Objects with three orthogonal symmetry planes: Oblique driving forces and Stokes flow motion

Stokes flow studies are fundamental to advancing medical and industrial technologies, particularly in areas such as drug targeting, cell studies, the optimization of drug carrier vehicles, high viscosity flows, and magnetic particle imaging. While previous research has focused on the motion of obliquely falling cylindrical rods and magnetic particle chains, a broader analytical framework is required to understand more complex particle-fluid migrations. In this paper, we first generalize the two-dimensional motion of an obliquely falling rod in a gravitational field to the three-dimensional motion of an object possessing three mutually perpendicular planes of symmetry falling through a viscous fluid in the Stokes limit. We derive a general formula for the three components of velocity-including both downward and sideways components-for objects of arbitrary orientation and uniform density. These analytical solutions are defined in terms of the object's orientation, specified via Euler angles, and the velocity of the object falling along each of its three principal axes, or the drag coefficient along each of those axes. We give a variety of examples of objects that satisfy this general formula. In addition, we apply the formula to a cuboid for which those velocity components along each of its principal axes have been measured experimentally by other researchers, thus giving both the downward and sideways components for arbitrary orientation. We then analyze the motion in a gradient magnetic field of elongated magnetic particles, such as nanorods and nanoellipsoids, for which the induced magnetic moment is along the long axis of the particle. We discuss the similarities and differences with the gravitational case. By providing a unified framework for predicting the trajectories of these symmetric bodies, this work enhances the understanding of the motion of inertial and magnetic particles under the influence of gravitational and gradient magnetic fields, respectively.