An inertial slender-body theory

We present a fully inertial slender-body theory (SBT) that incorporates the effect of fluid inertia on the scale of the length (the "outer" region) as well as the characteristic diameter (the "inner" region) of a steadily translating slender particle. This is achieved by matching the solution of the quasi-two-dimensional full Navier-Stokes equations in the inner region to an outer solution that consists of a superposition of a solution of the linearized Navier-Stokes equations driven by a line of forces and a potential flow solution driven by a line distribution of sources and source dipoles. The drag and lift forces result from the distribution of Oseen force singularities. These Oseenlets also predominantly govern the torque at small Reynolds numbers and large aspect ratios. However, the potential flow singularities play a crucial role in yielding a torque that grows with increasing Reynolds number at large Reynolds numbers and finite aspect ratios. By comparing the forces and torque on the steadily translating particle with those obtained from a finite difference Navier-Stokes solution, we demonstrate the accuracy of the resulting inertial SBT for $\mathrm{Re}_D$ up to 10, where $\mathrm{Re}_D$ is the Reynolds number based on the smallest dimension, i.e., the characteristic cross-sectional diameter of the slender particle.