Second-order inertial forces and torques on a spherical particle in a viscous linear flow

Making use of matched asymptotic expansions and of a coordinate system co-moving with the background flow, we compute the force and torque acting on a small rigid sphere translating and rotating unsteadily in a general linear flow field. The loads are determined up to second order with respect to the relevant small parameter, namely the square root of the shear Reynolds number. The outer solution (which at first order is responsible for the Basset-Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or rotational acceleration of the particle and fluid, while others take the form of quadratic terms combining the relative translational/rotational velocity between the particle and fluid with the rotation/strain rate of the background flow. The complete set of second-order contributions is obtained by adding the outer and inner contributions. Classical effects such as the added-mass force or the spin-induced lift force are recovered, and new effects involving the rotation/strain rate of the background flow are revealed. The resulting force and torque equations provide the first rational extension of the classical Basset-Boussinesq-Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow.