Orientation of Symmetric Bodies Falling in a Second-Order Liquid at Nonzero Reynolds Number

Galdi, G. P., Pokorny, M., Vaidya, A., Joseph, D. D. & Feng, J. J.

Mathematical Models and Methods in Applied Sciences Vol. 12, No. 11, 1653-1690 (2002).

Abstract - We study the steady translation 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 allows, 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 Ec for the elasticity number E=We/Re, where Ec depends only on the geometric properties of the body, such as size and shape, and on the quantity (F1+F2)/F1, where F1 and F2 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.