 Last surviving characteristic of a settling fan

If both shocks meet inside a settling rarefaction fan, $t_c=(\sqrt{A}+\sqrt{D}{)}^2/u_s$ and the last characteristic has concentration ${\varphi }_{\ast }=\sqrt{A}/(\sqrt{A}+\sqrt{D})$. It runs from the initial internal interface to the final deposit point. This construction requires $b\le {\varphi }_{\ast }\le a$; otherwise a front exits the fan before final deposition.