass red giants only if the horizontal turbulence is very weak .
In rotating stars the Rayleigh–Taylor and the shear instabilities need simultaneous treating .
This has for consequence that rotation laws of the form 1 / r ^ { \alpha } are predicted to be unstable for \alpha > 1.6568 , while the usual Rayleigh criterion predicts instability only for \alpha > 2 .
Also , the shear instabilities are somehow reduced in Main Sequence stars by the effect of the Rayleigh–Taylor criterion .
Various instability criteria should be expressed differently in rotating stars than in simplified geometries .

Conclusions :