Impurity in a Maxwellian Unforced Granular Fluid,

E. Ben-Naim and P.L. Krapivsky

We investigate velocity statistics of an impurity immersed in a uniform granular fluid. We consider the cooling phase, and obtain scaling solutions of the inelastic Maxwell model analytically. First, we analyze identical fluid-fluid and fluid-impurity collision rates. We show that light impurities have similar velocity statistics as the fluid background, although their temperature is generally different. Asymptotically, the temperature ratio increases with the impurity mass, and it diverges at some critical mass. Impurities heavier than this critical mass essentially scatter of a static fluid background. We then analyze an improved inelastic Maxwell model with collision rates that are proportional to the {\it average} fluid-fluid and fluid-impurity relative velocities. Here, the temperature ratio remains finite, and the system is always in the light impurity phase. Nevertheless, ratios of sufficiently high order moments $\langle v^n_{\rm impurity}\rangle/\langle v^n_{\rm fluid}\rangle$ may diverge, a consequence of the multiscaling asymptotic behavior.

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