Growth rate of the TDEI for a cylindrical shell of aspect ratio H/R = 2 with an elliptical cross section (η = 0.2,  E = 0.0036,  β = 0.47,  Pr = 1) and an ellipsoidal shell (η = 0.3,  E = 0.0029,  β = 0.317,  Pr = 1) with a rotation axis of length c = (a + b)/2. The figure compares the numerical growth rate in the autogravitating case where the gravity is given by the Poisson equation for the gravitational potential of a homogeneous fluid (see Cébron et al. 2010c, for details), and the case where the gravity is played by a centrifugal force (n = 0), as in the experiments of Lavorel & Le Bars (2010). The numerical growth rate is translated vertically in the figure, to match the inviscid growth rate 9/16 at , which corresponds to a surfacic damping term coefficient α = 3.24 for the cylindrical shell (squares and triangles) and α = 3.1 for the ellipsoidal shell (circles and diamonds). Theoretical growth rates are shown by a continuous red line for n = 0,  m = 1 and a dashed blue line for n = m = 1.