  We consider the hydrodynamic stability of homogeneous, incompressible and
rotating ellipsoidal fluid masses. The latter are the simplest models of fluid
celestial bodies with internal rotation and subjected to tidal forces. The
classical problem is the stability of Roche--Riemann ellipsoids moving on
circular Kepler orbits. However, previous stability studies have to be
reassessed. Indeed, they only consider global perturbations of large wavelength
or local perturbations of short wavelength. Moreover many planets and stars
undergo orbital motions on eccentric Kepler orbits, implying time-dependent
ellipsoidal semi-axes. This time dependence has never been taken into account
in hydrodynamic stability studies. In this work we overcome these stringent
assumptions. We extend the hydrodynamic stability analysis of rotating
ellipsoids to the case of eccentric orbits. We have developed two open-source
and versatile numerical codes to perform global and local inviscid stability
analyses. They give sufficient conditions for instability. The global method,
based on an exact and closed Galerkin basis, handles rigorously global
ellipsoidal perturbations of unprecedented complexity. Tidally driven and
libration-driven elliptical instabilities are first recovered and unified
within a single framework. Then we show that new global fluid instabilities can
be triggered in ellipsoids by tidal effects due to eccentric Kepler orbits.
Their existence is confirmed by a local analysis and direct numerical
simulations of the fully nonlinear and viscous problem. Thus a non-zero orbital
eccentricity may have a strong destabilising effect in celestial fluid bodies,
which may lead to space-filling turbulence in most of the parameters range.
