Orientation of symmetric bodies falling in a second-order liquid at nonzero reynolds number

We study the steady translational fall of a homogeneous body of revolution around an axis a, with fore-and-aft symmetry, in a second-order liquid at nonzero Reynolds (Re) and Weissenberg (We) numbers. We show that, at first order in these parameters, only two orientations are allowed, namely, those with a either parallel or perpendicular to the direction of the gravity g. In both cases the translational velocity is parallel to g. The stability of the orientations can be described in terms of a critical value E_c for the elasticity number E = We/Re, where E_c depends only on the geometric properties of the body, such as size or shape, and on the quantity (ψ₁ + ψ₂)/ψ₁, where ψ₁ and ψ₂ are the first and second normal stress coefficients. These results are then applied to the case when the body is a prolate spheroid. Our analysis shows, in particular, that there is no tilt-angle phenomenon at first order in Re and We.