Forces on a spherical particle with an arbitrary axis of rotation in a weak shear flow of a highly rarefied gas

In this paper, we derive analytical formulas for the force and torque on a spherical particle in a weak shear flow of a highly rarefied gas. The rotation axis of the particle is not restricted with regard to its direction. So this paper is an extension of our previous paper [N. Liu and D. Bogy, Phys. Fluids 20, 107102 (2008)] where the rotation axis is in the same direction as the gradient of the incoming shear flow. The contributions to the force from rotation of the particle and nonuniformity of the flow are shown to be decoupled. These two effects, however, do produce a coupled effect in the torque. Combined with the equation of motion, the trajectory of a spherical particle in a weak shear flow of a highly rarefied gas can be analyzed based on these analytical formulas together with appropriate initial conditions.