Viscous dynamics of vortices in a ferromagnetic film

We derive viscous forces for vortices in a thin-film ferromagnet. The viscous force acting on vortex $i$ is a linear superposition $\mathbf F_i = - \sum_{j} \hat{D}_{ij} \mathbf V_j$, where $\mathbf V_j$ is the velocity of vortex $j$. Thanks to the long-range nature of vortices, the mutual drag tensor $\hat{D}_{ij}$ is comparable in magnitude to the coefficient of self-drag $D_{ii}$.