On the viscous motion of a small particle in a rotating cylinder

Abstract

The dynamics of a non-neutrally buoyant particle moving in a rotating cylinder filled with a Newtonian fluid is examined analytically. The particle is set in motion from the centre of the cylinder due to the acceleration caused by the presence of a gravitational field. The problem is formulated in Cartesian coordinates and a relevant fractional Lagrangian equation is proposed. This equation is solved exactly by recognizing that the eigenfunctions of the problem are Mittag–Leffler functions. Virtual mass, gravity, pressure, and steady and history drag effects at low particle Reynolds numbers are considered and the balance of forces acting on the particle is studied for realistic cases. The presence of lift forces, both steady and unsteady, is taken into account. Results are compared to the exact solution of the Maxey–Riley equation for the same conditions. Substantial differences are found by including lift in the formulation when departing from the infinitesimal particle Reynolds number limit. For particles lighter than the fluid, an asymptotically stable equilibrium position is found to be at a distance from the origin characterized by X ≈ −V_τ/Ω and Y/X ≈ (C_S/3π√2) Re_s^1/2, where V_τ is the terminal velocity of the particle, Ω is the positive angular velocity of the cylinder, Re_s is the shear Reynolds number a²Ω/v, and C_S is a constant lift coefficient. To the knowledge of the authors this work is the first to solve the particle Lagrangian equation of motion in its complete form, with or without lift, for a non-uniform flow using an exact method.