Nonuniqueness of Tangential and Kinematic Restitution in Frictional Oblique Particle Impacts

Abstract

We systematically investigate the different coefficients of restitution of a dissipative, frictional disc as a function of the impact angle $\theta$, the friction coefficient $\mu$ and initial rotation $\omega_{\mathrm{in}}$. We observe a non-monotonic, non-linear behaviour of $e_{\mathrm{T}}$ and $e_{\mathrm{kin}}$ with a clear minimum at $\mu$-dependent values of $\theta$ and a convergence for all $\mu$ at large $\theta$, bounded by the cases of pure rolling and pure sliding. Changing the dissipative normal interaction affects the $\mu$ convergence at large $\theta$ and leads to a convergence at low $\theta$. The presence of initial angular velocity $\omega_{\mathrm{in}}$ can significantly alter the functional behaviour of $e_{\mathrm{T}}$ at steep impacts and slightly change the magnitude of $e_{\mathrm{kin}}$, with friction playing only a minor role. Overall, our results indicate that any specific value of $e_{\mathrm{T}}$ or $e_{\mathrm{kin}}$ is highly situational and can simultaneously describe entirely different contact configurations.