Curved settling fronts within a rarefaction fan ($z_u,z_d$)

For upper concentration $a$, lower concentration $b<a$ and $L=H-z_m$, define $A=aL$ and $D=(1-b)z_m$. Once inside the settling rarefaction fan, clearing and deposition shocks obey $z_u=z_m-u_{s}t+2\sqrt{u_{s}At}$ and $z_d=z_m+u_{s}t-2\sqrt{u_{s}Dt}$. These curves apply only while their neighboring concentration remains within the 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.