Rotation of a slender particle in a shear flow: influence of the rotary inertia and stability analysis

• Published in: Zeitschrift für angewandte Mathematik und Mechanik 2009-10, Vol.89 (10), p.823-832
• Main Authors: Altenbach, H., Brigadnov, I., Naumenko, K.
• Format: Article
• Language: English
• Subjects: Fiber orientation; global stability; nonlinear dynamical system; phase plane; quantization effect; shear flow
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.200900249

The rotation of an inertialess ellipsoidal particle in a shear flow of a Newtonian fluid has been firstly analyzed by Jeffery [17]. He found that the particle rotates such that the end of its axis of symmetry describes a closed periodic orbit. In the special case of a slender particle the Jeffery solution predicts the particle alignment parallel to the streamlines. In a recent publication [3] it was shown that the orbits are no longer observable if the rotary inertia is taken into account. Furthermore, in the case of a slender particle the inertia may cause the jump over the equilibrium alignment. In this paper we address a detailed analysis of the slender particle behavior in the shear flow. We recall the constitutive equation for the hydrodynamic moment and formulate equations of rotary motion. For a special initial condition we reduce the problem to a single second‐order ordinary differential equation with respect to the angle of rotation about a fixed axis. The phase portrait of this equation illustrates different cases of the particle behavior depending on the initial conditions and the “inertia” parameter. They include the particle alignment to a semi‐stable equilibrium position, the non‐uniform rotation about a fixed axis as well as the quantization effect (the particle locates in the neighborhood of the first equilibrium point over a relatively long time and then rotates towards the next equilibrium point). The rotation of an inertialess ellipsoidal particle in a shear flow of a Newtonian fluid has been firstly analyzed by Jeffery [Proc. R. Soc. London A 102, 161–179 (1922)]. He found that the particle rotates such that the end of its axis of symmetry describes a closed periodic orbit. In a recent publication [ZAMM 87, 81–93 (2007)] it was shown that the orbits are no longer observable if the rotary inertia is taken into account. In this paper the authors address a detailed dynamic analysis of the slender particle behavior in the shear flow. For a special initial condition they reduce the problem to a single second‐order ordinary differential equation with respect to the angle of rotation about a fixed axis. The phase portrait of this equation illustrates different cases of the particle behavior depending on the initial conditions and the “inertia” parameter. They include the particle alignment to a semi‐stable equilibrium position, the non‐uniform rotation about a fixed axis as well as the quantization effect (the particle locates in the neighborh